Modular Exercises for Four-Point Blocks - I
The well-known modular property of the torus characters and torus partition functions of (rational) vertex operator algebras (VOAs) and 2d conformal field theories (CFTs) has been an invaluable tool for studying this class of theories. In this work, we prove that sphere four-point chiral blocks of r...
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| description | The well-known modular property of the torus characters and torus partition functions of (rational) vertex operator algebras (VOAs) and 2d conformal field theories (CFTs) has been an invaluable tool for studying this class of theories. In this work, we prove that sphere four-point chiral blocks of rational VOAs are vector-valued modular forms for the groups Γ(2), Γ₀(2), or SL₂(ℤ). Moreover, we prove that the four-point correlators, combining the holomorphic and anti-holomorphic chiral blocks, are modular invariant. In particular, in this language, the crossing symmetries are simply modular. This gives the possibility of exploiting the available techniques and knowledge about modular forms to determine or constrain the physically interesting quantities, such as chiral blocks and fusion coefficients, which we illustrate with a few examples. We also highlight the existence of a sphere-torus correspondence equating the sphere quantities of certain theories ₛ with the torus quantities of another family of theories ₜ. A companion paper will delve into more examples and explore this sphere-torus duality.
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| first_indexed | 2026-03-21T18:31:12Z |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 21 (2025), 013, 61 pages
Modular Exercises for Four-Point Blocks – I
Miranda C.N. CHENG a, Terry GANNON b and Guglielmo LOCKHART cd
a) Korteweg–de Vries Institute for Mathematics and Institute of Physics,
University of Amsterdam, Amsterdam, The Netherlands
E-mail: mcheng@uva.nl
b) Department of Mathematics, University of Alberta, Canada
E-mail: tjgannon@ualberta.ca
c) Institute of Physics, University of Amsterdam, The Netherlands
E-mail: glockhar@uni-bonn.de
d) CERN, Theory Department, Geneva, Switzerland
Received May 08, 2024, in final form February 12, 2025; Published online February 28, 2025
https://doi.org/10.3842/SIGMA.2025.013
Abstract. The well-known modular property of the torus characters and torus partition
functions of (rational) vertex operator algebras (VOAs) and 2d conformal field theories
(CFTs) has been an invaluable tool for studying this class of theories. In this work we prove
that sphere four-point chiral blocks of rational VOAs are vector-valued modular forms for the
groups Γ(2), Γ0(2), or SL2(Z). Moreover, we prove that the four-point correlators, combining
the holomorphic and anti-holomorphic chiral blocks, are modular invariant. In particular,
in this language the crossing symmetries are simply modular symmetries. This gives the
possibility of exploiting the available techniques and knowledge about modular forms to
determine or constrain the physically interesting quantities such as chiral blocks and fusion
coefficients, which we illustrate with a few examples. We also highlight the existence of
a sphere-torus correspondence equating the sphere quantities of certain theories Ts with the
torus quantities of another family of theories Tt. A companion paper will delve into more
examples and explore more systematically this sphere-torus duality.
Key words: conformal field theory; vertex operator algebras; modularity
2020 Mathematics Subject Classification: 81T40; 17B69; 11F03
1 Introduction
Modular forms have been useful since they were known to humankind. Specifically, in the
study of two-dimensional conformal field theories (CFTs) and closely related vertex operator
algebras (VOAs, often referred to as chiral algebras), modular forms have played an important
role. On the VOA side, the torus characters of V-modules are well known to transform as
components of a vector-valued modular form (vvmf), when the VOA is rational and satisfies
the so-called C2 co-finite condition. Moreover, the S-matrix of the corresponding modular
group representation determines certain crucial data of the VOA – the fusion coefficients – via
the celebrated Verlinde formula (4.2). When combining the chiral and anti-chiral theories into
a physical CFT, the torus partition function is famously modular invariant. The role of modular
forms in this case can be understood through the role of the modular group SL2(Z) as the
mapping class group of the torus with one puncture. The modularity poses stringent constraints
on the spectrum of the theories and has been very valuable in exploring the existence of potential
theories [1, 8, 16, 17, 20, 33, 37, 41].
On the other hand, four-point conformal blocks (or chiral blocks more generally) are key
quantities of a VOA, as the braiding and fusing matrices associated to them essentially determine
mailto:mcheng@uva.nl
mailto:tjgannon@ualberta.ca
mailto:glockhar@uni-bonn.de
https://doi.org/10.3842/SIGMA.2025.013
2 M.C.N. Cheng, T. Gannon and G. Lockhart
the representation theory (i.e., the modular tensor category) of the VOA. For physical CFTs, the
crossing symmetry of the four-point correlation functions together with the modular invariance
on the torus guarantees the consistency of the theory on arbitrary Riemann surfaces. Again,
crossing symmetries pose valuable constraints on the theory and form the cornerstone of the
conformal bootstrap method.
In this work, we first focus on the chiral blocks and make explicit their relation to modular
forms. Namely, we prove that the corresponding core blocks, which we define as the part of the
chiral blocks that is not dictated by conformal symmetry and that is invariant under the Dehn
twists about punctures, form vector-valued modular forms. They are modular forms for the
modular group SL2(Z) when at least three of the external insertions are identical; when two of
the external insertions are identical, they are modular forms for its subgroup Γ0(2), and when
none are, they are modular forms for its subgroup Γ(2). To be more precise, for a given set of
operators ϕ1 ∈ M1, ϕ2 ∈ M2, ϕ3 ∈ M3, ϕ4 ∈ M4, inserted at the four punctures, we obtain
a vector-valued modular form, with each component labelled by a module P (the “internal
channel”) – cf. Figure 1.
Note that we consider not just the cases of Virasoro blocks but also general chiral blocks
when the chiral algebra is an extension of the Virasoro algebra, as is almost always the case.
In the case of Virasoro blocks, the relation between four-point conformal blocks and modular
forms is already implicit in the work by Zamolodchikov [77]. For generic CFTs, each chiral
block will involve the contributions from infinitely many Virasoro families (Virasoro primaries
and their Virasoro descendents) which are related through the action of the chiral algebra. Here,
the appearance of the modular group can again be understood via its relation with the relevant
mapping class group (cf. Section 3), or alternatively via the geometric picture of a torus as
a double cover of a sphere with four punctures (cf. Section 6). Subsequently, we consider the
four-point correlation function by assembling the chiral and anti-chiral sides and appropriately
summing over internal channels. Using the language of modular forms, what we do is to combine
holomorphic and anti-holomorphic vector-valued modular forms into a modular invariant, non-
holomorphic function. In other words, in the language of modular forms the crossing symmetry is
nothing but the modular symmetry of the correlation function (with a universal factor removed).
This is of course extremely reminiscent of the familiar relation between 2d CFT and modular
forms when torus blocks and torus partition functions are considered.
In fact, in some instances the analogy to torus modularity runs deeper. Namely, we find
that for certain VOAs Vs the four-point core blocks associated to a specific choice of external
operators M∗ coincide with the characters of a different VOA Vt. Moreover, one can combine
a chiral- and anti-chiral copy of the VOAs Vs and Vt into full CFTs Ts and Tt, and arrive at a CFT
version of the correspondence that is also outlined in Table 1, where we write ∗ :=
(
M∗, M̃∗
)
.
We stress that for the same Vs the core blocks for different choices of “twist field” M∗ may
be in correspondence with the characters of different VOAs Vt. This is the case for the Ising
model, for which the four-point core block associated to the h = 1
2 operator ϵ coincides with the
characters of the E8,1 Kac–Moody algebra, while the four-point blocks associated to the h = 1
16
operator σ coincide with the characters of the A1,1 Kac–Moody algebra.
Clearly, the versatile and powerful arsenal of modular forms should find applications in the
computation of sphere four-point chiral blocks and the four-point correlators, just like it has
been applied to the torus characters/partition functions to draw conclusions on a wide range of
properties of 2d VOAs and CFTs. Indeed, while different techniques such as Zamolodchikov’s
recursion relation and the BPZ/KZ equations can be employed to determine the conformal
blocks for different classes of theories, one of the major advantages of our approach is that it
applies uniformly to Virasoro minimal models as well as to any rational conformal field theory
with extended chiral algebras, and even to non-rational examples such as Liouville theory with
one degenerate external operator, as we will see in Section 8.
Modular Exercises for Four-Point Blocks – I 3
ϕ1
ϕ2
ϕ3
ϕ4
P
Figure 1. The four-point chiral block.
While we focus on laying out and proving the formalism in the present paper, in the com-
panion paper [3] we will give more examples to illustrate the consequences of the modularity
property. We will also examine the aforementioned sphere-torus connection between Tt and Ts
more systematically.
Table 1. A summary of the sphere-torus correspondence.
torus four-punctured sphere
torus chiral algebra Vt ↔ sphere chiral algebra Vs
irreducible module N ↔ irreducible module M(N)
characters χN (τ) = core blocks φ
M(N)
M∗,M∗,M∗,M∗
(τ)
torus S and T matrices T, S = sphere S and T matrices T , S (Theorem 5.3)
CFT Tt ↔ sphere CFT Ts
partition function ZT 2(τ, τ̄) = correlator F∗,∗,∗,∗(τ, τ̄) (7.7)
multiplicities ZN,Ñ = OPE coefficient (A∗,∗,∗,∗)M(N),M̃(Ñ) (7.6)
The plan of the paper is as follows. In Section 2, we list the various notations and symbols
used in the paper. In Section 3, we introduce the various discrete groups related to the mapping
class group of spheres with four punctures and clarify the relation among them. In Section 4, we
review the basic properties of chiral blocks. In Section 5, we define the notion of core blocks and
investigate their properties. In particular, in Theorem 5.1 we demonstrate that, unlike the chiral
blocks, the core blocks furnish representations of the mapping class group. In Section 6, we show
that the core blocks with given chiral data form vector-valued modular forms, with S and T
matrices determined by the braiding and fusing matrices (Theorem 6.1). In Section 7, we turn
our attention to the physical correlators and define the core correlators, capturing non-trivial
information of the correlators that is not just dictated by conformal symmetry. We then show
that the core correlators are non-holomorphic modular forms (see Corollary 7.2). In Section 8, we
provide some examples illustrating the relevance of modular forms in four-point chiral blocks and
correlators. In Section 9, we elaborate on the correspondence relating sphere and torus blocks.
We also include three appendices. In Appendix A, the relation between modular differential
equations and BPZ equations is exemplified by a detailed treatment in the case of the second-
order BPZ equation. In Appendix B, we collect several facts about the braiding and fusing
operators that we make use of in the main text. In Appendix C, we collect details about the
vector-valued modular forms for general chiral blocks.
While we were preparing this manuscript we learned that [56] has also developed an approach
that exploits modularity to constrain the correlators of Virasoro minimal models (the same
approach has recently also been extended to WZWmodels [55]). We thank the authors of [56] for
considering to coordinate the arXiv submission. While the basic idea of exploiting the relation of
4 M.C.N. Cheng, T. Gannon and G. Lockhart
the mapping class group to modular groups is the same as for us, the approach and scope of these
papers differs significantly. On the one hand, in this paper we develop an approach to determine
(holomorphic) conformal blocks of arbitrary rational vertex operator algebras by exploiting their
transformations as vector-valued modular forms; physical CFT correlators are then constructed
from bilinear combinations of conformal blocks by imposing modular invariance on them. On
the other hand, [55, 56] work directly at the level of the full CFT, and determine candidate,
crossing symmetry invariant, correlation functions by taking a specific choice of conformal blocks
as input data and averaging over its images under PSL(2,Z). The focus there is on determining
structure constants consistent with crossing symmetry. We comment further on the overlap
between these approaches in Section 8 when we consider specific examples.
2 Summary of notation
For convenience to the reader, we summarize our notation in the following table:
V The chiral algebra.
Υ = Υ(V) The set of irreducible V-modules, also referred to
as the conformal families.
M(n) The homogeneous component of V-module M of L0-grading n.
Σg,n The Riemann surface of genus g and with n punctures.
D = (g, n;M1, . . . ,Mn) The chiral datum for a surface Σg,n, with insertions
M1, . . . ,Mn ∈ Υ.
Tg,n The Teichmüller space of Σg,n.
Γg,n The pure mapping class group of Σg,n.
FΓg,n The full mapping class group of Σg,n.
PBrn The pure braid group on n strands.
Brn The full braid group on n strands.
Mg,n The Deligne–Mumford compactification of the moduli space
of Σg,n.
M̂g,n The extended moduli space.
Γ̂g,n The mapping class group extended by the action of Dehn twists.
T̂g,n The extended Teichmüller space of Σg,n.
ÊD The vector bundle on M̂g,n associated to the chiral datum D.
ED The vector bundle on Mg,n associated to the chiral datum D.
R̂D The representation of Γ̂g,n corresponding to the bundle ÊD.
RD The representation of Γg,n corresponding to the bundle ED.
Ψ̂D The space of meromorphic sections of ÊD.
ΨD The space of meromorphic sections of ED.
F̂D(ϕ1, . . . , ϕn) The space of chiral blocks associated to ϕ1 ∈M1, . . . , ϕn ∈Mn.
ND The rank of ÊD.
T, S The generators of SL2(Z).
T,S The T and S matrices acting on the torus blocks.
C The charge conjugation matrix acting on the torus blocks.
(0, n;M1, . . . ,Mn) The chiral datum for an n-punctured sphere. Also noted
as (M1, . . . ,Mn).
hM The conformal weight (lowest L0 eigenvalue) of M .
ΦM1
M2,M3
(z) An intertwiner from M2 ⊗M3 into M1.
YM1
M2,M3
The space of intertwiners from M2 ⊗M3 into M1.
Modular Exercises for Four-Point Blocks – I 5
N(0,3;M∗
1 ,M2,M3) The fusion coefficient.
M∗ The V-module dual to M .
Y (ϕ, z) The vertex operator in YV
V,V .
YM (ϕ, z) The vertex operator in YMV,M .
χM (ϕ, τ) The torus one-point block (graded character).
χ(ϕ, τ) The vector of graded characters.
hi The conformal weight of ϕi ∈Mi.
cΦϕ1ϕ2ϕ3 The structure constant. See (4.12).
w =
z12z34
z13z24
The cross-ratio.
w̃ The coordinate on the universal cover of P1\{0, 1,∞}.
h =
∑4
i=1 hi The sum of the conformal weights of external operators.
µij =
h
3 − hi − hj The specific linear combinations of conformal weights of external
operators.
FP
ϕ1,ϕ2,ϕ3,ϕ4
The four-point chiral block.
Ψ̂P
(M1,M2,M3,M4)
The subspace of Ψ̂(M1,M2,M3,M4) isomorphic to YM
∗
1
M2,P
⊗ YPM3,M4
.
φ The vector of core blocks.
σi The generator of FΓ0,4 moving the i-th strand over the
(i+ 1)-th strand.
σT , σS The generators of PSL2(Z) ∈ FΓ0,4. See (3.5).
σ̃i The generators of FΓ0,4, related to σi by conjugation (3.9).
BPQ
[
N2 N3
N1 N4
]
(ϵ) The braiding matrices.
FPQ
[
N2 N3
N1 N4
]
The fusing matrices.
ξN1
N2,N3
The skew symmetry (B.1) of the intertwiner spaces.
ζN1
N2,N3
The isomorphism (B.2) of the intertwiner spaces.
Ri The representation matrices (5.3) corresponding to the action
of σ̃i ∈ FΓ0,4 on the core blocks.
RT , RS The T and S matrices acting on the core blocks. See
(5.6) and (5.7).
S, T The S and T -matrices of PSL2(Z) acting on the core blocks.
PΓ0(2) The group Γ0(2)/{±}. See (5.10) and Section 6.1.3.
PΓ(2) The group Γ(2)/{±}. See (5.10) and Section 6.1.2.
R The representation matrix corresponding to the action
of
(
1 −1
2 −1
)
∈ PΓ0(2) on the core blocks.
U The representation matrix corresponding to the action
of
(
1 0
−2 1
)
∈ PΓ(2) on the core blocks.
j(τ) The modular j-function.
En(τ) The weight-n Eisenstein series.
∆(τ) The modular discriminant function.
η(τ) The Dedekind eta function.
D(k) :=
1
2πi
d
dτ − k
12E2(τ) The modular derivative acting on weight-k modular forms.
Dℓ
(0) The modular derivative taking weight-0 vvmf’s to weight-2ℓ
vvmf’s. See (6.4).
M!
0(ρ) The space of weakly-holomorphic weight-zero vvmf’s with
multiplier ρ.
M!
0(ρ;λ) The subspace of M!
0(ρ) whose growth as τ → i∞ is bounded
by qλ.
λ The modular lambda function. See Section 6.1.2.
θi The Jacobi theta functions.
6 M.C.N. Cheng, T. Gannon and G. Lockhart
κ The specific Hauptmodul for Γ0(2). See Section 6.1.3.
H The Hilbert space of a (non-chiral) CFT.
Ṽ The chiral algebra of right-movers of a (non-chiral) CFT.
ZM,M̃ The multiplicity of the module M ⊗ M̃ in H.
Ω The isomorphism between the maximally-extended chiral algebras V and Ṽ.
Φi
j,k A physical vertex operator of the RCFT.
dijk The OPE coefficients of the RCFT.
A1234 The crossing matrix with entries d12pd
p
34. See (7.6).
Fϕ1,ϕ2,ϕ3,ϕ4 The four-point core correlator. See (7.7).
ZT 2 The torus partition function of the (non-chiral) CFT.
3 Groups
The moduli space of genus g surfaces with n punctures is naturally the global orbifold Tg,n/Γg,n
where Γg,n is the (pure) mapping class group, and the universal cover Tg,n is the Teichmüller
space. The group Γg,n fixes the ordering of the n points; if we allow the points to be permuted,
then the appropriate group is the full mapping class group FΓg,n. Then Γg,n is to FΓg,n as the
pure braid group PBrn is to the braid group Brn: FΓg,n/Γg,n ∼= Brn/PBrn ∼= Sym(n). In this
introductory section we spell out the relation between the mapping class groups Γ0,4 and FΓ0,4
and various groups acting naturally on the upper-half plane H. From this relation we will
determine in Section 5 the modular properties of core blocks.
Recall the definitions of the full braid group Brn and the pure braid group PBrn on n strands.
The full braid group allows any braids. Each braid gives rise to a permutation. For example, in
the braid β ∈ Br3 shown in Figure 2 the strand in position 1 at the bottom of the diagram ends
up in position 3 at the top of the diagram and vice versa, while the one in position 2 ends up
in the same position. In other words, β gives rise to the permutation (13) ∈ Sym(3). The pure
braid group PBrn by definition consists of all braids in Brn giving rise to the trivial permutation.
It is a normal subgroup of Brn, with quotient Brn/PBrn ∼= Sym(n).
Figure 2. A braid β ∈ Br3 giving rise to the permutation (13) ∈ Sym(3).
We are mainly interested in the case n = 4:
Br4 = ⟨σ1, σ2, σ3|σiσi+1σi = σi+1σiσi+1, σ1σ3 = σ3σ1⟩, (3.1)
where i = 1, 2 in the above definition, as it is related to the full (unextended) mapping class
group of a sphere with four punctures in the following way. Consider the normal subgroup N
of Br4, which is the normal closure of the elements (σ1σ2σ3)
4 and σ1σ2σ
2
3σ2σ1. Then [15]
FΓ0,4 = Br4/N. (3.2)
Modular Exercises for Four-Point Blocks – I 7
This is the spherical braid group on four strands, quotiented by its centre (which is Z2). The
generator σi moves the i-th strand over the (i + 1)-th strand, and the relations σ1σ2σ
2
3σ2σ1 =
(σ1σ2σ3)
4 = 1 reflect the non-trivial topology of the sphere. On the Riemann sphere with four
punctures, σi acts by interchanging zi+2 ↔ zi+3 in the orientation
zi+2,i+3 7→ e
(
1
2
)
zi+2,i+3, (3.3)
while leaving other two points zj invariant, where we let the label of the four points zi be valued
in i ∈ Z4. For instance, σ2 : z41 7→ e
(
1
2
)
z41.
The above presentation facilitates translating between β ∈ FΓ0,4 and the braid group action.
Moreover, the mapping class group FΓ0,4 is isomorphic to (Z2×Z2)×PSL2(Z), where the group
multiplication is given by (u,A)(v,B) = (u + Av,AB). This fact will be crucial for the rest
of the paper. In what follows we write u, v ∈ Z2 × Z2 as column vectors and we recall that
PSL2(Z) = SL2(Z)/{±I}. The isomorphism [31]
FΓ0,4 ∼=−→ (Z2 × Z2)×PSL2(Z)
is realised by the map
σ1 7→
((
0
0
)
,
(
1 1
0 1
))
, σ2 7→
((
0
1
)
,
(
1 0
−1 1
))
, σ3 7→
((
1
0
)
,
(
1 1
0 1
))
. (3.4)
The above presentation of the full mapping class group FΓ0,4 is in some respects more convenient
than the one in terms of the braid group. For instance, it is immediate from (3.4) that σ21 = σ23,
a fact which is not manifest in the presentation (3.2).
The subgroup (0,PSL2(Z)) is generated by the T and S elements, given by
σT := σ1 7→
(
0,
(
1 1
0 1
))
, σS := σ3σ2σ3 7→
(
0,
(
0 −1
1 0
))
, (3.5)
where we denote simply by 0 the identity element of Z2 × Z2. This subgroup corresponds
to the elements of Γ0,4 which fix z1 while permuting the three points z2, z3, z4 arbitrarily:
σ1 permutes z3 ↔ z4 while σ3σ2σ3 permutes z2 ↔ z4. In terms of the cross ratio, the above
generators act as
σT : w 7→ w
w − 1
, σS : w 7→ 1− w. (3.6)
Of course, more important is the fact that σis also act on the universal cover of P1 \ {0, 1,∞}.
From this point of view, (3.6) says that, e.g., σT maps the fibre above w to that above w
w−1 .
We describe this explicitly in Section 6 below.
Finally, the pure mapping class group Γ0,4, which is the subgroup of FΓ0,4 that does not
permute the four points zi, is isomorphic to (0,PΓ(2)), where PΓ(2) = Γ(2)/{±I}. Recall
that PΓ(2) is the free group generated by T 2 and S−1T 2S, represented respectively by
σ21 = σ23 7→
(
0,
(
1 2
0 1
))
, σ22 7→
(
0,
(
1 0
−2 1
))
. (3.7)
Note that Sym(4) ∼= FΓ0,4/Γ0,4 is an extension of
Sym({z2, z3, z4}) ∼= Sym(3) ∼= PSL2(Z)/PΓ(2)
by Z2 × Z2 = ⟨(12)(34), (13)(24)⟩, which is generated by
σ−1
1 σ3 7→
((
1
0
)
, I
)
and σL := σ2σ
−1
1 σ3σ
−1
2 7→
((
1
1
)
, I
)
. (3.8)
8 M.C.N. Cheng, T. Gannon and G. Lockhart
Here we have made the slightly unconventional notational choice (3.3). This choice was made
so that our choice of PSL2(Z) ⊂ FΓ0,4, described in (3.5), leaves z1 fixed. This is related to the
more common convention in which σ̃i interchanges zi ↔ zi+1 via a simple conjugation using the
above σL. In other words, we have
σ̃i := σ−1
L σiσL. (3.9)
In particular,
σ̃1 = σ3, σ̃3 = σ1.
Finally, we note that σS is invariant under this conjugation, and we have
σS := σ3σ2σ3 = σ̃3σ̃2σ̃3.
The subgroup PSL2(Z) in FΓ0,4 is related to the SL2(Z) in Γ1,0 = Γ1,1 through a construction
which will play a large role in Section 9 below. Consider the torus C/(Zτ+Z), with fundamental
domain the parallelogram with vertices at 0, 1, τ +1, τ . Folding the torus by the map z 7→ −z,
we get a fundamental domain with vertices 0, 1, τ2 +1, τ2 , where the edge 0 → 1
2 is identified with
the edge 1
2 → 1, 0 → τ
2 is identified with 1 → τ
2 + 1, etc. Performing these identifications, we
obtain a sphere. There are 4 fixed-points of this Z2 action: the classes [0],
[
1
2
]
,
[
τ
2
]
and
[
τ
2 +1
]
.
Put another way, the torus is a double cover of the 4-punctured sphere.
An element γ ∈ SL2(Z) = FΓ1,0 commutes with that Z2-action, and so projects to an element
γ̄ ∈ PSL2(Z) ⊂ FΓ0,4. The subgroup Z2 × Z2 ⊂ FΓ0,4 is generated by the maps z 7→ z + 1
2
and z 7→ z + τ
2 ; these conformal maps permute the fixed points so certainly cannot be deformed
to the identity. They lift to the (conformal) maps on the torus given by the same formulas.
However, on the torus these translations can be continuously deformed to the trivial one, and
so Z2 × Z2 lift to the trivial element of the mapping class group of the torus.
4 The theory of chiral blocks
The core blocks of 2d conformal field theories which are the main object of this paper are closely
related to the more standard notion of chiral blocks,1 and roughly speaking are obtained from
them by isolating the part that is not dependent on Dehn twists around punctures. In this
section, we therefore provide a pedagogical review of the theory of chiral blocks, setting up
along the way our conventions and notations. As usual, we will focus on chiral blocks and
correlation functions of primary operators of the chiral algebra.
For now, we focus on the chiral half of the theory and let V be a rational vertex operator
algebra, or RCFT. Let Υ = Υ(V) denote the (finite) set of irreducible V-modules. For example,
we have V ∈ Υ, corresponding to the vacuum sector consisting of the identity operator and its
descendants. Any V-module M decomposes into a direct sum of L0-eigenstates. In other words,
we write
M =
∞⊕
n=0
M(n), L0|M(n) = (n+ hM )id, (4.1)
where M(0) ̸= 0 and hM is called the conformal weight of M . Here, L0 is an operator coming
from a copy of the Virasoro algebra contained in V.
1The chiral blocks are also often called conformal blocks, but we use the term chiral blocks to stress that they
are usually not Virasoro conformal blocks and can in general correspond to fields with respect to an extended
chiral algebra (VOA).
Modular Exercises for Four-Point Blocks – I 9
By a chiral datum we mean a choice D = (g, n;M1, . . . ,Mn), where g ≥ 0 refers to the genus
of a surface Σg,n, n ≥ 0 refers to the number of (distinct) punctures on Σg,n, which we name 1
to n, and where the “insertion”Mi ∈ Υ is assigned to the i-th point. The moduli space of genus g
surfaces with n punctures is compactified by including surfaces with nodal (“pinched”) singulari-
ties. This is called the Deligne–Mumford compactification, which we will denote by Mg,n. These
compactification points play the same role here as cusps do in the theory of modular forms, but
with the additional feature of the factorization formulas which describe what happens to the
chiral blocks as we tend to these “cusps”; we will return to this point later in this section.
In CFT, a fundamental operation is sewing together surfaces at the punctures. This requires
us to specify additional data. This can be done in many ways, but most convenient for the
VOA language is Vafa’s picture [72] (described in fuller detail in [42]), which includes a local
coordinate fi at each puncture. In this paper we focus on the surface Σ being the Riemann sphere
P1 = C ∪ {∞}, in which case the fi can be expressed as functions of the local coordinate z of
the sphere:
fi(z) =
∞∑
n=1
a(i)n (z − zi)
n when zi ∈ C and
fi(z) =
∞∑
n=1
a(i)n z
−n when zi = ∞,
where all a
(i)
n ∈ C and a
(i)
1 ̸= 0.
Conformal equivalences between these extended surfaces (i.e., surfaces with local coordinates
at punctures) can be defined in the obvious way, giving an extended moduli space M̂g,n. The rel-
evant mapping class group then becomes the extended mapping class group Γ̂g,n, which includes
the Dehn twists corresponding to rotating about the punctures. Thus Γ̂g,n is an extension of Γg,n
by n copies of Z. The extended Teichmüller space T̂g,n is now infinite-dimensional, namely Tg,n
with a copy of the set {
∑∞
n=1 anz
n | an ∈ C, a1 ̸= 0} attached to each puncture, and M̂g,n
is given by T̂g,n/Γ̂g,n.
To any chiral datum D, the CFT associates a finite rank holomorphic flat vector bundle
ÊD → M̂g,n, whose rank ND is given by Verlinde’s formula [73] which for arbitrary D reads [65]:
N(g,n;M1,M2,...,Mn) :=
∑
N∈Υ
(SVN )
2(1−g)SM1N
SVN
· · · SMnN
SVN
, (4.2)
where S is the S-matrix associated with the torus chiral blocks which will be defined shortly.
The bundle ÊD is trivial over T̂g,n, and corresponds to a representation R̂D of the extended
mapping class group Γ̂g,n. We denote by Ψ̂D the space of meromorphic sections of ÊD. These
sections can be lifted to T̂g,n, where they become a vector-valued meromorphic function on T̂g,n,
covariant (automorphic) with respect to Γ̂g,n. In fact, in this paper we will mostly focus on the
closely-related, but much simpler holomorphic flat vector bundle ED → Mg,n of the same rank,
corresponding to a representation RD of Γg,n whose meromorphic sections we denote by ΨD.
A chiral block is, among other things, a multilinear map M1⊗C · · ·⊗CMn → Ψ̂D. For a fixed
choice of states ϕi ∈ Mi, denote the space of chiral blocks at (ϕ1, . . . , ϕn) by F̂D(ϕ1, . . . , ϕn).
Hence the dimension of F̂D(ϕ1, . . . , ϕn) is bounded above (there may be accidental linear depen-
dencies depending on specific choices of ϕi) by the rank of the vector bundle
dim
(
F̂D(ϕ1, . . . , ϕn)
)
≤ rank
(
ÊD
)
= ND
computed as in (4.2). Chiral blocks obey certain factorization formulas that dictate the behavior
as Riemann surface approaches a nodal surface. A standard way to determine the space F̂D of
chiral blocks in a CFT is as solutions to differential equations such as the BPZ [14] or KZ [47]
10 M.C.N. Cheng, T. Gannon and G. Lockhart
equations, which result from the presence of null states in Mi and have the interpretation of
modular differential equations in our contexts as discussed in Section 6 and Appendix A. From
this point of view, the corresponding representation R̂D for the mapping class group is given by
the monodromy of the differential equations.
Let M be an irreducible V-module; as such, M restricts to a representation of the Virasoro
algebra contained in V. By a Virasoro-primary we mean a lowest-weight state ϕ ∈ M of some
Virasoro-subrepresentation of M , i.e., Lkϕ = 0 for all k > 0, and ϕ is an eigenvalue for L0.
Given a Virasoro-subrepresentation of M , the primary is uniquely determined up to a nonzero
scalar factor. By analogy, we say that ϕ ∈ M is a V-primary if it is a lowest conformal weight
state for the action by V, i.e., ϕ is a nonzero vector in M(0). Regardless of the chiral algebra V,
chiral blocks with arbitrary insertions ϕi ∈ Mi can be determined as linear combinations of
chiral blocks with Virasoro-primary insertions and their derivatives. For this reason it suffices
to restrict the insertions to be Virasoro-primaries. Note however that each M will have in-
finitely many inequivalent Virasoro primaries, with ever-increasing conformal weights, unless V
is a Virasoro minimal model. On the other hand, there are only finitely many linearly indepen-
dent V-primaries. Unfortunately, knowing the chiral blocks only for V-primaries is not sufficient
in general to know the chiral blocks for all insertions – we give an example of this in a few
paragraphs.
As mentioned earlier, the full mapping class group permutes the punctures. So if for example,
the modulesMi are pairwise distinct, it may be more convenient to restrict to the (pure) mapping
class group. On the other hand, in the special cases where allMi are equal, the relevant mapping
class group will be the full mapping class group FΓg,n, since in these cases the punctures can
clearly be taken as indistinguishable. More generally, when some but not all Mi will be equal,
the most natural group will be between the full and the pure.
The key ingredient of chiral blocks are intertwiners [32]. Choose any irreducible V-modules
Mi ∈ Υ, with conformal weights hMi respectively, for i ∈ {1, 2, 3}. Schematically, we can
represent the intertwiner ΦM1
M2,M3
(z) by the diagram of Figure 3. For each ϕ2 ∈ M2, it has
decomposition
ΦM1
M2,M3
(ϕ2, z) = zhM1
−hM3
−hM2
∑
n∈Z
(
ΦM1
M2,M3
(ϕ2)
)
n
z−n−1, (4.3)
where the mode
(
ΦM1
M2,M3
(ϕ2)
)
n
is a linear map M3(k) → M1(j + k − n− 1) when ϕ2 ∈ M2(j),
and where we have used the grading by the L0-eigenvalues (4.1). We denote by YM1
M2,M3
the space
of all such intertwiners.2 Its dimension is given by the rank NM1
M2,M3
:= N(0,3;M∗
1 ,M2,M3), which
bears the name fusion coefficient. The fusion coefficients define the associative, commutative
fusion ring on the Z-span of Υ, by
M ⊗M ′ =
∑
M ′′∈Υ
NM ′′
M,M ′M ′′,
and can be computed by the Verlinde formula (4.2):
NM ′′
M,M ′ =
∑
N∈Υ
SMNSM ′N
(
S−1
)
M ′′N
SVN
. (4.4)
Examples of intertwiners are the vertex operators Y (ϕ, z) ∈ YV
V,V and YM (ϕ, z) ∈ YMV,M cor-
responding to the VOA itself and to its modules M . Indeed, N V
V,V = NM
V,M = 1.
We will shortly see several examples of how chiral blocks are built from the intertwiners.
Here we recall some of their basic properties. Being (components of) quantum fields, they are
2To make the index conventions consistent throughout the paper, our definition of the intertwiners differs
slightly from the one given, e.g., in [65].
Modular Exercises for Four-Point Blocks – I 11
M2
M1 M3
Figure 3. Diagram corresponding to the intertwiner ΦM1
M2,M3
.
operator-valued distributions on the sphere, and as such cannot be evaluated at a point. For
instance, strictly speaking Φ(ϕ, 0) or Φ(ϕ,∞) does not exist. An important role here is played
by the local coordinates fi: an intertwiner Φ ∈ YM1
M2,M3
will arise in our expressions as the
formal expression Φ(ϕ2, f2(z)) where f2(z) is the chosen local coordinate about the puncture z2.
Creating the state ϕ2 at time t = −∞, which then propagates to the scattering virtual event,
corresponds to attaching to the world-sheet a semi-infinite tube; Vafa’s picture conformally
maps a world-sheet with n semi-infinite tubes, to a surface with n punctures. The end of the i-
th tube corresponds to the puncture zi, and fi is the local coordinate on the tube. Now, we
can relate Φ(ϕ2, f2(z)) to Φ(ϕ2, z − z2), using the transformation laws derived in, e.g., [34]:
Φ(ϕ2, f2(z))(f
′
2(z))
k = Tf2 ◦ Φ(ϕ2, z − z2) ◦ T−1
f2
. This transformation Tf2 induces a change of
VOA structure, including Virasoro representation, to a different but equivalent one – we will
recall the famous example with f(z) = e(z) − 1 in the next subsection. This ability to change
local coordinates allows us to restrict attention from now on to the standard choices of local
coordinates, namely fi(z) = z − zi when zi ∈ C and fi(z) = 1/z when zi = ∞. This should be
very natural and familiar: we study a function locally at a point z0 say through expansions in
powers of z − z0. Eventually, we will restrict attention to fi → 0.
In the remainder of this section, we review concrete examples of chiral blocks on the torus
and on the sphere.
Torus one-point functions. The most familiar example of chiral blocks are the 1-point
functions on the torus, i.e., the case D = (1, 1;V). The surface Σ here is the torus C/(Z+ τZ).
We can take the puncture z1 to be at [0] ∈ Σ, and take the local coordinate to be the standard
f1(z) = [z] ∈ Σ mapping from a small disc about 0 ∈ C to a small patch about [0] ∈ Σ.
Since hV = 0, the action of the Dehn twist about the single puncture is trivial, and as a result
the chiral blocks transform under a representation of the mapping class group Γ1,1 ∼= SL2(Z).
In this case, the dimension of the space of the chiral blocks is given by (4.2) to be
∑
M∈Υ 1 = ∥Υ∥.
This suggests that a basis of the space F̂1,1;V(ϕ), for each state ϕ ∈ V, is given as follows: for
each M ∈ Υ, we take the corresponding basis element to be the 1-point function
χM (ϕ, τ) = e(kz) trM YM (ϕ, e(z)− 1)qL0−c/24 = qhM−c/24
∞∑
n=0
trM(n) o(ϕ)q
n. (4.5)
Here and elsewhere, we write e(x) := e2πix for x ∈ C. Moreover, q = e(τ) where C/(Z+Zτ) is
our torus, c is the central charge of V, and hM is the conformal weight as before. YM is the vertex
operator. Implicit in (4.5) is the change of variables z 7→ e(z) − 1, which is the source of the
pervasive −c/24. Let L[n] denote the new Virasoro operators, whose corresponding conformal
vector is proportional to ω− c/24. Equation (4.5) requires ϕ to be an eigenvector of the energy
operator L[0], i.e., L[0]ϕ = kϕ. The zero-mode o(ϕ) = ϕk−1 of the state ϕ acts linearly on
each space M(n), and “tr” denotes its trace. The appearance of z in the middle of (4.5) is
just the ever-present dependence of the chiral blocks on the local coordinates, although here the
dependence turns out to be trivial.
These 1-point functions χM (ϕ, τ), also sometimes referred to as graded characters, are de-
scribed in more detail in [38, Section 5.3]: like finite group characters, they are built from traces
12 M.C.N. Cheng, T. Gannon and G. Lockhart
of matrices, and uniquely identify the modules M up to equivalence. They are the compo-
nents of a weakly holomorphic vector-valued modular form for SL2(Z) of weight k. This means
each χM (ϕ, τ) is holomorphic for τ ∈ H, grows at worst exponentially in Im(τ) as τ → i∞ (i.e.,
when q → 0), and transforms into a linear combination of χM ′(ϕ, τ), M ′ ∈ Υ, under SL2(Z).
Organising χM , M ∈ Υ into a vector χ, the resulting S-matrix S = R(1,1;V)
((
0 −1
1 0
))
, satisfying
χ(ϕ, τ) = τ−kS−1χ
(
ϕ,− 1
τ
)
,
is precisely that appearing in Verlinde’s formula (4.2), and the T -matrix T = R(1,1;V)
((
1 1
0 1
))
is
diagonal with M -th entry equal to e
(
hM − c
24
)
.
The familiar special case ϕ = 1 ∈ V, the vacuum, recovers the so-called characters (although
“graded dimensions” is arguably a more accurate name for them)
χM (τ) = χM (1, τ) = qhM−c/24
∞∑
n=0
dim(M(n))qn (4.6)
of the irreducible V-modules M . However, it is important to allow for other choices for ϕ. For
one thing, there is in general no way to recover the other χM (ϕ, τ) from the χM (τ), except in
the very special case of the Virasoro minimal models. Moreover, just as the dimension of a Lie
algebra or finite group representation rarely uniquely identifies it whereas its character always
does, it is the 1-point functions χM (ϕ, τ) and not the graded dimensions χM (τ) which uniquely
determine M as a V-module. For example, the representation R(1,1;V) decomposes into a direct
sum R+⊕R−, where R±(−I) = ±I, and χ(ϕ, τ) only sees R(−1)k , so the graded dimensions χ(τ),
like that of χ(ϕ, τ) for any fixed ϕ ∈ V, can only see roughly half of R(1,1;V). More precisely,
all χ(ϕ, τ) transform with respect to the same representation R(1,1;V), but there exists a basis
which diagonalizes the charge-conjugation matrix C = S2 (with dimR+ +1’s and dimR− −1’s
as eigenvalues). When expressed in this basis, the dimR− (resp. dimR+) components of χ(ϕ, τ)
will vanish when the insertion field ϕ has even (resp. odd) conformal weight.
The origin of the aforementioned modular property can be understood in the following way.
The extended mapping class group Γ̂1,1 is the braid group Br3 = ⟨σ1, σ2|σ1σ2σ1 = σ2σ1σ2⟩.
It relates to the modular group (P)SL2(Z) by Br3/⟨Z⟩ ∼= PSL2(Z) and Br3/
〈
Z2
〉 ∼= SL2(Z),
where Z = (σ1σ2)
3 generates the center of Br3. The Dehn twist about the puncture equals Z−2.
Because the Dehn twists act trivially on the chiral blocks, the action of Γ̂1,1 collapses to that
of Γ1,1. See Figure 2 and Section 3 for definitions and more detailed discussions about the braid
groups.
Let us consider the 2d Ising model as a concrete example. It has central charge c = 1/2 and
three irreducible modules, which we label Υ = {1, σ, ϵ}. Note that what we denote by 1 is really
the chiral algebra V itself. The nontrivial fusion coefficients of the Ising model are given by
ϵ⊗ ϵ = 1, ϵ⊗ σ = σ, σ ⊗ σ = 1 + ϵ.
For the Ising model, as for any Virasoro minimal model, it suffices to consider their graded
dimensions (as opposed to more general torus 1-point block), which are
χ(τ) =
χ1(τ)
χσ(τ)
χϵ(τ)
=
q−1/48
(
1 + q2 + q3 + 2q4 + 2q5 + 3q6 + · · ·
)
q1/24
(
1 + q + q2 + 2q3 + 2q4 + 3q5 + 4q6 + · · ·
)
q23/48
(
1 + q + q2 + q3 + 2q4 + 2q5 + 3q6 + · · ·
)
From this, we can read off the conformal weights h1 = 0, hσ = 1/16, hϵ = 1/2 (cf. (4.6)), and
the T -matrix is given by
T =
e
(
− 1
48
)
0 0
0 e
(
1
24
)
0
0 0 e
(
23
48
)
,
Modular Exercises for Four-Point Blocks – I 13
while the S-matrix is
S =
1
2
1
√
2 1√
2 0 −
√
2
1 −
√
2 1
.
By comparison, χ(ϕ, τ) = 0 for any ϕ of odd conformal weight, and χ(ω, τ) =
(
q d
dq +
c
24
)
χ(τ)
where ω ∈ V(2) is the conformal vector. Note that χ(ω, τ) is not a vector-valued modular form,
since ω is not an eigenstate of L[0].
Next we turn to the punctured spheres. To avoid clutter we write (M1, . . . ,Mn) instead
of (0, n;M1, . . . ,Mn) throughout the paper. As mentioned already, we restrict to the stan-
dard local coordinates z − zi and 1/z, where z is the global coordinate on C. As a vector
space, the homogeneous space M∗(k) is naturally identified with the dual vector space M(k)∗.
For x ∈M∗(k) and y ∈M(k), it is common to write ⟨y|x⟩ for the evaluation map x(y) ∈ C. The
permutation M 7→M∗ coincides with the involution C = S2 : Υ → Υ. Now define 1∗ ∈ V(0)∗ so
that 1∗(1) = 1 and 1∗(V(k)) = 0 for k > 0. Then for any composition F of intertwiners send-
ing V to V, we will write ⟨1, F1⟩ = 1∗(F (1)). Likewise, for any composition F of intertwiners
mapping M ′ to M∗, we also write ⟨y|F |x⟩ = (F (x))(y) for any x ∈M ′ and y ∈M .
Fix a choice of Ni ∈ Υ with N1 = M∗
1 , Nn−1 = Mn, and N V
M1,N1
,NN1
M2,N2
, . . . ,NNn−1
Mn,V all
nonzero (or else the chiral blocks are identically zero), and let
Φ1 ∈ YV
M1,N1
, Φi ∈ YNi−1
Mi,Ni
for 1 < i < n and Φn ∈ YNn−1
Mn,V .
With this choice, the chiral blocks for D = (M1, . . . ,Mn), with insertions ϕi ∈Mi(ki) at zi ∈ C,
can be written〈
1,Φ1
(
ϕ1, z
′
1
)
◦ Φ2
(
ϕ2, z
′
2
)
◦ · · · ◦ Φn
(
ϕn, z
′
n
)
1
〉
dz′ h̄11 · · · dz′ h̄nn , (4.7)
where z′i = z − zi and h̄i = hMi + ki. As always, if one of the punctures has zi = ∞,
then we replace z′i with 1/z. Note that Φ1 can be fixed by requiring that ⟨1,Φ1(ϕ1, z)x⟩ =
z−2h̄1x(ϕ1) for all x ∈M∗
1 (k1); similarly, Φn is fixed by requiring that ⟨y,Φn(ϕn, z)1⟩ = ⟨y, ϕn⟩
for all y ∈Mn(kn). Note that YV
M1,M∗
1
and YMn
Mn,V are both one-dimensional, with a canoni-
cal basis – namely ⟨1,Φ1(ϕ1, z)x⟩ = z−2h̄1x(ϕ1) for all x ∈M∗
1 (k1). Similarly, Φn chosen by
skew symmetry applied to the module vertex operator (see [32, equation (5.4.33)]).
With this canonical choice, (4.7) simplifies to
z′−2h̄1
1
〈
ϕ1
∣∣Φ2
(
ϕ2, z
′
2
)
◦ · · · ◦ Φn−1
(
ϕn−1, z
′
n−1
)
ϕn
〉
dz′ h̄11 · · · dz′ h̄nn . (4.8)
In VOA language, Φ(ϕ, z) is merely a formal expansion, though the combination (4.8) is a mero-
morphic multi-valued function of the z′i, with branch-points and poles only when some z′i = z′j
(equivalently, zi = zj). These singularities arise because the intertwiners, being distributions,
cannot be multiplied at the same points. The global conformal group PSL2(C) of the sphere
acts on (4.8) as follows: for a Virasoro-primary field ϕ ∈M(k), each γ =
(
a b
c d
)
gives
Φ
(
ϕ, z′
)
= Φ
(
ϕ, γ(z′)
)(
cz′ + d
)−2h̄
so (4.8) in the limit as the global parameter z → 0 is Möbius-invariant. This limit exists, and
indeed this is what is commonly meant by chiral blocks. This specialization is enough to recover
everything (including the full VOA structure and its module structures [35], hence the blocks
for any local coordinate fi(z)). For this reason, we will eventually restrict to z = 0, i.e., identify
the z′i with the punctures zi.
3
3The observant reader will have noticed that z = 0 gives z′i = −zi. But that is conformally equivalent to z′i = zi.
In our definition of core blocks we choose the specialization z′i = zi.
14 M.C.N. Cheng, T. Gannon and G. Lockhart
Factorization of chiral blocks on the sphere is called cluster decomposition:
lim
λ→∞
λ
∑k
i=1 hϕi (4.9)
× ⟨ϕ1|Φ2(ϕ2, λz2) ◦ · · · ◦ Φk(ϕk, λzk) ◦ Φk+1(ϕk+1, zk+1) ◦ · · · ◦ Φn−1(ϕn−1, zn−1)ϕn⟩
=
∑
ϕ
⟨ϕ1|Φ2(ϕ2, z2) ◦ · · · ◦ Φk(ϕk, zk)ϕ⟩⟨ϕ∗Φk+1(ϕk+1, zk+1) ◦ · · · ◦ Φn−1(ϕn−1, zn−1)ϕn⟩,
where the sum is over a basis ϕ of V-primaries (lowest conformal weight states) in the internal
module Nk, and ϕ∗ is the dual basis. We will use this in the following to relate four-point
functions at the cusp to structure constants. Let us now turn to concrete examples.
Sphere n-point functions with n < 4. The moduli spaces of spheres with n < 4 punctures
are trivial, i.e., single points, and their full mapping class group is Sym(n). This is because
the Möbius symmetry on P1 is triply transitive. Consider first n = 1, the 1-point function on
the sphere, with insertion ϕ ∈ M ∈ Υ. Without loss of generality we can take z1 = 0. Now,
N V
V,M = 0 unless M = V. When ϕ ∈ V(k),
⟨1, Y (ϕ, z)1⟩ =
〈
1,
∑
n
ϕnz
−n−11
〉
= z−k1∗(ϕk−1(1)) = 0
unless k = 0. Thus F̂(0,1;M)(ϕ) equals 0 unless M = V and k = 0, when it has the basis 1 and
the chiral block is a C number.
Slightly less trivial are the 2-point blocks. We keep z1 and z2 arbitrary, though one may fix
them to, e.g., z1 = ∞ and z2 = 0. Here and everywhere else we use the notation zij := zi − zj .
The extended mapping class group Γ̂0,2 is Z2, generated by Dehn twists around the two punc-
tures. The Dehn twist about z′1 sends dz′x1 to e(x)dz′x1 but fixes both dz′x2 and (z12)
y. Note
that N V
M1,M
NM
M2,V = 0 unless M = M2 = M∗
1 , in which case it equals 1. Choose ϕ ∈ M(k)
and ψ∗ ∈ M∗(ℓ); then we find F(M,M∗)(ϕ, ψ
∗) is 0, unless k = ℓ in which case the chiral block
becomes
z−2hM−2k
21 ψ∗(ϕ) dz′ h̄1 dz′ h̄2 . (4.10)
Consider next the case (g, n) = (0, 3). Using conformal symmetry, it is possible to fix the
punctures to be z1 = ∞, z2 = 1, z3 = 0, so z′1 = 1/z, z′2 = z−1, z′3 = z, but this will not be neces-
sary for our analysis. The chiral block with insertions M1, M2, M3 will be 0 unless NM∗
1
M2,M3
> 0.
As mentioned above, the moduli space M0,3 is a point and conformal symmetries dictate the
3-point chiral blocks corresponding to ϕi ∈Mi(ki) to have the following form:
Fϕ1,ϕ2,ϕ3
(
z′1, z
′
2, z
′
3
) 3∏
i=1
dz′ h̄ii = cϕ1,ϕ2,ϕ3
∏
1≤i<j≤3
z
h̄−h̄i−h̄j
ij
3∏
i=1
dz′ h̄ii , (4.11)
where h̄ = h̄1+ h̄2+ h̄3. As before, note that the above chiral block carries a nontrivial action of
the extended mapping class group Γ̂0,3 ∼= Z3 through the factors involving dz′i. Using (4.8), we
can compute the structure constants cϕ1,ϕ2,ϕ3 as follows. For any Φ ∈ YM
∗
1
M2,M3
, we get a core block
cΦϕ1,ϕ2,ϕ3 = z′h̄2+h̄3−h̄12
(
Φ
(
ϕ2, z
′
2
)
(ϕ3)
)
(ϕ1) = z′h̄2+h̄3−h̄12
〈
ϕ1
∣∣Φ(ϕ2, z′2)ϕ3〉. (4.12)
Choosing a basis of YM
∗
1
M2,M3
, we can collect these structure constants into a vector-valued one
cϕ1,ϕ2,ϕ3 ∈ CNM∗
1
M2,M3 ,
Modular Exercises for Four-Point Blocks – I 15
though there is a priori no canonical basis. Note that there is no well-defined structure constant
for a given choice of states ϕi ∈Mi, i = 1, 2, 3. Rather, there are precisely NM∗
1
M2,M3
independent
structure constants, one for each intertwiner Φ ∈ YM
∗
1
M2,M3
.
Sphere 4-point functions. Our main interest lies in the case (g, n) = (0, 4). The moduli
space M0,4 is one-complex dimensional, and is parametrized by the cross ratio
w :=
z12z34
z13z24
.
More precisely, there is a unique Möbius transformation which moves say z1 7→ ∞, z2 7→ 1,
and z4 7→ 0, in which case z3 is sent to the cross ratio w, which can take any value on P1 except
for 0, 1, ∞. The four-point blocks are multi-valued with respect to w, with monodromies around
the points 0, 1 and ∞. To make them single-valued, we have to replace w ∈ P1 \ {0, 1,∞} with
a parameter w̃ running over its universal cover. In Section 6 (cf. (6.11)), we will see that this
universal cover is most naturally identified with the upper-half plane H.
Fix any Virasoro-primaries ϕi ∈Mi(ki). Conformal symmetry dictates the following form for
a four-point chiral block:
Fϕ1,ϕ2,ϕ3,ϕ4
(
z′i
) 4∏
i=1
dz′ h̄ii = φϕ1,ϕ2,ϕ3,ϕ4(w̃)
∏
1≤j<k≤4
z
µjk
jk
4∏
i=1
dz′ h̄ii , (4.13)
where∑
j<i
µji +
∑
i<j
µij = −2h̄i. (4.14)
Choosing different solutions to the constraints (4.14) is equivalent to rescaling φϕ1,ϕ2,ϕ3,ϕ4(w̃) by
factors of the form
w̃a(1− w̃)b. (4.15)
We make the following natural choice:
µij =
h
3
− h̄i − h̄j , h =
4∑
i=1
h̄i, (4.16)
which is fixed by imposing that µij coincides with µi′j′ upon permuting h̄i ↔ h̄i′ , h̄j ↔ h̄j′ .
Using again the fact that S is a symmetric matrix, (4.2) reduces to
dim
(
ÊD
)
=
∑
P∈Υ
NM∗
1
M2,P
NP
M3,M4
, (4.17)
and analogous formulas hold for the sphere with n punctures. This dimension formula (4.17)
suggests a natural decomposition of the space Ψ̂(M1,M2,M3,M4) of sections of the bundle ÊD into
a direct sum of subspaces Ψ̂P
(M1,M2,M3,M4)
, for P ∈ Υ, each of which is naturally isomorphic to the
tensor product YM
∗
1
M2,P
⊗ YPM3,M4
of intertwiner spaces. For each P ∈ Υ with NM∗
1
M2,P
NP
M3,M4
> 0,
and each choice
Φ ∈ YM
∗
1
M2,P
and Φ′ ∈ YPM3,M4
,
we obtain a chiral block FP ;Φ,Φ′
ϕ1,ϕ2,ϕ3,ϕ4
(z′i) as in (4.8). We compute from (4.3):
FP ;Φ,Φ′
ϕ1,ϕ2,ϕ3,ϕ4
(
z′i
)
= z′−2h̄1
1 z′ h̄1−h̄2−hP2 z′hP−h̄3−h̄4
3
×
∞∑
m=0
(
z′3
z′2
)m〈
ϕ1|Φ(ϕ2)m+k2−k1−1 ◦ Φ′(ϕ3)k3+k4−m−1ϕ4
〉
. (4.18)
16 M.C.N. Cheng, T. Gannon and G. Lockhart
A basis for the space F̂M1,M2,M3,M4 of chiral blocks for the sphere with 4 punctures, labelled by
M1,M2,M3,M4 ∈ Υ(V), can be read off from the following description of the space Ψ̂M1,M2,M3,M4
of sections:⊕
P
YV
M1,M∗
1
⊗ YM
∗
1
M2,P
⊗ YPM3,M4
⊗ YM4
M4,V , (4.19)
where YM ′′
M,M ′ is the space of intertwiners introduced earlier. We can express the action of the
generators of the braid group Br4 on the space of chiral blocks in terms of braiding operators.
Here, the most convenient choice of generators are σ̃i, conjugates of σi, as given in (3.9):
σ̃1 7→
⊕
Q
BM∗
1Q
[
M1 M2
V P
]
(+)⊗ I ⊗ I,
σ̃2 7→
⊕
Q
I ⊗BPQ
[
M2 M3
M∗
1 M4
]
(+)⊗ I,
σ̃3 7→
⊕
Q
I ⊗ I ⊗BM4Q
[
M3 M4
P V
]
(+).
(4.20)
In the first line, the first factor acts on the first two factors in each summand in (4.19) and the
identity operators act on the third and fourth factors, and similarly for the rest. The braiding
operator is a standard quantity associated to any vertex operator algebra (see, e.g., [43] and
references therein, which follows from, but gives a more careful treatment than, [65]). See Ap-
pendix B for some of its key properties. It gives the isomorphism4
B
[
N2 N3
N1 N4
]
(+):
⊕
P
YN1
N2,P
⊗ YPN3,N4
∼=−→
⊕
Q
YN1
N3,Q
⊗ YQN2,N4
.
Note that the summands on the left- and right-hand side above are the spaces relevant for the
core blocks
φPϕ1,ϕ2,ϕ3,ϕ4(w̃) and φQϕ1,ϕ3,ϕ2,ϕ4(w̃),
respectively. Given a basis
{
ΦN1,a
N2,P
}
a
for YN1
N2,P
and similarly for YPN3,N4
, YN1
N3,Q
, YQN2,N4
, one can
express the above isomorphism in terms of the tensors BPQ
[
N2 N3
N1 N4
]
(+):
ΦN1,a
N2,P
(z)ΦP,bN3,N4
(z′) 7→
∑
Q
∑
c,d
BPQ
[
N2 N3
N1 N4
]a,b
c,d
(+) ΦN1,c
N3,Q
(z′)ΦQ,dN2,N4
(z), (4.21)
which holds when Im(z− z′) > 0; for Im(z− z′) < 0, one must use a different braiding operator
B
[
N2 N3
N1 N4
]
(−),
which obeys
B
[
N3 N2
N1 N4
]
(+)B
[
N2 N3
N1 N4
]
(−) = I.
4Note that, conforming with the vast amount of literature starting from [65], our B
[
N2 N3
N1 N4
]
(+) treats N1
differently from N2,3,4 as it has a reversed arrow, in accordance with the notation of Figure 3.
Modular Exercises for Four-Point Blocks – I 17
5 The core blocks
In Section 5.1, we introduce the notion of core blocks and explain their relation to the chiral
blocks, focussing primarily on the case of spheres with four punctures, given the fundamental
role four-point blocks and correlators play in conformal field theory [65]. In Section 5.2, we then
discuss the action of the mapping class groups Γ0,4 and FΓ0,4 on the core (resp. chiral) blocks
and prove that these give rise to ordinary (resp. projective) representations of the mapping class
groups. In Section 5.3, we then exploit the relation between mapping class groups and modular
groups of Section 3 to determine the modular transformation properties of core blocks.
5.1 Definition and basic properties
Given a chiral block, we take its corresponding core block to be the part capturing the non-
trivial information that only depends on the conformal equivalence classes of Riemann surfaces,
and is independent of the Dehn twists about the punctures. In particular, a core block will
be a multilinear map φ : M1 ⊗C · · · ⊗C Mn → ΨD to the space of meromorphic sections of ED.
In analogy to the chiral blocks, for a choice of states ϕi ∈Mi, we denote the space of core blocks
at (ϕ1, . . . , ϕn) by FD(ϕ1, . . . , ϕn). The core blocks are sensitive only to the most trivial choices
of local coordinates fi(z). In particular, they transform with respect to the mapping class group,
and do not see the extra parts of the extended mapping class group corresponding to the Dehn
twists.
As we have seen in Section 4, the action of Dehn twists is trivial on the torus one-point
functions, and therefore the definition of core and chiral blocks is identical in that case. The same
is true for the sphere one-point functions. We therefore focus on the cases (g, n) = (0, n)
with n ≥ 2. The effect on the chiral blocks of the Dehn twist about the i-th puncture will
be multiplication by e(hMi), where hMi is the conformal weight (the minimal L0-eigenvalue)
of Mi. These give rise to the factor
∏
i dz
hMi
i in the chiral blocks, which we will drop when
considering the core blocks. The (0, n)-chiral blocks moreover depend on the local coordinates zi
at the punctures through factors of the form (zi−zj)dij for quantities dij depending on the hMi ;
dropping those factors as well gives the core blocks. We will describe this more explicitly and
illustrate it with a few examples.
First of all, in the case n = 2, we have seen in Section 4 that the space of chiral blocks
F(M1,M2)(ϕ, ψ
∗) has dimension 1 only if M∗
1 = M2 = M , ϕ ∈ M(k) and ψ∗ ∈ M∗(k), in which
case the chiral block is given by equation (4.10), i.e.,
z−2hM−2k
21 ψ∗(ϕ) dz′ h̄1 dz′ h̄2 .
Otherwise, the space of chiral blocks has dimension zero. The Dehn twist-independent part,
which we take as our definition of core block, is then simply
φϕ,ψ∗ := ψ∗(ϕ).
Next, consider the case n = 3. Here, the only terms in the chiral block that do not depend
on Dehn twists are the structure constants (equation (4.12)), which we take as our definition of
the core blocks
φΦ
ϕ1,ϕ2,ϕ3 := cΦϕ1,ϕ2,ϕ3 = z′h̄2+h̄3−h̄12
〈
ϕ1
∣∣Φ(ϕ2, z′2)ϕ3〉.
We can combine these into a vector
φ
ϕ1,ϕ2,ϕ3
:= cϕ1,ϕ2,ϕ3 ∈ CNM∗
1
M2,M3
of three-point core blocks.
18 M.C.N. Cheng, T. Gannon and G. Lockhart
Finally, let us turn to the case of greatest interest, namely the four-point core blocks. Fixing
Virasoro-primaries ϕi ∈Mi(ki), the four-point chiral block is given by equation (4.18) as
FP ;Φ,Φ′
ϕ1,ϕ2,ϕ3,ϕ4
(
z′i
) 4∏
i=1
dz′ h̄ii
with
FP ;Φ,Φ′
ϕ1,ϕ2,ϕ3,ϕ4
(
z′i
)
= z′−2h̄1
1 z′ h̄1−h̄2−hP2 z′hP−h̄3−h̄4
3
×
∞∑
m=0
(
z′3
z′2
)m
⟨ϕ1|Φ(ϕ2)m+k2−k1−1 ◦ Φ′(ϕ3)k3+k4−m−1ϕ4⟩
and
µij =
h
3
− h̄i − h̄j , h =
4∑
i=1
h̄i.
Writing the chiral block as in equation (4.13),
FP ;Φ,Φ′
ϕ1,ϕ2,ϕ3,ϕ4
(
z′i
) 4∏
i=1
dz′ h̄ii = φϕ1,ϕ2,ϕ3,ϕ4(w̃)
∏
1≤j<k≤4
z
µjk
jk
4∏
i=1
dz′ h̄ii
we define the corresponding core block to be φP ;Φ,Φ′
ϕ1,ϕ2,ϕ3,ϕ4
(w̃). The choice (4.16) for the expo-
nents µij guarantees, among other things, that the core blocks for four identical insertions
transform as (vector valued) modular forms for the full modular group PSL2(Z); other choices
would lead to spurious multiplicative factors of the form (4.15) in the definition of the core
blocks which would spoil this modular transformation.
We can recover the core block of FP ;Φ,Φ′
as follows. First, one has
lim
u→∞
u2h̄1FP ;Φ,Φ′
(ϕi;u, z2, z3, 0) = ⟨ϕ1|Φ(ϕ2, z2)Φ′(ϕ3, z3)|ϕ4⟩,
and the (in this notation, multi-valued) core block is obtained by further specializing z2 = 1,
z3 = w:
φP ;Φ,Φ′
ϕ1,ϕ2,ϕ3,ϕ4
(w) = (1− w)−µ23w−µ34⟨ϕ1|Φ(ϕ2, 1)Φ′(ϕ3, w)|ϕ4⟩.
Of course it suffices to have Φ run over a basis of YM
∗
1
M2,P
and Φ′ run over one of YPM3,M4
; for generic
states ϕi this will yield a basis of F(ϕ1, ϕ2, ϕ3, ϕ4), though for some ϕi the dimension will be
smaller as there could be accidental linear dependencies. As before, we can then collect these
basis core blocks into a (multi-valued) vector φ(w). As we explicitly work out in later sections,
to make this single-valued we should lift this to w̃.
From equation (4.12) it follows that the w-expansion of the core blocks has the following
leading order behavior as w → 0:
φPϕ1,ϕ2,ϕ3,ϕ4(w) = w−h/3+hP
(
cP0 (ϕ1, ϕ2, ϕ3, ϕ4) +O(w)
)
, (5.1)
for some function cP0 (ϕ1, . . . , ϕ4) multilinear in the states ϕi. Using cluster decomposition (4.9),
the leading constant cP0 can be expressed in terms of structure constants, though this expression
will be complicated if the space P (0) of V-primaries in P is large. Likewise, we obtain
φPϕ1,ϕ2,ϕ3,ϕ4(w) = (1− w)−h/3+hP cP1 (ϕ1, ϕ2, ϕ3, ϕ4) + o
(
(1− w)−h/3+hP
)
,
φPϕ1,ϕ2,ϕ3,ϕ4(w) = wh/3−h
′
cP∞(ϕ1, ϕ2, ϕ3, ϕ4) + o
(
wh/3−h
′)
Modular Exercises for Four-Point Blocks – I 19
as the leading order terms in the expansions around w = 1 and w = ∞, respectively. Here cP1 , c
P
∞
are likewise multilinear, and h′ is the minimum of the hQ for Q satisfying NQ
M1,M4
NM3
Q,M2
̸= 0. Of
course, choosing as we did an s-channel basis broke the symmetry between the w = 0 expansions
and those at w = 1 and w = ∞; we could have instead diagonalized the expansions at w = 1 or
w = ∞ by instead using the t- or u-channels.
5.2 Mapping class group action
It has been shown in [65] that the action of the braiding matrices on the chiral blocks given
in (4.20) indeed satisfies the braid group relation (3.1). In other words, the space of four-point
chiral blocks
⊕
π∈Sym(4) F̂(ϕπ1, ϕπ2, ϕπ3, ϕπ4) furnishes a representation of Br4. In this subsection,
we will show that it moreover furnishes a projective representation of the full mapping class
group FΓ0,4 (cf. (3.2)). Furthermore, the space of core blocks
⊕
π∈Sym(4) F(ϕπ1, ϕπ2, ϕπ3, ϕπ4)
(cf. (4.13)) furnishes a (true) representation of FΓ0,4. For notational simplicity, we will sometimes
restrict in the following to the where
dimYM
∗
1
M2,P
dimYPM3,M4
≤ 1 for all P ∈ Υ(V)
in which case the braiding tensor component BPQ
[
M2 M3
M1 M4
]
(+) does not need the additional la-
bels a, b, c, d and is just a number (cf. (4.21)). This restriction is not significant: the generalisa-
tion to the more general case is straightforward. We will provide an explicit proof of Theorem 5.1
under this restriction, and explain after that how the proof works in the general case.
Theorem 5.1. The space of core blocks
⊕
π∈Sym(4) F(ϕπ1, ϕπ2, ϕπ3, ϕπ4) furnishes a representa-
tion of FΓ0,4.
Proof. We represent the action of σ̃i on the core blocks in terms of matrices Ri as follows:
φPϕ1,ϕ2,ϕ3,ϕ4(w̃) 7→
∑
Q
(Ri)PQ φ
Q
σ̃i(ϕ1,ϕ2,ϕ3,ϕ4)
(σ̃iw̃), (5.2)
where the action of the σ̃i on w̃ is determined (in part) by their action on the zi, and the
action on the labels φ1, . . . , φ4 (hence the sectors M1, . . . ,M4) is by the induced permutation.
Explicitly, one has
σ̃i(φ1, φ2, φ3, φ4) =
(φ2, φ1, φ3, φ4) for i = 1,
(φ1, φ3, φ2, φ4) for i = 2,
(φ1, φ2, φ4, φ3) for i = 3
and
(R1)PQ = δPQ e
(
µ12
2
)
BM∗
1M
∗
2
[
M1 M2
V P
]
(+),
(R2)PQ = e
(
µ23
2
)
BPQ
[
M2 M3
M∗
1 M4
]
(+),
(R3)PQ = δPQ e
(
µ34
2
)
BM4M3
[
M3 M4
P V
]
(+), (5.3)
where the phases arise from the factors of zij on the right-hand side of (4.13). Moreover, using
equations (B.4), one sees that R1 and R3 are given simply by
(R1)PQ = δPQζ
M∗
1
M2,P
e
(
h
6
− hP
2
)
, (R3)PQ = δPQξ
P
M3,M4
e
(
h
6
− hP
2
)
, (5.4)
20 M.C.N. Cheng, T. Gannon and G. Lockhart
where h is as in (4.16), and ξMk
MiMj
and ζMk
MiMj
are the phases defined in (B.1)–(B.2) and satisfy-
ing (B.3).
We need to verify that the following five relations, which follow from (3.2), are satisfied
(1) R1R3 = R3R1,
(2) R1R2R1 = R2R1R2,
(3) R2R3R2 = R3R2R3,
(4) R1R2R
2
3R2R1 = I,
(5) (R1R2R3)
4 = I.
The proof of relations (1)–(3) is straightforward since it has been shown in [65] that the braiding
matrices provide a representation of Br4, whose generators satisfy relations analogous to (1)–(3),
as we see in (3.1). Therefore, all we need to check is that the extra phases, present due to the
fact that we are considering the action on the core blocks and not the chiral blocks, are the
same on both sides of the equations. Indeed, in relation (1) the right- and left-hand sides
have an identical phase factor of e
(µ12+µ34
2
)
= e(−h/6), so (1) holds. Likewise, the left- and
right-hand side of (2) have an identical phase factor given by e
(µ12+µ23+µ13
2
)
, and that of (3) is
given by e
(µ23+µ34+µ24
2
)
.
To prove (4), we use (5.3) and (5.4) to write
(
R1R2R
2
3R2R1
)
PQ
=
∑
R
ζ
M∗
1
M2,P
(
ξRM1,M4
ξRM4,M1
)
ζ
M∗
2
M1,Q
e
(
h2 + h4 −
hP
2
− hR −
hQ
2
)
×BPR
[
M1 M3
M∗
2 M4
]
(+)BRQ
[
M3 M1
M∗
2 M4
]
(+),
which leads to (4) upon plugging in (B.3) and (B.6). Finally, to show that (5) holds we begin
by noting that the following relation holds:
(4′) R3R2R
2
1R2R3 = I,
the derivation is identical to that of (4). We now make repeated use of relations (1)–(4) and (4′)
as well as R2
1 = R2
3, which follows trivially from equations (5.4), to show that
(R1R2R3)
4 = (R1R2R1R3R2R3)
2 = (R2R1R2R3R2R3)
2
= R2R1R3R2R3R3R2R1R2R3R2R3 = R2R3
(
R1R2R
2
3R2R1
)
R2R3R2R3
= (R2R3)
3 = R3R2R3R3R2R3 = R3R2R
2
1R2R3 = I,
and therefore relation (5) holds as well. ■
Now let us explain why Theorem 5.1 holds in general. As shown in the proof of [9, Theo-
rem 5.4.1], our assignments give a (true) representation of the extended mapping class group
F Γ̂0,4 on the chiral blocks. To see that the core blocks carry a representation of FΓ0,4, it suffices
to verify that the Dehn twists about the punctures act trivially, as the quotient of the extended
mapping class group by those Dehn twists is FΓ0,4.
From the definition (4.13) of core blocks, it follows immediately.
Corollary 5.2. The space
⊕
π∈Sym(4) F̂(ϕπ1, ϕπ2, ϕπ3, ϕπ4) of chiral blocks furnishes a projective
representation of FΓ0,4.
Modular Exercises for Four-Point Blocks – I 21
5.3 Modular group
From the discussion in the previous section, it follows that the core blocks transform under the
generators σ̄T = σ̃3 and σ̄S = σ̃3σ̃2σ̃3 of PSL2(Z) as
φPϕ1,ϕ2,ϕ3,ϕ4(w̃) =
∑
Q
(RT )PQ φ
Q
ϕ1,ϕ2,ϕ4,ϕ3
(σ̄T w̃),
φPϕ1,ϕ2,ϕ3,ϕ4(w̃) =
∑
Q
(RS)PQ φ
Q
ϕ1,ϕ4,ϕ3,ϕ2
(σ̄Sw̃), (5.5)
where following (5.2) we write RT := R3, RS := R3R2R3, and denote their matrix elements by
TPQ
[
M2 M3
M1 M4
]
:= (RT )PQ = δPQξ
P
M3,M4
e
(
h
6
− hP
2
)
(5.6)
and
SPQ
[
M2 M3
M1 M4
]
:= (RS)PQ
= e
(
hM1 + hM4 − hP − hQ
2
)
ξPM3,M4
BPQ
[
M2 M4
M∗
1 M3
]
(+)ξQM2,M3
= FPQ
[
M2 M3
M∗
1 M4
]
ξ
M∗
1
Q,M4
ξQM2,M3
. (5.7)
In arriving at the second line of the last equation we have used the identity (B.5).
From Theorem 5.1 and since RT and RS provide a representation of the PSL2(Z) subgroup
of FΓ0,4, it follows that
R2
S = (RTRS)
3 = 1. (5.8)
Define the following vector of core blocks:
φϕ1,ϕ2,ϕ3,ϕ4(w̃) =
φ
P1
ϕ1,ϕ2,ϕ3,ϕ4
...
φPr
ϕ1,ϕ2,ϕ3,ϕ4
(w̃), (5.9)
where Pi ∈ Υ(V) runs along the set of internal channels. By making use of the Z2 × Z2
transformations (3.8) we can rearrange ϕ1, . . . , ϕ4 so that any given one of them is in the first
position. Thus in studying the action of PSL2(Z) on conformal blocks we can limit ourselves to
the following three cases:
(A) when at least three of the four states ϕi are the same. We can arrange this so that
D = (M1,M2,M2,M2) and ϕ1 ∈M1, ϕ2 = ϕ3 = ϕ4 ∈M2;
(B) when two of the four states are the same. We can arrange this so thatD=(M1,M2,M3,M3)
and ϕ1 ∈M1, ϕ2 ∈M2, ϕ3 = ϕ4 ∈M3;
(C) when all four states are distinct.
We want to generate actions of the following groups, with the following generating sets5
PSL2(Z) ±
(
1 1
0 1
)
,±
(
0 −1
1 0
)
PΓ0(2) ±
(
1 1
0 1
)
,±
(
1 −1
2 −1
)
PΓ(2) ±
(
1 2
0 1
)
,±
(
1 0
−2 1
)
.
(5.10)
5See Section 6.1 for general facts about these groups.
22 M.C.N. Cheng, T. Gannon and G. Lockhart
Corresponding to these generators, we define the matrices
±
(
1 1
0 1
)
7→ T := T
[
M2 M3
M1 M4
]
,
±
(
0 −1
1 0
)
7→ S := S
[
M2 M3
M1 M4
]
,
±
(
1 2
0 1
)
7→ T 2 := T
[
M2 M4
M1 M3
]
T
[
M2 M3
M1 M4
]
,
±
(
1 −1
2 −1
)
7→ R := T
[
M2 M3
M1 M4
]
S
[
M4 M3
M1 M2
]
T
[
M4 M2
M1 M3
]
T
[
M4 M3
M1 M2
]
S
[
M2 M3
M1 M4
]
,
±
(
1 0
−2 1
)
7→ U := S
[
M4 M3
M1 M2
]
T
[
M4 M2
M1 M3
]
T
[
M4 M3
M1 M2
]
S
[
M2 M3
M1 M4
]
.
In analogy with the definition of σ̄T and σ̄S in (3.5), we define σ̄U = σ̄S σ̄
2
T σ̄S and σ̄R = σ̄1σ̄U .
The following theorem then follows directly from the discussion above (see also Appendix C):
Theorem 5.3. The group PSL2(Z) acts on the vector-valued functions φA := φϕ1,ϕ2,ϕ2,ϕ2(w̃),
φB(w̃) := φB;ϕ1,ϕ2,ϕ3,ϕ3, and φC(w̃) := φC;ϕ1,ϕ2,ϕ3,ϕ4(w̃), defined in equations (5.9), (C.1),
and (C.2), as follows:
φA(w̃) = T −1 · φA(σ̄T w̃), φA(w̃) = S−1 · φA(σ̄Sw̃),
φB(w̃) = T −1
B · φB(σ̄T w̃), φB(w̃) = S−1
B · φB(σ̄Sw̃),
φC(w̃) = T −1
C · φC(σ̄T w̃), φC(w̃) = S−1
C · φC(σ̄Sw̃),
where the matrices TB, SB, TC , SC are defined in Appendix C.
In case (B), the group PΓ0(2) acts on the vector-valued function φ(w̃) = φϕ1,ϕ2,ϕ3,ϕ3(w̃) by
φ(w̃) = T −1 ·φ(σ̄T w̃) and φ(w̃) = R−1 ·φ(σ̄Rw̃). In case (C), the group PΓ(2) acts on the vector-
valued function φ(w̃) = φϕ1,ϕ2,ϕ3,ϕ4(w̃) by φ(w̃) =
(
T 2
)−1 ·φ(σ̄T σ̄T w̃) and φ(w̃) = U−1 ·φ(σ̄U w̃).
In cases (A) and (B), one verifies straightforwardly that these generators satisfy the PSL2(Z)
and PΓ0(2) relations
S2 = (T · S)3 = 1, R2 = 1
(the generators T 2 and U for PΓ(2) satisfy no relations; see Section 6.1), which follow from
equations (5.6)–(5.8). That SB, TB and SC , TC satisfy the PSL2(Z) relations is automatic from
the induction picture presented in Appendix C.
6 Modular forms
In this section, we review the basic properties of the relevant modular objects. Subsequently, we
discuss the properties of core blocks as modular forms and how such properties can be employed
to determine the core blocks.
6.1 Vector-valued modular forms
From Section 5.3, it follows that the type of modular object that is most relevant to us is the
vector-valued modular forms. In this subsection, we include some general comments on vvmf’s
that will be useful later.
Modular Exercises for Four-Point Blocks – I 23
6.1.1 Modular forms for SL2(Z)
The modular group SL2(Z), or its quotient PSL2(Z) = SL2(Z)/{±} which is what we will use,
is genus zero. We say that a discrete subgroup Γ of SL2(R) is genus zero if the quotient Γ\H is
a Riemann surface of genus zero minus finitely many points. This means that the (meromorphic)
modular functions form the field of rational functions C(j(τ)) in the j-function, with expansion
j(τ) = q−1 + 744 + 196884q + 21493760q2 + · · · , (6.1)
in terms of the local parameter q = e(τ) near the cusp τ = i∞. By a weakly holomorphic
modular form or function, we mean one which is holomorphic in H but with poles allowed at the
cusps. The weakly holomorphic modular functions form the ring C[j(τ)] of polynomials in j(τ).
The (holomorphic) modular forms form the polynomial ring C[E4(τ), E6(τ)], where
En(τ) = 1− 4n
B2n
∞∑
k=1
k2n−1qk
1− qk
is the weight-n Eisenstein series satisfying (for n > 2)
En(τ) = En(τ + 1), En(τ) = τ−nEn
(
− 1
τ
)
.
An important modular form is the discriminant form
∆(τ) = η(τ)24 =
E4(τ)
3 − E6(τ)
2
1728
,
where
η(τ) = q
1
24
∞∏
n=1
(1− qn)
is the Dedekind eta function, which transforms under PSL2(Z) as
η(τ) = e
(
− 1
24
)
η(τ + 1), η(τ) = (−iτ)−
1
2 η
(
− 1
τ
)
. (6.2)
The importance of ∆(τ) lies in the fact that it has no zeroes in H.
The modular transformations of E2(τ) are a little more complicated – it is a quasi-modular
form. It appears in the differential modular (Serre) derivative
D(k) =
1
2πi
d
dτ
− k
12
E2(τ), (6.3)
which takes weight-k (vector valued) modular forms to weight-(k+2) vvmf’s. In several equations
below we employ the differential operator
Dℓ
(0) := D(2ℓ−2) ◦ · · · ◦D(2) ◦D(0), (6.4)
which takes a weight-0 vvmf to a weight-2ℓ vvmf.
The Dedekind eta function is a modular form for PSL2(Z)
(
of weight 1
2
)
, but with nontrivial
multiplier (a projective representation of PSL2(Z) with values in 24th roots of 1). More generally,
we can speak of a vector-valued modular form f(τ) for PSL2(Z) of weight k and multiplier ρ:
here, ρ is a representation of PSL2(Z) (merely projective, if k ̸∈ 2Z) by d×d matrices for some d
called the rank, f(τ) takes values in Cd, and they obey
f(τ) = (cτ + d)−kρ
((
a b
c d
))−1
f
(
aτ + b
cτ + d
)
.
We call f(τ) weakly holomorphic if each component fi(τ) is weakly holomorphic.
24 M.C.N. Cheng, T. Gannon and G. Lockhart
Corresponding to a representation ρ of PSL2(Z), we can define a space of weight-zero weakly
holomorphic vvmf’s, which we will denote by M!
0(ρ). Furthermore, for any d-tuple λ ∈ Rd,
we denote by M0(ρ;λ) ⊂ M!
0(ρ) the subspace
M0(ρ;λ) :=
{
f ∈ M!
0(ρ) | q−λifi(τ) = O(1) for all i, as τ → i∞
}
, (6.5)
where T = ρ
(
1 1
0 1
)
is assumed to be diagonal. Note that there is a natural filtration among these
spaces: M0(ρ;λ
′) ⊂ M0(ρ;λ) whenever λ
′ ≥ λ component-wise.
The properties of vvmf’s can then be employed to help to compute the precise form of the
core blocks. In the remainder of this paper we will describe this application in some detail.
The scalar case dim(ρ) = 1 is particularly simple: any element of M!
0(ρ) can be expressed as
a polynomial in the j-function (6.1) multiplying one of the following functions [10]:
1,
E4(τ)
η(τ)8
,
E6(τ)
η(τ)12
,
E8(τ)
η(τ)16
,
E10(τ)
η(τ)20
,
E14(τ)
η(τ)28
. (6.6)
Because PSL2(Z) is generated by±
(
0 −1
1 0
)
and±
(
1 1
0 1
)
, the multiplier ρ is uniquely determined
by the values S := ρ
((
0 −1
1 0
))
and T := ρ
((
1 1
0 1
))
. We are primarily interested in the cases where
the corresponding T matrices are diagonal. Then the condition of weakly holomorphic reduces
to requiring that there is a diagonal matrix λ such that T = e(λ) and
f(τ) = qλ
∞∑
n=0
f [n]qn.
The coefficients f [n] ∈ Cd will in general not be integral – integrality is expected (see, e.g., [7])
only when the kernel of ρ contains a congruence subgroup
Γ(N) :=
{(
a b
c d
)
∈ SL2(Z) |
(
a b
c d
)
=
(
1 0
0 1
)
mod N
}
. (6.7)
6.1.2 Modular forms for Γ(2)
The congruence group Γ(2) (recall (6.7)), as well as its quotient PΓ(2) = Γ(2)/{±1}, are also
genus-0. The role of j(τ) is played in this case by
λ(τ) =
θ2(τ)
4
θ3(τ)4
= 16q1/2 − 128q + 704q3/2 − 3072q2 + 11488q5/2 − 38400q3 + · · · , (6.8)
where
θ2(τ) =
∑
n∈Z
q
1
2
(n+ 1
2
)2 , θ3(τ) =
∑
n∈Z
q
n2
2 , θ4(τ) =
∑
n∈Z
(−1)nq
n2
2 ,
transform under PSL2(Z) as a weight 1
2 vvmf:θ2θ3
θ4
(τ) =
e
(
−1
8
)
θ2
θ4
θ3
(τ + 1),
θ2θ3
θ4
(τ) = (−iτ)−
1
2
θ4θ3
θ2
(− 1
τ
)
. (6.9)
Recall that θ2, θ3, θ4 do not vanish in H, and (holomorphic) modular forms for PΓ(2) form the
polynomial ring C
[
θ43, θ
4
4
]
. A useful identity among these functions is θ42 = θ43 − θ44.
The group Γ(2) is index 6 in SL2(Z): SL2(Z)/Γ(2) ∼= PSL2(Z)/(PΓ(2)) ∼= Sym(3). The func-
tion λ(τ) is related to the j-function as follows:
j(τ) = 256
(
λ(τ)2 − λ(τ) + 1
)3
λ(τ)2(1− λ(τ))2
. (6.10)
Modular Exercises for Four-Point Blocks – I 25
The group PΓ(2) has 3 cusps, namely 0, 1 and i∞, and it is necessary to have control
on how the modular forms behave at all 3 cusps. One way to do this is to use τ 7→ −1/τ
and τ → −1/(τ − 1) to map the other cusps to i∞. For example, applying these to λ shows
that λ = 1 at τ = 0 and has a pole at τ = 1. For PΓ(2), weakly holomorphic functions are defined
as before: they are holomorphic in H and have at worse poles at the 3 cusps. For example, the
weakly holomorphic modular functions form the ring C
[
λ, λ−1, 1/(1−λ)
]
. Note that PΓ(2), like
any subgroup of PSL2(Z), inherits the same modular derivative D(k).
Next we discuss the multiplier for a PΓ(2) vector-valued modular form. The multiplier ρ is
a representation. Recall that PΓ(2) is freely generated by ±
(
1 2
0 1
)
and ±
(
1 0
−2 1
)
. As a result, ρ is
uniquely determined by its values ρ
(
±
(
1 2
0 1
))
and ρ
(
±
(
1 0
−2 1
))
. In the one-dimensional case,
ρ = ρα,β is specified by a pair α, β ∈ C×, where ρ
(
±
(
1 2
0 1
))
= α and ρ
(
±
(
1 0
−2 1
))
= β. Writ-
ing α = e(a) and β = e(b), we see that the space of weakly holomorphic vvmf of weight 0 for ρα,β
takes the form λa(1− λ)bC
[
λ, λ−1, 1/(1− λ)
]
. To understand this, note that θ2(τ), θ3(τ), θ4(τ)
have no zeros or poles in the simply-connected region H, and hence they have holomorphic
logarithms there. Being the composition of holomorphic functions, functions such as λ(τ)a =
exp(4a log(θ2(τ))− 4a log(θ3(τ))) are themselves holomorphic in H. Finally, in Appendix C we
explain how a vvmf for PΓ0(2) can induce up to one for PSL2(Z).
6.1.3 Modular forms for Γ0(2)
The group Γ0(2) =
{(
a b
c d
)
∈ SL2(Z) | c ∈ 2Z
}
is index 3 in SL2(Z), and is generated by ±
(
1 1
0 1
)
and an order two element ±
(
1 −1
2 −1
)
. As before, we are more interested in PΓ0(2). It is genus
zero, with the role of j(τ) being played by
κ(τ) =
θ3(τ)
4θ4(τ)
4
θ2(τ)8
=
1
256
q−1 − 3
32
+
69
64
q − 128q2 +
5601
128
q3 − · · · .
Holomorphic modular forms for PΓ0(2) form the polynomial ring C
[
E4, θ
4
3 + θ44
]
.
The group PΓ0(2) has 2 cusps, namely at 0 and i∞. Note that κ = 0 at τ = 0, and hence the
weakly holomorphic modular functions form the ring C
[
κ, κ−1
]
. The space of holomorphic vvmf
forms a module over that ring. The group PΓ0(2) is generated by ±
(
1 1
0 1
)
and ±
(
1 −1
2 −1
)
, where
the latter has order 2, so the multiplier ρ of any vvmf (of even weight) is uniquely determined by
the matrices ρ
(
±
(
1 1
0 1
))
and ρ
(
±
(
1 −1
2 −1
))
(the latter squaring to I). In Appendix C we explain
how a vvmf for PΓ0(2) induces up to one for PSL2(Z).
The one-dimensional representations are ρα,σ where α ∈ C× and σ = ±1: ρα,σ
(
±
(
1 1
0 1
))
= α
and ρ
(
±
(
1 −1
2 −1
))
= σ. Writing α = e(a) and σ = (−1)s for s ∈ {0, 1}, we find that the space of
weakly holomorphic vvmf of weight 0 for ρα,σ are κ−a
(
1 + 1
4κ
)s/2C[κ, κ−1
]
. To see this, note
that κ can be raised to arbitrary powers by the same argument as at the end of the last subsection;
the minus sign in the factor κ−a arises because κ has a pole and not a zero at τ = i∞. The elliptic
element ±
(
1 −1
2 −1
)
fixes τ = 1+i
2 and κ
(
1+i
2
)
= −1
4 , so κ+ 1
4 must have a double-zero at τ = 1+i
2 .
Being a uniformizing function for H/Γ0(2), the only zeros and poles of κ + 1
4 can be at the
obvious spots, namely the orbits of 1+i
2 and i∞. Thus
√
κ+ 1
4 is holomorphic in H.
6.2 Core blocks as vector-valued modular forms
As we have seen in the previous section, the core blocks transform in PSL2(Z) representations
which are completely determined provided that the braiding and fusing matrices of the RCFT
are given. This PSL2(Z) action is displayed very naturally by passing to the double-cover of
the four-punctured sphere, which is a torus. By a change of variable, one can express the chiral
blocks in terms of the torus complex structure τ = w̃ [77], which is related to the cross-ratio as
follows:
w = λ(τ). (6.11)
26 M.C.N. Cheng, T. Gannon and G. Lockhart
The function w = λ(τ) is the Hauptmodul identifying C\{0, 1}
(
P1 with three points removed
)
with Γ(2)\H, which maps the three cusps at τ = i∞, 0, 1 to the three points w = 0, 1,∞,
respectively.
It is easy to see that the identification (6.11) is consistent with the action of Γ0,4 ∼= PΓ(2)
via (3.7), when the action of PΓ(2) on H is taken to be the usual τ 7→ aτ+b
cτ+d . To see this, note
that σ̄21 sends z12 7→ e(1)z12 while leaving the other zij invariant, and similarly for σ̄22 and z41.
As alluded to above, the map (6.11) has a simple geometric origin in terms of the torus
that is the double cover of the four puncture sphere. Namely, the hyper-elliptic equation
y2 = x(x− 1)(x− λ) defines an isomorphism class of an elliptic curve together with a basis
of the group of its 2-torsion points, and hence an orbit [τ ] ∈ Γ(2)\H. Conversely, given two
periods ω1 and ω2 with τ = ω1/ω2, the points e1, e2, e3 ∈ C given by
e1 = ℘
(
ω2
2
)
, e2 = ℘
(
ω1 + ω2
2
)
, e3 = ℘
(
ω1
2
)
,
together with the point at infinity, have cross-ratio
λ(τ) =
e1 − e2
e1 − e3
.
A more direct way to motivate the identification (6.11) of w̃ with τ , is the realization that
the moduli space of 4-punctured spheres is the 3-punctured sphere (namely the location of the
4-th point, or the cross-ratio, after the first 3 are moved to 0, 1, ∞). The 3-punctured sphere
has universal cover H and mapping class group F2
∼= PΓ(2): PΓ(2) is genus zero with 3 cusps
and no elliptic points. Then Theorem 5.3 becomes:
Theorem 6.1. The core blocks φϕ1,ϕ2,ϕ3,ϕ4 are single-valued functions on H via the identifica-
tion (6.11). Moreover, they are weight zero vector-valued modular forms for multipliers given in
Theorem 5.3.
After establishing the single-valuedness as above, we will often write
φϕ1,ϕ2,ϕ3,ϕ4(w̃) = φϕ1,ϕ2,ϕ3,ϕ4(τ) and φPϕ1,ϕ2,ϕ3,ϕ4(w̃) = φPϕ1,ϕ2,ϕ3,ϕ4(τ)
to denote the (vector of) core blocks considered as (vector-valued) functions on H.
6.3 Methods for constructing vector-valued modular forms
For vvmf’s of general dimensions, three methods for constructing vector-valued modular forms
have appeared in the literature which we review next. In Section 6.3.4, we present a new
hybrid method which combines two of the existing techniques and is more effective in more
sophisticated examples. For concreteness, we will restrict our discussion to PSL2(Z), although
all the approaches we discuss should generalize. We also mention that for d ≤ 5 the (equivalence
classes of) irreducible representations are uniquely determined by their T eigenvalues [71], which
leads to useful simplifications in explicit calculations.
6.3.1 Rademacher sum
The simplest method is the Rademacher sum of [25, 57] (related are the Poincaré series of [48]).
Let ρ be a representation for SL2(Z) of dimension d, with diagonal ρ(T ) with entries ρ(T )j,j =
e(λj) for 0 ≤ λj < 1. Suppose
Xj(τ) = qλj
∞∑
n=−∞
Xj [n]q
n, 1 ≤ j ≤ d
Modular Exercises for Four-Point Blocks – I 27
are the components of a weakly holomorphic vector-valued modular form of weight 0 with
multiplier ρ, with each Xj [n] ∈ C and Xj [n] ̸= 0 for all but finitely many n < 0. For any
pair of coprime integers (c, d), there will be infinitely many γ ∈ SL2(Z) with bottom row (c d);
let γcd =
(
a b
c d
)
be any one of these. Define the principal part of Xj(τ) to be
P[Xj(τ)] = qλj
∑
n+λj<0
Xj [n]q
n. (6.12)
Given the principal part of Xj(τ), one has [48, 69, 74]
Xj [n] =
d∑
i=1
∑
m∈Z,
m+λi<0
−(m+ λi)
×
∞∑
c=1
4π2
c2
Kl(n, j,m, i; c)Xi[m]I
(
−4π2
c2
(m+ λi)(n+ λj)
)
, (6.13)
where I is given in terms of the Bessel function as
I(x) =
√
x
−1
I1
(
2
√
x
)
=
∞∑
k=0
xk
k!(k + 1)!
= 1 +
x
2
+
x2
12
+
x3
144
+ · · ·
and where the generalised Kloosterman sum Kl is given by
Kl(n, j,m, i; c) =
∑
0<−d<c, (d,c)=1
e
(
1
c (d(n+ λj) + a(m+ λi))
)
ρ(γc,d)
−1
j,i
for c > 1, while for c = 1 we define Kl(n, j,m, i; 1) = ρ(S)−1
j,i . It is easy to check that the above
quantity is independent of the choice of γcd for a given pair (c, d), and therefore the definition
makes sense.
The strength of the Rademacher sum method is that, for a given SL2(Z) representation,
the input is minimal (the principal part) and the formula is universal. The main computational
drawback is that it often converges rather slowly. For instance, even if one knows that the coef-
ficients Xj [n] are integers, one must often compute a large number of terms in the c-expansion
in (6.13) in order to predict their value. Also, in more general contexts there are restrictions to
the Rademacher method: special care must be taken for certain weights [19, 29, 49, 50, 51, 52, 67];
not all principal parts work, with the obstruction in the weight-zero cases given by weight-two
vector-valued cusp forms in the dual representation; and finally the multiplier system ρ is as-
sumed to be unitary. That said, in this paper we are interested in the weight zero unitary
representations with polar parts that necessarily correspond to modular forms, and hence these
restrictions do not affect us.
6.3.2 Riemann–Hilbert
The second method we discuss [10, 39] constructs explicit generators for the space M!
0(ρ) of
weakly holomorphic vector-valued modular forms with multiplier ρ (of dimension d).6 These
generators are determined once certain d × d matrices Λ and χ, satisfying a set of conditions
which we will discuss shortly, are found. These two matrices Λ and χ then determine a basis
6Here we focus on weight-zero vvmf’s for SL2(Z), though as explained in [39] the results also apply to non-zero
weight, and as explained in [36] the method extends with little change to any genus-0 group.
28 M.C.N. Cheng, T. Gannon and G. Lockhart
for M!
0(ρ). In more details, let Ξ[0] = I be the d × d identity matrix, and Ξ[1] = χ, and for
all n > 1 define Ξ[n] recursively by
[Λ,Ξ[n]] + nΞ[n] =
n−1∑
l=0
Ξ[l](fn−lΛ + gn−l(χ+ [Λ, χ])), (6.14)
where we write
(j(τ)− 984)
∆(τ)
E10(τ)
=
∞∑
n=0
fnq
n = 1 + 0q + 338328q2 + · · · ,
∆(τ)
E10(τ)
=
∞∑
n=0
gnq
n = q + 240q2 + 199044q3 + · · · .
Then the columns of the matrix
Ξ(τ) = qΛ
∞∑
n=0
Ξ[n]qn
are the desired generators. More precisely, any vvmf X(τ) ∈ M!
0(ρ) corresponds bijectively
to a vector-valued polynomial P (j(τ)) ∈ Cd[j(τ)], through X(τ) = Ξ(τ)P (j(τ)). This poly-
nomial can be read off from the principal part of X(τ), where principal part here means the
terms qΛ
∑
n≤0X[n]qn. In particular, the number of X[n] in the principal part of X(τ) coin-
cides with the dimension of M!
0(ρ), when dim(ρ) ≤ 5.7 Note that the definition for the principal
part employed in this method does not necessarily coincide with that in the Rademacher sum
method (cf. equation (6.12)); as opposed to the Rademacher case, here the principal part iso-
morphically identifies the vvmf. This method is called the Riemann–Hilbert method because
the recursion (6.14) comes from a Fuchsian differential equation solving the Riemann–Hilbert
problem for ρ on the sphere.
It remains to discuss the conditions on the pair of matrices Λ and χ. The method requires Λ
to be diagonal and to satisfy e(Λ) = ρ(T ) as well as
TrΛ = −7d
12
+
1
4
Tr ρ(S) +
2
3
√
3
Re
(
e(−1/12)Tr ρ
(
ST−1
))
. (6.15)
Additionally, the matrices Λ and χ must satisfy the equations
a2
(
a2 −
1
2
I
)
= 0,
a3
(
a3 −
1
3
I
)(
a3 −
2
3
I
)
= 0,
where a2 and a3 are d× d matrices given by
a2 = −31
72
Λ− 1
1728
(χ+ [Λ, χ]), a3 = −41
72
Λ +
1
1728
(χ+ [Λ, χ]).
Once a pair of matrices Λ and χ satisfying the above conditions is found, a basis for M!
0(ρ) can
be determined recursively.
7Additional subtleties must be taken into account for dim(ρ) > 5; we refer the reader to [39] for a careful
analysis of that case.
Modular Exercises for Four-Point Blocks – I 29
On the other hand, any X(τ) ∈ M!
0(ρ) whose components are linearly independent over C
gives us all of M!
0(ρ) (and hence also determines Λ and χ) via
M!
0(ρ) = C
[
j(τ),
E4E6
∆
D(0),
E2
4
∆
D2
(0),
E6
∆
D3
(0)
]
X(τ). (6.16)
Recall the space M0(ρ;λ) of vvmf defined in (6.5). We can assume e2πiλ = T with-
out loss of generality. The Riemann–Roch-like expression for the dimension of this space is
given in [39, equation (51)]. When ρ is irreducible and of dimension < 6, this simplifies to
(see [39, Proposition 3.3])
dimM0(ρ;λ) = max{0,Tr(Λ− λ) + d},
where Λ is any diagonal matrix satisfying e2πiΛ = T and (6.15).
One strength of this method is that, once successfully implemented, it determines d convenient
and explicit generators for M!
0(ρ); from them, arbitrarily many components of any X(τ) can
be rapidly computed with complete accuracy from the principal part of X(τ). In more general
contexts, the aforementioned subtlety occurring in the Rademacher method (namely that the
Rademacher sum procedure does not always yield a vvmf; it yields a vector-valued mock modular
form for some choices of polar parts) is avoided here because the principal part is defined with
respect to Λ. Moreover, this method does not assume ρ is unitary. The drawbacks are its
complexity and the fact that it is not always known how to explicitly determine Λ and χ.
At the present time, Λ, χ are both known whenever d = 2 (see [39, Section 4.2]), or when ρ
contains any congruence subgroup Γ(N), where the conductor N is a product of powers of
primes not exceeding 31 [36]. A nontrivial example of this method is included in Section 6.3.4.
6.3.3 Modular linear differential equation
The oldest available method (see, e.g., [4, 58, 60, 62]) is the method of Wronskians, now more
commonly known as the method of modular linear differential equations (MLDE). The idea is
that, if one has a d-dimensional vector-valued modular form X(τ) (which for the purposes of
our paper we take to be weakly holomorphic and of weight 0), then the vanishing of
det
f f ′ f ′′ · · · f (d)
X1 X ′
1 X ′′
1 · · · X
(d)
1
X2 X ′
2 X ′′
2 · · · X
(d)
2
...
...
Xd X ′
d X ′′
d · · · X
(d)
d
gives an order-d differential equation adf
(d)+ad−1f
(d−1)+ · · ·+a1f ′+a0f = 0 satisfied by the d
components Xj . The coefficients aj(τ) are quasi-modular (i.e., they also involves powers of the
second Eisenstein series E2(τ)), but we get something simpler by replacing d
dτ and its powers by
the modular derivative (6.3) and its powers. Replacing the dl
dτ l
in the Wronskian with Dl
(0) only
changes it by a factor of (2πi)d, so the expansion
bdD
d
(0)f + bd−1D
d−1
(0) f + · · ·+ b1D(0)f + b0f = 0,
where we have used the notation (6.4), is also satisfied by all components, but now the coef-
ficients bk(τ) will be weakly holomorphic modular forms of weight d(d + 1) − 2k with multi-
plier det(ρ), with principal parts controlled by the Xj . The spaces of such modular forms are
finite-dimensional, so partial information on the Xj (for instance the first few terms of their
30 M.C.N. Cheng, T. Gannon and G. Lockhart
q-expansions) will suffice to determine the coefficients bj(τ) exactly. We work out a nontrivial
example of this method in the next subsection on our hybrid method.
The strength of this method is its simplicity: any vvmf X(τ) is associated to a differential
equation with scalar modular forms as coefficients, and the solution space is the span over C
of the components Xj(τ). Furthermore, the representation ρ need not be unitary, and this
method can be generalized to any Fuchsian group. A drawback of this method is that, at best,
this method determines vector-valued modular forms which transform with some multiplier
equivalent to ρ; moreover it is unclear in general how one can obtain enough partial information
of the Xj to pin down the coefficients bj(τ).
We end this section by commenting on the role of modular differential equations in the context
of core blocks. It is very well known that conformal blocks involving primaries which have a null
state among their descendants at level-n is are solutions to an order-n differential equation (the
most famous examples being the BPZ equation [14] for Virasoro conformal blocks, and the KZ
equation in WZW models [47]). On the other hand, as we just reviewed, ℓ-dimensional vector-
valued modular forms are solutions to an MLDE of order ℓ; one therefore expects that differential
equations following from the presence of null states can be rephrased as MLDEs for the core
blocks. In Appendix A, we analyse explicitly this relation in the case of Virasoro blocks, and for
a null state at level 2 we rewrite the BPZ equation for the chiral blocks as a second-order MLDE.
6.3.4 A hybrid method
For simple examples, any of these three methods are effective. For more complicated methods,
where ρ does not contain a congruence group in the kernel, we propose a hybrid of the second
and third methods, which we argue is effective. We will describe this method with an example
we introduce in Section 8.1.2. There, we wish to find a core block with expansion8
φ(τ) = 2−
31
3
q−31/24(1 +O(q))
q−7/8
(
n(1,3) +O(q)
)
q11/24
(
n(1,5) +O(q)
)
q65/24
(
n(1,7) +O(q)
)
,
whose transformations under PSL2(Z) are given by
T =
e
(
− 7
24
)
0 0 0
0 e
(
−7
8
)
0 0
0 0 e
(
11
24
)
0
0 0 0 e
(
− 7
24
)
and
S =
1√
2
− 1√
6
√
1√
3
− 1
3
1√
3
1√
3√
1√
3
− 1
3
√
6
2 − 1√
2
0 −
√
2
3
(√
3− 1
)
1√
3
0 − 1√
2
1√
6
1√
3
−
√
2
3
(√
3− 1
)
1√
6
1√
2
−
√
6
3
.
The Rademacher sum method is not very helpful in this case, as it converges very slowly. This
is because the desired leading powers are not very equitable: one component starts with q−31/24,
whilst another starts with q65/24, which forces some leading coefficients to be very large for some
components, adversely affecting convergence.
8The overall factor of 2
31
3 guarantees that the vacuum chiral block is correctly normalized (see (8.3)).
Modular Exercises for Four-Point Blocks – I 31
Our hybrid method begins with the MLDE method. The components φi will be solutions
to the MLDE b0D
4
(0)f + b1D
3
(0)f + b2D
2
(0)f + b3D(0)f + b4f = 0. Here, bj is a (weakly holomor-
phic) modular form for SL2(Z) of weight 12+2j with leading term q1
(
since the products of the
components of φ has leading power q1
)
and trivial multiplier
(
since det(T ) = 1, which also
guarantees det(S) = det(T )−3 = 1
)
. Thus b0(τ) = a∆(τ), b1(τ) = 0, b2(τ) = bE4(τ)∆(τ),
b3(τ) = cE6(τ)∆(τ), and b4(τ) = dE4(τ)
2∆(τ). The constants a, b, c, d ∈ C satisfy the condition
that there are solutions fi(τ) to
aD4
(0)f + bE4(τ)D
2
(0)f + cE6(τ)D(0)f + dE4(τ)
2f = 0
with expansions f1(τ) = q−31/24(1 +O(q)), f2(τ) = q−7/8(1 +O(q)), f3(τ) = q11/24(1 +O(q)),
f4(τ) = q65/24(1+O(q)). Conveniently, there is a unique (up to rescaling by a constant) MLDE
with the required properties: take coefficients a = 1, b = −1381
288 , c = −1435
864 , d = 155155
110592 . This
uniquely determines the normalized solutions fi(τ), except that an arbitrary scalar multiple
of f4 can be added to f1. Choose any such f1.
We are not yet done, as until now we have completely ignored the S-transformation: the
MLDE method only retain information about det(ρ) and the solution f transforms according
to S̃ = P−1SP and T̃ = P−1T P under the S- and T -transformation. In the above we explicitly
choose f to so that T̃ = T , but it remains to determine S and from there the change of basis P .
In other words, we need to determine how the fi(τ) transform under τ 7→ −1/τ , as this will then
tell us how to obtain the desired φi from these fi. This is where the second (Riemann–Hilbert)
method appears. First, we need a diagonal matrix Λ compatible with T and with trace −2 (re-
call (6.15)): the choice Λ = diag
(
−31
24 ,−
15
8 ,−
13
24 ,
41
24
)
lets us recycle f as the first column of Ξ. The
other columns of the corresponding fundamental matrix Ξ can be obtained from f by repeatedly
applying to it the differential operators E4E6
∆ D(0),
E2
4
∆ D2
(0),
E6
∆ D3
(0) and taking linear combina-
tions over j(τ) (recall (6.16)). This step is just linear algebra. Each of these columns is simply
required to have a leading q-expansion compatible with leading term Ξ[0] = I. In particular,
the remaining three columns of Ξ are linear combinations of f , E4E6
∆ D(0)f ,
E2
4
∆ D2
(0)f ,
E6
∆ D3
(0)f .
It suffices to record their first nontrivial coefficients:
Ξ[1] = χ =
41633
21 −9068526720
2107 −93109120
189 −163826816
1161
1 −501885
301
520
9 −608440
11997
1 −577368
301 −5369
9 −121688
1935
1 −1994544
301 −1520
63
16031
387
.
Given χ and Λ, the full q-expansion of Ξ is easily obtained by the recursion (6.14). Again,
obtaining χ and hence Ξ is elementary if a little tedious.
The point is that Ξ(τ) = S̃−1 Ξ(−1/τ). Choose any τ0 close to but different from
√
−1,
then Ξ(τ0) will be invertible and S̃ = Ξ(−1/τ0)Ξ(τ0)
−1. So if we compute the first several terms
in the q-expansion of Ξ, and evaluate it at both τ0 and −1/τ0, then we get an approximation
of S̃. Both |e(τ0)| and |e(−1/τ0)| can be less than 0.002, so convergence of the q-expansion for Ξ
should be quite fast.
We know that
2
31
3 φ1(τ) = f1(τ) + αf4(τ),
2
31
3 φ2(τ) = βf2(τ),
2
31
3 φ3(τ) = γf3(τ),
2
31
3 φ4(τ) = δf4(τ),
32 M.C.N. Cheng, T. Gannon and G. Lockhart
for constants α, β, γ, δ ∈ C such that the corresponding φ(τ) transforms with respect to S.
Computing Ξ up to q9, and choosing τ0 = 1.0002i, we obtain n(1,1) = 1, n(1,3) ≈ 26.7262690,
n(1,5) ≈ 102595.534, n(1,7) ≈ 1.79592621× 109. We will see shortly how accurate this is.
This example illustrates the following point. The weakness of the second (Riemann–Hilbert)
method is in obtaining one vvmf, but that is provided by the third (MLDE) method. Once one
vvmf is obtained, the second method quickly determines all (equivalently, the generators Ξ).
The weakness of the third method is that it has no way to accurately and effectively determine
the S-matrix, but this can be obtained to whatever desired accuracy using Ξ. The hybrid
method successfully blends the strengths of the second and third methods.
Note that the eigenvalue e(−7/24) of T has multiplicity 2. If the dimension of an SL2(Z)-
representation is ≤ 5, and some eigenvalue of T has multiplicity > 1, then ρ will be reducible [71].
Here, P−1ρP is a direct sum of two 2-dimensional representations, where
P =
0
√
3
√
3− 3 0 2
√
3
3 0 0 0
0 0 3 0
0 −
√
6
√
3− 6 0
√
6
.
Because of this, we can solve this example exactly. From [39, Section 4.2] with the choices
Λ1 = diag
(
−15
8 ,
17
24
)
and Λ2 = diag
(
−13
24 ,−
7
24
)
, we obtain matrices Ξ1 and Ξ2, and the columns
of the matrix product P
(
Ξ1 0
0 Ξ2
)
are free generators (over the polynomial ring C[j(τ)]) of the
full space M!(ρ) of vvmf, which includes the desired φ. In this way we can quickly compute the
exact values
n(1,3) =
346921
32805
2
1
6
√
3 + 5/
√
3
Γ(7/12)3
πΓ(3/4)
≈ 26.72643460,
n(1,5) =
376960
9
3
3
4
Γ(11/12)Γ(7/12)
Γ(3/4)2
≈ 102595.517,
n(1,7) =
491455971328
387
√
2 ≈ 1.79592687× 109.
We see that the convergence of our hybrid method is quite fast.
7 Physical correlators
We now turn our attention to the correlation functions of the rational CFT, built out of the
physical operators which involve both chiral and anti-chiral degrees of freedom. Correlation
functions can always be expressed as sesquilinear combinations of chiral and anti-chiral blocks.
In a rational CFT, the physical Hilbert space H can be decomposed as
H =
⊕
M∈Υ
M̃∈Υ̃
ZM,M̃M ⊗ M̃,
where the sum is over the finitely many irreducible modules of the left- and right-moving chiral
algebras, V and Ṽ, and ZM,M̃ are non-negative integers corresponding to the multiplicities. Here
and everywhere else we use the tilde notation to indicate quantities associated to the right chiral
algebra. The multiplicities determine the modular invariant partition function of the RCFT:
ZT 2(τ, τ̄) =
∑
M,M̃
ZM,M̃χM (τ)χM̃ (τ),
Modular Exercises for Four-Point Blocks – I 33
where χM (τ) are the graded dimensions defined in (4.6). The modular invariance of ZT 2 (un-
der PSL2(Z)) requires the matrix ZM,M̃ to satisfy
ZM,M̃ =
(
SZS̃†)
M,M̃
=
(
TZT̃†)
M,M̃
, (7.1)
where S,T
(
resp. S̃, T̃
)
are the SL2(Z) modular transformation matrices of the chiral (anti-
chiral) sectors of the CFT. In particular, for the case V ∼= Ṽ, the unitary of the T and S
matrices imply that Z commutes with S and T.
It was shown in [64] that if V and Ṽ are the maximally-extended chiral algebras, then there
exists an isomorphism Ω of the fusion algebra of Ṽ with that of V such that
ZM,M̃ = δM,Ω(M̃). (7.2)
In fact, Ω is much more: it is a tensor equivalence between the corresponding modular tensor
categories of V and Ṽ, which means that it establishes equivalences between all mapping class
group representations, matches up intertwiners of V and Ṽ, etc. Although V and Ṽ do not have to
be isomorphic as VOAs, Ω establishes an equivalence of their combinatorial structures. Without
loss of generality, in the following we assume that V and Ṽ are maximally-extended chiral
algebras, which lead to simpler expressions.
In this paper, we will mainly focus on n-point correlation functions on the sphere. Recall the
discussion of chiral blocks on the sphere from Section 4. For each i = 1, . . . , n, fix any sectorMi⊗
Ω(Mi) ∈ Υ(V)×Υ(Ṽ) as in (7.2). Choose any quasi-primary states ϕi = ϕi ⊗ ϕ̃i ∈Mi ⊗ Ω(Mi).
We say ϕ is quasi-primary if it is annihilated by the Virasoro generator L1 and is an eigenvector
for L0, the eigenvalue being its conformal weight hϕi (similarly for ϕ̃). We write
〈
ϕ1(z1, z̄1) · · ·ϕn(zn, z̄n)
〉 n∏
i=1
dz
hϕi
i dz̄
hϕ̃i
i
for the associated correlator. This will be a sesquilinear combination of chiral blocks,
⟨1,Φ1(ϕ1, z1) ◦ · · · ◦ Φn(ϕn, zn)1⟩
〈
1, Φ̃1
(
ϕ̃1, z1
)
◦ · · · ◦ Φ̃n
(
ϕ̃n, zn
)
1
〉 n∏
i=1
dz
hϕi
i dz̄
hϕ̃i
i ,
where the intertwiners Φi and Φ̃i = Ω(Φi) run through a basis of YN
∗
i+1
Ni,Mi
and ỸΩ(Ni+1)
∗
Ω(Ni),Ω(Mi)
respectively.
In the rest of this section, we focus on the case where the states ϕi and ϕ̃i are V- and Ṽ-
primaries, i.e., states with conformal weight hMi and hΩ(Mi), respectively. As a result, we will
often label the correlators simply by the choices of the modules Mi, M̃i
(
as opposed to ϕi ∈Mi,
ϕ̃i ∈ M̃i
)
. In contrast with the chiral blocks, the correlation functions for the full CFT are
invariant under the action of the mapping class group. Importantly, this statement of crossing
symmetry also applies to non-rational 2d CFTs.
Consider first the two-point correlation functions. These are just given by
〈
ϕ1(z1, z̄1)ϕ
2(z2, z̄2)
〉 2∏
i=1
dz
hMi
i dz̄
hM̃i
i = ϕ2(ϕ1) ϕ̃2
(
ϕ̃1
)δM1,M∗
2
δM̃1,M̃∗
2
z
2hM1
12 z̄
2hM̃1
12
2∏
i=1
dz
hMi
i dz̄
hM̃i
i ,
and by a choice of normalization we can set
ϕ2(ϕ1) = ϕ̃2
(
ϕ̃1
)
= 1.
34 M.C.N. Cheng, T. Gannon and G. Lockhart
The two-point correlation functions are invariant under the action of the extended mapping class
group generated by the Dehn twist, which is just Z, if and only if
hMi = hΩ(Mi) mod Z.
Note that this is also required by the invariance of ZT 2 under τ 7→ τ + 1.
Next we look at three-point correlation functions. We can introduce physical vertex operators
which take the form
Φi
j,k(z, z̄) = dijkΦ
Mi
Mj ,Mk
(ϕk, z)Φ̃
M̃i
M̃j ,M̃k
(
ϕ̃k, z̄
)
, (7.3)
where
M̃i = Ω(Mi), M̃j = Ω(Mj), M̃k = Ω(Mk),
ΦMi
Mj ,Mk
∈ YMi
Mj ,Mk
, ΦMi
Mj ,Mk
= Ω
(
ΦMi
Mj ,Mk
)
∈ YM̃i
M̃j ,M̃k
,
and ϕk ∈Mk, ϕ̃k ∈ M̃k. Denote the module V ⊗ Ṽ by the label 0. The OPE coefficients satisfy,
among other things,
di0i = dii0 = 1 and d0i∗i = ±1,
where the sign can be shown to be always positive for unitary RCFTs [64]. Using d0i∗i to raise
and lower indices, we can then write the three-point correlator as follows (cf. (4.11)):
⟨Φ1(z1, z̄1)Φ2(z2, z̄2)Φ3(z3, z̄3)⟩
= ⟨0|Φ0
1,1∗(z1, z̄1)Φ
1∗
2,3(z2, z̄2)Φ
3
3,0(z3, z̄3)|0⟩
= d123FM1,M2,M3(z1, z2, z3)F̃M̃1,M̃2,M̃3
(z̄1, z̄2, z̄3)
=
( ∏
1≤i<j≤3
z
h−2hi−2hj
ij z̄
h̃−2hi−2hj
ij
)
d123φM1,M2,M3φ̃M̃1,M̃2,M̃3
. (7.4)
In the above, h = hM1+hM2+hM3 and similarly h̃ = hM̃1
+hM̃2
+hM̃3
. Notice that in principle one
has the freedom to rescale the left and right chiral structure constants φ, φ̃, while simultaneously
rescaling the OPE coefficients d, such that the three-point correlator (7.4) remains the same.
A natural choice for the normalisation of the intertwiners is to choose Φ
M∗
1
M2,M3
∈ YM
∗
1
M2,M3
such
that (cf. (4.11) and (4.12))
lim
u→∞
u2hM1FM1,M2,M3(u, z, 0) = ⟨M1|Φ(z)|M3⟩,
in which case one has φΦ
M1,M2,M3
= cΦM1,M2,M3
(4.12) and similarly for φ̃Φ̃
M̃1,M̃2,M̃3
. Finally, let
us turn to the four-point correlation functions; using the physical vertex operators, they can be
expanded as follows:9〈
4∏
i=1
ϕi(zi, z̄i)
〉
=
∑
p=P⊗P̃
d12p d
p
34F
P
M1,M2,M3,M4
(z1, z2, z3, z4)F̃ P̃
M̃1,M̃2,M̃3,M̃4
(z̄1, z̄2, z̄3, z̄4)
= z4−pt
∑
p=P⊗P̃
(A1234)PP̃ φ
P
M1,M2,M3,M4
(τ)φ̃ P̃
M̃1,M̃2,M̃3,M̃4
(τ̄), (7.5)
9For notational convenience, we will make the simplifying assumption that all NM1
P,M2
NP
M3,M4
≤ 1. More
generally, one needs to sum over inequivalent choices of physical vertex operators Φ (7.3).
Modular Exercises for Four-Point Blocks – I 35
where (cf. (4.16))
z4−pt :=
∏
1≤i<j≤4
z
µij
ij z̄
µ̃ij
ij
and we have defined the matrix A1234 by setting
(A1234)PP̃ := d12p d
p
34. (7.6)
In the above, p = P ⊗ P̃ runs through all internal channels, corresponding to physical states for
which the spaces of chiral and anti-chiral blocks both have nonzero dimension. Based on the
zi-dependence of the correlator (7.5), it is natural to define the core correlator
Fϕ1,ϕ2,ϕ3,ϕ4(τ, τ̄) :=
〈∏4
i=1 ϕ
i(zi, z̄i)
〉
z4−pt
. (7.7)
The fact that the physical correlators are single-valued functions of the location zi of the in-
sertions, combined with the above and the identification (6.11) between the λ-function and the
cross ratio, then immediately leads to the fact that
〈∏4
i=1 ϕ
i(zi, z̄i)
〉
/z4−pt is a non-holomorphic
function on H invariant under the action of PΓ(2), and is moreover invariant under PΓ0(2) when
two of the four insertions are identical, and under the full modular group PSL2(Z) when at least
three insertions are.
More generally, one has the following relation analogous to (7.1) for the torus partition
functions:
Theorem 7.1. The OPE coefficients (7.6) satisfy
A1234 = (ρT )
−1A1243ρ̃T = (ρS)
−1A1423ρ̃S ,
where ρT and ρS are as given in (5.6) and (5.7) and similarly for ρ̃.
Proof. Alternatively to the “s-channel” expansion above, we can choose to expand the corre-
lator in the “t-channel” and “u-channel” and obtain
Fϕ1,ϕ2,ϕ3,ϕ4(τ, τ̄) =
∑
p=P⊗P̃
(A1234)PP̃φ
P
M1,M2,M3,M4
(τ)φ̃P̃
M̃1,M̃2,M̃3,M̃4
(τ̄)
=
∑
p=P⊗P̃
(A1243)PP̃φ
P
M1,M2,M4,M3
(τ + 1)φ̃P̃
M̃1,M̃2,M̃3,M̃4
(τ̄ + 1)
=
∑
p=P⊗P̃
(A1423)PP̃φ
P
M1,M4,M3,M2
(
−1
τ
)
φ̃P̃
M̃1,M̃4,M̃3,M̃2
(
−1
τ̄
)
.
The desired statement then follows from the unitarity of ρ and ρ̃, and upon applying (5.5). ■
Corollary 7.2. The core correlator satisfies
Fϕ1,ϕ2,ϕ3,ϕ4(τ, τ̄) = Fϕ1,ϕ2,ϕ3,ϕ4
(
aτ + b
cτ + d
,
aτ + b
cτ + d
)
for all
(
a b
c d
)
∈ Γ(2). Moreover, the above holds for all
(
a b
c d
)
∈ SL2(Z) if ϕ2 = ϕ3 = ϕ4.
36 M.C.N. Cheng, T. Gannon and G. Lockhart
8 Examples
In this section, we present a few examples to illustrate the theorems in the main text. While
we organize the examples into various classes, we stress that our techniques apply uniformly
to all examples, regardless of the class of theories they belong to. For all examples we choose,
the conformal blocks will turn out to be elements of a one-dimensional space of vvmf’s, and
therefore we will be able to determine them with the braiding matrices of the CFT as the sole
input. More general examples and more detailed analysis will be presented in the companion
paper [3].
8.1 Virasoro minimal models
Our first class of examples arises in Virasoro minimal models. It should be emphasized that these
are misleadingly simple: there is no difference between Virasoro-primary and VOA-primaries,
the spaces of VOA-primaries are all 1-dimensional, and all fusion coefficients are either 0 or 1;
it is not necessary for these properties to hold in order for our methods to be applicable.
Recall that Virasoro minimal models are parametrized by two positive integers p > p′.
The left and right chiral algebras are just two copies of the Virasoro algebra at central charge
c = 1− 6
(p− p′)2
pp′
,
possibly extended. More precisely, the left and right chiral algebras are isomorphic to the
canonical VOA structure on the irreducible Virasoro module with h = 0 and the given c. We
denote this chiral algebra as M(p, p′). A minimal model is unitary when p = p′ + 1.
The minimal model M(p, p′) possesses the following collection of irreducible modules:
Mr,s with 1 ≤ r < p′, 1 ≤ s < p, (8.1)
subject to the identification Mr,s =Mp′−r,p−s. The conformal weight of Mr,s is given by
hr,s =
(pr − p′s)2 − (p− p′)2
4p′p
.
Furthermore one has Mr,s = M∗
r,s, that is, the charge conjugation matrix C is the identity
matrix. We denote the corresponding Virasoro primaries by ϕr,s = ϕp′−r,p−s.
The fusion rules for the M(p, p′) minimal model are given as follows:
Mr,s ⊗Mm,n =
kmax⊕
k=1+|r−m|
k+r+m=1 mod 2
lmax⊕
l=1+|s−n|
l+s+n=1 mod 2
Mk,l, (8.2)
where
kmax = min(r +m− 1, 2p′ − 1− r −m),
lmax = min(s+ n− 1, 2p− 1− s− n).
In particular, all fusion coefficients are either 0 or 1.
Explicit expressions are known for the braiding and fusing matrices of the minimal mod-
els [27, 28]. In the following examples, we employed the Mathematica code provided in [30]
to compute the braiding matrices required to determine the PSL2(Z) matrices SD.
For the M(p, p′), clearly chiral blocks and conformal blocks coincide. It hence suffices to
consider here the insertions to be ϕi = ϕri,si . Virasoro conformal blocks have been determined
Modular Exercises for Four-Point Blocks – I 37
by a variety of techniques, including by making explicit use of the Virasoro algebra, by exploiting
the Zamolodchikov recursion relations [76], or for minimal models by solving the BPZ equation
(see Appendix A). More recently, Perlmutter has found an explicit solution to Zamolodchikov’s
recursion relations [68] which provides expressions at arbitrary orders in the perturbation series
of the Virasoro conformal blocks. For the discussion that follows, it will be sufficient to know the
leading order term in the q-expansion of the core blocks. From (6.8) and (5.1) we immediately
see that
φNϕ1,ϕ2,ϕ3,ϕ4(w) = Cq−
h
6
+
hN
2 (1 +O(q)), (8.3)
where C = cM1,M2,NcN∗,M3,M42
8(−h
6
+
hN
2
), and Mi =Mri,si and N =MrN ,sN .
The examples below demonstrate how the technology developed in the previous sections
efficiently determines the Virasoro blocks in terms of vector-valued modular forms, often leading
to exact, analytic expressions.
8.1.1 M(4, 3)
The Ising model M(4, 3) has three independent primaries, whose properties are summarized in
the following table:
Primary (r, s) h
1 (1, 1) 0
σ (2, 2) 1
16
ϵ (2, 1) 1
2
Given this chiral theory, the only modular-invariant way to combine left- and right-movers is
to take the diagonal theory:
ZT 2 = |χ1|2 + |χσ|2 + |χϵ|2.
We can arrange all nontrivial core blocks into three vector-valued modular forms:
1. D = (ϵ, ϵ, ϵ, ϵ). The space of core blocks is one-dimensional and has the following leading
order behavior:
φ1
ϵ,ϵ,ϵ,ϵ(τ) = q−
1
3 (N1 +O(q)).
It transforms as a one-dimensional representation of PSL2(Z) with
T =
(
e
(
−1
3
))
, S = (1).
The space of vvmf’s transforming under this representation, and with prescribed singular
behavior, M!
0
(
ρD,−1
3
)
, is itself one-dimensional. A straightforward way to see this is by
applying the method of Section 6: from equation (6.15), we obtain
TrΛ = −1
3
=⇒ Λ =
(
−1
3
)
.
Therefore, the principal part of φ1
ϵ,ϵ,ϵ,ϵ is
PΛφ
1
ϵ,ϵ,ϵ,ϵ(τ) = q−
1
3N1
38 M.C.N. Cheng, T. Gannon and G. Lockhart
and so φ1
ϵ,ϵ,ϵ,ϵ is uniquely determined by its normalisation. The generator of M!
0(ρD) is
a famous modular form:
j(τ)
1
3 =
E4(τ)
η(τ)8
= q−
1
3
(
1 + 248q + 4124q2 + 34752q3 + · · ·
)
= χ
E8,1
1 (τ),
which coincides with the unique character χ
E8,1
1 (τ) at level 1 of the E8 Kac–Moody algebra
corresponding to the trivial representation 1. As we will see in Section 9 and in more details
in [3], this is not a mere coincidence.
A straightforward application of the Rademacher sum (where we restrict the sum to
c ≤ 500) reproduces this result to very good approximation:
φ1
ϵ,ϵ,ϵ,ϵ(τ) ≃ N1q
− 1
3
(
1.001 + 247.991q + 4124.003q2 + 34752.002q3 + · · ·
)
.
The normalization of the conformal block is uniquely fixed according to equation (8.3):
φ1
ϵ,ϵ,ϵ,ϵ(τ) = 2−8/3c1ϵϵc
1
ϵϵj(τ)
1/3 =
(
j(τ)
256
)1/3
,
where the structure constants c1ϵϵ = c1ϵϵ = 1 are fixed by the normalization of the ϵ
primary. Indeed, if we re-write the core block as a function of the cross-ratio w = λ(τ)
using equation (6.10) we recover the well-known expression for the conformal block [2]:
F1
ϵ,ϵ,ϵ,ϵ(z1, z2, z3, z4) =
(∏
i<j
z
− 1
3
ij
)
1− w + w2
w2/3(1− w)2/3
. (8.4)
For the physical correlator (for the diagonal theory), one obtains
⟨ϵ(z1, z̄1) · · · ϵ(z4, z̄4)⟩ = 2−
16
3
(∏
i<j
|zij |−
2
3
)
Z
E8,1
T 2 (τ, τ̄).
In other words, the corresponding core correlator coincides up to a numerical prefactor
with the torus partition function of the level-1 E8 WZW model (cf. (7.7)):
Fϵ,ϵ,ϵ,ϵ(τ, τ̄) = 2−
16
3 Z
E8,1
T 2 (τ, τ) = 2−
16
3
∣∣χE8,1
1 (τ)
∣∣2.
2. D = (σ, σ, σ, σ). In this case, the core blocks assemble into a two-dimensional vvmf:
φ(τ) =
(
φ1
σ,σ,σ,σ(τ)
φϵσ,σ,σ,σ(τ)
)
=
(
q−
1
24 (N1 +O(q))
q
5
24 (Nϵ +O(q))
)
.
The vvmf transforms as a representation ρD of PSL2(Z) determined by the matrices
TD =
(
e
(
− 1
24
)
0
0 e
(
5
24
)) , SD =
1√
2
(
1 1
1 −1
)
.
Again, according to the discussion in Section 6, dim
(
M!
0
(
ρD,
(
− 1
24 ,
5
24
)))
= 1: indeed,
equation (6.15) gives
TrΛ = −5
6
.
Modular Exercises for Four-Point Blocks – I 39
We can therefore pick
Λ =
(
− 1
24 0
0 −19
24
)
so that
PΛφ(τ) = e(Λτ)
(
N1
0
)
,
and again the vvmf is determined up to an overall factor. The single basis vector of
M!
0
(
ρD,
(
− 1
24 ,
5
24
))
is a well-known vector-valued modular form: its components are the
two characters at level 1 of the A1 Kac–Moody algebra corresponding to the trivial (1)
and fundamental (2) representations:
(
χ
A1,1
1 (τ)
χ
A1,1
2 (τ)
)
=
θ3(2τ)
η(τ)
θ2(2τ)
η(τ)
=
(
q−
1
24
(
1 + 3q + 4q2 + 7q3 + · · ·
)
q
5
24
(
2 + 2q + 6q2 + 8q3 + · · ·
) ) .
As a result, the normalization of the core blocks is again determined straightforwardly by
looking at the vacuum channel
φ1
σ,σ,σ,σ(τ) = 2−
1
3 q−
1
24 (1 +O(q)),
which gives
φ(τ) = 2−
1
3
(
χ
A1,1
1 (τ)
χ
A1,1
2 (τ)
)
.
Again, we could have reproduced this result to good approximation by performing the
Rademacher sum up to c ≤ 500, which gives
φ(τ) ≃ 2−1/3
(
q−
1
24
(
1.002 + 2.999q + 4.001q2 + 7.000q3 + · · ·
)
q
5
24
(
2.001 + 2.000q + 5.998q2 + 7.999q3 + · · ·
) ). (8.5)
Using equation (8.5), from the second entry of φ(τ) one obtains the well-known value (up
to a sign) of the structure constant
(cσσϵ)
2 =
1
2
.
It can be checked that the core blocks we obtained are equivalent to the standard expres-
sions for the conformal blocks in terms of the cross-ratio w:
F1
σ,σ,σ,σ(z1, z2, z3, z4) =
(∏
i<j
z
− 1
24
ij
) (
1 +
√
1− w
) 1
2
√
2w− 1
12 (1− w)−
1
12
,
F ϵ
σ,σ,σ,σ(z1, z2, z3, z4) =
(∏
i<j
z
− 1
24
ij
) (
1−
√
1− w
) 1
2
√
2w− 1
12 (1− w)−
1
12
. (8.6)
Finally, we note that the physical correlator
⟨σ(z1, z̄1) · · ·σ(z4, z̄4)⟩ = 2−
2
3
( ∏
1≤i<j≤4
|zij |−
1
12
)
Z
A1,1
T 2 (τ, τ)
coincides, up to a scalar prefactor, to the torus partition function of the level-1 A1 WZW
model:
Z
A1,1
T 2 (τ, τ) =
∣∣χA1,1
1
∣∣2 + ∣∣χA1,1
2
∣∣2.
40 M.C.N. Cheng, T. Gannon and G. Lockhart
3. D = (σ, σ, ϵ, ϵ). The last set of core blocks involves unequal external operators, and
therefore according to the discussion in case (B) of Section 5.3 and to Appendix C they
can be organized in terms of the following vvmf for PSL2(Z):
φ(τ) =
φ1
σ,σ,ϵ,ϵ(τ)
φσσ,ϵ,σ,ϵ(τ)
φσσ,ϵ,ϵ,σ(τ)
=
q−
3
16 (N1 +O(q))
q−
5
32 (Nσ +O(q))
q−
5
32 (N ′
σ +O(q))
.
The corresponding PSL2(Z) representation has
TD =
e
(
− 3
16
)
0 0
0 0 e
(
− 5
32
)
0 e
(
− 5
32
)
0
and SD =
0 0 1
0 −1 0
1 0 0
.
To apply the methods of Section 6.3.2, one first needs to diagonalize TD, which leads to
the equivalent representation ρ′D for which
T ′
D =
e
(
− 3
16
)
0 0
0 e
(
11
32
)
0
0 0 e
(
− 5
32
)
and S ′
D =
1
2
0
√
2
√
2√
2 −1 1√
2 1 −1
.
Equation (6.15) then implies that dim
(
M!
0
(
ρ′D,
(
− 3
16 ,
11
32 ,−
5
32
)))
= 1.
By using the explicit transformation properties of the theta functions, equations (6.9)
and (6.2), one can easily check that the core blocks are given in terms of the following
expression:
φ(τ) = 2−
3
2
(
2η(τ)
θ2(τ)
) 1
4 θ3(τ)
4 + θ4(τ)
4
2η(τ)4(
2η(τ)
θ3(τ)
) 1
4 −θ2(τ)4 + θ4(τ)
4
2η(τ)4(
2η(τ)
θ4(τ)
) 1
4 θ2(τ)
4 + θ3(τ)
4
2η(τ)4
, (8.7)
where the normalization is again fixed by imposing the correct normalization for the vac-
uum channel. Note that the fractional powers here and elsewhere are not a problem, as
explained in Section 6.1.2.
The expressions appearing in (8.7) are equivalent to the well-known expressions for the
conformal blocks [2]
φ1
σ,σ,ϵ,ϵ(w) =
1− w/2
w3/8(1− w)5/16
,
φσσ,ϵ,σ,ϵ(w) =
1− 2w
2(w(1− w))5/16
,
φσσ,ϵ,ϵ,σ(w) =
1 + w
2w5/16(1− w)3/8
. (8.8)
The corresponding physical correlator is given simply by
⟨σ(z1, z̄1)σ(z2, z̄2)ϵ(z3, z̄3)ϵ(z4, z̄4)⟩ = |z4−pt|2φ1
σ,σ,ϵ,ϵ(w)φ
1
σ,σ,ϵ,ϵ(w),
and similarly for other permutations of external operators.
Modular Exercises for Four-Point Blocks – I 41
This example can be handled more directly as a one-dimensional Γ0(2) vvmf. Recall the
discussion in Section 6.1.3. The PΓ0(2) representation here is ρe(−3/16),−1, and we require
leading power to be q−3/16 near i∞ and leading power to be q
−5/16
0 near τ = 0, where q0 =
e(−πi/τ) is the local coordinate for Γ0(2) at τ = 0 (which is the square-root of the local
coordinate there for PSL2(Z)). The core block will be κ−5/16
√
κ+ 1
4 times some Laurent
polynomial in κ. The leading power of q is correct in κ−5/16
√
κ+ 1
4 , so no positive powers
of κ can appear in the polynomial. Likewise, the leading power of q0 is also correct, so
no negative powers of κ can appear either. Thus the polynomial is constant, and the core
block is a scalar multiple of κ−5/16
√
κ+ 1
4 . The reader can verify this agrees with the first
component of φ appearing in (8.7).
Finally we remark that it is accidental that the expressions such as (8.4), (8.6) and (8.8) for
conformal blocks are so simple as functions of the cross-ratio w, and does not hold for larger
values of p and p′. In general, the expressions as vvmf in τ will be much more accessible than
the multivalued functions of w.
8.1.2 M(12, 11)
Our next example highlights the construction of physical correlators out of core blocks. We con-
sider the M(12, 11) Virasoro minimal model and focus on the correlators associated to the chiral
datum D = (ϕ1,4, ϕ1,4, ϕ1,4, ϕ1,4), in the notation of (8.1). An important feature of this example
is that there exists more than one crossing-symmetry-invariant physical correlator that can be
constructed by combining left- and right-movers in inequivalent ways (corresponding to inequiv-
alent physical theories). As discussed in [56], in such cases the method of modular averages
cannot be applied straightforwardly. On the other hand, working at the chiral level we will
find that the conformal blocks are once again completely determined by their PSL2(Z) trans-
formation, and we will find that in each of the two physical theories built by combining left-
and right-movers the physical correlators are also uniquely determined by imposing modular
invariance on them. Write χr,s(τ) for the character (graded dimension) of Mr,s. According to
the classification of [16, 17], one can construct three inequivalent physical theories by combining
two copies of M(12, 11): the diagonal one of type (A10, A11), which has torus partition function
ZT 2(τ, τ̄) =
1
2
10∑
r=1
11∑
s=1
|χr,s(τ)|2;
the one of type (A10, D7), which has
ZT 2(τ, τ̄) =
10∑
r=1
∑
s∈{1,3,5}
|χr,s(τ)|2 +
10∑
r=1
∑
s∈{2,4,6}
χr,s(τ)χr,12−s(τ);
and the one of type (A10, E6), which has
ZT 2(τ, τ̄) =
1
2
10∑
r=1
(
|χr,1(τ) + χr,7(τ)|2 + |χr,4(τ) + χr,8(τ)|2 + |χr,5(τ) + χr,11(τ)|2
)
.
As in [56], we consider the study the (non-chiral) four-point function corresponding to the chiral
datum D = (ϕ1,4, ϕ1,4, ϕ1,4, ϕ1,4)
⟨ϕe(z1, z̄1)ϕe(z2, z̄2)ϕe(z3, z̄3)ϕe(z4, z̄4)⟩,
where
ϕe(z, z̄) ≡ ϕ1,4(z)ϕ1,4(z̄).
42 M.C.N. Cheng, T. Gannon and G. Lockhart
This four point function only arises in the (A10, A11) and (A10, E6) models, since the field ϕe(z, z̄)
is not part of the spectrum of the (A10, D7) model.
Applying (8.2) gives
M1,4 ⊗M1,4 =M1,1 ⊕M1,3 ⊕M1,5 ⊕M1,7.
At the level of chiral theory, there is therefore a four-dimensional vector of core blocks:
φ(τ) =
φ
M1,1
ϕ1,4,ϕ1,4,ϕ1,4,ϕ1,4
(τ)
φ
M1,3
ϕ1,4,ϕ1,4,ϕ1,4,ϕ1,4
(τ)
φ
M1,5
ϕ1,4,ϕ1,4,ϕ1,4,ϕ1,4
(τ)
φ
M1,7
ϕ1,4,ϕ1,4,ϕ1,4,ϕ1,4
(τ)
=
q−
31
24
(
N(1,1) +O(q)
)
q−
7
8
(
N(1,3) +O(q)
)
q
11
24
(
N(1,5) +O(q)
)
q
65
24
(
N(1,7) +O(q)
)
,
whose transformations under PSL2(Z) are given in terms of
TD =
e
(
− 7
24
)
0 0 0
0 e
(
−7
8
)
0 0
0 0 e
(
11
24
)
0
0 0 0 e
(
− 7
24
)
and
SD =
1√
2
− 1√
6
√
1√
3
− 1
3
1√
3
1√
3√
1√
3
− 1
3
√
6
2 − 1√
2
0 −
√
2
3
(√
3− 1
)
1√
3
0 − 1√
2
1√
6
1√
3
−
√
2
3
(√
3− 1
)
1√
6
1√
2
−
√
6
3
.
The method of Section 6.3.2 shows that the space of vvmf’s transforming under this PSL2(Z)
representation and with the appropriate singular behavior is once again one-dimensional; as
discussed in Section 6.3.4, one may then efficiently determine the core blocks either recursively
by our hybrid technique, or exactly by noticing that it corresponds to a reducible PSL2(Z)
representation. The goal of this example, however, is to illustrate the construction of PSL2(Z)-
invariant four-point functions.
In the diagonal theory, the four-point function is the usual
⟨ϕe(z1, z̄1)ϕe(z2, z̄2)ϕe(z3, z̄3)ϕe(z4, z̄4)⟩(A10,A10) = |z4−pt|2
3∑
i=0
∣∣φM1,1+2i
ϕ1,4,ϕ1,4,ϕ1,4,ϕ1,4
(τ)
∣∣2,
which is obviously modular invariant.
On the other hand, in the non-diagonal (A10, E6) theory, only the following internal channels
appear:
p1 =M1,1 × M̃1,1, p2 =M1,5 × M̃1,5, p3 =M1,3 × M̃1,7,
p4 =M1,7 × M̃1,3, p5 =M1,7 × M̃1,7,
so we must find a different modular-invariant four-point function that only involves these con-
formal families. In other words, we ask that the four-point correlator take the form
⟨ϕe(z1, z̄1)ϕe(z2, z̄2)ϕe(z3, z̄3)ϕe(z4, z̄4)⟩(A10,E6) = |z4−pt|2 φ(τ) ·D · φ(τ),
Modular Exercises for Four-Point Blocks – I 43
where
D =
1 0 0 0
0 0 0 d2eep4
0 0 d2eep2 0
0 d2eep3 0 d2eep5
.
Invariance under TD is automatic. On the other hand, invariance under SD implies that
S−1
D ·D · SD = D,
which determines for us the OPE coefficients
d2eep2 =
3
2
, d2eep3 = d2eep4 =
√
1
2
, d2eep5 =
1
2
.
8.1.3 M(10, 7)
Our next example is the non-unitary minimal model M(10, 7), whose central charge is c = 8/35.
Choosing chiral datum D = (ϕ1,2, ϕ1,2, ϕ1,2, ϕ1,2), we obtain two core blocks:
φ(τ) =
φM1,1
ϕ1,2,ϕ1,2,ϕ1,2,ϕ1,2
(τ)
φ
M1,3
ϕ1,2,ϕ1,2,ϕ1,2,ϕ1,2
(τ)
=
(
q−
1
60
(
N(1,1) +O(q)
)
q
11
60
(
N(1,3) +O(q)
) ) .
The PSL2(Z) data is
TD =
(
e
(
− 1
60
)
0
0 e
(
11
60
)) and SD =
√
5+
√
5
10
√
5−
√
5
10√
5−
√
5
10 −
√
5+
√
5
10
.
The space of vvmf’s M!
0
(
ρD,
(
− 1
60 ,
11
60
))
is one-dimensional. The core blocks are solutions of
the two-dimensional MLDE (A.1) with µ = 11/900. That is, they coincide, up to a numerical
prefactor, with the level-1 characters of the A 1
2
, the intermediate vertex subalgebra [70] of A1:
φ(τ) = 2
22
15
χ1
A 1
2 ,1
(τ)
χ1′
A 1
2 ,1
(τ)
= 2
2
15
(
q−
1
60
(
1 + q + q2 + q3 + 2q4 + 2q5 + 3q6 +O
(
q7
))
q
11
60
(
1 + q2 + q3 + q4 + q5 + 2q6 +O
(
q7
)) )
.
8.1.4 M(13, 10)
Finally we look at the non-unitary minimal model M(13, 10), of central charge c = 38/65. Again
we choose chiral datum D = (ϕ2,1, ϕ2,1, ϕ2,1, ϕ2,1) and obtain two core blocks:
φ(τ) =
φM1,1
ϕ2,1,ϕ2,1,ϕ2,1,ϕ2,1
(τ)
φ
M3,1
ϕ2,1,ϕ2,1,ϕ2,1,ϕ2,1
(τ)
=
(
q−
19
60
(
N(1,1) +O(q)
)
q
29
60
(
N(3,1) +O(q)
) ) .
The PSL2(Z) data is
TD =
(
e
(
−19
60
)
0
0 e
(
29
60
)) and SD =
√
5+
√
5
10
√
5−
√
5
10√
5−
√
5
10 −
√
5+
√
5
10
.
44 M.C.N. Cheng, T. Gannon and G. Lockhart
The space of vvmf’s M!
0
(
ρD,
(
−19
60 ,
29
60
))
is one-dimensional. The core blocks are solutions of
the two-dimensional MDE (A.1) with µ = 551/900, and coincide with the level-1 characters of
the E7+ 1
2
intermediate vertex subalgebra of E8 as given in [46]! That is,
φ(τ) = 2−
38
15
χ1
E
7+1
2 ,1
(τ)
χ57
E
7+1
2 ,1
(τ)
= 2−
38
15
(
q−
19
60
(
1 + 190q + 2831q2 + 22306q3 + 129276q4 + 611724q5 +O
(
q6
))
q
29
60
(
57 + 1102q + 9367q2 + 57362q3 + 280459q4 + 1181838q5 +O
(
q6
))) .
8.2 RCFTs at c = 1
For our next example, we consider the Z2-orbifold of the compact boson at radius R =
√
2N ,
for N ∈ Z. This family of models possesses the following spectrum of chiral algebra pri-
maries [26]:
hΦ Φ
0 1
1 j
N
4 ϕiN i = 1, 2
k2
4N ϕk k = 1, . . . , N − 1
1
16 σi i = 1, 2
9
16 τi i = 1, 2
We will consider the case where N is even, in which case the S-matrix for this CFT is
given by10
S = 1√
8N
1 j ϕ1N ϕ2N ϕk′ σ1 σ2 τ1 τ2
1 1 1 1 2
√
N
√
N
√
N
√
N 1
1 1 1 1 2 −
√
N −
√
N −
√
N −
√
N j
1 1 1 1 2(−1)k
′ √
N −
√
N
√
N −
√
N ϕ1N
1 1 1 1 2(−1)k
′ −
√
N
√
N −
√
N
√
N ϕ2N
2 2 2(−1)k 2(−1)k 4 cos
(
πkk′
N
)
0 0 0 0 ϕk√
N −
√
N
√
N −
√
N 0
√
2N 0 −
√
2N 0 σ1√
N −
√
N −
√
N
√
N 0 0
√
2N 0 −
√
2N σ2√
N −
√
N
√
N −
√
N 0 −
√
2N 0
√
2N 0 τ1√
N −
√
N −
√
N
√
N 0 0 −
√
2N 0
√
2N τ2
For definiteness, let us focus on the four-point functions between four ϕN/2 operators:
D = (ϕN/2, ϕN/2, ϕN/2, ϕN/2).
From the S-matrix and (4.4), one obtains the fusion rule
ϕN/2 × ϕN/2 = 1+ j + ϕ1N + ϕ2N ,
so the chiral blocks have four internal channels. (In the above equation, by slight abuse of
notation we denote the modules by the same name as the primaries.) However, as the two
10We have corrected a factor of 2 with respect to [26].
Modular Exercises for Four-Point Blocks – I 45
operators ϕ1N and ϕ2N are charge conjugate, the corresponding conformal blocks will be identical.
We can therefore assemble the core blocks into a three dimensional representation of PSL2(Z):
φ =
φ1
ϕN/2,ϕN/2,ϕN/2,ϕN/2
(τ)
φjϕN/2,ϕN/2,ϕN/2,ϕN/2
(τ)
φ
ϕiN
ϕN/2,ϕN/2,ϕN/2,ϕN/2
(τ)
.
The following T-transformation for the core blocks is readily computed:
TD =
e
(
−N
24
)
0 0
0 e
(
−N
24 + 1
2
)
0
0 0 e
(
−N
24 + N
8
)
.
As dimφ ≤ 5, SD is also uniquely determined and is given by
SD =
1
2
1 1
√
2
1 1 −
√
2√
2 −
√
2 0
which indeed satisfies
(SD · TD)3 = SD · SD = 1.
The space of vvmf’s associated to this class of representations is one-dimensional. We find that
it is given by
φ = 2−
N
3
1
2
(
θ3(τ)
N
η(τ)N
+
θ4(τ)
N
η(τ)N
)
1
2
(
θ3(τ)
N
η(τ)N
− θ4(τ)
N
η(τ)N
)
1√
2
θ2(τ)
N
η(τ)N
= 2−
N
3
χ
DN,1
1 (τ)
χ
DN,1
N (τ)
√
2χ
DN,1
S (τ)
, (8.9)
in other words, we find that up to normalization they coincide with the characters χ
DN,1
R (τ) of
the DN = SO(2N) Kac–Moody algebra at level 1. More precisely, the DN,1 Kac–Moody algebra
possesses four inequivalent primaries, corresponding to the trivial (1), vector (N), spinor (S+)
and conjugate spinor (S−)) representations, but the (unrefined) characters of the two spinor
representations coincide:
χ
DN,1
S+ (τ) = χ
DN,1
S− (τ) = χ
DN,1
S (τ),
allowing us to organize them into a three-dimensional representation of PSL2(Z), in complete
analogy with the sphere core blocks.
8.3 WZW models
We illustrate our techniques by applying them to the G = SU(2) WZW model at level 1.
This model admits a free-field realization in terms of a compact chiral boson ϕ(x). The chiral
algebra is generated by the operators H = i∂ϕ and E± = :e±i
√
2ϕ:, which have the following
non-vanishing OPE’s:
H(z)H(w) ∼ 1
(z − w)2
, H(z)E±(w) ∼ ±
√
2E±(w)
z − w
,
E+(z)E−(w) ∼ 1
(z − w)2
+
√
2H(w)
z − w
.
46 M.C.N. Cheng, T. Gannon and G. Lockhart
For a generic group G, the chiral algebra primaries are labeled by highest weights ω of the
Lie algebra of G. The conformal weight of a primary is given by
hω =
C2(ω)
h∨G + k
,
where C2(ω) is the Casimir invariant of the representation of highest weight ω. For a chiral
block to be nonzero, the tensor product of the corresponding representations must contain a
singlet.
For G = SU(2), at level 1 the only primaries are the vacuum and the WZW-primary
ξω1 ≡ :e
i√
2
ϕ
: corresponding to the highest weight ω1 of the fundamental representation of SU(2).
The core block corresponding to the chiral datum D = (ξω1 , ξω1 , ξω1 , ξω1) would be a (scalar)
PSL2(Z) modular form with TD =
(
e
(
−1
6
))
, since the leading order q-power in the corre-
sponding core blocks is q−
1
6 ; however, no such modular form exists (see the discussion around
equation (6.6)), and therefore the conformal block vanishes. In order to obtain non-vanishing
blocks, we must take some of the external operators to be the operator ξ−ω1 = E−ξω1 , which is
a Virasoro-primary but not a WZW-primary.
The core block D = (ξ−ω1 , ξω1 , ξω1 , ξω1) still vanishes for the same reason; on the other hand,
for D = (ξω1 , ξω1 , ξ−ω1 , ξ−ω1) one obtains the three-dimensional PSL2(Z) representation
φ(τ) =
φ1
ξω1 ,ξω1 ,ξ−ω1 ,ξ−ω1
φ1
ξω1 ,ξ−ω1 ,ξω1 ,ξ−ω1
φ1
ξω1 ,ξ−ω1 ,ξ−ω1 ,ξω1
(τ)
(alternatively, each component of φ(τ) can be viewed as scalar-valued modular form for Γ(2)).
Determining the modular transformation properties requires some care. The OPE between ξω1
and ξ−ω1 contains the identity operator, and as a result the leading order q-power in the
core blocks φ1
ξω1 ,ξ−ω1 ,ξω1 ,ξ−ω1
(τ) is q−
1
6
(4hω1 ) = q−
1
6 . On the other hand, the OPE of ξ±ω1 with
itself contains descendants of the identity but not the identity itself:
ξ±ω1(z)ξ±ω1(w) = E±(z)(z − w)
1
2 +O
(
(z − w)
3
2
)
.
As a consequence, for the core block φ1
ξω1 ,ξω1 ,ξ−ω1 ,ξ−ω1
(τ) the operator of lowest dimension ap-
pearing in the vacuum channel is the operator E+ which has conformal weight 1. With this
information, we obtain
TD =
e
(
1
3
)
0 0
0 0 e
(
−1
6
)
0 e
(
−1
6
)
0
.
As the dimension of the representation is less than 6, there is a unique SD compatible with TD.
This is easily found to be
SD =
0 0 1
0 1 0
1 0 0
.
The relevant space of vvmf’s is again one-dimensional, as can be checked by applying equa-
tion (6.15) (after applying the appropriate conjugation to TD and SD to make TD diagonal).
It is easy to recognize that the conformal blocks must be given by
φ(τ) =
2−
4
3
η(τ)4
θ2(τ)4θ4(τ)
4
θ3(τ)
4
,
which, once expressed in terms of the cross ratio w, is in agreement with the expressions for the
conformal blocks obtained by Knizhnik and Zamolodchikov [47].
Modular Exercises for Four-Point Blocks – I 47
8.4 Liouville theory
In our last example, we show how our approach also extends to the case of non-rational conformal
field theories which contain null states. In particular we apply our method to the Liouville theory
with central charge
c = 1 + 6Q2, where Q = b+
1
b
, b ∈ R.
As is well known, the spectrum of Liouville theory primaries consists of a continuum of states ϕp
(see [22]) of momentum
αp =
b
2
+
1
2b
+ p, p ∈ iR
as well as degenerate operators ϕm,n of momentum
αm,n = − b
2
m− 1
2b
n, m, n ∈ Z+.
The conformal weight of an operator is given in terms of its momentum α as
h = α(Q− α).
For definiteness, let us consider a core block involving one degenerate operator of momentum
α1,0 = − b
2 and three non-degenerate operators of momentum Q
2 + p:
D = (ϕ1,0, ϕp, ϕp, ϕp).
In this case, there is a two-dimensional space of core blocks corresponding to the modulesMp±b/2
with primary ϕp±b/2. One finds
TD =
(
e
(
1
12 − bp
2
)
0
0 e
(
1
12 + bp
2
)) , SD =
csc(πbp)
2
√
1− csc(πbp)2
4√
1− csc(πbp)2
4
−csc(πbp)
2
,
where SD is uniquely determined by requiring
S4
D = 1, S2
D = (SD · TD)3
and by working with a basis of conformal blocks such that SD is symmetric (cf. equation (A.3)).
The conformal blocks are the solutions to the two-dimensional MLDE, namely equation (A.1)
with µ = − 1
36 +b
2p2. They are given in terms of hypergeometric functions as in equations (A.2):
φ
M
p∓ b
2
ϕ1,0,ϕp,ϕp,ϕp
(τ) = N∓
(
2−4λ(τ)
) 1
6
±bp
(1− λ(τ))
1
6
±bp
2F1
(
1
2
± bp,
1
2
± 3bp, 1± 2bp;λ(τ)
)
,
where
N− = 1, N+ = 26bp
√
Γ(−bp)Γ(1− 2bp)Γ
(
1
2 + 3bp
)
Γ(bp)Γ(1 + 2bp)Γ
(
1
2 − 3bp
) .
The core blocks have the following series expansion:
φ
M
p∓ b
2
ϕ1,0,ϕp,ϕp,ϕp
(τ)
= N∓q
1
12
± bp
2
(
1− 2(1± 6bp)(1∓ 5bp)
1± bp
q +
2(1± 6bp)
(
−2± 9bp− 25b2p2 ± 300b3p3
)
(1± bp)(2± bp)
q2
+ 2
(1± 6bp)
(
6∓ 25bp− 274b2p2 ± 1255b3p3 + 550b4p4 ± 3000b5p5
)
(1± bp)(2± bp)(3± bp)
q3 +O
(
q4
))
.
Of course, as for the minimal models the core blocks obtained by solving the modular differential
equation are consistent with the chiral blocks obtained from the BPZ equation.
48 M.C.N. Cheng, T. Gannon and G. Lockhart
Figure 4. Schematic depiction of the torus-sphere correspondence between Tt and Sym2Tt.
9 A sphere-torus correspondence
In the previous section, we have encountered a number of examples in which the core blocks
coincide with the characters of different VOAs. Note that this relation bears some similarity to
another better-known relation between a theory Tt on the torus and the corresponding symmetric
product theory Sym2Tt = (Tt × Tt)/Z2 on the sphere [40, 54, 75]. We now briefly review the
main ingredients of this correspondence. On one side, one considers the partition function
of a torus theory Tt of central charges c = c̃
(
which for simplicity we take to be diagonal
theory with chiral algebras Vt = Ṽt
)
, and on the other side one considers a specific four-point
correlator ⟨O(z1, z1), . . . ,O(z4, z4)⟩Sym2Tt in the symmetric product theory Sym2Tt of central
charges (2c, 2c). This correspondence can be understood geometrically by realizing the torus
as the double cover of the four-punctured sphere, as mentioned in Section 6.2. The theory on
the sphere can then be described starting from two copies of Tt, one for each of the two sheets.
The Z2 orbifold reflects the fact that in circling one of the branch points one moves between the
two sheets, thus exchanging the two theories. This is implemented on the sphere side by inserting
at every branch point a twist operator O(z, z) = O(z)⊗Õ(z), which is the twisted sector ground
state of Sym2Tt of conformal weights
(
hO, h̃O
)
=
(
c
16 ,
c
16
)
.11 It can be shown that the OPE of O
with itself includes every conformal family of Tt
(
labeled by p =M ⊗ M̃ ∈ Υt ⊗ Υ̃t
)
[40]. This
can be used to argue that the torus partition function of Tt, which involves a sum over all p,
can be equivalently viewed as a sphere four-point correlator of Sym2Tt, which involves a sum
over the internal channels, with the conformal family labeled by p⊗p∗ appearing as the internal
channel (see Figure 4). More precisely, one obtains the relation
ZTt(τ, τ) = 2
2c
3 z4pt ⟨O(z1, z1), . . . ,O(z4, z4)⟩Sym2Tt = 2
2c
3 FO,O,O,O(τ, τ)Sym2Tt ,
where the factor in parentheses arises from the Weyl transformation between the variables τ
and z which parametrize the sphere and the torus respectively. If one takes Tt to be the diagonal
theory built out of two copies of a chiral algebra At, one can refine this correspondence and show
that it holds at the chiral level, where it relates the characters of Vt to the core blocks of Sym2Vt:
χM (τ) = 2
c
3φM⊗M∗
O,O,O,O(τ)Sym2At
. (9.1)
We now turn to the chiral torus-sphere correspondence encountered in the present work. On
the torus side, we take Vt to be a Kac–Moody algebra at level 1 and Tt to be the corresponding
(non-chiral) diagonal theory. On the sphere side, we consider a theory Ts which as we explain
below is closely related to the coset theory G1 ×G1/G2, and has central charges
cs = c̃s = 2
dim(G)
h∨G + 1
− dim(G) · 2
h∨G + 2
=
2dim(G)(
h∨G + 1
)(
h∨G + 2
) .
11States in the twisted sector are in one-to-one correspondence with states of Tt, so the twisted sector ground
state O is unique as long as the CFT Tt possesses a single vacuum.
Modular Exercises for Four-Point Blocks – I 49
Many of the ingredients of the correspondence appear to be strikingly similar to the Tt−Sym2Tt
correspondence. However, note that the geometric intuition detailed in the previous paragraphs
and in Figure 4 for the Tt−Sym2Tt correspondence is not readily applicable in the present case.
Let us flesh out the ingredients of this chiral correspondence between Tt and Ts observables
(see also Table 1 in the introduction for a summary). First, let us denote the respective chiral
algebras as Vt and Vs. We observe that there exists a module of Vs whose primary ϕ∗ ∈ M∗
has conformal weight h = c
16 ; in analogy with the Tt − Sym2Tt correspondence we will call this
primary the twist operator. To each irreducible module N of Vt, of conformal weight hN , we
associate a moduleM(N) of Vs of conformal weight 2hN . Then, in analogy with equation (9.1),
we find an identity between the characters of the modules of Vt and the core blocks of Vs
corresponding to chiral datum D = (ϕ∗, ϕ∗, ϕ∗, ϕ∗):
χN (τ) = 2
c
3φ
M(N)
ϕ∗,ϕ∗,ϕ∗,ϕ∗
(τ),
where the quantity on the left-hand (resp. right-hand) side is defined for the chiral algebra Vt
(resp. Vs). The above relation also implies that the SL2(Z) matrices S, T acting on the characters
of Vt coincide with the matrices SD, TD acting on the core blocks of theory Vs. Finally, passing
to the physical theories Tt and Ts obtained by combining the left- and right-movers, we find an
identity between torus partition function and core correlators:∑
N,Ñ
ZN,ÑχN (τ)χN (τ) = 2
2c
3
∑
N,Ñ
(AD)M(N),M̃(Ñ)φ
M(N)
ϕ∗,ϕ∗,ϕ∗,ϕ∗
(τ)φ
M̃(Ñ)
ϕ̃∗,ϕ̃∗,ϕ̃∗,ϕ̃∗
(τ),
that is,
ZT 2(τ, τ) = 2
2c
3 Fϕ∗×ϕ̃∗,...,ϕ∗×ϕ̃∗(τ, τ),
where the quantity on the left-hand (resp. right-hand) side is defined for the chiral algebra Tt
(resp. Ts). Note that the OPE coefficients (AD)M(N),M̃(Ñ) are identified with the multiplici-
ties ZN,Ñ .
The sphere-torus correspondence equates the level-1 Kac–Moody algebra characters to spe-
cific core blocks of several well-known theories. For example, when G belongs to the Deligne–
Cvitanović family of exceptional Lie algebras [23, 24] and G ̸= D4, the theory Ts is a unitary
Virasoro minimal model, as summarized in the following table, where we also provide the la-
bels (r, s) of the operators ϕr,s playing the role of the twist operator in the Tt − Sym2Tt corre-
spondence:
G A1 A2 G2 F4 E6 E7 E8
Ts M(3, 4) M(5, 6) M(9, 10) M(10, 11) M(6, 7) M(4, 5) M(3, 4)
(r, s) (1, 2) (1, 2) (1, 2) (2, 1) (2, 1) (2, 1) (2, 1)
c 1
2
4
5
14
15
52
55
6
7
7
10
1
2
The missing entry for G = D4 would correspond to a CFT at c = 1 on the sphere and will be
discussed shortly.
Note that the field content of (the chiral half of) the coset theory G1 × G1/G2 in some
cases consists of only a subset of the minimal model field content, and in particular the “twist
operator” ϕr,s does not necessarily belong to the spectrum of the coset theory. This is the
case for example when G = A2, where the field content of the coset theory consists only of
the Virasoro primaries ϕ1,1, ϕ1,3, ϕ1,5, ϕ2,1, ϕ2,3, ϕ2,5, and their descendants. The remaining
modules, including the one associated to the primary ϕ1,2, can nevertheless still be realized as
twisted modules in the coset theory.
50 M.C.N. Cheng, T. Gannon and G. Lockhart
As another example, we noted in Section 8.2 that the torus characters for G = DN with N
even coincide with core blocks of the Z2 orbifold of the boson at radius R =
√
2N , for the chiral
datum D = (ϕN/2, ϕN/2, ϕN/2, ϕN/2). See (8.9). This in particular also includes the missing
case G = D4 in the Deligne–Cvitanović sequence.
There are a few ways in which the correspondence appears to generalize. First of all, it is possi-
ble to extend the Deligne–Cvitanović sequence by including the two intermediate Lie algebras A 1
2
and E7+ 1
2
, which have dimensions 1 and 190 and dual Coxeter numbers 3
2 and 24, respectively.
Mimicking the coset construction for ordinary Lie algebras suggests a relation between the char-
acters of the two corresponding intermediate VOAs [46] and a pair of RCFTs on the sphere of
central charge respectively
2 · 1(
3
2 + 1
)(
3
2 + 2
) =
8
35
and
2 · 190
(24 + 1)(24 + 2)
=
38
65
,
which are respectively the central charges of the Virasoro minimal models M(10, 7), M(13, 10).
Indeed, for the former minimal model we have found in Section 8.1.3 that for the choice of chiral
datum D = (ϕ1,2, ϕ1,2, ϕ1,2, ϕ1,2) one has
2
2
15φ
M1,1
ϕ1,2,ϕ1,2,ϕ1,2,ϕ1,2
(τ) = χ1
A 1
2 ,1
(τ), 2
2
15φ
M1,3
ϕ1,2,ϕ1,2,ϕ1,2,ϕ1,2
(τ) = χ2
A 1
2 ,1
(τ),
while for the latter minimal model we found in Section 8.1.4 that for D = (ϕ2,1, ϕ2,1, ϕ2,1, ϕ2,1)
one has
2
38
15φ
M1,1
ϕ2,1,ϕ2,1,ϕ2,1,ϕ2,1
(τ) = χ1
E
7+1
2 ,1
(τ), 2
38
15φ
M3,1
ϕ2,1,ϕ2,1,ϕ2,1,ϕ2,1
(τ) = χ57
E
7+1
2 ,1
(τ).
To the best of our knowledge, this is the first, albeit quite indirect, appearance of the intermediate
algebra E7+ 1
2
in a physical theory!
Another intriguing relation is obtained if one takes the torus theory to be the affine algebra G
belonging to the Deligne–Cvitanović sequence, at the negative level
k = −
h∨G
6
− 1.
Notably, this class of theories, which has been the object of considerable interest in the context of
representation theory [5, 6], also appears prominently in the context of 4d N = 2 SCFT, where
it captures the chiral algebra of the rank-one SCFTs describing a D3 brane probing a singular
F -theory background [11, 12, 13].
The characters for this class of CFTs are solutions to a two-dimensional MLDE. In the ca-
ses G = A 1
2
, A1, A2, G2 and F4, both characters transform as modular forms, and admit a well-
defined q-series expansion. For G = D4, E6, E7, E7+ 1
2
, and E8, the solution corresponding to
the vacuum character admits a q-series expansion, while the other solution includes logarithmic
terms, and the two characters transform as a quasi-modular form [45]. The appearance of
logarithms can be interpreted as resulting from the regularization of the refined character (which
also depends on a parameter m ∈ hGC valued in the complexification of the Cartan subalgebra
of G), which in the unrefined limit is divergent [13, 21].
We find that the characters of Gk coincide with certain Virasoro core blocks at central charge
c = 2
dim(G) · k
h∨G + k
− dim(G) · 2k
h∨G + 2k
=
2dim(G)k2(
h∨G + k
)(
h∨G + 2k
) ,
which we compute by using the solution of [68] to Zamolodchikov’s recurrence formula. Specif-
ically, in the core blocks we take the external operators to have conformal weight he = −h∨G+1
8
and the internal operator to have conformal weight 0 or hp = −h∨G
3 . For
G ∈
{
G2, D4, F4, E6, E7, E7+ 1
2
, E8
}
,
Modular Exercises for Four-Point Blocks – I 51
these values are realized in Liouville theory at
b =
√
2
1
3h
∨
G − 1
, (9.2)
if we take all external operators to be the degenerate operator ϕ1,0 and the intermediate pri-
maries to be the vacuum and ϕ2,0. For G = A 1
2
, A1, b as given in equation (9.2) is imaginary
and the core blocks are realized in timelike Liouville theory. Finally, G = A2 corresponds to
taking c→ ∞, he = −1
2 and hp = 0 or 1 in the Virasoro core blocks.
For all G, the vacuum character χvac
Gk
(τ) is non-singular, and we find the following relation:
φ
M(vac)
ϕ∗,ϕ∗,ϕ∗,ϕ∗
(τ) = 2
16
3
heχvac
Gk
(τ).
For G = A 1
2
, A1, A2, G2, F4, we also find agreement between the non-vacuum character
χnon vac
Gk
(τ)
and the other conformal block:
φ
M(non vac)
ϕ∗,ϕ∗,ϕ∗,ϕ∗
(τ) = 2
16
3
heχnon vac
Gk
(τ).
For G = D4, E6, E7, E7+ 1
2
, and E8, as discussed above, the q-series expansion of the non-va-
cuum character involves logarithmic terms. Consistent with this, we find that certain coeffi-
cients of the solution of [68] to Zamolodchikov’s recurrence formula have poles at hp = −h∨G
3
and therefore the q-series expansion is ill-defined. We provide some details for this sphere-torus
relation in Table 2.
Finally, additional relations appear to exist between the level-1 characters of affine Kac–
Moody algebras and core blocks of minimal models with three identical external operators. For
instance, corresponding to the chiral datum D = (ϕ2,1, ϕ2,3, ϕ2,3, ϕ2,3) in the M(4, 5) minimal
model are two core blocks associated to the internal channels M2,1 and M2,3. It turns out that
these core blocks are again proportional to the characters of A1,1:φM2,1
ϕ2,1,ϕ2,3,ϕ2,3,ϕ2,3
φ
M2,3
ϕ2,1,ϕ2,3,ϕ2,3,ϕ2,3
(τ) = N
(
χ3
A1,1
χ1
A1,1
)
(τ)
for some N ∈ C. These additional relations will also be addressed in more detail in the com-
panion paper [3].
We conclude this section with a few additional remarks on the physical interpretation of the
sphere-torus correspondence. Given an affine algebra G1 at level 1, whose central charge is
given by c = dimG
h∨G+1
, one can construct the following two distinct VOAs: the symmetric prod-
uct, Sym2G1, and the diagonal coset G1×G1
G2
. Both theories contain twist operators η of the same
conformal dimension hη =
c
16 , and these turn out to have identical fusion rules
η × η =
∑
i
ϕi
in the two theories, involving primaries ϕi of conformal dimension hi = 2hGi , where h
G
i are the
dimensions of certain G1 primaries. This observation is sufficient, given the low dimensionality
of the SL(2,Z) representations involved, to guarantee that the four-point core blocks with four
insertions of η are identical in the two theories Sym2G1 and
G1×G1
G2
, although their interpretation
in the two theories is different since they are defined with respect to the respective extended
chiral algebras, which are distinct. In turn, the Sym2G1 core blocks coincide with G1 characters
by virtue of the relation (9.1), which has a clear physical interpretation. At present, however, it
remains unclear to us whether a direct physical interpretation of the sphere-torus correspondence
can be given, and whether the correspondence generalizes to further classes of examples.
52 M.C.N. Cheng, T. Gannon and G. Lockhart
Torus VOA b2 c he hp 2−
16
3
heq
2
3
he− 1
2
hpf
hp
he,he,he,he
(τ)
A 1
2
,−5/4 −4 −25
2 − 5
16
0 1 + q + 3q2 + 4q3 + 7q4 + 10q5 + · · ·
−1
2 1 + 3q + 4q2 + 7q3 + 13q4 + 19q5 + · · ·
A1,−4/3 −6 −24 −3
8
0 1 + 3q + 9q2 + 19q3 + 42q4 + 81q5 + · · ·
−2
3 1 + 8q + 17q2 + 46q3 + 98q4 + 198q5 + · · ·
A2,−3/2 ∞ ∞ −1
2
0 1 + 8q + 36q2 + 128q3 + 394q4 + 1088q5 + · · ·
−1 1 + 28q + 134q2 + 568q3 + 1809q4 + 5316q5 + · · ·
G2,−5/3 6 50 −5
8
0 1 + 14q + 92q2 + 456q3 + 1848q4 + 6580q5 + · · ·
−4
3 1 + 78q + 729q2 + 4382q3 + 19917q4 + 77274q5 + · · ·
D4,−2 2 28 −7
8
0 1 + 28q + 329q2 + 2632q3 + 16380q4 + 85764q5 + · · ·
−2 −
F4,−5/2 1 25 −5
4
0 1 + 52q + 1106q2 + 14808q3 + 147239q4 + 1183780q5 + · · ·
−3 1−272q−34696q2−1058368q3−17332196q4−197239456q5+· · ·
E6,−3
2
3 26 −13
8
0 1 + 78q + 2509q2 + 49270q3 + 698425q4 + 7815106q5 + · · ·
−4 −
E7,−4
2
5
152
5 −19
8
0 1+133q+7505q2+254885q3 + 6093490q4 + 112077998q5 + · · ·
−6 −
E7+ 1
2
,−5
2
7
250
7 −25
8
0 1+190q+15695q2+783010q3+27319455q4+725679750q5+· · ·
−8 −
E8,−6
2
9
124
3 −31
8
0 1+248q+27249q2+1821002q3+85118126q4+3017931282q5+· · ·
−10 −
Table 2. Correspondence between the characters of Kac–Moody algebras at level k = −h∨
G
6 − 1
and Virasoro core blocks, normalized here so that the leading order term is 1. Note that for G =
D4, E6, E7, E7+ 1
2
, E8 the q-expansion of the Virasoro blocks corresponding to the non-vacuum character
is ill-defined.
A MLDE and BPZ
The components f1(τ), . . . , fn(τ) of a weight-zero PSL2(Z) vector-valued modular form are
necessarily solutions to a degree-n modular differential equation (MLDE), that is, an equation
which takes the form(
Dn
(0) + gn−1D
n−1
(0) + · · ·+ g1D(0) + g0
)
f = 0
using notation explained in section 6 (cf. (6.4)). The coefficient gk(τ) (namely bk/bn in the
notation of Section 6.3.3) is a meromorphic modular form of weight 2(n−k) for PSL2(Z). When
the Wronskian bd(τ) is a power of the η-function [58], then the gk will be holomorphic and hence
could be expressed as polynomials in E4(τ), E6(τ). We will restrict to this case in the following.
Modular differential equations have been applied to the context of RCFT since the 1980s [4];
in particular, characters of a RCFT are solutions to MLDEs whose q-expansion coefficients
are positive integers, and this strategy was used early on to classify unitary RCFTs with
one or two characters [59, 66] – see also [18] and references therein for more recent develop-
ments.
Order 2 MLDEs in particular depend on a single parameter µ:{
D2 − E2(τ)
6
D − µ
4
E4(τ)
}
f(τ) = 0. (A.1)
Modular Exercises for Four-Point Blocks – I 53
where D = 1
2πi
d
dτ . Two solutions to this MLDE can in general be expressed in terms of hyper-
geometric functions [61]:
f1(τ) =
(
2−4λ(τ)(1− λ(τ))
) 1−x
6
2F1
(
1
2
− x
6
,
1
2
− x
2
, 1− x
3
, λ(τ)
)
= q
1−x
12 (1 +O(q)), (A.2)
f2(τ) = N
(
2−4λ(τ)(1− λ(τ))
) 1+x
6
2F1
(
1
2
+
x
6
,
1
2
+
x
2
, 1 +
x
3
, λ(τ)
)
= Nq
1+x
12 (1 +O(q)),
where x =
√
1 + 36µ and
N = 2x
√
Γ
(
1− x
3
)
Γ
(
−x
6
)
Γ
(
1
2 + x
2
)
Γ
(
1 + x
3
)
Γ
(
x
6
)
Γ
(
1
2 − x
2
) .
The normalization is such that the modular S matrix is symmetric:
S =
Γ
(
1− x
3
)
Γ
(
x
3
)
Γ
(
1
2−
x
6
)
Γ
(
1
2+
x
6
) √
Γ
(
1+ x
3
)
Γ
(
x
3
)
Γ
(
1− x
3
)
Γ
(
−x
3
)
Γ
(
1
2+
x
6
)
Γ
(
1
2+
x
2
)
Γ
(
1
2−
x
6
)
Γ
(
1
2−
x
2
)
√
Γ
(
1+ x
3
)
Γ
(
x
3
)
Γ
(
1− x
3
)
Γ
(
−x
3
)
Γ
(
1
2+
x
6
)
Γ
(
1
2+
x
2
)
Γ
(
1
2−
x
6
)
Γ
(
1
2−
x
2
) Γ
(
1+ x
3
)
Γ
(
−x
3
)
Γ
(
1
2−
x
6
)
Γ
(
1
2+
x
6
)
. (A.3)
The values of µ for which solutions to equation (A.1) have a q-expansion with positive integer
coefficients were determined in [59]; the solutions to the MLDE equation in these cases are
the characters of a sequence of RCFTs labeled by a Lie algebra G belonging to the Deligne–
Cvitanović sequence:
µ 11
900
5
144
1
12
119
900
2
9
299
900
5
12
77
144
551
900
2
3
G A 1
2
A1 A2 G2 D4 F4 E6 E7 E7 1
2
E8
For G a simple Lie algebra, the RCFT is simply the level-1 G Kac–Moody algebra, while
in the cases G = A 1
2
and E7 1
2
, one finds the characters of the intermediate vertex subalgebra
associated to the intermediate Lie algebra G [53, 70].
In Section 6, we have argued that the core blocks are components of a weight-zero weakly
holomorphic vector-valued modular form; in particular, they must be solutions of a MLDE. On
the other hand, four-point chiral blocks of Virasoro minimal models are known to be solutions
to the BPZ differential equation [14], due to the presence of null states. In what follows, we
verify explicitly that the BPZ equation satisfied by the chiral block (in the simplest case where
the null descendant occurs at level 2) can be turned into the order-2 MLDE for the core block.
Let us recall briefly how the BPZ equation arises. Consider a chiral block between Virasoro
primaries ϕ1(z1), ϕ2(z2), . . . of conformal weight h1, h2, . . . , and assume ϕ1(z1) has a null Vira-
soro descendant at level n. A generic level-n descendant of a Virasoro primary operator ϕ1(z1)
can written in the form
ϕ′(z) =
∑
Y
|Y |=n
αY L−Y ϕ(z)
for αY ∈ C; here,
Y = {r1, . . . , rk}, 1 ≤ r1 ≤ r2 ≤ · · · ≤ rk
54 M.C.N. Cheng, T. Gannon and G. Lockhart
is a Young diagram of size n, and L−Y = L−r1L−r2 . . . L−rk . When such a descendant is a null
state, any physical correlator involving it vanishes:〈(
ϕ1
)′
(z1, z̄1)ϕ
2(z2, z̄2) . . .
〉
= 0,
where ϕi(z, z̄) = ϕi(z) × ϕ̃i(z̄). This implies that physical correlators involving the primary
field ϕ1(z, z̄) satisfy a holomorphic order n differential equation:∑
Y
|Y |=n
αY L−Y (z1)⟨ϕ(z1, z̄1)ϕ1(z2, z̄2) . . . ⟩ = 0, (A.4)
where
L−Y (z1) = L−r1(z1)L−r2(z1) . . .L−rk(z1)
and
L−r(z1) =
∑
i≥2
(
(r − 1)hi
zri1
− 1
zr−1
i1
∂zi
)
.
Let us from now on focus on four-point blocks. The BPZ equation (A.4) then translates to the
following differential equation for the chiral blocks:[
∂2z1 − t
4∑
i=2
(
hi
z2i1
− 1
zi1
∂zi
)]
FP
ϕ1,ϕ2,ϕ3,ϕ4(z1, z2, z3, z4) = 0, (A.5)
where P is an internal channel. We claim that the BPZ equation lifts to a modular differential
equation for the core of the conformal block. Let us explicitly verify this in the case where the
null state is at level 2. Such a null state is given by ϕ′1(z1) =
(
L2
−1 − tL−2
)
ϕ(z1), where
t =
4
3
h1 +
2
3
.
We now express the holomorphic cross ratio w as w = λ(τ) in terms of the modular parameter
on the torus. The relation between conformal and core block is given by
FP
ϕ1,ϕ2,ϕ3,ϕ4(z1, z2, z3, z4) = φPϕ1,ϕ2,ϕ3,ϕ4(w)
∏
1≤j<k≤4
z
µjk
jk , (A.6)
where µjk is defined in equation (4.16). To obtain the differential equation for the core block, we
express the differential operator appearing in equation (A.5) in terms of derivatives with respect
to the cross ratio w = z12z34
z13z24
. Thus, for example
∂z1 = ∂z1(w)∂w +
µ12
z12
+
µ13
z13
+
µ14
z14
,
where
∂z1(w) =
z23z34
z213z24
,
and similarly for ∂z2(w), ∂z3(w), ∂z4(w). Likewise, one finds
∂2z1 = −
(
µ12
z212
+
µ13
z213
+
µ14
z214
)
+
(
µ12
z12
+
µ13
z13
+
µ14
z14
)2
+ 2
(
µ12
z12
+
µ13
z13
+
µ14
z14
)
∂z1(w)∂w + ∂2z1(w)∂w + ∂z1(w)
2∂2w.
Modular Exercises for Four-Point Blocks – I 55
Taking the limit (z1, z2, z3, z4) → (∞, 1, w, 0) and using relation (A.6), we obtain the following
differential equation for the core block:{
d2
dw2
+
[
2
(
µ12
w
+
µ14
w − 1
)
+ t
2w − 1
w(w − 1)
]
d
dw
+
[(
µ12
w
+
µ14
w − 1
)2
− µ12
w2
− µ14
(w − 1)2
]
+ t
(
µ12 − h2
w2
+
µ14 − h4
(w − 1)2
− µ24
w(w − 1)
)}
φPϕ1,ϕ2,ϕ3,ϕ4(w) = 0. (A.7)
Next, we use w = λ(τ) =
( θ2(τ)
θ3(τ)
)4
to rewrite equation (A.7) in terms of the variable q = e(τ).
In particular, we find
d
dw
=
2
λ(τ)(1− λ(τ))θ3(τ)4
D,
and
d2
dw2
=
4
λ(τ)2(1− λ(τ))2θ3(τ)8
{
D2 +
(
2λ(τ)− 1
2
θ3(τ)
4 −
D
(
θ3(τ)
4
)
θ3(τ)4
)
D
}
.
Making also use of the identities(
λ(τ)2 − λ(τ) + 1
)
θ3(τ)
8 = E4(τ),
2λ(τ)− 1
6
θ3(τ)
4 −
D
(
θ3(τ)
4
)
θ3(τ)4
= −1
6
E2(τ),
we can rewrite equation (A.7) as{
D2 +
[
−E2(τ)
6
+
(
θ2(τ)
4h4 − h3
3
+ θ3(τ)
4h4 − h2
3
+ θ4(τ)
4h3 − h2
3
)]
D
− 1
18
E4(τ)(2h1(h− 1) + h)− 1
36
4∑
i=2
Λiθi(τ)
8
}
φPϕ1,ϕ2,ϕ3,ϕ4(λ(τ)) = 0,
where
Λi = (4h1 + 1)(3µ1i + 2h1) +
4∏
j=2
j ̸=i
(3µ1j + 2h1).
Let us now further specialize to the case ϕ2 = ϕ3 = ϕ4 with h2 = h3 = h4 = h∗, corresponding
to PSL2(Z) symmetry. Note that in this case Λi = 0, so we obtain{
D2 − E2(τ)
6
D − µ(h1, h∗)
4
E4(τ)
}
φPϕ1,ϕ2,ϕ2,ϕ2(λ(τ)) = 0,
which takes precisely the form of the modular differential equation (A.1), with
µ(h1, h∗) =
2
9
(
2h21 + 6h1h∗ + 3h∗ − h1
)
.
This establishes that the second-order BPZ equations for four-point chiral blocks are equivalent
to modular linear differential equations satisfied by the core blocks. It should be straightforward
to establish analogous relations for the higher-order cases.
56 M.C.N. Cheng, T. Gannon and G. Lockhart
B Braiding and fusing
In this appendix, we collect a few useful properties of the braiding and fusing operators (the
latter are also called 6j-symbols)
B
[
N2 N3
N1 N4
]
(ε) :
⊕
P
YN1
N2,P
⊗ YPN3,N4
−→
⊕
Q
YN1
N3,Q
⊗ YQN2,N4
,
F
[
N2 N3
N1 N4
]
:
⊕
P
YN1
N2,P
⊗ YPN3,N4
−→
⊕
Q
YN1
Q,N4
⊗ YQN2,N3
,
from the classic papers [63, 65], to which we refer for details. The choice of sign ε = ± is
explained in Section 4, and admits a natural generalization to any ε ∈ Z. Picking bases ΦN1,a
N2,N3
,
for the spaces YN1
N2,N3
, we write the action of braiding and fusing as
ΦN1,a
N2,P
ΦN1,b
N2,P
=
∑
Q
∑
c,d
BP,Q
[
N2 N3
N1 N4
]a,b
c,d
(ε)ΦN1,c
N3,Q
ΦQ,dN2,N4
,
ΦN1,a
N2,P
ΦN1,b
N2,P
=
∑
Q
∑
c,d
FPQ
[
N2 N3
N1 N4
]a,b
c,d
ΦN1,c
Q,N4
ΦQ,dN2,N3
.
Exchange of N2, N3 in the intertwiner ΦN1
N2,N3
defines an isomorphism, which is the skew sym-
metry relation:
ξN1
N2,N3
: YN1
N2,N3
→ YN1
N3,N2
, (B.1)
which squares to the identity operator. We write its action on the basis ΦN1,a
N2,N3
as
ΦN1,a
N2,N3
=
∑
b
(
ξN1
N2,N3
)a
b
ΦN1,b
N3,N2
.
Similarly, exchange of N1, N2 defines an isomorphism which sends an intertwiner to its adjoint:
ζN1
N2,N3
: YN1
N2,N3
→ YN
∗
2
N∗
1 ,N3
, (B.2)
and we write its action on the basis ΦN1,a
N2,N3
as
ΦN1,a
N2,N3
=
∑
b
(
ζN1
N2,N3
)a
b
Φ
N∗
2 ,b
N∗
1 ,N3
.
These maps are carefully studied in [44, Section 7].
It is clear that the matrices ζ and ξ are involutions in the sense that
ξNk
Ni,Nj
ξNk
Nj ,Ni
= ζNk
Ni,Nj
ζ
N∗
i
N∗
k ,Nj
= I. (B.3)
When NN1
N2,N3
= 1, as will be the case in the main text, ζ and ξ are simply phases.
When one of the Ni is the vacuum module one obtains simple expressions for the braiding
matrices:
BQR
[
N1 N2
V P
]·,b
·,d
(+) =
(
ζ
N∗
1
N2,P
)b
d
δN∗
1 ,Q
δN∗
2 ,R
e
(
hN1 + hN2 − hP
2
)
,
BQR
[
N3 N4
P V
]a,·
c,·
(+) =
(
ξPN3,N4
)a
c
δN4,QδN3,R e
(
hN3 + hN4 − hP
2
)
. (B.4)
Modular Exercises for Four-Point Blocks – I 57
One can further show that braiding and fusing are related as follows:
BPQ
[
N2 N3
N1 N4
]a,b
c,d
(ε)
=
∑
e,f
(
ξPN3,N4
)b
e
(
ξN1
Q,N3
)f
c
e
(
ε
2
(hP + hQ − hN1 − hN4)
)
FPQ
[
N2 N4
N1 N3
]a,e
f,d
. (B.5)
As a consequence,∑
Q
∑
c,d
BPQ
[
N2 N3
N1 N4
]a,b
c,d
(ε) e
(
ε
(
hN1 + hN4 −
hP
2
− hQ − hR
2
))
BQR
[
N3 N2
N1 N4
]c,d
e,f
(ε)
= δP,Rδ
a
e δ
b
f (B.6)
and ∑
Q
∑
b,c,d,e,h,i
FPQ
[
N2 N3
N1 N4
]a,b
c,d
FQR
[
N3 N4
N1 N2
]h,i
e,f
(
ξPN4,N3
)g
b
(
ξN1
Q,N4
)c
h
(
ξQN4,N2
)d
i
(
ξN1
R,N4
)e
j
= δP,Rδ
a
j δ
g
f .
C Modular group representations for general cases
According to Theorem 5.3, in cases (B) respectively (C) the vector of core blocks φϕ1,ϕ2,ϕ3,ϕ3
(w̃)
in (5.9) only transforms with respect to PΓ0(2) respectively PΓ(2). At the cost of increasing
the dimensions, we can repackage the core blocks in those cases so that they too transform
with respect to PSL2(Z). In representation theory terminology, this corresponds to inducing
the representation from the subgroups PΓ0(2) and PΓ(2) up to PSL2(Z). In practice though,
for these cases it is usually easier to work with the core blocks as vvmf’s for PΓ0(2) or PΓ(2),
though both options are given in Theorem 5.3.
(B) D = (M1,M2,M3,M3), with ϕ1 ∈M1, ϕ2 ∈M2, and ϕ3 = ϕ4 ∈M3.
We can also arrange the core blocks in a vector as follows:
φ
B;ϕ1,ϕ2,ϕ3
=
φP1
ϕ1,ϕ2,ϕ3,ϕ3
...
φ
PiP
ϕ1,ϕ2,ϕ3,ϕ3
φQ1
ϕ1,ϕ3,ϕ2,ϕ3
...
φ
QiQ
ϕ1,ϕ3,ϕ2,ϕ3
φR1
ϕ1,ϕ3,ϕ3,ϕ2
...
φ
RiR
ϕ1,ϕ3,ϕ3,ϕ2
, (C.1)
where Pi, Qi, Ri ∈ Υ(V). We also define the following matrices, written in block form
TB =
T [ 2 3
1 3
]
0 0
0 0 T
[
3 2
1 3
]
0 T
[
3 3
1 2
]
0
, SB =
0 0 S
[
2 3
1 3
]
0 S
[
3 2
1 3
]
0
S
[
3 3
1 2
]
0 0
.
Then φB;ϕ1,ϕ2,ϕ3
has three times the number of rows of φ
ϕ1,ϕ2,ϕ3,ϕ3
, and transforms with
respect to PSL2(Z).
58 M.C.N. Cheng, T. Gannon and G. Lockhart
(C) D = (M1,M2,M3,M4) with theMi arbitrary (e.g., some may be equal), but with all states
ϕi ∈ Mi distinct. We can arrange the core blocks corresponding to various permutations
of the external operators as follows:
φ
C;ϕ1,ϕ2,ϕ3,ϕ4
=
φP1
ϕ1,ϕ2,ϕ3,ϕ4
...
φ
PiP
ϕ1,ϕ2,ϕ3,ϕ4
φQ1
ϕ1,ϕ2,ϕ4,ϕ3
...
φ
QiQ
ϕ1,ϕ2,ϕ4,ϕ3
φR1
ϕ1,ϕ3,ϕ2,ϕ4
...
φ
RiR
ϕ1,ϕ3,ϕ2,ϕ4
φU1
ϕ1,ϕ3,ϕ4,ϕ2
...
φ
UiU
ϕ1,ϕ3,ϕ4,ϕ2
φV1ϕ1,ϕ4,ϕ2,ϕ3
...
φ
ViV
ϕ1,ϕ4,ϕ2,ϕ3
φW1
ϕ1,ϕ4,ϕ3,ϕ2
...
φ
WiW
ϕ1,ϕ4,ϕ3,ϕ2
(C.2)
and define
TC =
0 T
[
2 3
1 4
]
0 0 0 0
T
[
2 4
1 3
]
0 0 0 0 0
0 0 0 T
[
3 2
1 4
]
0 0
0 0 T
[
3 4
1 2
]
0 0 0
0 0 0 0 0 T
[
4 2
1 3
]
0 0 0 0 T
[
4 3
1 2
]
0
,
SC =
0 0 0 0 0 S
[
2 3
1 4
]
0 0 0 S
[
2 4
1 3
]
0 0
0 0 0 0 S
[
3 2
1 4
]
0
0 S
[
3 4
1 2
]
0 0 0 0
0 0 S
[
4 2
1 3
]
0 0 0
S
[
4 3
1 2
]
0 0 0 0 0
.
Then φC;ϕ1,ϕ2,ϕ3,ϕ4
has six times the number of rows of φ
ϕ1,ϕ2,ϕ3,ϕ4
, and transforms with
respect to PSL2(Z).
Acknowledgements
We thank Vassilis Anagiannis, Christopher Beem, Francesca Ferrari, Alex Maloney, Greg Moore,
and Gim Seng Ng for useful conversations. We are also grateful to the anonymous referees for
their helpful suggestions. This project has received funding from the European Union’s Horizon
2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement
Modular Exercises for Four-Point Blocks – I 59
No 708045. The work of M.C. and G.L. is supported by ERC starting grant H2020 #640159.
The work of M.C. has also received support from NWO vidi grant (number 016.Vidi.189.182).
The work of T.G. is supported by an NSERC Discovery grant.
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1 Introduction
2 Summary of notation
3 Groups
4 The theory of chiral blocks
5 The core blocks
5.1 Definition and basic properties
5.2 Mapping class group action
5.3 Modular group
6 Modular forms
6.1 Vector-valued modular forms
6.1.1 Modular forms for SL_2(Z)
6.1.2 Modular forms for Gamma(2)
6.1.3 Modular forms for Gamma_0(2)
6.2 Core blocks as vector-valued modular forms
6.3 Methods for constructing vector-valued modular forms
6.3.1 Rademacher sum
6.3.2 Riemann–Hilbert
6.3.3 Modular linear differential equation
6.3.4 A hybrid method
7 Physical correlators
8 Examples
8.1 Virasoro minimal models
8.1.1 M(4,3)
8.1.2 M(12,11)
8.1.3 M(10,7)
8.1.4 M(13,10)
8.2 RCFTs at c=1
8.3 WZW models
8.4 Liouville theory
9 A sphere-torus correspondence
A MLDE and BPZ
B Braiding and fusing
C Modular group representations for general cases
References
|
| id | nasplib_isofts_kiev_ua-123456789-212878 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-21T18:31:12Z |
| publishDate | 2025 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Cheng, Miranda C.N. Gannon, Terry Lockhart, Guglielmo 2026-02-13T13:49:43Z 2025 Modular Exercises for Four-Point Blocks - I. Miranda C.N. Cheng, Terry Gannon and Guglielmo Lockhart. SIGMA 21 (2025), 013, 61 pages 1815-0659 2020 Mathematics Subject Classification: 81T40; 17B69; 11F03 arXiv:2002.11125 https://nasplib.isofts.kiev.ua/handle/123456789/212878 https://doi.org/10.3842/SIGMA.2025.013 The well-known modular property of the torus characters and torus partition functions of (rational) vertex operator algebras (VOAs) and 2d conformal field theories (CFTs) has been an invaluable tool for studying this class of theories. In this work, we prove that sphere four-point chiral blocks of rational VOAs are vector-valued modular forms for the groups Γ(2), Γ₀(2), or SL₂(ℤ). Moreover, we prove that the four-point correlators, combining the holomorphic and anti-holomorphic chiral blocks, are modular invariant. In particular, in this language, the crossing symmetries are simply modular. This gives the possibility of exploiting the available techniques and knowledge about modular forms to determine or constrain the physically interesting quantities, such as chiral blocks and fusion coefficients, which we illustrate with a few examples. We also highlight the existence of a sphere-torus correspondence equating the sphere quantities of certain theories ₛ with the torus quantities of another family of theories ₜ. A companion paper will delve into more examples and explore this sphere-torus duality. We thank Vassilis Anagiannis, Christopher Beem, Francesca Ferrari, Alex Maloney, Greg Moore, and Gim Seng Ng for useful conversations. We are also grateful to the anonymous referees for their helpful suggestions. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curiegrantagreement No 708045. The work of M.C. and G.L. is supported by an ERC starting grant H2020 #640159. The work of M.C. has also received support from an NWO Vidi grant (number 016. Vidi. 189.182). The work of T.G. is supported by an NSERC Discovery grant. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Modular Exercises for Four-Point Blocks - I Article published earlier |
| spellingShingle | Modular Exercises for Four-Point Blocks - I Cheng, Miranda C.N. Gannon, Terry Lockhart, Guglielmo |
| title | Modular Exercises for Four-Point Blocks - I |
| title_full | Modular Exercises for Four-Point Blocks - I |
| title_fullStr | Modular Exercises for Four-Point Blocks - I |
| title_full_unstemmed | Modular Exercises for Four-Point Blocks - I |
| title_short | Modular Exercises for Four-Point Blocks - I |
| title_sort | modular exercises for four-point blocks - i |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212878 |
| work_keys_str_mv | AT chengmirandacn modularexercisesforfourpointblocksi AT gannonterry modularexercisesforfourpointblocksi AT lockhartguglielmo modularexercisesforfourpointblocksi |