Virtual Classes of Representation Varieties of Upper Triangular Matrices via Topological Quantum Field Theories
In this paper, we use a geometric technique developed by González-Prieto, Logares, Muñoz, and Newstead to study the -representation variety of surface groups G(Σ) of arbitrary genus for being the group of upper triangular matrices of fixed rank. Explicitly, we compute the virtual classes in the Röt...
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| description | In this paper, we use a geometric technique developed by González-Prieto, Logares, Muñoz, and Newstead to study the -representation variety of surface groups G(Σ) of arbitrary genus for being the group of upper triangular matrices of fixed rank. Explicitly, we compute the virtual classes in the Röthendieck ring of varieties of the -representation variety and the moduli space of -representations of surface groups for being the group of complex upper triangular matrices of rank 2, 3, and 4 via constructing a topological quantum field theory. Furthermore, we show that in the case of upper triangular matrices, the character map from the moduli space of -representations to the -character variety is not an isomorphism.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 18 (2022), 095, 38 pages
Virtual Classes of Representation Varieties
of Upper Triangular Matrices
via Topological Quantum Field Theories
Márton HABLICSEK and Jesse VOGEL
Mathematical Institute, Niels Bohrweg 1, 2333 CA Leiden, The Netherlands
E-mail: m.hablicsek@math.leidenuniv.nl, j.t.vogel@math.leidenuniv.nl
Received February 28, 2022, in final form November 28, 2022; Published online December 06, 2022
https://doi.org/10.3842/SIGMA.2022.095
Abstract. In this paper, we use a geometric technique developed by González-Prieto, Loga-
res, Muñoz, and Newstead to study the G-representation variety of surface groups XG(Σg) of
arbitrary genus for G being the group of upper triangular matrices of fixed rank. Explicitly,
we compute the virtual classes in the Grothendieck ring of varieties of the G-representation
variety and the moduli space of G-representations of surface groups for G being the group
of complex upper triangular matrices of rank 2, 3, and 4 via constructing a topological
quantum field theory. Furthermore, we show that in the case of upper triangular matrices
the character map from the moduli space of G-representations to the G-character variety is
not an isomorphism.
Key words: representation variety; character variety; topological quantum field theory; Gro-
thendieck ring of varieties
2020 Mathematics Subject Classification: 14D23; 14D21; 14C30; 14D20; 14D07; 57R56
1 Introduction
Let X be a closed connected manifold with finitely generated fundamental group π1(X), and G
an algebraic group over a field k. The set of group representations ρ : π1(X)→ G,
XG(X) = Hom(π1(X), G),
carries a natural structure of an algebraic variety, and is called the G-representation variety
of X. Indeed, given a set of generators γ1, . . . , γn of π1(X), the morphism
XG(X)→ Gn, ρ 7→ (ρ(γ1), . . . , ρ(γn))
identifies the G-representation variety XG(X) with a subvariety of Gn, and this structure can be
shown to be independent of the chosen generators. When X = Σg is a closed oriented surface
of genus g, the G-representation variety is the closed subvariety of G2g given by
XG(Σg) =
{
(A1, B1, . . . , Ag, Bg) ∈ G2g
∣∣∣∣ g∏
i=1
[Ai, Bi] = 1
}
. (1.1)
The algebraic group G acts by conjugation on the variety XG(X), so one can look at the
categorical quotient (the affine GIT quotient)
MG(X) = XG(X) � G,
which is known as the moduli space of G-representations.
mailto:m.hablicsek@math.leidenuniv.nl
mailto:j.t.vogel@math.leidenuniv.nl
https://doi.org/10.3842/SIGMA.2022.095
2 M. Hablicsek and J. Vogel
When the group G is a linear algebraic group (G ≤ GLr(k) for some r > 0), there exists
another natural variety that parametrizes G-representations up to conjugation, which is defined
as follows. To a representation ρ : π1(X)→ G we can associate its character
χρ : π1(X)→ k, γ 7→ tr(ρ(γ)).
The image of the so-called character map
χ : XG(X)→ kΓ, ρ 7→ χρ
is called the G-character variety and is denoted by χG(X). By results from [6], there ex-
ists a finite set of elements γ1, . . . , γa ∈ π1(X) such that χρ is determined by the images
(χρ(γ1), . . . , χρ(γa)) for any ρ. This way, χG(X) can be identified with the image of the map
XG(X)→ ka given by ρ 7→ (χρ(γ1), . . . , χρ(γa)), which indeed provides χG(X) a natural struc-
ture of a variety. Again, this structure is independent of the chosen γi.
Note that the character map χ is a G-invariant morphism: indeed the trace map is invariant
under conjugation. Since the affine GIT quotient is a categorical quotient, we obtain a natural
morphism
χ : MG(X)→ χG(X). (1.2)
The map χ is an isomorphism, for instance, in the case of G = SLn(C), Sp2n(C) or SO2n+1(C)
[6, 8, 20].
The study of G-character varieties received a lot of attention. For instance, when X is
the underlying topological space of a smooth complex projective variety, the moduli space of G-
representations,MG(X) is one of the moduli spaces studied in non-abelian Hodge theory. In the
case of a smooth complex projective curve C and the algebraic group G = GLn(C), the character
varietyMG(C) parametrizes vector bundles over C of rank n and degree zero equipped with a flat
connection. With this identification, the Riemann–Hilbert correspondence [32] provides a real
analytic isomorphism between the character variety and the moduli space of G-flat connections
on the curve C. Moreover, the Hitchin–Kobayashi correspondence [31] gives a real analytic
isomorphism between MG(C) and the moduli space of semistable G-Higgs bundles of rank n
and degree zero on C. These correspondences were used by Hitchin [18] to compute the Poincaré
polynomial of twisted character varieties for G = GL2(C).
However, these correspondences are far from being algebraic. As a result, the mixed Hodge
structures of the above-mentioned moduli spaces have been extensively studied, for instance, via
the virtual Deligne–Hodge polynomial, or E-polynomial
e(X) =
∑
k,p,q
(−1)khk;p,qc (X)upvq ∈ Z[u, v],
encoding the dimensions hk;p,qc (X) = dimCHk;p,q
c (X) of the associated graded components with
respect to the weight and Hodge filtrations of the mixed Hodge structures on the compactly sup-
ported cohomology of a complex variety X. It follows from the excision long-exact sequence on
cohomology and from the Künneth formula (see [7]) that the E-polynomial extends to a motivic
measure
e : K(VarC)→ Z[u, v],
from the Grothendieck ring of varieties to the polynomial ring in two variables Z[u, v], that maps
the class of a complex variety [X] ∈ K(VarC) to its E-polynomial e(X).
Inspired by the Weil conjectures, an arithmetic approach was introduced by Hausel and
Rodŕıguez–Villegas [16] to compute the E-polynomial of twisted GLn(C)-character varieties by
Virtual Classes of Representation Varieties of Upper Triangular Matrices via TQFTs 3
counting its number of points over finite fields. In fact, using a theorem of Frobenius [10, 36]
relating the number of points of a G-representation variety over a finite field Fq to the dimen-
sions of the irreducible representations of G over Fq, Hausel and Rodŕıguez–Villegas show that
the number of points of the twisted character variety over the finite fields Fq of q elements is
a polynomial p(q) ∈ Z[q] in q, which in turn by a theorem of Katz, computes the E-polynomial
of the complex twisted character variety by setting q = uv. This method was extended to
the cases of SLr(C)-character varieties [26], GLr(C)-character varieties with a generic parabolic
structure [25], or non-orientable surfaces [21].
Recently, a geometric approach was introduced by Logares, Muñoz, and Newstead [22] by
dividing the representation variety into pieces and computing the E-polynomial piecewise. Us-
ing this technique, Mart́ınez and Muñoz [24] gave an explicit expression for the E-polynomial
of the SL2(C)-representation variety. Moreover, by combining the arithmetic and geometric ap-
proaches, Baraglia and Hekmati [2] gave explicit expressions for the cases G = GL3(C),SL3(C).
The recursive patterns in these computations lead González-Prieto, Logares, and Muñoz [14]
to develop a new method using topological quantum field theories (TQFTs) to compute the vir-
tual class of the representation varieties in the Grothendieck ring of varieties K(VarC). TQFTs,
originated from physics, were first introduced by Witten [35] and axiomatized by Atiyah [1]:
a TQFT is given by a monoidal functor
Z : Bdn → R-Mod
from the category of bordisms to the category of R-modules for some commutative ring R. In
particular, any closed manifold X can be seen as a bordism X : ∅ → ∅, so we obtain an R-
module map Z(X) : R → R, since Z(∅) = R by monoidality. As a consequence, any closed
manifod X has an associated invariant Z(X)(1) ∈ R. In [14], González-Prieto, Logares and
Muñoz construct a lax monoidal TQFT with R = K(VarC) such that the associated invariant
for a closed manifold X is the virtual class of the representation variety XG(X). Then, a closed
surface X = Σg of genus g can be considered as a composition of bordisms
Σg = ◦ ◦ · · · ◦︸ ︷︷ ︸
g times
◦
so that computing the TQFT for these smaller bordisms will yield the virtual class of XG(Σg)
for all g. This method was used in [11] and [13] to compute the virtual class of the (parabolic)
SL2(C)-character variety in K(VarC). A clear advantage of this method is that it not only
computes the E-polynomial of the representation variety, but more generally, its class in the
Grothendieck ring of varieties.
1.1 Main results
Virtual classes of representation varieties. In this paper, we focus on the groups of
complex upper triangular matrices Un of rank n = 2, 3, 4. We compute the virtual classes of the
corresponding representation varieties in a suitably localized Grothendieck ring of varieties (see
Theorems 3.3, 3.10 and 3.12).
Theorem 1.1. Let q =
[
A1
C
]
be the class of the affine line in the Grothendieck ring of varieties
over C. Then
(i) the virtual class of the U2-representation variety XU2(Σg) is
[XU2(Σg)] = q2g−1(q − 1)2g+1
(
(q − 1)2g−1 + 1
)
,
4 M. Hablicsek and J. Vogel
(ii) the virtual class of the U3-representation variety XU3(Σg) is
[XU3(Σg)] = q3g−3(q − 1)2g
(
q2(q − 1)2g+1 + q3g(q − 1)2
+ q3g(q − 1)4g + 2q3g(q − 1)2g+1
)
,
(iii) the virtual class of the U4-representation variety XU4(Σg) is
[XU4(Σg)] = q12g−6(q − 1)8g + q12g−6(q − 1)2g+3 + q10g−4(q − 1)2g+3
+ q10g−3(q − 1)4g+1 + q8g−2(q − 1)6g+1 + q8g−2(q − 1)4g+2
+ 2q10g−4(q − 1)6g+1 + 3q12g−6(q − 1)6g+1 + 3q12g−6(q − 1)4g+2
+ q10g−4(q − 1)4g+1(2q − 3).
By setting q = uv in the above formulae, we obtain the E-polynomials of the representation
varieties XU2(Σg), XU3(Σg) and XU4(Σg).
We remark that in an independent work [15], González-Prieto, Logares, and Muñoz computed
the virtual class of the AGL1-representation varieties, where AGL1 is the general affine group
of the line. Their result can be deduced from our result on XU2(Σg) through the isomorphism
of groups Gm ×AGL1
∼−→ U2 (see Remark 3.4).
We also provide a more general result with parabolic structures involved in the case of U2,
see Theorem 3.5 of the paper.
Theorem 1.2. Let Σg be a compact oriented surface of genus g, with parabolic data
Q = {(S1,Jλ1), . . . , (Sk,Jλk
), (Sk+1,Mµ1,σ1), . . . , (Sk+ℓ,Mµℓ,σℓ
)}.
(i) If
∏k
i=1 λi
∏ℓ
j=1 µj ̸= 1 or
∏k
i=1 λi
∏ℓ
j=1 σj ̸= 1, then
[XU2(Σg, Q)] = 0.
(ii) Otherwise, and if ℓ = 0, then
[XU2(Σg, Q)] = q2g−1(q − 1)2g
(
(−1)k(q − 1) + (q − 1)2g+k
)
,
(iii) and if ℓ > 0, then
[XU2(Σg, Q)] = q2g+ℓ−1(q − 1)4g+k.
In particular, we obtain that the representation varieties for U2, U3, and U4, and the rep-
resentation varieties with parabolic structures for U2 are of balanced type, meaning that their
virtual classes are generated by the virtual class of the affine line. This can be seen in relation
to Higman’s conjecture [17], which states that the number Cn(q) of conjugacy classes of Un(Fq)
over finite fields Fq is polynomial in q, for all n. Indeed, using Burnside’s lemma, one can relate
the number of Fq-points of the Un-representation variety of a surface of genus 1 to the number
of conjugacy classes:
|XUn(Fq)(Σ1)| = |Un(Fq)|Cn(q).
Now, Higman’s conjecture along with Katz’s theorem imply that the E-polynomial of XUn(C)(Σ1)
is a polynomial in uv. We ask whether a motivic version of Higman’s conjecture holds.
Conjecture 1.3. Let Σg be a closed oriented surface of genus g. The class [XUn(C)(Σg)] of the
Un-representation varieties in the Grothendieck ring of varieties is of balanced type for all n
and g.
Virtual Classes of Representation Varieties of Upper Triangular Matrices via TQFTs 5
The moduli space of G-representations and G-character varieties. In this paper,
we study the moduli space of G-representations of compact oriented surfaces Σg for the linear
groups Un with n ≥ 2. As these groups are non-reductive, there is no guarantee, a priori, that
the categorical quotientMUn(Σg) is of finite type over C. Nevertheless, we show the following
result.
Theorem 1.4. For all n ≥ 1 and g ≥ 0, there exists an isomorphism of varieties
MUn(Σg) ∼=
(
A1
C \ {0}
)2ng
.
Furthermore, we show that the map χ of (1.2) fails to be an isomorphism.
Theorem 1.5. For n ≥ 2 and g ≥ 1, the natural morphism
χ : MUn(Σg)→ χUn(Σg)
is not an isomorphism.
The paper is organized as follows. In Section 2, we construct a topological quantum field
theory (TQFT) computing the virtual classes of the representation varieties XG(Σg) in the
Grothendieck ring of varieties. We mainly follow [12, 13, 14, 33]. The novelty of this section
lies in Proposition 2.17, which can be used to ‘reduce’ the TQFT. While not strictly necessary
to perform computations, it does allow for a simplification of the computations. In Section 3,
we will apply this theory to the groups of upper triangular matrices G = Un of rank n = 2, 3, 4.
We prove our main theorems, Theorems 1.1 and 1.5. Explicitly, we compute the classes of the
representation varieties XG(Σg) in a suitably localized Grothendieck ring of varieties for the
groups of upper triangular matrices U2, U3, U4. Moreover, we show that the natural morphism
χ : MUn(Σg)→ χUn(Σg)
fails to be an isomorphism for Un when n ≥ 2 and g ≥ 1.
2 TQFTs and representation varieties
In this section, we follow [12, 13, 14, 33] to construct a topological quantum field theory (TQFT)
that computes the virtual classes of the representation varieties XG(Σg) in the Grothendieck
ring of varieties K(Vark). More precisely, we construct a lax monoidal TQFT Z over the ring
K(Vark) such that the invariant associated to a closed manifold X is Z(X)(1) = [XG(X)] ∈
K(Vark). This construction allows to solve a more general problem: if Λ is the set of conjugacy-
closed subsets of G, one can put a parabolic structure Q = {(S1, E1), . . . , (Ss, Es)} with data
in Λ on Σg, such that the invariant associated to (Σg, Q) is the class of the variety
XG(Σg, Q) =
{
(A1, B1, . . . , Ag, Bg, C1, . . . , Cs) ∈ G2g+s
∣∣∣∣
g∏
i=1
[Ai, Bi]
s∏
i=1
Ci = 1 and Ci ∈ Ei
}
.
The novelty of this section is Proposition 2.17, which allows us to modify the TQFT and
simplify the computations. The rest is added for the sake of completeness. We begin by defining
the categories involved in the construction of the TQFT.
6 M. Hablicsek and J. Vogel
2.1 The 2-category of bordisms
Let i : M → ∂W be an inclusion, where W is an n-dimensional oriented manifold with boundary,
and M an (n−1)-dimensional closed oriented manifold. (All manifolds we consider are assumed
to be smooth.) Take a point x ∈ i(M), let {v1, . . . , vn−1} be a positively oriented basis for Txi(M)
with respect to the orientation induced by M , and pick some w ∈ Txi(M) that points inwards
compared to W . Then if {v1, . . . , vn−1, w} is a positively oriented basis for TxW , we say x is an
in-boundary point, and an out-boundary point otherwise. Note that this is independent of the
chosen vectors vi and w. If all x ∈ i(M) are in-boundary (resp. out-boundary) points, we say i
is an in-boundary (resp. out-boundary).
Definition 2.1. Given two (n−1)-dimensional closed oriented manifolds M and M ′, a bordism
from M to M ′ is an n-dimensional oriented manifold W (with boundary) with maps
M ′ W M,i′ i
where i is an in-boundary, i′ an out-boundary and ∂W = i(M) ⊔ i′(M ′). Two such bor-
disms W , W ′ are said to be equivalent if there exists an orientation-preserving diffeomorphism
W
∼−→W ′ such that
W
M ′ M
W ′
≀
commutes.
For a more precise definition of bordisms, see [27] or [19].
Suppose we have bordisms W : M → M ′ and W ′ : M ′ → M ′′. One can glue W and W ′ as
topological spaces by identifying the images of M ′, which we denote by W ⊔M ′ W ′. By [27,
Theorem 1.4], there exists a smooth manifold structure on W ⊔M ′ W ′ such that the inclusions
W →W ⊔M ′W ′ and W ′ →W ⊔M ′W ′ are diffeomorphisms onto their images, which is unique up
to (non-unique) diffeomorphism. Hence W ⊔M ′ W ′ belongs to a well-defined equivalence class,
and moreover this class only depends on the classes of W and W ′. Namely, if W̃ : M →M ′ and
W̃ ′ : M ′ → M ′′ are equivalent to W and W ′, respectively, then any such manifold structure on
W ⊔M ′W ′ induces such a manifold structure on W̃ ⊔M ′ W̃ ′ via the homeomorphism W ⊔M ′W ′ →
W̃ ⊔M ′ W̃ ′, showing W ⊔M ′ W ′ and W̃ ⊔M ′ W̃ ′ are equivalent. This implies that equivalence
classes of bordisms can be composed to obtain an equivalence class of bordisms M →M ′′.
The discussion above gives rise to the following definition.
Definition 2.2. The category of n-bordisms, denoted Bdn, is defined as follows. Its objects are
(n− 1)-dimensional closed oriented manifolds, and morphisms M →M ′ are equivalence classes
of bordisms from M to M ′. Composition is given by gluing along the common boundary: if
W : M →M ′ and W ′ : M ′ →M ′′, then W ′ ◦W = W ⊔M ′ W ′ : M →M ′′.
Definition 2.3. The category of pointed n-bordisms, denoted Bdpn, is the 2-category consisting
of:
� Objects: pairs (M,A) with M an (n−1)-dimensional closed oriented manifold, and A ⊂M
a finite set of points intersecting each connected component of M .
� 1-morphisms: a map (M1, A1) → (M2, A2) is given by a class of pairs (W,A) with
W : M1 → M2 a bordism, and A ⊂ W a finite set intersecting each connected compo-
nent of W such that A∩M1 = A1 and A∩M2 = A2. Two such pairs (W,A) and (W ′, A′)
Virtual Classes of Representation Varieties of Upper Triangular Matrices via TQFTs 7
are in the same class if there is a diffeomorphism F : W → W ′ such that F (A) = A′ and
such that the diagram
W
M2 M1
W ′
≀ (2.1)
commutes.
The composition of (W,A) : (M1, A1) → (M2, A2) and (W ′, A′) : (M2, A2) → (M3, A3) is
(W ⊔M2 W
′, A ∪A′) : (M1, A1)→ (M3, A3).
� 2-morphisms: a map (W,A) → (W ′, A′) is given by a diffeomorphism F : W → W ′ such
that F (A) ⊂ A′ and (2.1) commutes.
Note that so far, no identity morphism exists for (M,A), unless M = A = ∅. For this reason,
we loosen the definition of a bordism a bit, and allow M itself to be seen as a bordism M →M ,
so that (M,A) will be the identity morphism for (M,A).
In this paper, we also consider manifolds which carry a so-called parabolic structure. Fix a set
Λ and call it the parabolic data. We say a parabolic structure on a manifold M is a finite set
Q = {(S1, E1), . . . , (Ss, Es)} with Ei ∈ Λ and the Si are pairwise disjoint compact submanifolds
of M of codimension 2 with a co-orientation (i.e. an orientation of its normal bundle) such that
Si ∩ ∂M = ∂Si transversally.
Definition 2.4. Let Λ be a set. The 2-category of pointed n-bordisms with parabolic structures
over Λ, denoted Bdpn(Λ), is the 2-category consisting of:
� Objects: triples (M,A,Q) with M an (n−1)-dimensional closed oriented manifold, Q a pa-
rabolic structure on M , and A ⊂ M a finite set of points not intersecting any of the Si
of Q.
� 1-morphisms: a map (M1, A1, Q1) → (M2, A2, Q2) is given by a class of triples (W,A,Q)
where W : M1 → M2 is a bordism, Q a parabolic structure on W , and A ⊂ W a finite
set intersecting each connected component of W but not intersecting any of the Si of Q,
such that A ∩M1 = A1, A ∩M2 = A2, Q|M1 = Q1 and Q|M2 = Q2. Here we use the
notation Q|Mi = {(Sj∩Mi, Ej) | (Sj , Ej) ∈ Λ}. Two such triples (W,A,Q) and (W ′, A′, Q′)
are in the same class if there is a diffeomorphism F : W → W ′ such that F (A) = A′ and
(S, E) ∈ Q if and only if (F (S), E) ∈ Q′ and such that the diagram
W
M2 M1
W ′
≀ (2.2)
commutes.
The composition of bordisms (W,A,Q) : (M1, A1, Q1) → (M2, A2, Q2) and (W ′, A′, Q′) :
(M2, A2, Q2)→ (M3, A3, Q3) is given by (W,A,Q)◦(W ′, A′, Q′) = (W⊔M2W
′, A∪A′, Q⊔M2
Q′), where Q ⊔M2 Q
′ denotes the union of Q and Q′ but where we glue pairs (S, E) ∈ Q
and (S′, E) ∈ Q′ that have a common boundary (in M2).
� 2-morphisms: a map (W,A,Q) → (W ′, A′, Q′) is given by a diffeomorphism F : W → W ′
such that F (A) ⊂ A′ and (F (S), E) ∈ Q′ for each (S, E) ∈ Q and such that (2.2) commutes.
8 M. Hablicsek and J. Vogel
Actually, Bdpn can be seen as a particular case of Bdpn(Λ) for Λ = ∅. The category
Bdpn(Λ) (and thus Bdpn as well) is a monoidal category. The tensor product is given by
taking disjoint unions:
(M,A,Q) ⊔ (M ′, A′, Q′) = (M ⊔M ′, A ∪A′, Q ∪Q′)
for objects, and similarly for bordisms. The unital object is (∅,∅,∅), which we also denote
simpy by ∅.
Note that (non-empty) parabolic structures can only exist on manifolds of dimension ≥ 2. In
particular for Bdp2(Λ), its 1-dimensional objects have Q = ∅ and the parabolic structures of
its 2-dimensional bordisms are of the form {(p1, E1), . . . , (ps, Es)} with pi points on the interior
of the bordism that have a preferred orientation of small loops around them.
2.2 The Grothendieck ring of varieties
Definition 2.5. Let S be a variety over a field k (i.e., a reduced separated scheme of finite type
over k). The Grothendieck ring of varieties over S, denoted K(Var/S), is defined as the quotient
of the free abelian group on the set of isomorphism classes of varieties over S, by relations of
the form
[X] = [X\Z] + [Z],
where Z is a closed subvariety of X and X\Z is its open complement. Multiplication is dis-
tributively induced by
[X] · [Y ] = [(X ×S Y )red],
which is indeed associative and commutative. It follows that [∅] = 0 and [S] = 1 in K(Var/S).
When S is the base field k we denote the Grothendieck ring of varieties by K(Vark). To distin-
guish between the classes of different rings, we will write [X]S for the class of X in K(Var/S)
and for the class of X in K(Vark) we will simply write [X].
Notice that K(Var/S) is a monoid object in the category of K(Vark)-modules. Indeed,
K(Var/S) has a natural K(Vark)-module structure induced by
[T ] · [X]S = [T ×k X]S
for T a variety over k and X a variety over S, such that the multiplication map [X]S · [Y ]S =
[(X ×S Y )red]S is K(Vark)-bilinear.
Now, we describe module maps K(Var/X)→ K(Var/Y ) which will be used in defining the
TQFT. Let f : X → Y be a morphism of varieties over k. Composition with f yields a functor
f! : Var/X → Var/Y(
V
g−→ X
)
7→
(
V
fg−→ Y
)
.
As f! sends isomorphisms to isomorphisms, and also T ×k f!V = f!(T ×k V ) for any variety T
over k, we have that f! induces a K(Vark)-module morphism
f! : K(Var/X)→ K(Var/Y ).
Note that this map will in general not be a ring morphism. For example, the unit [X]X ∈
K(Var/X) need not be sent to the unit [Y ]Y ∈ K(Var/Y ).
Virtual Classes of Representation Varieties of Upper Triangular Matrices via TQFTs 9
Similarly, pulling back along f yields a functor
f∗ : Var/Y → Var/X
sending W
h−→ Y to W ×Y X
f∗h−→ X. Also f∗ induces a map,
f∗ : K(Var/Y )→ K(Var/X),
which is a K(Vark)-module morphism as T ×k f∗(V ) = T ×k (V ×Y X) ∼= (T ×k V ) ×Y X =
f∗(T ×k V ) for any variety T over k. In contrast to f!, the map f∗ is a ring morphism as
(V ×Y W )×Y X ∼= (V ×Y X)×X (W ×Y X) for any V , W over Y .
Example 2.6. The rings K(Var/X) are objects of the category of K(Vark)-Mod. Moreover,
given a variety Z over k with morphisms f : Z → X and g : Z → Y , we have an induced
K(Vark)-module morphism g! ◦ f∗ : K(Var/X)→ K(Var/Y ).
K(Var/X) K(Var/Z) K(Var/Y ).
f∗ g!
Remark 2.7. The functors f∗ and f! are adjoint, as for any varieties V
v−→ X and W
w−→ Y
there is a bijection
HomVar/Y (f!V,W ) ∼= HomVar/X(V,W ×Y X)
natural in V and W . Namely, by the universal property of the fiber product, to give a morphism
φ : V → W ×Y X is to give morphisms V
r−→ W and V
s−→ X such that w ◦ r = f ◦ s, and
requiring φ to be over X means to have s = v. Hence, to give φ over X is to give V
r−→W such
that w ◦ r = v, i.e. a morphism V
r−→W over Y . The naturality of this bijection is easily seen.
In this paper, the target category of the TQFT is the 2-category of K(Vark)-modules with
twists.
Definition 2.8. Let R be a commutative ring. Given R-module morphisms f, g : M → N ,
we say g is an immediate twist of f if there exists an R-module P and R-module morphisms
f1 : M → P , f2 : P → N and h : P → P such that f = f2 ◦ f1 and g = f2 ◦ h ◦ f1:
M P N.
f1
g
h
f2
We say a twist from f to g is a finite sequence f = f0, f1, . . . , fn = g : M → N of R-module
morphisms such that fi+1 is an immediate twist of fi.
Now the 2-category of R-modules with twists, denoted R-Modt, is the category whose objects
are R-modules, its 1-morphisms are R-module morphisms, and its 2-morphisms are twists.
2.3 The 2-category of spans
Definition 2.9. Given a category C with pullbacks, we define the 2-category Span(C) as follows:
� Its objects are the objects of C.
10 M. Hablicsek and J. Vogel
� An arrow from C to C ′ is given by a diagram C ← D → C ′ in C, called a span. Composition
of the spans C ← D → C ′ and C ′ ← D′ → C ′′ is given by the span C ← E → C ′′ such
that the square in
E
D D′
C C ′ C ′′
is a pullback square.
� A 2-morphism from C ← D → C ′ to C ← D′ → C ′ is given by an arrow D → D′ such
that the following diagram commutes:
D
C C ′.
D′
The category Spanop(C) is defined analogously on categories with pushouts, where we reverse
all arrows.
If C is a monoidal category, then Span(C) naturally has the structure of a monoidal category
as well. The tensor product and unital object naturally carry over, the associator will be given
by the span
A⊗ (B ⊗ C) A⊗ (B ⊗ C) (A⊗B)⊗ Cid αA,B,C
and the left and right unitor by
I ⊗A I ⊗A Aid λA and A⊗ I A⊗ I A.id ρA
We are ready to construct the TQFT computing the classes of the representation varieties
XG(Σg) in K(Vark).
2.4 Constructing the TQFT
Definition 2.10. Let (X,A) be a pair of topological spaces. The fundamental groupoid of X
w.r.t. A denoted Π(X,A) is the groupoid category whose objects are elements of A, and an arrow
a → b for each homotopy class of paths from a to b. Composition of morphisms is given by
concatenation of paths. Indeed this construction only depends on the homotopy type of (X,A).
In particular, if A = {x0} is a single point, we obtain the fundamental group π1(X,x0).
Note that if f : (X,A)→ (X ′, A′) is a map of pairs of topological spaces, there is an induced
functor Π(X,A)→ Π(X ′, A′) between groupoids, mapping an object a ∈ A to f(a) ∈ A′ and an
arrow γ : a→ b to f ◦ γ. This allows us to construct the following functor.
Definition 2.11. The geometrization functor is a 2-functor Π: Bdpn → Spanop(Grpd), with
Grpd the category of groupoids, defined as follows:
� To each object (X,A) we assign the fundamental groupoid Π(X,A).
� For each 1-morphism (W,A) : (X1, A1)→ (X2, A2) we assign the cospan
Π(X1, A1)
i1−→ Π(W,A)
i2←− Π(X2, A2)
with i1 and i2 induced by inclusions.
Virtual Classes of Representation Varieties of Upper Triangular Matrices via TQFTs 11
� For each 2-morphism (W,A) → (W,A′) given by diffeomorphism F : W → W ′ with
F (A) ⊂ A′, we obtain a groupoid morphism ΠF yielding the commutative diagram
Π(W,A)
Π(X1, A1) Π(X2, A2),
Π(W ′, A′)
ΠF
i1
i′1
i2
i′2
which is a 2-morphism in Spanop(Grpd).
The Seifert–van Kampen theorem for fundamental groupoids [5] provides that Π defined
above is indeed a functor (for more details see [12]).
Suppose X is a compact connected manifold (possibly with boundary), A ⊂ X a finite set
of points, and denote G = Π(X,A). We write Ga = HomG(a, a) for a in G. Since a compact
connected manifold has the homotopy type of a finite CW-complex, every Ga = π1(X, a) is
a finitely generated group.
The groupoid G has finitely many connected components, where we say objects a, b ∈ G
are connected if HomG(a, b) is non-empty. Pick a subset S = {a1, . . . , as} ⊂ A such that each
connected component of G contains exactly one of the ai. Also pick an arrow fa : ai → a for
each a ∈ A (with ai ∈ S in the connected component of a) such that fai = idai for each ai ∈ S.
Now if G is a group, then a morphism of groupoids ρ : G → G is uniquely determined by the
group morphisms ρi : Gai → G and a choice of ρ(fa) ∈ G. Namely, any γ : a → b in G can be
written as γ = fb ◦ γ′ ◦ (fa)−1 for some γ′ ∈ Gai (with ai ∈ S in the connected component of a
and b). The elements ρ(fa) can take any value for a ̸∈ A\S (and ρ(fai) = 1 ∈ G for ai ∈ S), so
if G has n objects and s connected components, we have
HomGrpd(G, G) ∼= Hom(Ga1 , G)× · · · ×Hom(Gas , G)×Gn−s. (2.3)
If G is an algebraic group, each of these factors naturally carries the structure of an algebraic
variety. Namely, each Gai is finitely generated, so Hom(Gai , G) can be identified with a subvariety
of Gm for some m > 0. This gives Hom(G, G) the structure of an algebraic variety, and this
structure can be shown not to depend on the choices.
Definition 2.12. Let X be a compact connected manifold (possibly with boundary) and A ⊂ X
a finite set. Then we define the G-representation variety of (X,A) to be
XG(X,A) = HomGrpd(Π(X,A), G).
Note that the functor HomGrpd(−, G) sends pushouts to pullbacks, so we obtain an induced
2-functor
F : Bdpn → Span(Vark),
which we refer to as the field theory. This functor sends an object (M,A) to XG(M,A), a bordism
(W,A) : (M1, A1)→ (M2, A2) to the span
XG(M1, A1)←− XG(W,A) −→ XG(M2, A2),
and a 2-morphism given by diffeomorphism F : (W,A)→ (W ′, A′) with F (A) ⊂ A′ to an inclu-
sion of the corresponding varieties.
Recall that Vark is a monoidal category, the tensor product being ×k and the unital object
Spec k, so the category Span(Vark) is monoidal as well. Following the above construction,
12 M. Hablicsek and J. Vogel
one can see that F is a monoidal functor. Indeed, XG(∅) = HomGrpd(∅, G) is a point, and
XG(X ⊔ X ′, A ∪ A′) ∼= XG(X,A) ×k XG(X
′, A′) as can be easily shown from (2.3). Also, F is
seen to be symmetric.
The last step in constructing the TQFT is the quantization functor
Q : Span(Vark)→ K(Vark)-Modt,
which assigns to an object X the K(Vark)-module K(Var/X), and to a span X
f←− Z
g−→ Y
the morphism g! ◦ f∗ : K(Var/X)→ K(Var/Y ) (see Example 2.6). Given a 2-morphism
Z1
X Y
Z2
h
f1 g1
f2 g2
we see that (g1)!◦(f1)∗ = (g2)!◦h!◦h∗◦f∗
2 , which defines an (immediate) twist from (g1)!◦(f1)∗ to
(g2)! ◦ (f2)∗. For a proof that Q is a lax (symmetric) monoidal 2-functor, see [11, Theorem 4.13].
Remark 2.13. Contrary to F , the quantization functor Q : Span(Vark) → K(Vark)-Modt is
not a monoidal functor. Namely, even though there is a natural map
Q(X)⊗Q(Y ) = K(Var/X)⊗K(Var/Y )→ K(Var/(X × Y )) = Q(X × Y ),
[V → X]⊗ [W → Y ] 7→ [V ×W → X × Y ],
this map need not be an isomorphism. Relaxing the condition of this natural map to be an
isomorphism, we obtain the notion of a lax monoidal functor.
At last, we define the symmetric lax monoidal TQFT as the composition of the field theory
and the quantization functor
Z = Q ◦ F : Bdpn → K(Vark)-Modt.
Now, any closed connected oriented manifold X of dimension n, with a point ⋆ on X, can be
viewed as a bordism (X, ⋆) : ∅→ ∅. Then F(X, ⋆) is the span
⋆
t←− XG(X, ⋆) = XG(X)
t−→ ⋆
and Z(X, ⋆)(1) = t!t
∗(⋆) = t!
(
[XG(X)]XG(X)
)
= [XG(X)] as desired.
2.5 Parabolic structures
Let Λ be a set of conjugacy-closed subsets of G. We will slightly modify the construction above
to obtain a (lax symmetric) TQFT ZΛ : Bdpn(Λ)→ K(Vark)-Modt. This will be an extension
of Z in the sense that it yields the same modules and morphisms as Z in the absence of parabolic
structures, i.e., ZΛ(X,A,∅) = Z(X,A).
Let X be a compact manifold (possibly with boundary) with a parabolic structure Q given
by
Q = {(S1, E1), . . . , (Ss, Es)},
and A ⊂ X a finite set intersecting each connected component of X, but not intersecting
S = ∪iSi. Then the representation variety of (X,A,Q) is defined as
XG(X,A,Q) =
ρ : Π(X − S,A)→ G
∣∣∣∣ ρ(γ) ∈ Ei for all loops γ around Si
positive w.r.t. the co-orientation,
for all (Si, Ei) ∈ Q
, (2.4)
Virtual Classes of Representation Varieties of Upper Triangular Matrices via TQFTs 13
where ‘γ around Si’ means a non-zero loop γ in Π(X−S,A) which is zero in Π(X− (S−Si), A).
Since the Ei are conjugacy-closed, the condition on the loops γ around Si is independent on the
chosen base point. Indeed, this definition of XG(X,A,Q) agrees with Definition 2.12 for Q = ∅.
When X is connected, we write
XG(X,Q) = XG(X, ⋆,Q).
In the particular case of X = Σg with parabolic structure Q = {(S1, E1), . . . , (Ss, Es)} (co-
orientation induced from orientation on Σg), we find
XG(Σg, Q) =
{
(A1, B1, . . . , Ag, Bg, C1, . . . , Cs) ∈ G2g+s
∣∣∣∣
g∏
i=1
[Ai, Bi]
s∏
i=1
Ci = 1 and Ci ∈ Ei
}
.
Consider the modified field theory FΛ : Bdpn(Λ)→ Span(Vark) that maps an object (M,A,Q)
to XG(M,A,Q), a bordism (W,A,Q) : (M1, A1, Q1)→ (M2, A2, Q2) to the span
XG(M1, A1, Q1)←− XG(W,A,Q) −→ XG(M2, A2, Q2)
induced by the inclusions and a 2-morphism given by diffeomorphism F : W→W ′ with F (A)⊂A′
to an inclusion of the corresponding varieties (see [12]). It is easy to see that this functor is still
monoidal. We obtain the resulting (lax symmetric) TQFT
ZΛ = Q ◦ FΛ : Bdpn(Λ)→ K(Vark)-Modt.
To a closed connected oriented manifold X with parabolic structure Q is now associated the
invariant
ZΛ(X,A,Q)(1) = [XG(X,A,Q)].
Since ZΛ is understood to be an extension of the earlier TQFT Z : Bdpn → K(Vark)-Modt,
and since it is clear what set Λ we consider, we will just write Z for ZΛ.
2.6 Field theory in dimension 2
We focus on the case of dimension n = 2. Let X = Σg be a closed oriented 2-dimensional surface
of genus g, possibly with a parabolic structure Q. Now Σg can be considered as a bordism
∅ → ∅, and after taking a suitable finite set A ⊂ Σg, be written as a composition of the
following bordisms:
D† :
(
S1, ⋆
)
→ ∅ L :
(
S1, ⋆
)
→ (S1, ⋆) LE :
(
S1, ⋆
)
→
(
S1, ⋆
)
D : ∅→
(
S1, ⋆
) (2.5)
Here LE denotes the cylinder with parabolic structure {(⋆, E)} with E ∈ Λ. Now indeed, if we
write Q = {(p1, E1), . . . , (ps, Es)} for the parabolic structure on Σg, we have
(Σg, A,Q) = D† ◦ Lg ◦ LE1 ◦ · · · ◦ LEs ◦D. (2.6)
Of course, the category Bdp2(Λ) consists of more objects and morphisms than just the ones
mentioned in (2.5). However, as we are only interested in closed connected surfaces (possibly
14 M. Hablicsek and J. Vogel
with a parabolic structure), we will restrict our attention to a subcategory of Bdp2(Λ): we say
a strict tube is any composition of the bordisms in (2.5), and let Tb2(Λ) be the subcategory of
Bdp2(Λ) whose objects are disjoint copies of
(
S1, ⋆
)
and bordisms are disjoint unions of strict
tubes. Note that Tb2(Λ) is still monoidal (with the same monoidal structure as Bdp2(Λ)). We
refer to Tb2(Λ) as the category of tubes.
We restrict Z to a functor Tb2(Λ)→ K(Vark)-Modt, and explicitly describe what the TQFT
does to our objects and bordisms in (2.5).
The fundamental groups π1(D) and π1
(
D†) are trivial, implying XG(D) = XG
(
D†) = ⋆. Since
π1
(
S1, ⋆
)
= Z, we have XG
(
S1, ⋆
)
= Hom(Z, G) = G and since Π(∅) is the empty groupoid, we
have XG(∅) = ⋆. Hence the field theory for D and D† is given by
F(D) : ⋆ ←− ⋆ −→ G
⋆ ←[ ⋆ 7→ 1
and
F
(
D†) : G ←− ⋆ −→ ⋆
1 ←[ ⋆ 7→ ⋆.
For the bordism L, call its two basepoints a and b. The surface of L is homotopic to a torus
with two punctures, so its fundamental group (w.r.t. a) is the free group F3. We pick generators
γ, γ1, γ2 as depicted in the following image, and a path α connecting a and b:
γ
γ1
γ2
α
a b
According to (2.3) we can now identify
XG(L) ∼= Hom(F3, G)×G ∼= G4, ρ 7→ (ρ(γ), ρ(γ1), ρ(γ2), ρ(α)).
A generator for π1(S
1, b) is given by αγ[γ1, γ2]α
−1, and so the field theory for L is found to be
F(L) : G ←− G4 −→ G
g ←[ (g, g1, g2, h) 7→ hg[g1, g2]h
−1.
(2.7)
Finally for the bordism LE , call its two basepoints a and b. The fundamental group (w.r.t. a) of
the cylinder with a puncture is the free group F2. We pick generators γ, γ′ as depicted in the
following image, and a path α connecting a and b:
γ
γ′
α
a b
Using (2.4) we can now identify
XG(LE) ∼= G2 × E , ρ 7→ (ρ(γ), ρ(α), ρ(γ′)).
A generator for π1(S
1, b) is given by αγγ′α−1, and so the field theory for LE is found to be
F(Lλ) : G ←− G2 × E −→ G
g ←[ (g, h, ξ) 7→ hgξh−1.
Finally, using (2.3) and (2.6) we can express the class of the character variety XG(Σg, Q) in
terms of the TQFT (see also [11, Theorem 4.11]).
Virtual Classes of Representation Varieties of Upper Triangular Matrices via TQFTs 15
Theorem 2.14. Let Σg be the closed oriented surface of genus g with parabolic structure Q =
{(S1, E1), . . . , (Ss, Es)}. Then,
[XG(Σg, Q)] =
1
[G]g+s
[XG(Σg, {g + s+ 1 points}, Q)]
=
1
[G]g+s
Z
(
D† ◦ Lg ◦ LE1 ◦ · · · ◦ LEs ◦D
)
(1). (2.8)
Note that [G] might be a zero-divisor. Indeed, in this paper, we focus on the groups of upper
triangular matrices Un of rank n = 2, 3, 4 whose classes are products of the class of the punctured
affine line
[
A1
k \ {0}
]
and the class of the affine line
[
A1
k
]
, where the latter is a zero divisor in
K(Vark) [4, 23]. However, one can consider a suitable localization of K(Vark), the localization
with the classes of
[
A1
k
]
and
[
A1
k \ {0}
]
, in which the computation described by (2.8) holds.
2.7 Reduction of the TQFT
Let Z : B → R-Modt be a (lax monoidal) TQFT, where B is some kind of bordism category
(e.g., Bdpn, Bdpn(Λ) or Tb2(Λ)) and R is a commutative ring. For some TQFTs there is
a symmetry, such as a group action, which can be used to ‘reduce’ the TQFT, allowing for
a simplification of the computations. In this section, we will show how such a reduction can be
obtained.
For each object M ∈ B, let
Z(M) NM
αM
βM
be R-module morphisms, with NM an R-module. Assume that N∅ = Z(∅) with α∅ and β∅ the
identity maps. Let VM be a submodule of NM such that (αM ′ ◦ Z(W ) ◦ βM )(VM ) ⊂ VM ′ for all
bordisms W : M →M ′ in B. In particular, (αM ◦ βM )(VM ) ⊂ VM for any M .
Lemma 2.15. Suppose that the map αM ◦ βM : VM → VM is invertible for all M ∈ B. Then
for every W : M → M ′ there exists a unique R-linear map Z̃(W ) : VM → VM ′ such that the
following diagram commutes:
βM (VM ) Z(W )(βM (VM ))
VM VM ′ .
Z(W )
αM αM′
Z̃(W )
Proof. Indeed, the above diagram is well-defined by the assumption that (αM ′ ◦ Z(W ) ◦
βM )(VM ) ⊂ VM ′ . We are looking for an R-linear map Z̃(W ) : VM → VM ′ such that Z̃(W )◦αM =
αM ′ ◦ Z(W ). Precomposing this equality with βM ◦ (αM ◦ βM )−1 gives
Z̃(W ) = αM ′ ◦ Z(W ) ◦ βM ◦ (αM ◦ βM )−1. (2.9)
This shows there is a unique choice of Z̃(W ), and it is easy to see that this choice makes the
diagram commute: any x ∈ βM (VM ) can be written as x = βM (y) for some y ∈ VM , so
Z̃(W ) ◦ αM (x) = Z̃(W ) ◦ (αM ◦ βM )(y) = αM ′ ◦ Z(W ) ◦ βM (y) = αM ′ ◦ Z(W )(x). ■
16 M. Hablicsek and J. Vogel
Now our goal is to construct a 2-functor Z̃ : B → R-Modt with Z̃(M) = VM and Z̃(W ) : VM
→ VM ′ for any W : M → M ′ as above. From (2.9) we see that a twist from Z(W1) to Z(W2)
induces a twist from Z̃(W1) to Z̃(W2). Note that Z̃ preserves compositions of 1-morphisms if
the following additional assumption holds:
Z(W )(βM (VM )) ⊂ βM ′(VM ′) for any bordism W : M →M ′.
Namely if so, let W : M →M ′ and W ′ : M ′ →M ′′ be bordisms. Then
βM (VM ) Z(W )(βM (VM )) Z(W ′ ◦W )(βM (VM ))
VM VM ′ VM ′′
Z(W )
αM
Z(W ′)
αM′ αM′′
Z̃(W ) Z̃(W ′)
is a commutative diagram by the previous lemma. We have Z̃(W ′)◦Z̃(W )◦αM = αM ′′ ◦Z(W ′)◦
Z(W ) = αM ′′ ◦Z(W ′ ◦W ), so uniqueness implies that Z̃(W ′ ◦W ) = Z̃(W ′)◦ Z̃(W ), and hence Z̃
is a functor.
Summarizing, we obtain the following definition.
Definition 2.16. For each object M in B, let Z(M) NM
αM
βM
be R-module morphisms
with NM an R-module, and VM ⊂ NM a submodule. If
(i) N∅ = Z(∅) and α∅, β∅ are identity maps,
(ii) (αM ′ ◦ Z(W ) ◦ βM )(VM ) ⊂ VM ′ for all bordisms W : M →M ′,
(iii) the restriction αM ◦ βM : VM → VM is invertible for all M ,
(iv) Z(W )(βM (VM )) ⊂ βM ′(VM ′) for all bordisms W : M →M ′,
then we speak of a reduction of the TQFT, and call the functor Z̃ of Lemma 2.15 the reduced
TQFT.
The whole point of the reduced TQFT Z̃ is that it computes the same invariants as Z for
closed manifolds, while allowing for easier computations. Indeed, if W : ∅ → ∅ is a bordism,
then
Z̃(W )(1) = Z̃(W ) ◦ α∅(1) = α∅ ◦ Z(W )(1) = Z(W )(1).
In Section 3, we will apply this to the category B = Tb2(Λ) and TQFT of Section 2.5, as follows.
We have Z
(
S1, ⋆
)
= K(Var/G), and there is an action of G on itself by conjugation. Suppose
there are conjugacy-closed strata C1, . . . , Cn for G, with maps πi : Ci → Ci whose fibers are
precisely the orbits of G. Then we have an induced decomposition K(Var/G) = K(Var/C1)⊕
· · · ⊕K(Var/Cn), and the maps (πi)! and (πi)
∗ induce maps
K(Var/C1)⊕ · · · ⊕K(Var/Cn) K(Var/C1)⊕ · · · ⊕K(Var/Cn),
π!
π∗
(2.10)
which by slight abuse of notation we denote by π∗ and π!. For bordisms W :
(
S1, ⋆
)
→ (S1, ⋆),
we write Zπ(W ) = π! ◦ Z(W ) ◦ π∗ as a shorthand. Let V ⊂
⊕n
i=1K(Var/Ci) be a submodule
on which η = π!π
∗ is invertible. Now, not any such stratification will satisfy the conditions for
a reduction. To make this precise, consider the variety ∆ =
{
(g1, g2) ∈ G2 | g1 ∼ g2
}
of the
pairs of conjugate elements of G, and the conjugation map
c : G2 → ∆, (g, h) 7→
(
g, hgh−1
)
.
Virtual Classes of Representation Varieties of Upper Triangular Matrices via TQFTs 17
The stratification of G naturally induces a stratification of ∆, whose strata we denote by
∆i =
{
(g1, g2) ∈ C2i : g1 ∼ g2
}
, and the map c!c
∗ naturally restricts to a map K(Var/∆i) →
K(Var/∆i).
Proposition 2.17. Let G be an algebraic group, stratified by conjugacy-closed strata Ci, with
maps πi : Ci → Ci whose fibers are precisely the orbits of G, and let V ⊂
⊕n
i=1K(Var/Ci) be a
submodule. Assume that
(i) Zπ(W )(V ) ⊂ V for all bordisms W :
(
S1, ⋆
)
→
(
S1, ⋆
)
,
(ii) the map η = π!π
∗ : V → V is invertible,
(iii) [{1}] ∈ π∗V and whenever X ∈ V then Xi = X|Ci ∈ V as well,
(iv) for each stratum Ci, the stabilizers Stabi of the points are all isomorphic and special.1
Then the maps in (2.10) and the submodule V yield a reduction of the TQFT.
Proof. The only remaining condition to show is (iv) of Definition 2.16, and it suffices to show
this holds for the bordisms D, D†, L and LE . This holds for D by assumption (iii), and for D†
trivially because V∅ = K(Vark). For L, we consider the associated span F(L) given by
G
p←− G4 q−→ G,
g ←[ (g, g1, g2, h) 7→ hg[g1, g2]h
−1,
and also the modified span
G
p̃←− G3 q̃−→ G,
g ←[ (g, g1, g2) 7→ g[g1, g2].
It is not hard to see that
q!p
∗ = (p2)!c!c
∗(p1)
∗q̃!p̃
∗, (2.11)
with p1, p2 : ∆→ G the projections, as both sides of the equality send X
f−→ G to{
(x, g, g1, g2, h) ∈ X ×G4 | g = f(x)
}
−→ G,
(x, g, g1, g2, h) 7→ hg[g1, g2]h
−1.
Let us describe the restrictions (c!c
∗)i : K(Var/∆i) → K(Var/∆i). For any X
f−→ ∆i, the
pullback c∗X → X is a Stabi-torsor, so by assumption (iv), it is Zariski-locally trivial, yielding
[c∗X]X = [Stabi][X]X . In particular, the restrictions (c!c
∗)i are given by scalar multiplication
with [Stabi].
Now, take any X ∈ V , let Y = q̃!p̃
∗π∗X ∈ K(Var/G), and decompose Y =
∑n
i=1 Yi
with each Yi ∈ K(Var/Ci) according to the stratification of G. Then π!Y = π!q̃!p̃
∗π∗X =
1
[G]Zπ(L)(X) lies in V by (i), so π!Yi = (π!Y )i lies in V by (iii). Furthermore, (p2)!c!c
∗(p1)
∗Yi =
[Stabi](p2)!(p1)
∗Yi = [Stabi]π
∗π!Yi ∈ π∗V , so it follows using (2.11) that
Z(L)(π∗X) =
∑
i
π∗π!Yi ∈ π∗V.
A completely similar argument shows that Z(LE)(π
∗V ) ⊂ π∗V as well. ■
1We say that a linear algebraic group G is special if every G-torsor (locally trivial in the étale topology) is
locally trivial in the Zariski topology.
18 M. Hablicsek and J. Vogel
In this paper, we consider the groups of upper triangular matrices U2, U3 and U4 over C.
For these groups, there exists a natural stratification given by the types of conjugacy classes.
In particular, for each stratum, the stabilizers of the points are isomorphic and special, hence
satisfying condition (iv) of the proposition above.
Now we see that the reduced TQFT leads to a simplification of the computation, as it allows
us to work over K(Var/Ci) instead of K(Var/Ci). This is reflected in the computations by the
fact that we can get rid of the conjugation by h as in the span of F(L) (2.7).
Remark 2.18. For the groups G and strata Ci we consider in the next sections, it might be that
the map η = π!π
∗ is not invertible as a K(Vark)-module morphism. However, one can replace
the ring K(Vark) by a suitable localization (often it suffices to invert [G]), to make η is invertible.
As a consequence, the resulting classes [XG(X,Q)] will only be defined in that localization. This
is not unreasonable, since [G] needs to be invertible anyway in order to apply (2.8). Also in
many cases we can still extract algebraic data from the localized class: given a multiplicative
system S ⊂ K(Vark) and an element x ∈ S−1K(Vark) that admits a lift x ∈ K(Vark), this lift
is defined up to a sum of annihilators of elements of S. If φ : K(Vark)→ R is a ring morphism
with R a domain such that φ(s) ̸= 0 for all s ∈ S, then φ(a) = 0 for any annihilator a of any
s ∈ S. Hence φ(x) is independent on the choice of lift. The example to have in mind here is
the E-polynomial e : K(VarC) → Z[u, v]. Since e(q) = uv ̸= 0, where q =
[
A1
C
]
∈ K(VarC),
to compute the E-polynomial of some variety X over C it is sufficient to know its class in the
localized ring S−1K(VarC) for S =
{
1, q, q2, . . .
}
. (Similarly we could invert q − 1 or q + 1.)
3 Applications
In this section, we apply the technique developed in Sections 2.6 and 2.7 to compute the class
of the G-representation varieties XG(Σg) in the Grothendieck ring K(VarC), with G being the
groups of complex upper triangular n×n matrices for n = 2, 3, 4. We prove our main theorems,
Theorems 1.1 and 1.5. We also discuss generalizations of Theorem 1.1 to representation varieties
with parabolic structures.
3.1 Upper triangular 2 × 2
We denote the group of 2× 2 upper triangular matrices over C by
U2 =
{(
a b
0 c
) ∣∣∣∣ a, c ̸= 0
}
.
It is easily seen that the class of U2 in the Grothendieck ring of varieties is
[
A1
C
][
A1
C \ {0}
]2
=
q(q−1)2. Moreover, this group contains the following three types of conjugacy classes according
to their orbit with respect to conjugation.
1. All scalar matrices
(
λ 0
0 λ
)
have a singleton orbit.
2. All matrices of the form
(
λ b
0 λ
)
with b ̸= 0 are conjugate to the Jordan block
(
λ 1
0 λ
)
, and
thus have an orbit isomorphic to A1
C \ {0}.
3. All remaining matrices in U2 are of the form
(
λ b
0 µ
)
, with λ, µ ̸= 0, which are conjugate if
and only if they have the same diagonal. Hence, these matrices have an orbit isomorphic
to A1
C.
We denote these three types of conjugacy classes by
S =
{(
λ 0
0 λ
) ∣∣∣∣λ ̸= 0
}
, J =
{(
λ b
0 λ
) ∣∣∣∣λ, b ̸= 0
}
,
Virtual Classes of Representation Varieties of Upper Triangular Matrices via TQFTs 19
M =
{(
λ b
0 µ
) ∣∣∣∣λ, µ ̸= 0, λ ̸= µ, b ∈ C
}
and the orbits of their elements by
Sλ =
{(
λ 0
0 λ
)}
, Jλ =
{(
λ b
0 λ
) ∣∣∣∣ b ̸= 0
}
, Mλ,µ =
{(
λ b
0 µ
) ∣∣∣∣ b ∈ C
}
for any λ, µ ̸= 0 with λ ̸= µ. It is easy to see that the classes of these varieties in K(VarC) are
given by
[S] = q − 1, [J ] = (q − 1)2, [M] = q(q − 1)(q − 2),
[Sλ] = 1, [Jλ] = q − 1, [Mλ,µ] = q.
We denote the orbit spaces by S, J and M = {(λ, µ) ∈ C∗×C∗ | λ ̸= µ} respectively. We remark
that both S and J are isomorphic to A1
C \ {0} as varieties. The quotient maps that identify the
orbits are
πS : S → S :
(
λ 0
0 λ
)
7→ λ, πJ : J → J :
(
λ b
0 λ
)
7→ λ,
πM : M→M :
(
λ b
0 µ
)
7→ (λ, µ).
These maps induce the morphisms
K(Var/S)⊕K(Var/J )⊕K(Var/M) K(Var/S)⊕K(Var/J)⊕K(Var/M)
π!
π∗
.
To obtain a reduction of the TQFT for G = U2 with stratification given by S, J andM, let
us show that condition (iv) of Proposition 2.17 holds.
Lemma 3.1. The stabilizer subgroup of any g ∈ G is isomorphic to
StabS = U2, StabJ = Gm ×Ga, or StabM = Gm ×Gm,
depending whether g lies in S, J orM, respectively. Moreover, all of these subgroups are special.
Proof. The proof is straightforward, we leave it to the reader. ■
As can be seen in the proof of Proposition 2.17, the restriction of c!c
∗ to ∆E is now just
multiplication by [StabE ] for each stratum E = S,J orM.
Remark 3.2. Alternatively, one could show even more explicitly that the restriction of c!c
∗
to ∆E is multiplication by [StabE ]. Consider the representatives of the conjugacy classes
ξSλ =
(
λ 0
0 λ
)
, ξJλ =
(
λ 1
0 λ
)
, ξMλ,µ =
(
λ 0
0 µ
)
.
Their stabilizers are explicitly given by
Stab
(
ξSλ
)
= U2, Stab
(
ξJλ
)
=
{(
α β
0 α
)
| α ̸= 0
}
, Stab
(
ξMλ,µ
)
=
{(
α 0
0 β
)
| α, β ̸= 0
}
.
For all E = S,J ,M it is straightforward to come up with a map σ : E → G such that g =
σ(g)ξEπE(g)
σ(g)−1 for any g ∈ E . For example, for E = J we can take σ
(
λ b
0 λ
)
=
(
1 0
0 1/b
)
, and
for E =M one can take σ
(
λ b
0 µ
)
=
(
1 b/(µ−λ)
0 1
)
. Now, for any variety X
(f1,f2)−→ ∆E we have an
isomorphism
X × StabE
∼−→ c!c
∗X =
{
(x, h) ∈ X ×G | f2(x) = hf1(x)h
−1
}
,
(x, s) 7→
(
x, σ(f2(x))sσ(f1(x))
−1
)
,
which shows that c!c
∗[X]∆E = [StabE ][X]∆E .
20 M. Hablicsek and J. Vogel
Write TSλ
∈ K(Var/S), TJλ ∈ K(Var/J) and TMλ,µ
∈ K(Var/M) for the classes of the points
{λ} → S, {λ} → J and {(λ, µ)} →M . We consider the submodule V generated by these classes
V = ⟨TSλ
, TJλ , TMλ,µ
⟩. From the computations that follow, it will be clear that V is invariant
under η = π! ◦ π∗ and Zπ. Hence all conditions from Proposition 2.17 are satisfied, so we have
a reduction of the TQFT.
Since all fibrations S → S, J → J andM→M are trivial, we immediately find that
η(TSλ
) = [Sλ]TSλ
= TSλ
, η(TJλ) = [Jλ]TJλ = (q − 1)TJλ ,
η(TMλ,µ
) = [Mλ,µ]TMλ,µ
= qTMλ,µ
,
that is,
η =
TSλ
TJλ TMλ,µ
TSλ
1 0 0
TJλ 0 q − 1 0
TMλ,µ
0 0 q
.
For computing Zπ(L), recall from (2.7) that F(L) is given by
U2 U4
2 U2,
g (g, g1, g2, h) hg[g1, g2]h
−1.
p q
First we compute Zπ(L)(TSλ
). We have π∗(TSλ
) = [Sλ]U2 . Note that for any group elements
g1 =
(
a1 b1
0 c1
)
and g2 =
(
a2 b2
0 c2
)
, the commutator [g1, g2] = ( 1 x
0 1 ) with
x =
a1b2 − a2b1 + b1c2 − b2c1
c1c2
.
Hence q(p∗(Sλ)) ⊂ Sλ ∪Jλ, and thus Zπ(L)(TSλ
) are generated by TSλ
and TJλ . Then, we have
that
Zπ(L)(TSλ
)|TSλ
=
[
Sλ × U3
2 ∩ q−1(Sλ)
]
= [{g1, g2 ∈ U2 | [g1, g2] = 1}] · [U2]
=
[{
a1, b1, c1, a2, b2, c2 ∈ C | a1b2−a2b1+b1c2−b2c1=0
and a1c1a2c2 ̸=0
}]
· [U2].
We cut up the variety
[{a1, b1, c1, a2, b2, c2 ∈ C | a1b2 − a2b1 + b1c2 − b2c1 = 0 and a1c1a2c2 ̸= 0}]
into three pieces given by extra conditions: a1 ̸= c1; a1 = c1, a2 ̸= c2; and finally a1 = c1,
a2 = c2. In the first case, the equation a1b2 − a2b1 + b1c2 − b2c1 = 0 can be solved for b2 given
the values of the other variables, hence
[{a1b2 − a2b1 + b1c2 − b2c1 = 0, a1c1a2c2 ̸= 0 and a1 ̸= c1}] = q(q − 1)3(q − 2).
In the second case, the equations a1 = c1 and a2 ̸= c2 yield b1 = 0, hence
[{a1b2 − a2b1 + b1c2 − b2c1 = 0, a1c1a2c2 ̸= 0, a1 = c1 and a2 ̸= c2}] = q(q − 1)2(q − 2).
Finally, in the third case, the equations a1 = c1 and a2 = c2 imply that
a1b2 − a2b1 + b1c2 − b2c1 = 0
Virtual Classes of Representation Varieties of Upper Triangular Matrices via TQFTs 21
providing that
[{a1b2 − a2b1 + b1c2 − b2c1 = 0, a1c1a2c2 ̸= 0, a1 = c1 and a2 = c2}] = q2(q − 1)2.
As a consequence, we obtain
[{a1b2 − a2b1 + b1c2 − b2c1 = 0 and a1c1a2c2 ̸= 0}] = q2(q − 1)3
implying that
Zπ(L)(TSλ
)|TSλ
= q2(q − 1)3 · [U2] = q3(q − 1)5.
Now it follows that Zπ(L)(TSλ
)|TJλ
is simply
[
Sλ × U3
2
]
− q3(q − 1)5 = q3(q − 1)5(q − 2).
Next we compute Zπ(L)(TJλ). We have π∗(TJλ) = [Jλ]U2 . By the same observation as above
about the commutator, we see that Zπ(L)(TJλ) is also generated by TSλ
and TJλ . Note that(
λ b
0 λ
)
[g1, g2] =
(
λ 0
0 λ
)
if and only if [g1, g2] =
(
1 −b/λ
0 1
)
,
which implies that
Zπ(L)(TJλ)|TSλ
= Zπ(L)(TSλ
)|TJλ
= q3(q − 1)5(q − 2).
Now it follows that
Zπ(L)(TJλ)|TJλ
=
[
Jλ × U3
2
]
− q3(q − 1)5(q − 2) = q3(q − 1)5
(
q2 − 3q + 3
)
.
Lastly we compute Zπ(L)(TMλ,µ
). We have π∗(TMλ,µ
) = [Mλ,µ]U2 . By the observation about
the commutator, we immediately see that Zπ(L)(TMλ,µ
) must be generated by TMλ,µ
. Therefore,
Zπ(L)(TMλ,µ
) =
[
Mλ,µ × U3
2
]
TMλ,µ
= q4(q − 1)6TMλ,µ
.
In summary,
Zπ(L) = q3(q − 1)5
TSλ
TJλ TMλ,µ
TSλ
1 q − 2 0
TJλ q − 2 q2 − 3q + 3 0
TMλ,µ
0 0 q(q − 1)
.
Now, we turn our attention to the cylinder with parabolic structures and we compute Zπ(LSλ
),
Zπ(LJλ
) and Zπ(LMλ,µ
). Recall that F(LE) is given by
U2 U2
2 × E U2,
g (g, h, ξ) hgξh−1.
r s
Let g ∈ Sλ, and note that if ξ ∈ Sσ then gξ ∈ Sλσ, if ξ ∈ Jσ then gξ ∈ Jλσ, and if ξ ∈ Mσ,ρ
then gξ ∈Mλσ,λρ. Hence we have that Zπ(LSλ
) is given by
TSσ TJσ TMσ,ρ
TSλσ
[Sσ × U2] 0 0
TJλσ 0 [Jσ × U2] 0
TMλσ,λρ
0 0 [Mσ,ρ × U2]
= q(q − 1)2
TSσ TJσ TMσ,ρ
TSλσ
1 0 0
TJλσ 0 q − 1 0
TMλσ,λρ
0 0 q
.
22 M. Hablicsek and J. Vogel
Now let g ∈ Jλ. We see that if ξ ∈ Sσ then gξ ∈ Jλσ, and if ξ ∈ Mσ,ρ then gξ ∈ Mλσ,λρ. If
ξ ∈ Jσ, then gξ ∈ Sλσ precisely if g = λσξ−1 and otherwise gξ ∈ Jλσ. Hence we have
Zπ(LJλ
) =
TSσ TJσ TMσ,ρ
TSλσ
0 [Jσ × U2] 0
TJλσ [Sσ × U2][Jλ] [Jσ × U2]([Jλ]− 1) 0
TMλσ,λρ
0 0 [Mσ,ρ × U2][Jλ]
= q(q − 1)2
TSσ TJσ TMσ,ρ
TSλσ
0 q − 1 0
TJλσ q − 1 (q − 1)(q − 2) 0
TMλσ,λρ
0 0 q(q − 1)
.
Lastly, let g ∈ Mλ,µ. If ξ ∈ Sσ then gξ ∈ Mλσ,µσ, and if ξ ∈ Jσ then gξ ∈ Mλσ,µσ as well.
If ξ ∈ Mσ,ρ, then gξ ∈ Mλσ,µρ if λσ ̸= µρ and otherwise gξ ∈ Sλσ precisely for g = λσξ−1 and
else gξ ∈ Jλσ. Hence we see that Zπ(LMλ,µ
) is given by
TSσ TJσ TMσ,ρ TMσ′,ρ′
TSλσ
0 0 0 [Mσ′,ρ′ × U2]
TJλσ 0 0 0 [Mσ′,ρ′×U2]([Mλ,µ]− 1)
TMλσ,λρ
[Sσ×U2][Mλ,µ] [Jσ×U2][Mλ,µ] [Mσ,ρ×U2][Mλ,µ] 0
= q(q − 1)2
TSσ TJσ TMσ,ρ TMσ′,ρ′
TSλσ
0 0 0 q
TJλσ 0 0 0 q(q − 1)
TMλσ,λρ
q q(q − 1) q2 0
,
with λσ ̸= µρ and λσ′ = µρ′.
Since the conditions of Proposition 2.17 are satisfied, we can consider the reduced TQFT, Z̃.
We have Z̃(L) = Zπ(L) ◦ η−1, so
Z̃(L) = q3(q − 1)4
TSλ
TJλ TMλ,µ
TSλ
q − 1 q − 2 0
TJλ (q − 2)(q − 1) q2 − 3q + 3 0
TMλ,µ
0 0 (q − 1)2
.
We can diagonalize this matrix as
Z̃(L) = q3(q − 1)4A
1 0 0
0 (q − 1)2 0
0 0 (q − 1)2
A−1 with A =
1 1 0
−1 q − 1 0
0 0 1
,
which yields
Z̃(Lg) = q3g−1(q − 1)4g
×
TSλ
TJλ TMλ,µ
TSλ
(q − 1)((q − 1)2g−1 + 1) (q − 1)2g − 1 0
TJλ (q − 1)((q − 1)2g − 1) (q − 1)2g+1 + 1 0
TMλ,µ
0 0 q(q − 1)2g
. (3.1)
In particular, we proved Theorem 1.1 for the group G = U2.
Theorem 3.3. The class of the representation variety XU2(Σg) in the localized Grothendieck
ring of varieties is
[XU2(Σg)] =
1
[U2]g
Z̃(Lg)(TS1)|TS1
= q2g−1(q − 1)2g+1
(
(q − 1)2g−1 + 1
)
. (3.2)
Virtual Classes of Representation Varieties of Upper Triangular Matrices via TQFTs 23
Remark 3.4. For small values of g, we find
[XU2(Σ1)] = q2(q − 1)3,
[XU2(Σ2)] = q4(q − 1)5
(
q2 − 3q + 3
)
,
[XU2(Σ3)] = q6(q − 1)7
(
q4 − 5q3 + 10q2 − 10q + 5
)
,
[XU2(Σ4)] = q8(q − 1)9
(
q6 − 7q5 + 21q4 − 35q3 + 35q2 − 21q + 7
)
.
Note that [XU2(Σg)] has a factor (q − 1)2g+1, which can be explained as follows. There is a free
action of G2g
m on XU2(Σg) given by scaling the Ai, Bi (notation as in (1.1)). This yields a G2g
m -
torsor XU2(Σg)→ XU2(Σg) � G2g
m , which by is trivial in the Zariski topology as G2g
m is a special
group. Hence, [XU2(Σg)] is divisible by [G2g
m ] = (q − 1)2g. For the remaining factor (q − 1), let
D ⊂ XU2(Σg) be the subvariety where all Ai, Bi are diagonal. Then [D] = (q− 1)4g and there is
a free action of Gm on XU2(Σg)\D given by conjugation with ( 1 0
0 x ) for x ∈ C∗.
In fact, the affine GIT quotient XU2(Σg)�G2g
m can be identified with the representation variety
[XAGL1(Σg)], where
AGL1 =
{(
a b
0 1
)
: a ̸= 0
}
is the general affine group of the line. Therefore, we obtain that
[XAGL1(Σg)] =
[XU2(Σg)]
(q − 1)2g
= q2g−1(q − 1)
(
(q − 1)2g−1 + 1
)
,
recovering a result of González-Prieto, Logares, and Muñoz [15]. Alternatively, this result can
also be obtained from the isomorphism of algebraic groups Gm × AGL1
∼−→ U2 which maps(
t,
(
a b
0 1
))
to
(
ta tb
0 t
)
, as it implies
[XU2(Σg)] = [XGm(Σg)][XAGL1(Σg)],
where [XGm(Σg)] = (q − 1)2g.
Now, we compute the classes of the twisted representation varieties XU2(Σg, Q). Similar
calculations as before shows that
Z̃(LSλ
) = Zπ(LSλ
) ◦ η−1 = q(q − 1)2
TSσ TJσ TMσ,ρ
TSλσ
1 0 0
TJλσ 0 1 0
TMλσ,µσ
0 0 1
,
Z̃(LJλ
) = Zπ(LJλ
) ◦ η−1 = q(q − 1)2
TSσ TJσ TMσ,ρ
TSλσ
0 1 0
TJλσ q − 1 q − 2 0
TMλσ,µσ
0 0 q − 1
, (3.3)
Z̃(LMλ,µ
) = Zπ(LMλ,µ
) ◦ η−1 = q(q − 1)2
TSσ TJσ TMσ,ρ TMσ′,ρ′
TSλσ
0 0 0 1
TJλσ 0 0 0 q − 1
TMλσ,µρ
q q q 0
, (3.4)
with λσ ̸= µρ but λσ′ = µρ′.
As a consequence, we obtain the classes of the twisted representation varieties XU2(Σg, Q).
Theorem 3.5. Let Σg be a surface of genus g, with parabolic data Q = {(S1,Jλ1), . . . , (Sk,Jλk
),
(Sk+1,Mµ1,σ1), . . . , (Sk+ℓ,Mµℓ,σℓ
)}.
24 M. Hablicsek and J. Vogel
(i) If
∏k
i=1 λi
∏ℓ
j=1 µj ̸= 1 or
∏k
i=1 λi
∏ℓ
j=1 σj ̸= 1, then
[XU2(Σg, Q)] = 0.
(ii) Otherwise, and if ℓ = 0, then
[XU2(Σg, Q)] = q2g−1(q − 1)2g
(
(−1)k(q − 1) + (q − 1)2g+k
)
,
(iii) and if ℓ > 0, then
[XU2(Σg, Q)] = q2g+ℓ−1(q − 1)4g+k.
Proof. First note that (Σg, Q) can be seen as the composition
D† ◦ Lg ◦ LJλ1
◦ · · · ◦ LJλk
◦ LMµ1,σ1
◦ · · ·LMµℓ,σℓ
◦D.
(i) From expressions (3.1), (3.3) and (3.4), we can see that
Z
(
Lg ◦ LJλ1
◦ · · · ◦ LJλk
◦ LMµ1,σ1
◦ · · · ◦ LMµℓ,σℓ
)
(TS1)|TS1
= 0,
and hence [XU2(Σg, Q)] = 0.
(ii) Using (3.3) and the diagonalization 0 1 0
q − 1 q − 2 0
0 0 q − 1
= A
−1 0 0
0 q − 1 0
0 0 q − 1
A−1 with A =
−1 1
q−1 0
1 1 0
0 0 1
,
we find that
Z̃(LJλ1
◦ · · · ◦ LJλk
)(TS1)
= qk−1(q − 1)2k
(
(−1)k(q − 1) + (q − 1)k
)
TSλ
+
(
(−1)k+1(q − 1) + (q − 1)k+1
)
TJλ ,
where λ =
∏k
i=0 λi. Then, using (3.1) and that λ = 1, we have
Z̃
(
Lg ◦ LJλ1
◦ · · · ◦ LJλk
)
(TS1)|TS1
= q3g+k−1(q − 1)4g+2k
(
(−1)k(q − 1) + (q − 1)2g+k
)
.
So finally
[XU2(Σg, Q)] =
1
[U2]g+k
Z̃
(
Lg ◦ LJλ1
◦ · · · ◦ LJλk
)
(TS1)|TS1
= q2g−1(q − 1)2g
(
(−1)k(q − 1) + (q − 1)2g+k
)
.
Note that this is in accordance with (3.2) for k = 0.
(iii) Note that
∏ℓ
i=0 µi =
∏ℓ
i=0 σi. In combination with (3.4) it follows that
Z̃
(
LMµ1,σ1
◦ · · · ◦ LMµℓ,σℓ
)
(TS1) = q2ℓ−1(q − 1)2ℓ(TSµ + (q − 1)TJµ),
where µ =
∏ℓ
i=0 µi. Similar as before, we use (3.3) to obtain
[XU2(Σg, Q)]
=
1
[U2]g+k+ℓ
Z̃(Lg ◦ LJλ1
◦ · · · ◦ LJλk
◦ LMµ1,σ1
◦ · · · ◦ LMµℓ,σℓ
)(TS1)|TS1
= q2g+ℓ−1(q − 1)4g+k. ■
Virtual Classes of Representation Varieties of Upper Triangular Matrices via TQFTs 25
3.2 Upper triangular 3 × 3 matrices
Now consider the case where G = U3, the group of upper triangular 3× 3 matrices, that is,
U3 =
a b c
0 d e
0 0 f
∣∣∣∣ a, d, f ̸= 0
.
It is easy to see that the class of U3 in the Grothendieck ring of varieties is
[
A1
C
]3[A1
C \ {0}
]3
=
q3(q − 1)3.
For simplicity we will just consider the representation varieties without parabolic data. As
any commutator [g1, g2] in U3 has ones on the diagonal, we only need to consider the conjugacy
classes of such elements in order to compute the classes of representation varieties XU3(Σg).
There are five such conjugacy classes are given by
C1 =
1 0 0
0 1 0
0 0 1
, C2 =
1 α β
0 1 γ
0 0 1
∣∣∣∣α, γ ̸= 0
,
C3 =
1 α β
0 1 0
0 0 1
∣∣∣∣α ̸= 0
, C4 =
1 0 β
0 1 α
0 0 1
∣∣∣∣α ̸= 0
,
C5 =
1 0 α
0 1 0
0 0 1
∣∣∣∣α ̸= 0
with representatives given by
ξ1 =
1 0 0
0 1 0
0 0 1
, ξ2 =
1 1 0
0 1 1
0 0 1
, ξ3 =
1 1 0
0 1 0
0 0 1
,
ξ4 =
1 0 0
0 1 1
0 0 1
, ξ5 =
1 0 1
0 1 0
0 0 1
.
Hence, we obtain maps πi : Ci → Ci mapping each conjugacy class to the space of orbits under
conjugaction. In this case, there is only one orbit for each conjugacy class, hence all the Ci are
points. Technically one should stratify G\ ∪i Ci as well, but as everything happens over ∪iCi we
omit this. To obtain a reduction of the TQFT for G = U3 with the above stratification, let us
show that condition (iv) of Proposition 2.17 holds.
Lemma 3.6. The stabilizer subgroup of any g ∈ G is isomorphic to
Stab1 = U3, Stab2 =
x y z
0 x y
0 0 x
∣∣∣∣x ̸= 0
,
Stab3 =
x y z
0 x 0
0 0 w
∣∣∣∣x,w ̸= 0
, Stab4 =
x 0 z
0 w y
0 0 w
∣∣∣∣x,w ̸= 0
or
Stab5 =
x y z
0 w v
0 0 x
∣∣∣∣x,w ̸= 0
,
depending on whether g lies in C1, C2, . . . , or C5, respectively. Moreover, all of these subgroups
are special.
26 M. Hablicsek and J. Vogel
Proof. The proof is straightforward. Note that all stabilizers are extensions of copies of Gm
and Ga, which are special. ■
We write Ti = [Ci]Ci ∈ K(Var/Ci), and consider V = ⟨T1, . . . , T5⟩. In what follows, all
matrices and vectors will be written with respect to the basis {T1, . . . , T5}.
Since, all the fibrations π : Ci → Ci are trivial, the map η = π!π
∗ is simply given by
η =
[C1]
[C2]
[C3]
[C4]
[C5]
=
1
q(q − 1)2
q(q − 1)
q(q − 1)
q − 1
.
Now we compute Zπ(L) = π! ◦ Z(L) ◦ π∗, starting with Zπ(L)(T1). Since the commutator
[g1, g2] has ones on the diagonal for all g1, g2 ∈ U3, indeed we have that Zπ(L)(T1) ∈ ⟨T1, . . . , T5⟩.
We write gi =
(
ai bi ci
0 di ei
0 0 fi
)
.
We have that Zπ(L)(T1)|T1 is the class of
{
(g1, g2, h) ∈ U3
3 | g1g2 = g2g1
}
. Since g1g2 and
g2g1 have the same elements on the diagonal for every pair of group elements g1 and g2, we only
need to check on three entries whether g1g2 = g2g1. This gives us three equations in the entries
of g1 and g2. Explicitly, we obtain
a1b2 − a2b1 + b1d2 − b2d1 = 0,
a1c2 − a2c1 + b1e2 − b2e1 + c1f2 − c2f1 = 0,
d1e2 − d2e1 + e1f2 − e2f1 = 0.
Lemma 3.7. The variety in the affine space A12
C (with coordinates being the ai, bi, ci, di, ei, fi
for i = 1 and i = 2) cut out by the three equations above has class q3(q − 1)4
(
q2 + q − 1
)
in the
Grothendieck ring of varieties K(VarC).
Proof. We cut the variety in pieces as follows.
� a1 ̸= d1, a1 ̸= f1, d1 ̸= f1: In this case, the first equation can be solved for b2, the second
for c2 and the third for e2 yielding that the class of this piece is (q−1)(q−2)(q−3)(q−1)3q3.
� a1 = d1, a1 ̸= f1, a2 ̸= d2: In this case, the first equation can be solved for b1, the second
for c2 and the third for e2 yielding that the class of this piece is (q−1)(q−2)(q−1)2(q−2)q3.
� a1 = d1, a1 ̸= f1, a2 = d2: In this case, the first equation is always satisfied, moreover,
the second can be solved for c2 and the third for e2 yielding that the class of this piece is
(q − 1)(q − 2)(q − 1)2q4.
� a1 ̸= d1, d1 = f1, d2 ̸= f2: In this case, the first equation can be solved for b2, the second
for c2 and the third for e1 yielding that the class of this piece is (q−1)(q−2)(q−1)2(q−2)q3.
� a1 ̸= d1, d1 = f1, d2 = f2: In this case, the first equation can be solved for b2, the second
for c2 and the third equation is always satisfied yielding that the class of this piece is
(q − 1)(q − 2)(q − 1)2q4.
� a1 ̸= d1, a1 = f1, a2 ̸= f2: In this case, the first equation can be solved for b2, the second
for c1 and the third for e2 yielding that the class of this piece is (q−1)(q−2)(q−1)2(q−2)q3.
� a1 ̸= d1, a1 = f1, a2 = f2: In this case, the first equation can be solved for b2, the third
for e2, and the second is then satisfied yielding that the class of this piece is (q − 1)(q −
2)(q − 1)2q4.
� a1 = d1 = f1: In this case, we separate again into cases:
Virtual Classes of Representation Varieties of Upper Triangular Matrices via TQFTs 27
– b1 = e1 = 0, then c1(a2 − f2) = 0 yielding that the class is 2(q − 1)4q3
– b1 = 0, e1 ̸= 0, then d2 = f2 and the second equation can be solved for b2 yielding
that the class is (q − 1)3(q − 1)q3,
– b1 ̸= 0, e1 = 0, then a2 = d2 and the second equation can be solved for e2 yielding
that the class is (q − 1)3(q − 1)q3,
– b1 ̸= 0, e1 ̸= 0, then a2 = d2 = f2 and the second equation can be solved for e2
yielding that the class is (q − 1)2(q − 1)2q3.
So in total, we have that the class is (q − 1)3q3(5q − 5).
Adding the pieces together we obtain that the class of this variety is q3(q − 1)4(q2 + q − 1). ■
As a consequence, we obtain that class of
{
(g1, g2, h) ∈ U3
3 | g1g2 = g2g1
}
is q3(q − 1)4(q2 + q −
1)[U3] in K(VarC).
Remark 3.8. In general, in the computations of Zπ(L)(T1), we follow the same strategy as we
explained in the proof above. We cut the variety into pieces given by some variables being 0
or some variables being equal to each other. This idea can be made into an Algorithm A.1
(see Appendix A) which we use to compute the other classes. We added the proof of the above
lemma for sake of completeness.
We have that Zπ(L)(T1)|T2 is the class of
{
(g1, g2, h) ∈ U3
3 | [g1, g2] ∈ C2
}
, which is given by
the equations
a1b2 − a2b1 + b1d2 − b2d1 ̸= 0,
d1e2 − d2e1 + e1f2 − e2f1 ̸= 0.
This evaluates to q6(q − 2)2(q − 1)4[U3] using Algorithm A.1 or a similar calculation as in
Lemma 3.7.
We have that Zπ(L)(T1)|T3 is the class of
{
(g1, g2, h) ∈ U3
3 | [g1, g2] ∈ C3
}
, which is given by
the equations
a1b2 − a2b1 + b1d2 − b2d1 ̸= 0,
d1e2 − d2e1 + e1f2 − e2f1 = 0.
This evaluates to q6(q − 2)(q − 1)4[U3] using Algorithm A.1 or a similar calculation as in
Lemma 3.7.
We have that Zπ(L)(T1)|T4 is the class of
{
(g1, g2, h) ∈ U3
3 | [g1, g2] ∈ C4
}
, which is given by
the equations
a1b2 − a2b1 + b1d2 − b2d1 = 0,
d1e2 − d2e1 + e1f2 − e2f1 ̸= 0.
This is symmetric to the previous case, so it also evaluates to q6(q − 2)(q − 1)4[U3].
We have that Zπ(L)(T1)|T5 is the class of
{
(g1, g2, h) ∈ U3
3 | [g1, g2] ∈ C5
}
, which is given by
the equations
a1b2 − a2b1 + b1d2 − b2d1 = 0,
d1e2 − d2e1 + e1f2 − e2f1 = 0,
−a1b2d2e1 − a1b2e2f1 + a1c2d1d2 + a2b1d2e1 + a2b1e2f1 − a2c1d1d2
+ b1d1d2e2 − b1d
2
2e1 − b1d2e2f1 + b2d1e2f1 + c1d1d2f2 − c2d1d2f1 ̸= 0.
28 M. Hablicsek and J. Vogel
The latter inequality can be simplified to
a1c2 − a2c1 + b1e2 − b2e1 + c1f2 − c2f1 ̸= 0.
This evaluates to q3(q − 1)6(q + 1)[U3] using Algorithm A.1 or a similar calculation as in
Lemma 3.7.
So far we have computed the first column of the matrix of Zπ(L). As a check, indeed we have
that the sum of the entries of this column equals [U3]:
q3(q − 1)4
(
q2 + q − 1
)
[U3] + q6(q − 2)2(q − 1)4[U3] + q6(q − 2)(q − 1)4[U3]
+ q6(q − 2)(q − 1)4[U3] + q3(q − 1)6(q + 1)[U3] = [U3]
3.
We use a similar strategy as in the previous section to determine Zπ(L)(Ti) for i = 2, 3, 4, 5
from the case i = 1. We have
Zπ(L)(Tj)|Ti = [Xij ] · [U3] with Xij =
{
(g, g1, g2) ∈ Cj × U2
3 | g[g1, g2] ∈ Ci
}
.
We can stratify Xij by
Xijk =
{
(g, g1, g2) ∈ Cj × U2
3 | g[g1, g2] ∈ Ci and [g1, g2] ∈ Ck
}
for k = 1, . . . , 5.
Note that for each conjugacy class Ck we have an algebraic normal subgroup N so that every
element g ∈ Ck is given by nξkn
−1. In other words, there exists a (non-unique) morphism of
varieties σk : Ck → U3 such that σk(g)ξkσk(g)
−1 = g for all g ∈ Ck. Moreover, if [g1, g2] = nξkn
−1,
then g[g1, g2] ∈ Ci if and only if n−1gnξ ∈ Ci. Thus, for each i, j, k we have an isomorphism of
varieties
Xijk
∼−→ {g ∈ Cj | gξk ∈ Ci} ×
{
(g1, g2) ∈ G2 | [g1, g2] ∈ Ck
}
,
(g, g1, g2) 7→ (σk([g1, g2])
−1gσk([g1, g2]), g1, g2),
so we find that
Zπ(L)(Tj)|Ti =
5∑
k=1
Fijk · Zπ(L)(T1)|Tk
with Fijk = [{g ∈ Cj | gξk ∈ Ci}]. (3.5)
Although there are about 53 = 125 computations to be done to determine the coefficients Fijk,
all of them are quite simple. For instance, it is clear that Fi,1,k = δik, the Kronecker delta. For
j = 2, take any g =
(
1 α β
0 1 γ
0 0 1
)
∈ C2. Then gξ1, gξ5 ∈ C2. We have gξ3 ∈ C4 if α = −1 and gξ3 ∈ C2
otherwise. Similarly, gξ4 ∈ C3 if γ = −1 and gξ4 ∈ C2 otherwise. Finally,
gξ2 ∈
C1 if α, γ = −1 and β = 0,
C2 if α, γ ̸= −1,
C3 if α ̸= −1 and γ = −1,
C4 if α = −1 and γ ̸= −1,
C5 if α, γ = −1 and β ̸= 0.
This gives
Fi,2,k =
0 1 0 0 0
q(q − 1)2 q(q − 2)2 q(q − 2)(q − 1) q(q − 2)(q − 1) q(q − 1)2
0 q(q − 2) 0 q(q − 1) 0
0 q(q − 2) q(q − 1) 0 0
0 q − 1 0 0 0
,
Virtual Classes of Representation Varieties of Upper Triangular Matrices via TQFTs 29
with i the row index and k the column index. By completely similar arguments, one can show
that
Fi,3,k =
0 0 1 0 0
0 q(q − 2) 0 q(q − 1) 0
q(q − 1) 0 q(q − 2) 0 q(q − 1)
0 q 0 0 0
0 0 q − 1 0 0
,
Fi,4,k =
0 0 0 1 0
0 q(q − 2) q(q − 1) 0 0
0 q 0 0 0
q(q − 1) 0 0 q(q − 2) q(q − 1)
0 0 0 q − 1 0
,
Fi,5,k =
0 0 0 0 1
0 q − 1 0 0 0
0 0 q − 1 0 0
0 0 0 q − 1 0
q − 1 0 0 0 q − 2
.
Using (3.5) we obtain the matrix representing Zπ(L). Precomposing this map with η−1 gives
Z̃(L), which can be diagonalized as follows:
Z̃(L) = q6(q − 1)5A
q3
q(q − 1)2
q3(q − 1)2
q3(q − 1)2
q3(q − 1)4
A−1
with A =
1 1 0 1 1
q 0 0 −q(q − 1) q(q − 1)2
−q 0 1 0 q(q − 1)
−q 0 −1 q(q − 2) q(q − 1)
q − 1 −1 0 q − 1 q − 1
.
Remark 3.9. We see that Zπ(L) is symmetric. This can be explained by the fact that for each i
and j we have an isomorphism:{
(g, g1, g2) ∈ Ci ×G2 | g[g1, g2] ∈ Cj
} ∼←→
{
(g, g1, g2) ∈ Cj ×G2 | g[g1, g2] ∈ Ci
}
,
(g, g1, g2) 7→ (g[g1, g2], g2, g1),
(g[g1, g2], g2, g1)←[ (g, g1, g2).
Hence the classes of both sides are equal in K(VarC), and so Zπ(L)(Ti)|Tj = Zπ(L)(Tj)|Ti . This
does not hold for any TQFT, but it relies on the fact that the Ci are points.
Theorem 3.10. For any g ≥ 0, the virtual class of the U3-representation variety XU3(Σg) is
[XU3(Σg)] = q3g−3(q − 1)2g
(
q2(q − 1)2g+1 + q3g(q − 1)2 + q3g(q − 1)4g + 2q3g(q − 1)2g+1
)
.
Proof. One can check that
A−1 =
1
q3
(q − 1)2 1 1− q 1− q (q − 1)2
q2(q − 1) 0 0 0 −q2
q(q − 2)(q − 1) −q(q − 2) q3 − 2q2 + 2q −2q(q − 1) q3 − 3q2 + 2q
2q − 2 −2 q − 2 q − 2 2q − 2
1 1 1 1 1
.
30 M. Hablicsek and J. Vogel
By matrix multiplication, we find that
XU3(Σg) =
1
[U3]g
Z̃(L)g(T1)|T1 = q3g−3(q − 1)2g
×
(
q2(q − 1)2g+1 + q3g(q − 1)2 + q3g(q − 1)4g + 2q3g(q − 1)2g+1
)
. ■
Remark 3.11. In particular, for small values of g, we find
[XU3(Σ1)] = q3(q − 1)4
(
q2 + q − 1
)
,
[XU3(Σ2)] = q7(q − 1)6
(
q8 − 6q7 + 15q6 − 18q5 + 9q4 + q3 − 3q2 + 3q − 1
)
,
[XU3(Σ3)] = q11(q − 1)8
(
q14 − 10q13 + 45q12 − 120q11 + 210q10 − 250q9 + 200q8
− 100q7 + 25q6 + q5 − 5q4 + 10q3 − 10q2 + 5q − 1
)
.
As in Remark 3.4, the factor (q − 1)2g+2 can be explained from the actions of G2g
m (given by
scaling the Ai, Bi) and G2
m (given by conjugating with
(
1
x
y
)
, x, y ∈ C∗).
3.3 Upper triangular 4 × 4 matrices
The last case we will treat is the group U4 of upper triangular 4× 4 matrices. We can use the
same strategies as in the previous case of U3, but all computations are done using Algorithm A.1.
Source code for these computations is given in [34]. The group U4 contains sixteen unipotent
conjugacy classes [3]. We consider the following representatives of these classes
1
1
1
1
,
1 1
1
1
1
,
1 1
1
1
1
,
1 1
1
1
1
,
1
1 1
1
1
,
1
1 1
1
1
,
1
1
1 1
1
,
1 1
1 1
1
1
,
1 1
1 1
1
1
,
1 1
1
1 1
1
,
1 1
1 1
1
1
,
1 1
1
1 1
1
,
1
1 1
1 1
1
,
1 1
1 1
1
1
,
1 1
1 1
1 1
1
,
1 1 1
1
1 1
1
,
which we denote in order by ξ1, . . . , ξ16. Explicitly, the conjugacy classes are given by
C1 = {a0,1 = a0,2 = a0,3 = a1,2 = a1,3 = a2,3 = 0},
C2 = {a1,2 = a1,3 = a2,3 = 0, a0,1 ̸= 0},
C3 = {a0,1 = a1,2 = a1,3 = a2,3 = 0, a0,2 ̸= 0},
C4 = {a0,1 = a0,2 = a1,2 = a1,3 = a2,3 = 0, a0,3 ̸= 0},
C5 = {a0,1 = a2,3 = a0,3a1,2 − a0,2a1,3 = 0, a1,2 ̸= 0},
C6 = {a0,1 = a0,2 = a1,2 = a2,3 = 0, a1,3 ̸= 0},
C7 = {a0,1 = a0,2 = a1,2 = 0, a2,3 ̸= 0},
Virtual Classes of Representation Varieties of Upper Triangular Matrices via TQFTs 31
C8 = {a2,3 = 0, a0,1 ̸= 0, a1,2 ̸= 0},
C9 = {a1,2 = a2,3 = 0, a0,1 ̸= 0, a1,3 ̸= 0},
C10 = {a1,2 = a0,2a2,3 + a0,1a1,3 = 0, a0,1 ̸= 0, a2,3 ̸= 0},
C11 = {a0,1 = a1,2 = a2,3 = 0, a0,2 ̸= 0, a1,3 ̸= 0},
C12 = {a0,1 = a1,2 = 0, a0,2 ̸= 0, a2,3 ̸= 0},
C13 = {a0,1 = 0, a1,2 ̸= 0, a2,3 ̸= 0},
C14 = {a0,1 = a2,3 = 0, a1,2 ̸= 0, a0,3a1,2 − a0,2a1,3 ̸= 0},
C15 = {a0,1 ̸= 0, a1,2 ̸= 0, a2,3 ̸= 0},
C16 = {a1,2 = 0, a0,1 ̸= 0, a2,3 ̸= 0, a0,2a2,3 + a0,1a1,3 ̸= 0}.
As before, we write Ti = [Ci]Ci ∈ K(Var/Ci) and consider V = ⟨T1, . . . , T16⟩. We wish to
compute Zπ(L)(T1) first and deduce the other columns from this column as we did before. We
have
Zπ(L)(T1)|Ti = [{(g1, g2) ∈ U4 | [g1, g2] ∈ Ci}].
In terms of coordinates, this will yield systems of equations in 20 variables. Using Algorithm A.1
we obtain
Zπ(L)(T1) = q6(q − 1)2
q(q − 1)3
(
q2 + 3q − 2
)
q(q − 2)(q − 1)2
(
q2 + q − 1
)
q(q − 1)2
(
q3 − 3q + 1
)
q(q − 1)3
(
q2 + 2q − 2
)
(q − 2)(q − 1)2
(
q3 + q − 1
)
q(q − 1)2
(
q3 − 3q + 1
)
q(q − 2)(q − 1)2
(
q2 + q − 1
)
q3(q − 2)2(q − 1)
q(q − 2)(q − 1)3(q + 1)
q(q − 2)(q − 1)2
(
q2 − q − 1
)
q(q − 1)
(
q4 − 2q3 − q2 + 4q − 1
)
q(q − 2)(q − 1)3(q + 1)
q3(q − 2)2(q − 1)
(q − 2)(q − 1)3
(
q2 + q + 1
)
q3(q − 2)3
q(q − 2)
(
q4 − 3q3 + 2q2 − 1
)
.
Completely analogous to the previous section, the other columns can be computed from this
result via
Zπ(L)(Tj)|Ti =
16∑
k=1
Fijk · Zπ(L)(T1|Tk
),
where
Fijk = [{g ∈ Cj | gξk ∈ Ci}].
Again, see [34] for the actual computations. As usual, the map η = π!π
∗ is diagonal, with
η(Ti) = [Ci]Ti. Composing Zπ(L) with η−1 gives us Z̃(L), which can be found in Appendix B.
After diagonalizing the reduced TQFT Z̃(L) = Zπ(L) ◦ η−1, we obtain the following result.
32 M. Hablicsek and J. Vogel
Theorem 3.12. For any g ≥ 0, the virtual class of the U4-representation variety XU4(Σg) is
[XU4(Σg)] = q12g−6(q − 1)8g + q12g−6(q − 1)2g+3 + q10g−4(q − 1)2g+3 + q10g−3(q − 1)4g+1
+ q8g−2(q − 1)6g+1 + q8g−2(q − 1)4g+2 + 2q10g−4(q − 1)6g+1
+ 3q12g−6(q − 1)6g+1 + 3q12g−6(q − 1)4g+2 + q10g−4(q − 1)4g+1(2q − 3).
Remark 3.13. For small values of g, we have
[XU4(Σ1)] = q7(q − 1)5
(
q2 + 3q − 2
)
,
[XU4(Σ2)] = q15(q − 1)7
(
q2 − 3q + 3
)(
q10 − 6q9 + 15q8 − 18q7 + 9q6 + 2q5 − 6q4 + 7q3
− 4q2 + 3q − 1
)
,
[XU4(Σ3)] = q23(q − 1)9
(
q4 − 5q3 + 10q2 − 10q + 5
)(
q18 − 10q17 + 45q16 − 120q15
+ 210q14 − 250q13 + 200q12 − 100q11 + 25q10 + 2q9 − 10q8 + 20q7 − 20q6
+ 11q5 − 6q4 + 10q3 − 10q2 + 5q − 1
)
.
3.4 Moduli space of representations and character variety
Let X be a path-connected topological space with finitely generated fundamental group and G
a linear algebraic group over an algebraically closed field k. There is a natural action of G on
the affine representation variety XG(X) given by conjugation. If G is reductive, one can look at
the affine geometric invariant theory (GIT) quotient
MG(X) = XG(X) � G,
which is defined as the spectrum of the ring of invariants Spec(OXG(X))
G. This scheme is known
as the moduli space of G-representations.
A theorem of Nagata [28] shows that the affine GIT quotient is finitely generated over k,
using that G is reductive. However, specializing to the non-reductive groups G = Un, the ring of
invariants is no longer guaranteed to be finitely generated over k. Hence, we will instead focus
on the categorical quotient of XG(X) by G, which coincides with the affine GIT quotient for G
reductive [29]. We begin with the following lemma.
Lemma 3.14. Let X be a variety over an algebraically closed field k equipped with a G-action.
Let π : X → Y be a G-equivariant morphism of varieties over k, such that the action of G on Y
is trivial. Assume that there exists a G-equivariant morphism σ : Y → X such that π ◦ σ = idY .
If for any x ∈ X the Zariski-closure of the G-orbit of x contains σ(π(x)), then π is a categorical
quotient.
Proof. Let f : X → Z be any G-equivariant morphism, where Z has trivial G-action. We need
to show there exists a unique G-invariant morphism g : Y → Z such that f = g ◦ π. If such
a morphism exists, it must be given by g = f ◦ σ since f ◦ σ = (g ◦ π) ◦ σ = g. This already
shows uniqueness. Now we show that for g = f ◦ σ we have f = g ◦ π. Take x ∈ X and note
that as f is G-equivariant, so f(x̃) = f(x) for any x̃ in the orbit of x. By continuity, we find
that f(σ(π(x))) = f(x) finishing the proof. ■
Let us apply this lemma to the case of G = U2.
Lemma 3.15. There exists an isomorphism of varietiesMU2(Σg) ∼=
(
A1
C \ {0}
)4g
over C.
Proof. Consider
M =
{
(A1, B1, . . . , Ag, Bg) ∈ XU2(Σg) | all Ai, Bi are diagonal
}
,
Virtual Classes of Representation Varieties of Upper Triangular Matrices via TQFTs 33
which is isomorphic to
(
A1
C \ {0}
)4g
. Let π : XU2(Σg) → M be the morphism that sends every
matrix
(
ai bi
0 ci
)
to the matrix
(
ai 0
0 ci
)
, and take σ : M → XU2(Σg) to be the inclusion. Indeed π
is U2-invariant, and for any
A =
((
a1 b1
0 c1
)
, . . . ,
(
a2g b2g
0 c2g
))
∈ XU2(Σg)
we find that
lim
x→0
(
x 0
0 1
)
A
(
x 0
0 1
)−1
=
((
a1 0
0 c1
)
, . . . ,
(
a2g 0
0 c2g
))
= σ(π(A)),
so σ(π(A)) lies in the analytic-closure (and thus Zariski-closure) of the orbit of A. By Lem-
ma 3.14, we conclude that π : XU2(Σg)→M is the categorical quotient of XU2(Σg) by the action
of U2 providing the required isomorphismMU2(Σg) ∼=
(
A1
C \ {0}
)4g
. ■
We remark that the moduli space of G-representationsMU2(Σg) is a variety even though U2
is a non-reductive linear group.
Now, we turn our attention to the G-character variety of X. We write Γ = π1(X), which is
assumed to be finitely generated. The character of a representation ρ ∈ XG(X) is defined as
the map
χρ : Γ→ k : γ 7→ tr(ρ(γ)),
and the character map as
χ : XG(X)→ kΓ : ρ 7→ χρ.
The image of χ is called the G-character variety, denoted χG(X). By the results from [6], there
exists a finite set of elements γ1, . . . , γa ∈ π1(X) such that χρ is determined by the characters
χρ(γ1), . . . , χρ(γa) for any ρ. This way χG(X) can be identified with the image of the map
XG(X) → ka : ρ 7→ (χρ(γ1), . . . , χρ(γa)), which gives the G-character variety the structure of
a variety. This structure is independent of the chosen γi.
Note that the character map χ is a G-invariant morphism: indeed the trace map is invariant
under conjugation. By the universal property of the categorical quotient, we obtain an induced
map
χ : MG(X)→ χG(X).
In the case of G = SLn(C), Sp2n(C) or SO2n+1(C) this map is an isomorphism [6, 8, 20].
However, χ fails to be an isomorphism for the surfaces Σg and the non-reductive groups G = U2
with g ≥ 1.
Theorem 3.16. The map
χ :MU2(Σg)→ χU2(Σg)
is not an isomorphism.
Proof. Arguing as in Lemma 3.14, the map χ must be given by χ(A) = χA. But this cannot
be an isomorphism: for general A ∈ MU2(Σg) one can consider B = ( 0 1
1 0 )A ( 0 1
1 0 )
−1
(where the
diagonal entries of A are interchanged), and we have χA = χB, even though in general A ̸= B.
Therefore, the moduli spaceMU2(Σg) is not isomorphic to the character variety χU2(Σg) through
the natural map. ■
34 M. Hablicsek and J. Vogel
All of the above can easily be generalized to the case G = Un for any n ≥ 2. Namely, similar
to before, let
M =
{
(A1, B1, . . . , Ag, Bg) ∈ XUn(Σg) | all Ai, Bi are diagonal
}
,
and π : XUn(Σg) → M the map that sets all off-diagonal entries to zero, and σ : M → XUn(Σg)
the inclusion. Then σ ◦π = id, and one easily checks that σ(π(A)) lies in the closure of the orbit
of A for any A ∈ XUn(Σg), e.g., by conjugating A with
xn−1
xn−2
. . .
1
and taking the limit
x → 0. This shows that MUn(Σg) ∼=
(
A1
C \ {0}
)2gn
, proving Theorem 1.4. Again, note that
the natural map χ :MUn(Σg) → χUn(Σg) cannot be an isomorphism as there are symmetries
(permuting diagonal entries) that are invariant under the character map, proving Theorem 1.5.
Remark 3.17. In fact, as pointed out by the reviewers, the discussion above implies that the
Un-character variety χUn(Σg) is the GIT quotient of the moduli space of Un-representations
MUn(Σg) under the symmetric group Sn,
χUn(Σg) ∼=MUn(Σg) � Sn =
(
A1
C \ {0}
)2gn � Sn,
where Sn permutes the eigenvalues of the 2g generators simultaneously. Indeed, the traces of
the products of the powers of the 2g generators determine the eigenvalues of the 2g generators
up to a simultaneous Sn-action, which in turn, determine the character of any representation
χρ : Γ→ k.
In this way, the Un-character variety χUn(Σg) can be understood geometrically. The variety(
A1
C \ {0}
)n
is identified with variety of diagonal matrices of GLn(C), and thus, with a maximal
torus T . The symmetric group Sn can be identified with the Weyl group W acting on the
maximal torus, and thus, one gets isomorphisms
χUn(Σg) ∼=
(
A1
C \ {0}
)2gn � Sn
∼= T 2g � Sn
∼= Hom
(
Z2g, T
)
� W.
Now the map
T 2g → Hom
(
Z2g, T
)
→ Hom
(
Z2g,GLn(C)
)
→ Hom
(
Z2g,GLn(C)
)
� GLn(C)
factors through T 2g �W , so we obtain a map from the Un-character variety to the moduli space
of GLn(C)-representations of the free abelian group Z2g (which is the abelianization of π1(Σg))
χUn(Σg) ∼= Hom
(
Z2g, T
)
� W → Hom
(
Z2g,GLn(C)
)
� GLn(C),
which is, in fact, an isomorphism, see [9, 30] for more details.
A Algorithmically computing virtual classes
In this appendix, we describe our algorthim for computing classes of affine varieties over C in
the Grothendieck ring of varieties K(VarC) in terms of q =
[
A1
C
]
.
Let S = {x1, . . . , xn} be a finite set (of variables), and F , G be finite subsets of C[S]. Then
we write X(S, F,G) for the (reduced) subvariety of An
C given by f = 0 for all f ∈ F and g ̸= 0
for all g ∈ G. For example,
An
C = X({x1, . . . , xn},∅,∅) and GL2(C) = X({a, b, c, d},∅, {ad− bc}).
Virtual Classes of Representation Varieties of Upper Triangular Matrices via TQFTs 35
For convenience we will write evx(f, u) with f ∈ C[S] for the polynomial where x ∈ S in f is
substituted for u ∈ C[S]. Then we write
evx(F, u) = {evx(f, u) | f ∈ F}
and
evx(F, u, v) =
{
vdegx(f) · evx(f, u/v) | f ∈ F
}
for a set of polynomials F and x ∈ S and u, v ∈ C[S]. Note that for evx(F, u, v) the substituted
polynomials are multiplied by a suitable number of factors v, in order to clear denominators.
We describe our recursive algorithm which computes the class [X(S, F,G)] in terms of
q =
[
A1
C
]
.
Algorithm A.1. Let X = X(S, F,G) for some S, F and G as above.
1. If F contains a non-zero constant or if 0 ∈ G, then X = ∅, so [X] = 0.
2. If F = ∅ and G = ∅, then X = A#S
C , so [X] = q#S .
3. If some x ∈ S ‘does not appear’ in any f ∈ F and any g ∈ G, then we can factor
X ≃ A1
C ×X ′ with X ′ = X(S \ {x}, F,G). We have [X] = q[X ′].
4. If f = un (with n > 1) for some f ∈ F and u ∈ C[S], then we can replace f with u, not
changing X. That is, X = X(S, (F \ {f}) ∪ {u}, G). Similarly, if g = un (with n > 1) for
some g ∈ G and u ∈ C[S], then X = X(S, F, (G \ {g}) ∪ {u}).
5. If some f ∈ F is univariate in x ∈ S, we write f = (x − α1) · · · (x − αm), and we have
[X] =
∑m
i=1[Xi] with Xi = X(S \ {x}, evx(F \ {f}, αi), evx(G,αi)).
6. If f = uv for some f ∈ F and u, v ∈ C[S] (both not constant), then X1 = X(S, (F \{f})∪
{u}, G) = X ∩ {u = 0} and X2 = X(S, (F \ {f}) ∪ {v}, G ∪ {u}) = X ∩ {u ̸= 0, v = 0}
define a stratification for X, and thus [X] = [X1] + [X2].
7. If f = xu+v for some f ∈ F , x ∈ S and u, v ∈ C[S] with x not appearing in u and v, then
we consider the following cases. For any point p of X, either u(p) = 0, implying v(p) = 0
as well, or u(p) ̸= 0, implying x(p) = −v(p)/u(p). Therefore [X] = [X1] + [X2] with
X1 = X(S, (F \{f})∪{u, v}, G) and X2 = X(S, evx(F \{f},−v, u)), evx(G,−v, u)∪{u}).
8. Suppose f = x2u+ xv+w for some f ∈ F , x ∈ S and u, v, w ∈ C[S] with x not appearing
in u, v and w. Moreover, suppose that the discriminant D = v2 − 4uw is a square, i.e., we
can write D = h2 for some h ∈ C[S]. Then for any point p of X, we consider the following
cases. Either u(p) = 0, in which case (xv+w)(p) = 0. If u(p) ̸= 0, we distinguish between
D(p) = 0 and D(p) ̸= 0. In the first case we find that x(p) =
(−v
2u
)
(p), and in the latter
case we have the two possibilities x(p) =
(−v±h
2u
)
(p). Hence [X] = [X1]+[X2]+[X3]+[X4],
with
X1 = X(S, (F \ {f}) ∪ {u, xv + w}, G),
X2 = X(S, evx(F \ {f},−v, 2u) ∪ {D}, evx(G,−v, 2u) ∪ {u}),
X3 = X(S, evx(F \ {f},−v − h, 2u), evx(G,−v − h, 2u) ∪ {u,D}),
X4 = X(S, evx(F \ {f},−v + h, 2u), evx(G,−v + h, 2u) ∪ {u,D}).
9. If G ̸= ∅, pick any g ∈ G. We have [X] = [X1] − [X2] where X1 = X(S, F,G \ {g}) and
X2 = X(S, F ∪ {g}, G).
An implementation of this algorithm in Python can be found at [34], together with the code
for the computations done Sections 3.2 and 3.3.
36 M. Hablicsek and J. Vogel
B Matrix Z̃(L) for U4
The map Z̃(L) for G = U4 is represented by the following matrix:
T1 T2 T3 T4
T1 q7(q − 1)5
(
q2 + 3q − 2
)
q9(q − 2)(q − 1)5
(
q2 + q − 1
)
q8(q − 1)5
(
q3 − 3q + 1
)
q7(q − 1)6
(
q2 + 2q − 2
)
T2 q7(q − 2)(q − 1)4
(
q2 + q − 1
)
q9(q − 1)4
(
q2 − 3q + 3
) (
q2 + q − 1
)
q8(q − 2)(q − 1)5
(
q2 + q − 1
)
q7(q − 2)(q − 1)5
(
q2 + q − 1
)
T3 q7(q − 1)4
(
q3 − 3q + 1
)
q9(q − 2)(q − 1)5
(
q2 + q − 1
)
q8(q − 1)4
(
q4 − q3 − 2q2 + 4q − 1
)
q7(q − 1)5
(
q3 − 3q + 1
)
T4 q7(q − 1)5
(
q2 + 2q − 2
)
q9(q − 2)(q − 1)5
(
q2 + q − 1
)
q8(q − 1)5
(
q3 − 3q + 1
)
q7(q − 1)5
(
q3 + q2 − 3q + 2
)
T5 q6(q − 2)(q − 1)4
(
q3 + q − 1
)
q11(q − 2)2(q − 1)4 q10(q − 2)(q − 1)5 q6(q − 2)(q − 1)6
(
q2 + q + 1
)
T6 q7(q − 1)4
(
q3 − 3q + 1
)
q9(q − 2)(q − 1)6 (q + 1) q8(q − 1)4
(
q4 − 2q3 − q2 + 4q − 1
)
q7(q − 1)5
(
q3 − 3q + 1
)
T7 q7(q − 2)(q − 1)4
(
q2 + q − 1
)
q11(q − 2)2(q − 1)4 q8(q − 2)(q − 1)6 (q + 1) q7(q − 2)(q − 1)5
(
q2 + q − 1
)
T8 q9(q − 2)2(q − 1)3 q11(q − 2)(q − 1)3
(
q2 − 3q + 3
)
q10(q − 2)2(q − 1)4 q9(q − 2)2(q − 1)4
T9 q7(q − 2)(q − 1)5 (q + 1) q9(q − 1)5 (q + 1)
(
q2 − 3q + 3
)
q8(q − 2)(q − 1)6 (q + 1) q7(q − 2)(q − 1)6 (q + 1)
T10 q7(q − 2)(q − 1)4
(
q2 − q − 1
)
q11(q − 2)(q − 1)3
(
q2 − 3q + 3
)
q8(q − 2)(q − 1)3
(
q4 − 3q3 + 2q2 − 1
)
q7(q − 2)(q − 1)5
(
q2 − q − 1
)
T11 q7(q − 1)3
(
q4 − 2q3 − q2 + 4q − 1
)
q9(q − 2)(q − 1)6 (q + 1) q8(q − 1)3
(
q5 − 3q4 + 2q3 + 3q2 − 5q + 1
)
q7(q − 1)4
(
q4 − 2q3 − q2 + 4q − 1
)
T12 q7(q − 2)(q − 1)5 (q + 1) q11(q − 2)2(q − 1)4 q8(q − 2)(q − 1)4
(
q3 − q2 + 1
)
q7(q − 2)(q − 1)6 (q + 1)
T13 q9(q − 2)2(q − 1)3 q11(q − 2)3(q − 1)3 q10(q − 2)2(q − 1)4 q9(q − 2)2(q − 1)4
T14 q6(q − 2)(q − 1)5
(
q2 + q + 1
)
q11(q − 2)2(q − 1)4 q10(q − 2)(q − 1)5 q6(q − 2)(q − 1)4
(
q4 − q3 + 1
)
T15 q9(q − 2)3(q − 1)2 q11(q − 2)2(q − 1)2
(
q2 − 3q + 3
)
q10(q − 2)3(q − 1)3 q9(q − 2)3(q − 1)3
T16 q7(q − 2)(q − 1)2
(
q4 − 3q3 + 2q2 − 1
)
q11(q − 2)(q − 1)3
(
q2 − 3q + 3
)
q8(q − 2)(q − 1)2
(
q5 − 4q4 + 5q3 − 2q2 + 1
)
q7(q − 2)(q − 1)3
(
q4 − 3q3 + 2q2 − 1
)
T5 T6 T7 T8
q8(q − 2)(q − 1)5
(
q3 + q − 1
)
q8(q − 1)5
(
q3 − 3q + 1
)
q9(q − 2)(q − 1)5
(
q2 + q − 1
)
q12(q − 2)2(q − 1)5
q11(q − 2)2(q − 1)4 q8(q − 2)(q − 1)6 (q + 1) q11(q − 2)2(q − 1)4 q12(q − 2)(q − 1)4
(
q2 − 3q + 3
)
q11(q − 2)(q − 1)5 q8(q − 1)4
(
q4 − 2q3 − q2 + 4q − 1
)
q9(q − 2)(q − 1)6 (q + 1) q12(q − 2)2(q − 1)5
q8(q − 2)(q − 1)6
(
q2 + q + 1
)
q8(q − 1)5
(
q3 − 3q + 1
)
q9(q − 2)(q − 1)5
(
q2 + q − 1
)
q12(q − 2)2(q − 1)5
q7(q − 1)4
(
q2 − 3q + 3
) (
q4 + q − 1
)
q10(q − 2)(q − 1)5 q11(q − 2)2(q − 1)4 q12(q − 2)(q − 1)4
(
q2 − 3q + 3
)
q11(q − 2)(q − 1)5 q8(q − 1)4
(
q4 − q3 − 2q2 + 4q − 1
)
q9(q − 2)(q − 1)5
(
q2 + q − 1
)
q12(q − 2)2(q − 1)5
q11(q − 2)2(q − 1)4 q8(q − 2)(q − 1)5
(
q2 + q − 1
)
q9(q − 1)4
(
q2 − 3q + 3
) (
q2 + q − 1
)
q12(q − 2)3(q − 1)4
q11(q − 2)(q − 1)3
(
q2 − 3q + 3
)
q10(q − 2)2(q − 1)4 q11(q − 2)3(q − 1)3 q12(q − 1)3
(
q2 − 3q + 3
)2
q11(q − 2)2(q − 1)4 q8(q − 2)(q − 1)4
(
q3 − q2 + 1
)
q11(q − 2)2(q − 1)4 q12(q − 2)(q − 1)4
(
q2 − 3q + 3
)
q11(q − 2)3(q − 1)3 q8(q − 2)(q − 1)3
(
q4 − 3q3 + 2q2 − 1
)
q11(q − 2)(q − 1)3
(
q2 − 3q + 3
)
q12(q − 2)2(q − 1)3
(
q2 − 3q + 3
)
q11(q − 2)(q − 1)5 q8(q − 1)3
(
q5 − 3q4 + 2q3 + 3q2 − 5q + 1
)
q9(q − 2)(q − 1)6 (q + 1) q12(q − 2)2(q − 1)5
q11(q − 2)2(q − 1)4 q8(q − 2)(q − 1)6 (q + 1) q9(q − 1)5 (q + 1)
(
q2 − 3q + 3
)
q12(q − 2)3(q − 1)4
q11(q − 2)(q − 1)3
(
q2 − 3q + 3
)
q10(q − 2)2(q − 1)4 q11(q − 2)(q − 1)3
(
q2 − 3q + 3
)
q12(q − 2)2(q − 1)3
(
q2 − 3q + 3
)
q7(q − 1)5 (q + 1)
(
q2 + 1
) (
q2 − 3q + 3
)
q10(q − 2)(q − 1)5 q11(q − 2)2(q − 1)4 q12(q − 2)(q − 1)4
(
q2 − 3q + 3
)
q11(q − 2)2(q − 1)2
(
q2 − 3q + 3
)
q10(q − 2)3(q − 1)3 q11(q − 2)2(q − 1)2
(
q2 − 3q + 3
)
q12(q − 2)(q − 1)2
(
q2 − 3q + 3
)2
q11(q − 2)3(q − 1)3 q8(q − 2)(q − 1)2
(
q5 − 4q4 + 5q3 − 2q2 + 1
)
q11(q − 2)(q − 1)3
(
q2 − 3q + 3
)
q12(q − 2)2(q − 1)3
(
q2 − 3q + 3
)
T9 T10 T11 T12
q9(q − 2)(q − 1)7 (q + 1) q9(q − 2)(q − 1)6
(
q2 − q − 1
)
q8(q − 1)5
(
q4 − 2q3 − q2 + 4q − 1
)
q9(q − 2)(q − 1)7 (q + 1)
q9(q − 1)6 (q + 1)
(
q2 − 3q + 3
)
q11(q − 2)(q − 1)4
(
q2 − 3q + 3
)
q8(q − 2)(q − 1)7 (q + 1) q11(q − 2)2(q − 1)5
q9(q − 2)(q − 1)7 (q + 1) q9(q − 2)(q − 1)4
(
q4 − 3q3 + 2q2 − 1
)
q8(q − 1)4
(
q5 − 3q4 + 2q3 + 3q2 − 5q + 1
)
q9(q − 2)(q − 1)5
(
q3 − q2 + 1
)
q9(q − 2)(q − 1)7 (q + 1) q9(q − 2)(q − 1)6
(
q2 − q − 1
)
q8(q − 1)5
(
q4 − 2q3 − q2 + 4q − 1
)
q9(q − 2)(q − 1)7 (q + 1)
q11(q − 2)2(q − 1)5 q11(q − 2)3(q − 1)4 q10(q − 2)(q − 1)6 q11(q − 2)2(q − 1)5
q9(q − 2)(q − 1)5
(
q3 − q2 + 1
)
q9(q − 2)(q − 1)4
(
q4 − 3q3 + 2q2 − 1
)
q8(q − 1)4
(
q5 − 3q4 + 2q3 + 3q2 − 5q + 1
)
q9(q − 2)(q − 1)7 (q + 1)
q11(q − 2)2(q − 1)5 q11(q − 2)(q − 1)4
(
q2 − 3q + 3
)
q8(q − 2)(q − 1)7 (q + 1) q9(q − 1)6 (q + 1)
(
q2 − 3q + 3
)
q11(q − 2)(q − 1)4
(
q2 − 3q + 3
)
q11(q − 2)2(q − 1)3
(
q2 − 3q + 3
)
q10(q − 2)2(q − 1)5 q11(q − 2)3(q − 1)4
q9(q − 1)4
(
q2 − 3q + 3
) (
q3 − q2 + 1
)
q11(q − 2)(q − 1)4
(
q2 − 3q + 3
)
q8(q − 2)(q − 1)5
(
q3 − q2 + 1
)
q11(q − 2)2(q − 1)5
q11(q − 2)(q − 1)4
(
q2 − 3q + 3
)
q9(q − 1)3
(
q2 − 3q + 3
) (
q4 − 3q3 + 3q2 + 1
)
q8(q − 2)(q − 1)3
(
q5 − 4q4 + 5q3 − 2q2 + 1
)
q11(q − 2)(q − 1)4
(
q2 − 3q + 3
)
q9(q − 2)(q − 1)5
(
q3 − q2 + 1
)
q9(q − 2)(q − 1)3
(
q5 − 4q4 + 5q3 − 2q2 + 1
)
q8(q − 1)3
(
q6 − 4q5 + 6q4 − 2q3 − 5q2 + 6q − 1
)
q9(q − 2)(q − 1)5
(
q3 − q2 + 1
)
q11(q − 2)2(q − 1)5 q11(q − 2)(q − 1)4
(
q2 − 3q + 3
)
q8(q − 2)(q − 1)5
(
q3 − q2 + 1
)
q9(q − 1)4
(
q2 − 3q + 3
) (
q3 − q2 + 1
)
q11(q − 2)3(q − 1)4 q11(q − 2)2(q − 1)3
(
q2 − 3q + 3
)
q10(q − 2)2(q − 1)5 q11(q − 2)(q − 1)4
(
q2 − 3q + 3
)
q11(q − 2)2(q − 1)5 q11(q − 2)3(q − 1)4 q10(q − 2)(q − 1)6 q11(q − 2)2(q − 1)5
q11(q − 2)2(q − 1)3
(
q2 − 3q + 3
)
q11(q − 2)(q − 1)2
(
q2 − 3q + 3
)2
q10(q − 2)3(q − 1)4 q11(q − 2)2(q − 1)3
(
q2 − 3q + 3
)
q11(q − 2)(q − 1)4
(
q2 − 3q + 3
)
q9(q − 1)2
(
q2 − 3q + 3
) (
q5 − 4q4 + 6q3 − 3q2 − 1
)
q8(q − 2)(q − 1)2
(
q6 − 5q5 + 9q4 − 7q3 + 2q2 − 1
)
q11(q − 2)(q − 1)4
(
q2 − 3q + 3
)
T13 T14 T15 T16
q12(q − 2)2(q − 1)5 q8(q − 2)(q − 1)7
(
q2 + q + 1
)
q12(q − 2)3(q − 1)5 q9(q − 2)(q − 1)5
(
q4 − 3q3 + 2q2 − 1
)
q12(q − 2)3(q − 1)4 q11(q − 2)2(q − 1)5 q12(q − 2)2(q − 1)4
(
q2 − 3q + 3
)
q11(q − 2)(q − 1)5
(
q2 − 3q + 3
)
q12(q − 2)2(q − 1)5 q11(q − 2)(q − 1)6 q12(q − 2)3(q − 1)5 q9(q − 2)(q − 1)4
(
q5 − 4q4 + 5q3 − 2q2 + 1
)
q12(q − 2)2(q − 1)5 q8(q − 2)(q − 1)5
(
q4 − q3 + 1
)
q12(q − 2)3(q − 1)5 q9(q − 2)(q − 1)5
(
q4 − 3q3 + 2q2 − 1
)
q12(q − 2)(q − 1)4
(
q2 − 3q + 3
)
q7(q − 1)6 (q + 1)
(
q2 + 1
) (
q2 − 3q + 3
)
q12(q − 2)2(q − 1)4
(
q2 − 3q + 3
)
q11(q − 2)3(q − 1)5
q12(q − 2)2(q − 1)5 q11(q − 2)(q − 1)6 q12(q − 2)3(q − 1)5 q9(q − 2)(q − 1)4
(
q5 − 4q4 + 5q3 − 2q2 + 1
)
q12(q − 2)(q − 1)4
(
q2 − 3q + 3
)
q11(q − 2)2(q − 1)5 q12(q − 2)2(q − 1)4
(
q2 − 3q + 3
)
q11(q − 2)(q − 1)5
(
q2 − 3q + 3
)
q12(q − 2)2(q − 1)3
(
q2 − 3q + 3
)
q11(q − 2)(q − 1)4
(
q2 − 3q + 3
)
q12(q − 2)(q − 1)3
(
q2 − 3q + 3
)2
q11(q − 2)2(q − 1)4
(
q2 − 3q + 3
)
q12(q − 2)3(q − 1)4 q11(q − 2)2(q − 1)5 q12(q − 2)2(q − 1)4
(
q2 − 3q + 3
)
q11(q − 2)(q − 1)5
(
q2 − 3q + 3
)
q12(q − 2)2(q − 1)3
(
q2 − 3q + 3
)
q11(q − 2)3(q − 1)4 q12(q − 2)(q − 1)3
(
q2 − 3q + 3
)2
q9(q − 1)3
(
q2 − 3q + 3
) (
q5 − 4q4 + 6q3 − 3q2 − 1
)
q12(q − 2)2(q − 1)5 q11(q − 2)(q − 1)6 q12(q − 2)3(q − 1)5 q9(q − 2)(q − 1)3
(
q6 − 5q5 + 9q4 − 7q3 + 2q2 − 1
)
q12(q − 2)(q − 1)4
(
q2 − 3q + 3
)
q11(q − 2)2(q − 1)5 q12(q − 2)2(q − 1)4
(
q2 − 3q + 3
)
q11(q − 2)(q − 1)5
(
q2 − 3q + 3
)
q12(q − 1)3
(
q2 − 3q + 3
)2
q11(q − 2)(q − 1)4
(
q2 − 3q + 3
)
q12(q − 2)(q − 1)3
(
q2 − 3q + 3
)2
q11(q − 2)2(q − 1)4
(
q2 − 3q + 3
)
q12(q − 2)(q − 1)4
(
q2 − 3q + 3
)
q7(q − 1)4
(
q2 − 3q + 3
) (
q5 − q4 + 1
)
q12(q − 2)2(q − 1)4
(
q2 − 3q + 3
)
q11(q − 2)3(q − 1)5
q12(q − 2)(q − 1)2
(
q2 − 3q + 3
)2
q11(q − 2)2(q − 1)3
(
q2 − 3q + 3
)
q12(q − 1)2
(
q2 − 3q + 3
)3
q11(q − 2)(q − 1)3
(
q2 − 3q + 3
)2
q12(q − 2)2(q − 1)3
(
q2 − 3q + 3
)
q11(q − 2)3(q − 1)4 q12(q − 2)(q − 1)3
(
q2 − 3q + 3
)2
q9(q − 1)2
(
q2 − 3q + 3
) (
q6 − 5q5 + 10q4 − 9q3 + 3q2 + 1
)
Acknowledgements
The authors thank Bas Edixhoven and David Holmes for reading a previous version of this paper
and giving valuable comments; Ángel González-Prieto whose papers were the starting point of
this research and who was kind enough to answer any questions; and Sean Lawton for pointing
out an error regarding χ. The authors also thank the reviewers for their detailed feedback and
invaluable comments. The paper is part of the master’s thesis [33] of the second author.
Virtual Classes of Representation Varieties of Upper Triangular Matrices via TQFTs 37
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38 M. Hablicsek and J. Vogel
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1 Introduction
1.1 Main results
2 TQFTs and representation varieties
2.1 The 2-category of bordisms
2.2 The Grothendieck ring of varieties
2.3 The 2-category of spans
2.4 Constructing the TQFT
2.5 Parabolic structures
2.6 Field theory in dimension 2
2.7 Reduction of the TQFT
3 Applications
3.1 Upper triangular 2 times 2
3.2 Upper triangular 3 times 3 matrices
3.3 Upper triangular 4 times 4 matrices
3.4 Moduli space of representations and character variety
A Algorithmically computing virtual classes
B Matrix Z(L) for U_4
References
|
| id | nasplib_isofts_kiev_ua-123456789-211809 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-21T05:20:19Z |
| publishDate | 2022 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Hablicsek, Márton Vogel, Jesse 2026-01-12T10:14:33Z 2022 Virtual Classes of Representation Varieties of Upper Triangular Matrices via Topological Quantum Field Theories. Márton Hablicsek and Jesse Vogel. SIGMA 18 (2022), 095, 38 pages 1815-0659 2020 Mathematics Subject Classification: 14D23; 14D21; 14C30; 14D20; 14D07; 57R56 arXiv:2008.06679 https://nasplib.isofts.kiev.ua/handle/123456789/211809 https://doi.org/10.3842/SIGMA.2022.095 In this paper, we use a geometric technique developed by González-Prieto, Logares, Muñoz, and Newstead to study the -representation variety of surface groups G(Σ) of arbitrary genus for being the group of upper triangular matrices of fixed rank. Explicitly, we compute the virtual classes in the Röthendieck ring of varieties of the -representation variety and the moduli space of -representations of surface groups for being the group of complex upper triangular matrices of rank 2, 3, and 4 via constructing a topological quantum field theory. Furthermore, we show that in the case of upper triangular matrices, the character map from the moduli space of -representations to the -character variety is not an isomorphism. The authors thank Bas Edixhoven and David Holmes for reading a previous version of this paper and giving valuable comments; Άngel González-Prieto, whose papers were the starting point of this research, and who was kind enough to answer any questions; and Sean Lawton for pointing out an error regarding χ. The authors also thank the reviewers for their detailed feedback and invaluable comments. The paper is part of the master’s thesis [33] of the second author. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Virtual Classes of Representation Varieties of Upper Triangular Matrices via Topological Quantum Field Theories Article published earlier |
| spellingShingle | Virtual Classes of Representation Varieties of Upper Triangular Matrices via Topological Quantum Field Theories Hablicsek, Márton Vogel, Jesse |
| title | Virtual Classes of Representation Varieties of Upper Triangular Matrices via Topological Quantum Field Theories |
| title_full | Virtual Classes of Representation Varieties of Upper Triangular Matrices via Topological Quantum Field Theories |
| title_fullStr | Virtual Classes of Representation Varieties of Upper Triangular Matrices via Topological Quantum Field Theories |
| title_full_unstemmed | Virtual Classes of Representation Varieties of Upper Triangular Matrices via Topological Quantum Field Theories |
| title_short | Virtual Classes of Representation Varieties of Upper Triangular Matrices via Topological Quantum Field Theories |
| title_sort | virtual classes of representation varieties of upper triangular matrices via topological quantum field theories |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211809 |
| work_keys_str_mv | AT hablicsekmarton virtualclassesofrepresentationvarietiesofuppertriangularmatricesviatopologicalquantumfieldtheories AT vogeljesse virtualclassesofrepresentationvarietiesofuppertriangularmatricesviatopologicalquantumfieldtheories |