Triply Periodic Monopoles and Difference Modules on Elliptic Curves
We explain the correspondences between twisted monopoles with Dirac-type singularity and polystable twisted mini-holomorphic bundles with Dirac-type singularity on a 3-dimensional torus. We also explain that they are equivalent to polystable parabolic twisted difference modules on elliptic curves.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 16 (2020), 048, 23 pages
Triply Periodic Monopoles and Difference Modules
on Elliptic Curves
Takuro MOCHIZUKI
Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 606-8502, Japan
E-mail: takuro@kurims.kyoto-u.ac.jp
Received October 29, 2019, in final form May 18, 2020; Published online June 03, 2020
https://doi.org/10.3842/SIGMA.2020.048
Abstract. We explain the correspondences between twisted monopoles with Dirac type
singularity and polystable twisted mini-holomorphic bundles with Dirac type singularity on
a 3-dimensional torus. We also explain that they are equivalent to polystable parabolic
twisted difference modules on elliptic curves.
Key words: twisted monopoles; twisted difference modules; twisted mini-holomorphic bun-
dles; Kobayashi–Hitchin correspondence
2020 Mathematics Subject Classification: 53C07; 58E15; 14D21; 81T13
1 Introduction
We studied the Kobayashi–Hitchin correspondences for singular monopoles with periodicity in
one direction [4] or two directions [5]. In this paper, we study singular monopoles with periodicity
in three directions. In the analytic aspect, this case is much simpler than the other cases because
a 3-dimensional torus is compact. But, there still exist interesting correspondences with algebro-
geometric objects. Moreover, everything is generalized to the twisted case. (See Section 2 for the
twisted objects.) Though we also study a generalization to the twisted case, this introduction
is devoted to explain the results in the untwisted case.
1.1 Triply periodic monopoles with Dirac type singularity
Let Y be an oriented 3-dimensional R-vector space with an Euclidean metric gY . Let Γ be
a lattice of Y . We set M := Y/Γ, which is equipped with the induced metric gM. Let Z be
a finite subset of M. Let E be a C∞-vector bundle on M \ Z with a Hermitian metric h,
a unitary connection ∇ and an anti-self-adjoint endomorphism φ. A tuple (E, h,∇, φ) is called
a monopole on M\ Z if the Bogomolny equation
F (∇) = ∗∇φ
is satisfied, where F (∇) denotes the curvature of ∇, and ∗ denotes the Hodge star operator with
respect to gM. A point of P ∈ Z is called a Dirac type singularity of the monopole (E, h,∇, φ) if
|φQ|h = O
(
d(Q,Z)−1
)
for any Q ∈M\Z, where φQ denotes the element of the fiber End(E)|Q
over Q induced by φ, and d(Q,Z) denotes the distance between Q and Z. Note that the notion
of Dirac type singularity was originally introduced by Kronheimer [3]. The above condition is
equivalent to the original definition, according to [6].
This paper is a contribution to the Special Issue on Integrability, Geometry, Moduli in honor of Motohico Mu-
lase for his 65th birthday. The full collection is available at https://www.emis.de/journals/SIGMA/Mulase.html
mailto:takuro@kurims.kyoto-u.ac.jp
https://doi.org/10.3842/SIGMA.2020.048
https://www.emis.de/journals/SIGMA/Mulase.html
2 T. Mochizuki
1.2 Mini-holomorphic bundles with Dirac type singularity
Let us explain a correspondence between monopoles with Dirac type singularity and polystable
mini-holomorphic bundles with Dirac type singularity on a 3-dimensional torus. (See Section 2
below for more details on the notions of mini-complex structures and mini-holomorphic bundles
with Dirac type singularity on 3-dimensional manifolds.) It was formulated by Kontsevich and
Soibelman [2].
1.2.1 Mini-complex structure
We take a linear coordinate system (x1, x2, x3) on Y compatible with the orientation such that
gY =
∑
dxi dxi, and we set t := x1 and w = x2 +
√
−1x3. The coordinate system induces
a mini-complex structure on M\ Z. A C∞-function f on an open subset of M is called mini-
holomorphic if ∂tf = ∂wf = 0. Let OM\Z denote the sheaf of mini-holomorphic functions
on M\ Z.
1.2.2 Mini-holomorphic bundles with Dirac type singularity
Let V be a locally free OM\Z-module. Let P be a point of Z. We take a lift (t0, w0) ∈ Y
of P . Let ε and δ denote small positive numbers. Set Bw0(δ) :=
{
w ∈ C
∣∣ |w − w0| < δ
}
and
B∗w0
(δ) :=
{
w ∈ C
∣∣ 0 < |w−w0| < δ
}
. For any t0 − ε < t < t0 + ε, the restriction V|{t}×B∗w0
(δ) is
naturally a locally free OB∗w0
(δ)-module. If t 6= t0, they extend to locally free OBw0 (δ)-modules
V|{t}×Bw0 (δ). Because mini-holomorphic functions are constant in the t-direction, we obtain
an isomorphism of OB∗w0
(δ)-modules V|{t0−ε1}×B∗w0
(δ) ' V|{t0+ε1}×B∗w0
(δ) for 0 < ε1 < ε. If it
is meromorphic at w0, then P is called a Dirac type singularity of V. If every point of Z is
Dirac type singularity, then V is called a mini-holomorphic bundle with Dirac type singularity
on (M;Z).
1.2.3 Stability condition
Kontsevich and Soibelman [2] introduced a sophisticated way to define a stability condition for
mini-holomorphic bundles with Dirac type singularity on (M;Z).
Let Hj(M\Z) denote the j-th cohomology group ofM\Z with R-coefficient. Let Hj(M, Z)
denote the relative j-th homology group of (M, Z) with R-coefficient. Note that there exists
the natural isomorphism
ΦZ : H2(M\ Z) ' H1(M, Z).
Let T denote the space of left invariant vector fields onM, and let T∨ denote the left invariant
1-forms on M. Let σ denote the image of 1 via the canonical morphism R −→ T ⊗ T∨. It is
described as σ =
∑
i=1,2,3 ∂xi ⊗ dxi.
For any mini-holomorphic bundle with Dirac type singularity V on (M;Z), we obtain c1(V) ∈
H2(M\ Z), and hence ΦZ(c1(V)) ∈ H1(M, Z). Then, we obtain the following invariant vector
field ∫
ΦZ(c1(V))
σ =
∑
i=1,2,3
(∫
ΦZ(c1(V))
dxi
)
∂xi ∈ T.
Kontsevich and Soibelman discovered that
∫
ΦZ(c1(V)) σ is a scalar multiplication of ∂t = ∂x1 , and
they define the degree degKS(V) for V as follows∫
ΦZ(c1(V))
σ = degKS(V)∂t.
Triply Periodic Monopoles and Difference Modules on Elliptic Curves 3
They introduced the following stability condition.
Definition 1.1. A mini-holomorphic bundle with Dirac type singularity V on (M;Z) is called
stable (resp. semistable) if
degKS(V ′)/ rank(V ′) < degKS(V)/ rank(V)(
resp. degKS(V ′)/ rank(V ′) ≤ degKS(V)/ rank(V)
)
for any locally free OM\Z-submodule V ′ of V such that 0 < rank(V ′) < rank(V). It is called
polystable if it is semistable and a direct sum of stable submodules.
1.2.4 Kobayashi–Hitchin correspondence
Let (E, h,∇, φ) be a monopole with Dirac type singularity on M\ Z. We set ∂E,w := ∇w and
∂E,∂t := ∇t −
√
−1φ. Let V be the sheaf of sections s of E such that ∂E,ws = ∂E,ts = 0. It
is a standard fact that V is a mini-holomorphic bundle with Dirac type singularity on (M;Z).
The following theorem was formulated by Kontsevich and Soibelman [2].
Theorem 1.2 (the untwisted case in Theorem 3.16, Proposition 4.2). The procedure induces
an equivalence between monopoles with Dirac type singularity on M \ Z and polystable mini-
holomorphic bundles with Dirac type singularity of degree 0 on (M;Z).
We shall relate the degree of Kontsevich and Soibelman with the analytic degree defined in
terms of Hermitian metrics (Proposition 4.2). Then, Theorem 1.2 follows from the fundamental
theorem due to Simpson [7] as we shall explain in the proof of Theorem 3.16, which is an analogue
of a result due to Charbonneau and Hurtubise [1] for singular monopoles on 3-dimensional
manifolds obtained as the product of S1 and a compact Riemann surface. See also the work of
Yoshino [8] on the Kobayashi–Hitchin correspondence for monopoles with Dirac type singularity
on mini-complex 3-dimensional manifolds.
1.3 Parabolic difference modules on elliptic curves
Let us give a complement on correspondences between mini-holomorphic bundles with Dirac
type singularity on a 3-dimensional torus and parabolic difference modules on elliptic curves.
Remark 1.3. After completing the first version of this paper, the author was informed that [2]
also already contains the correspondence with difference modules on elliptic curves.
1.3.1 Parabolic difference modules on elliptic curves and a stability condition
Let Γ0 be a lattice of C. We set T := C/Γ0. Let a ∈ C. Let Φ: T −→ T be the morphism
induced by Φ(z) = z + a. Let D ⊂ T be a finite subset. Let OT (∗D) denote the sheaf of
meromorphic functions on T which may have poles along D. For any OT -module F , we set
F(∗D) := F ⊗OT OT (∗D). A parabolic a-difference module on T consists of the following data
V∗ =
(
V, (τP ,LP )P∈D
)
:
• A locally free OT -module V .
• An isomorphism of OT (∗D)-modules V (∗D) ' (Φ∗)−1(V )(∗D).
• A sequence 0 ≤ τP,1 < τP,2 < · · · < τP,m(P ) < 1 for each P ∈ D.
• Lattices LP,i (i = 1, . . . ,m(P ) − 1) of the stalk V (∗D)P at each P ∈ D. We formally set
LP,0 := VP and LP,m(P ) := (Φ∗)−1(V )P at each P ∈ D.
When we fix (τP )P∈D, it is called a parabolic a-difference module on (T, (τP )P∈D).
4 T. Mochizuki
The degree of a parabolic a-difference module (V, (τP ,LP )P∈D) is defined as follows
deg
(
V, (τP ,LP )P∈D
)
:= deg(V ) +
∑
P∈D
m(P )∑
i=1
(1− τP,i) deg(LP,i,LP,i−1). (1.1)
Here, we set deg(LP,i,LP,i−1) := length
(
LP,i/LP,i−1 ∩LP,i
)
− length
(
LP,i−1/LP,i−1 ∩LP,i
)
. The
degree can be rewritten as
deg
(
V, (τP ,LP )P∈D
)
:= deg(V )−
∑
P∈D
m(P )∑
i=1
τP,i deg(LP,i,LP,i−1),
because
∑
P∈D
∑m(P )
i=1 deg(LP,i,LP,i−1) = 0. The slope is defined in the standard way
µ(V, (τP ,LP )P∈D) := deg(V, (τP ,LP )P∈D)/ rankV.
For any OT (∗D)-submodule 0 6= V ′ ⊂ V such that V ′(∗D) ' (Φ∗)−1(V ′)(∗D), we obtain
lattices L′P,i of V ′(∗D)P by setting L′P,i := LP,i∩V ′(∗D)P in V (∗D)P , and we obtain a parabolic
a-difference module (V ′, (τP ,L′P )P∈D). Such (V ′, (τP ,L′P )P∈D) is called a parabolic a-difference
submodule of (V, (τP ,LP )P∈D).
Definition 1.4. (V, (τP ,LP )P∈D) is called stable (resp. semistable) if
µ(V ′, (τP ,L′P )P∈D) < µ(V, (τP ,LP )P∈D)(
resp. µ(V ′, (τP ,L′P )P∈D) ≤ µ(V, (τP ,LP )P∈D)
)
for any parabolic a-difference submodules such that 0 < rankV ′ < rankV . It is called polystable
if it is semistable and a direct sum of stable objects.
1.3.2 Equivalence
We return to the situation in Section 1.2. We take a generator ei = (ai, αi) (i = 1, 2, 3) of
Γ ⊂ Rt × Cw = Y , which is compatible with the orientation of Y . We also assume that α1
and α2 generate a lattice in C and compatible with the orientation of C. Let Γ0 denote the
lattice, and we set T := C/Γ0. We set
γ := − a1α2 − a2α1
α1α2 − α2α1
, t := a3 + 2 Re(γα3), a := α3.
It is easy to see that t > 0. We define the isomorphism F : Rt × Cw ' Rs × Cu by
s = t+ 2 Re(γw), u = w.
Note that the induced action of Γ on Rs × Cu is expressed as follows:
ei(s, u) = (s, u+ αi) (i = 1, 2), e3(s, u) = (s+ t, u+ a).
We set [0, t[ := {0 ≤ s < t}. Let ZY be the pull back of Z by Y −→ M. Let D denote the
image of the composite of the following maps:
F (ZY ) ∩
(
[0, t[×Cu
)
⊂ Rs × Cu −→ Cu −→ T.
For any P ∈ D, we take u0 ∈ C which is mapped to P . We obtain a sequence 0 ≤ sP,1 <
sP,2 < · · · < sP,m(P ) < t by the condition:
{(sP,i, u0) | i = 1, . . . ,m(P )} = F (ZY ) ∩
(
[0, t[×{u0}
)
.
It is independent of the choice of u0. We set τP,i := sP,i/t.
Triply Periodic Monopoles and Difference Modules on Elliptic Curves 5
Proposition 1.5 (the untwisted case in Propositions 3.13 and 3.14). There exists an equivalence
between parabolic difference modules on (T, (τP )P∈D) and mini-holomorphic bundles with Dirac
type singularity on (M;Z). The equivalence preserves the degree up to the multiplication of
a positive constant. As a result, the equivalence preserves the (poly)stability condition.
See Section 3.2.2 for the explicit correspondence. As a consequence of Theorem 1.2 and
Proposition 1.5, we obtain the following theorem.
Theorem 1.6. We have the equivalence of the following objects:
• Monopoles with Dirac type singularity on M\ Z.
• Polystable mini-holomorphic bundles with Dirac type singularity of degree 0 on (M;Z).
• Polystable parabolic difference modules of degree 0 on (T, (τP )P∈D).
Here, Z and (τP )P∈D are related as above.
This study is partially motivated by the holomorphic Floer theory [2] of Kontsevich and
Soibelman. Among other things, they revisit the Riemann–Hilbert correspondence for D-
modules from the viewpoint of symplectic topology, and they extend it to the context of difference
modules of various types. Moreover, they propose an analogue of the non-abelian Hodge theory
in the context of difference modules, where the role of harmonic bundles should be played by
monopoles as in Theorem 1.6.
Though the untwisted case is explained in this introduction, we shall study the twisted case,
i.e., equivalences of twisted mini-holomorphic bundles, twisted difference modules, and twisted
monopoles. We should note that Kontsevich and Soibelman suggested that there should exist
a twisted version of of Theorem 1.6.
2 Preliminary
We introduce the notions of twisted mini-holomorphic bundles and twisted monopoles as ge-
neralizations of the notions of mini-holomorphic bundles [4] and monopoles. We are interested
only in the case where the base manifolds are 3-dimensional torus. We also introduce twisted
difference modules on elliptic curves.
2.1 Mini-complex structure on 3-dimensional manifolds
Let (t, w) denote the standard coordinate system on R×C. Let M be an oriented 3-dimensional
C∞-manifold. A mini-complex coordinate system on M is a family of open subsets Uλ (λ ∈ Λ)
equipped with an oriented embedding ϕλ : Uλ −→ R× C satisfying the following conditions.
• M =
⋃
λ∈Λ Uλ.
• Let Fλ,µ : ϕµ(Uλ∩Uµ) −→ ϕλ(Uλ∩Uµ) denote the induced diffeomorphism of open subsets
in R × C. Note that Fλ,µ is expressed as ((Fλ,µ)t(t, w), (Fλ,µ)w(t, w)) in terms of the
coordinate systems. Then, it holds that ∂t(Fλ,µ)w = 0 and ∂w(Fλ,µ)w = 0.
Two mini-complex coordinate systems {(Uλ, ϕλ)}λ∈Λ and {(Vµ, ψµ)}µ∈Γ are called equivalent if
their union is also a mini-complex coordinate system. A mini-complex structure on M is an
equivalence class of mini-complex coordinate systems. We shall not distinguish a mini-complex
structure and a mini-complex coordinate system contained in the mini-complex structure.
Suppose that M is equipped with a mini-complex structure. On a mini-complex coordinate
neighbourhood (U ; t, w), let TSU denote the subbundle of the tangent bundle TU generated
6 T. Mochizuki
by ∂t. By patching TSU for any mini-complex coordinate neighbourhoods (U ; t, w) we obtain
the subbundle TSM ⊂ TM .
Let T ∗SM denote the dual bundle of TSM . Let T ∗QM denote the kernel of the natural surjection
T ∗M −→ T ∗SM . It is naturally equipped with a complex structure J . Let Ω1,0
Q M ⊂ T ∗QM ⊗ C
(resp. Ω0,1
Q M) denote the eigen subbundle with respect to J corresponding to
√
−1 (resp. −
√
−1).
We set Ω0,1M := (T ∗M ⊗ C)/Ω1,0
Q M and Ω0,iM :=
∧i Ω0,1M for i = 0, 1, 2. Similarly, we set
Ω1,0M := (T ∗M ⊗ C)/Ω0,1
Q M and Ωi,0M :=
∧i Ω1,0M for i = 0, 1, 2.
Let ∂M denote the differential operator C∞(M,C) −→ C∞
(
M,Ω0,1M
)
induced by the exte-
rior derivative and the projection T ∗M ⊗ C −→ Ω0,1M . The induced operator
C∞
(
M,Ω0,1M
)
−→ C∞
(
M,Ω0,2M
)
is also denoted by ∂M . Similarly, we obtain the operator
∂M : C∞
(
M,Ωi,0M
)
−→ C∞
(
M,Ωi+1,0M
)
.
2.1.1 Riemannian case
Suppose that M is also equipped with a Riemannian metric gM . Let T ∗S,gMM denote the or-
thogonal complement of T ∗QM . We shall naturally identify T ∗S,gMM and T ∗SM .
Because T ∗M and T ∗QM are oriented, T ∗S,gMM is also oriented. Let η be the unique section
of T ∗S,gMM in the positive direction such that the norm of η is 1. By η, T ∗S,gMM is identified
with R × M . If there exists a mini-complex coordinate system (U ; t, w) such that gM |U =
dt dt+ dw dw, then η|M = dt.
We obtain a decomposition
T ∗M ⊗ C = Ω1,0
Q M ⊕ Ω0,1
Q M ⊕ T ∗S,gMM ⊗ C. (2.1)
We also obtain the isomorphisms
Ω1,0
Q M ⊕ T ∗S,gMM ⊗ C ' Ω1,0M, Ω0,1
Q M ⊕ T ∗S,gMM ⊗ C ' Ω0,1M.
If the complex structure J on T ∗QM is an isometry with respect to gM , the decomposition (2.1)
is orthogonal.
2.2 Twisted mini-holomorphic bundles
Let M be a mini-complex 3-dimensional manifold. Let E be a C∞-vector bundle on M . We
shall always assume that the rank of E is finite. Let % ∈ C∞
(
M,Ω0,2M
)
.
Definition 2.1. A %-twisted mini-holomorphic structure of E is a differential operator ∂E :
C∞(M,E) −→ C∞
(
M,Ω0,1M ⊗ E
)
such that the following conditions are satisfied.
• ∂E(fs) = f∂E(s) + (∂Mf)⊗ s holds for any f ∈ C∞(M,C) and s ∈ C∞(M,E).
• The induced operator C∞
(
M,Ω0,1M ⊗E
)
−→ C∞
(
M,Ω0,2M ⊗E
)
is also denoted by ∂E .
Then, ∂E ◦ ∂E = % idE holds.
Such (E, ∂E) is called a %-twisted mini-holomorphic vector bundle. If % = 0, we shall omit the
adjective “0-twisted”.
Triply Periodic Monopoles and Difference Modules on Elliptic Curves 7
Remark 2.2. A C∞-function f on an open subset U ⊂ M is called mini-holomorphic if
∂Mf = 0. Let OM denote the sheaf of mini-holomorphic functions. In the case % = 0, mini-
holomorphic bundles are naturally identified with locally free OM -modules of finite rank. Let
(E, ∂E) be a mini-holomorphic bundle on M . A local section s of E is called mini-holomorphic if
∂E(s) = 0. Let Ẽ denote the sheaf of mini-holomorphic sections of E. Then, it is easy to observe
that Ẽ is a locally free OM -module of finite rank. This correspondence induces an equivalence
between mini-holomorphic bundles and locally free OM -modules of finite rank.
2.2.1 Scattering map
Let (E, ∂E) be a %-twisted mini-holomorphic vector bundle on M . Let γ : [0, 1] −→M be a C∞-
path such that Tγ(T [0, 1]) ⊂ TSM . Then, γ−1(E) is equipped with a connection induced by
the %-twisted mini-holomorphic structure ∂E , and hence we obtain the induced isomorphism
Eγ(0) ' Eγ(1). It is called the scattering map in [1].
Let (U ; t, w) be a mini-complex coordinate neighbourhood of M . Let ∂E,t (resp. ∂E,w) denote
the differential operators of E|U induced by ∂E and ∂t (resp. ∂w). We have the expression
% = %0 dt dw. Then, the condition ∂E ◦ ∂E = % idE on U is equivalent to
[
∂E,t, ∂E,w
]
= %0 idE .
Assume that there exists ν = νt dt + νwdw ∈ C∞(U,Ω0,1) such that ∂ν = % on U . Note
that such ν always exists locally. On U , we set ∂
ν
E = ∂E − ν idE . Then, (E|U , ∂
ν
E) is clearly
a mini-holomorphic bundle.
Suppose that U is isomorphic to {t0 < t < t1} × Bδ, where Bδ = {w ∈ C | |w| < δ}. Take
t0 < b1 < b2 < t1. We obtain the scattering map F : E|{t=b1}×Bδ ' E|{t=b2}×Bδ . Let ∂E,w,bi
denote the operators on E|{t=bi}×Bδ by ∂E,w.
Lemma 2.3. F ∗(∂E,w,b2) = ∂E,w,b1 +
(∫ b2
b1
%0 dt
)
id.
Proof. Take ν = νw dw such that ∂ν = %, i.e., ∂tνw = %0. Then, F ∗(∂νE,w,b2) = ∂νE,w,b1 because
of [∂νE,t, ∂
ν
E,w] = 0. Then, the claim of the lemma follows. �
2.2.2 Twisted mini-holomorphic bundles with Dirac type singularity
Let Z ⊂M be a discrete subset. Let (E, ∂E) be a %-twisted mini-holomorphic bundle on M \Z.
Let P be a point of Z. Let (U ; t, w) be a mini-complex coordinate neighbourhood around P ∈ Z.
We may assume (t(P ), w(P )) = (0, 0). By shrinking U , we assume that U ' {−2ε < t < 2ε}×Bδ
by the mini-complex coordinate system for some ε > 0 and δ > 0. Set B∗δ := Bδ \ {0}. We
obtain the scattering map F : E|{−ε}×B∗δ ' E|{ε}×B∗δ .
Definition 2.4. P is a Dirac type singularity of (E, ∂E) if F and F−1 are O(|w|−N ) for some
N > 0 with respect to C∞-frames of E|{±ε}×B∗δ . If each point of Z is Dirac type singularity of
(E, ∂E), we say that (E, ∂E) is a %-twisted mini-holomorphic bundle with Dirac type singularity
on (M ;Z).
Take ν = νt dt + νwdw ∈ C∞
(
U,Ω0,1
)
such that ∂ν = %. We set ∂
ν
E := ∂E|U − ν id so that(
E|U , ∂
ν
E
)
is mini-holomorphic. The scattering map F ν : E|{−ε}×B∗δ ' E|{ε}×B∗δ for ∂
ν
E is holo-
morphic with respect to ∂νE,w. Note that F ν = exp
( ∫ ε
−ε νt
)
F . The condition in Definition 2.4
is satisfied if and only if F ν extends to a meromorphic isomorphism
(
E|{−ε}×Bδ , ∂
ν
E,w,−ε
)
(∗0) '
(E|{ε}×Bδ , ∂
ν
E,w,ε)(∗0), i.e., P is Dirac type singularity of
(
E|U , ∂
ν
E
)
in the sense of [4, Section 2.2].
We regard U as an open subset of R × C by the coordinate system (t, w). Let ϕ : C2 −→
R × C be given by ϕ(z1, z2) =
(
|z1|2 − |z2|2, 2z1z2
)
. Let Ũ be the pull back of U by ϕ. The
mini-holomorphic bundle
(
E, ∂
ν
E
)
|U\{P} induces an S1-equivariant holomorphic vector bundle
8 T. Mochizuki(
Ẽ′P , ∂
ν
Ẽ′P
)
on Ũ \ {(0, 0)}, which uniquely extends to an S1-equivariant holomorphic vector
bundle
(
ẼνP , ∂
ν
ẼP
)
on Ũ . (See [6, Section 2.2] for a more detailed explanation.)
Lemma 2.5. Suppose that νi = νi,t dt + νi,wdw ∈ C∞
(
U,Ω0,1
)
(i = 1, 2) satisfy ∂νi = %.
Then, the natural identification Ẽν1
P |Ũ\{(0,0)}
= Ẽ′P = Ẽν2
P |Ũ\{(0,0)}
uniquely extends to a C∞-
isomorphism Ẽν1P ' Ẽ
ν2
P .
Proof. Set ν0 = ν0,t dt+ ν0,w dw := ν2 − ν1. We have ∂
ν2
E = ∂
ν1
E − ν0 idE . By the construction
(see [6, Section 2.2]), we have ∂
ν2
Ẽ′P
= ∂
ν1
Ẽ′P
−
(
ϕ∗(ν0,t)∂ϕ
∗(t) + ϕ∗(ν0,w)∂ϕ∗(w)
)
id. Then, the
claim of the lemma is clear. �
We set ẼP := ẼνP for ν ∈ C∞
(
U,Ω0,1
)
such that ∂ν = %, which is called the Kronheimer
resolution of
(
E, ∂E
)
at P .
Definition 2.6. A Hermitian metric h of E is called adapted at P if the induced metric h̃P
of Ẽ′P extends to a C∞-metric of the Kronheimer resolution ẼP . If h is adapted at any point
of Z, then h is called an adapted metric of
(
E, ∂E
)
.
2.2.3 Chern connections and Higgs fields
Suppose that we are given a splitting TM/TSM −→ TM . It induces the following decomposi-
tions:
T ∗M ⊗ C ' Ω1,0
Q M ⊕ Ω0,1
Q M ⊕ T ∗SM ⊗ C, (2.2)
Ω0,1M ' Ω0,1
Q M ⊕ T ∗SM ⊗ C, (2.3)
Ω1,0M ' Ω0,1
Q M ⊕ T ∗SM ⊗ C. (2.4)
Let (E, ∂E) be a %-twisted mini-holomorphic bundle on M . By (2.3), we obtain a decompo-
sition ∂E = ∂
S
E ⊕ ∂
Q
E , where ∂
S
E(s) ∈ C∞
(
X, (TSM ⊗ C)∨
)
and ∂
Q
E(s) ∈ C∞
(
X,Ω0,1
Q M
)
.
Let h be a Hermitian metric of E. We obtain the differential operator ∂E,h : C∞(X,E) −→
C∞
(
X,Ω1,0M ⊗E
)
satisfying the condition ∂Mh(u, v) = h(∂Eu, v) + h(u, ∂E,hv) for any u, v ∈
C∞(X,E). We also obtain the decomposition ∂E,h = ∂QE,h + ∂SE,h induced by (2.4). For a mini-
complex coordinate neighbourhood (U ; t, w), we obtain the operators ∂E,h,w (resp. ∂E,h,t) on E
induced by ∂E,h and ∂w (resp. ∂t).
Remark 2.7. In [4], ∂E,h,t is denoted as ∂′E,h,t.
By using (2.2), we set
∇h := ∂
Q
E + ∂QE,h +
1
2
(
∂
S
E + ∂SE,h
)
, φh :=
√
−1
2
(
∂
S
E − ∂SE,h
)
.
They are called the Chern connection and the Higgs field of (E, ∂E , h). Note that they depend
on the choice of a splitting TM/TSM −→ TM .
If M is also equipped with a Riemannian metric gM , we shall use the splitting TM/TSM −→
TM induced by gM . Moreover, by the section η in Section 2.1.1, T ∗S,gMM is identified with the
product bundle R ×M . Hence, we regard φh as an anti-Hermitian endomorphism of E. In
particular, if gM = dt dt + dw dw on a mini-complex coordinate neighbourhood (U ; t, w), the
following holds for any s ∈ C∞(U,E):
∇h(s) = (∂E,ws) dw + (∂E,h,ws) dw +
1
2
(
∂E,ts+ ∂E,h,ts
)
dt,
φh(s) =
√
−1
2
(
∂E,ts− ∂E,h,ts
)
.
Triply Periodic Monopoles and Difference Modules on Elliptic Curves 9
2.3 Twisted monopoles in the locally Euclidean case
2.3.1 Twisted monopoles
Let (M, gM ) be an oriented Riemannian 3-dimensional manifold. Let B be a real 2-form on M .
Let E be a vector bundle on M equipped with a Hermitian metric h, a unitary connection ∇,
and an anti-Hermitian endomorphism φ.
Definition 2.8. Such a tuple (E, h,∇, φ) is called a B-twisted monopole if the following B-
twisted Bogomolny equation is satisfied:
F (∇) = ∗∇φ+
√
−1B idE .
Here F (∇) denotes the curvature of ∇, and ∗ denotes the Hodge star operator.
Let A and f be a real 1-form and an R-valued C∞-function on M , respectively. We set
∇̃ := ∇ +
√
−1A id, φ̃ := φ +
√
−1f id and B̃ := B + dA − ∗(df). Then, the following is easy
to see.
Lemma 2.9. (E, h,∇, φ) is a B-twisted monopole if and only if
(
E, h, ∇̃, φ̃
)
is a B̃-twisted
monopole.
Remark 2.10. If M is compact, any real 2-form B on M is expressed as B = dA− ∗df + B0,
where A is a real 1-form, f is a R-valued C∞-function, and B0 is a harmonic 1-form. Indeed,
let G denote the Green operator for the Laplace-Beltrami operator on the space of 2-forms on M .
Then, B − (d∗d + dd∗)G(B) is a harmonic 2-form, and G(B) is C∞. We can also deduce that
for any point P ∈M , there exists a neighbourhood MP of P such that B|MP
= dAP − ∗dfP for
a real 1-form AP and an R-valued C∞-function fP on MP .
2.3.2 Twisted monopoles and twisted mini-holomorphic bundles
in the locally Euclidean case
Suppose that M is also equipped with a mini-complex structure. Moreover, we assume that M is
a locally Euclidean, i.e., for each P ∈M , there exists a mini-complex coordinate neighbourhood
(U ; t, w) of P such that the Riemannian metric of M on U is dt dt+dw dw. Note that dη = 0 for
the global trivialization η of T ∗S,gMM in Section 2.1.1. By (2.1), for any complex vector bundle
V on M , we obtain the decomposition
V ⊗
2∧(
T ∗M ⊗ C
)
=
(
V ⊗ Ω1,0
Q M ∧ η
)
⊕
(
V ⊗ Ω0,1
Q M ∧ η
)
⊕
(
V ⊗ Ω1,1
Q M
)
, (2.5)
where Ω1,1
Q M := Ω1,0
Q M ∧ Ω0,1
Q M . For any section s of V ⊗
∧2(T ∗M ⊗ C
)
, we obtain the
decomposition s = s(1,0),η + s(0,1),η + s(1,1) according to (2.5). In particular, we obtain the
decomposition B = B(1,0),η+B(0,1),η+B(1,1). Because B is real, B(1,1) is also real, and B(1,0),η =
B(0,1),η holds.
We can check the following lemma by a direct computation.
Lemma 2.11. Let (E, h,∇, φ) be a B-twisted monopole on M . We have the decomposition
∇ = ∇1,0
Q +∇0,1
Q +∇S induced by (2.1). We set ∂E := ∇0,1
Q +∇S −
√
−1φη. Then, (E, ∂E) is a√
−1B(0,1),η-twisted mini-holomorphic bundle.
Conversely, let (E, ∂E) be a %-twisted mini-holomorphic bundle on M . Let h be a Hermitian
metric of E. We obtain the Chern connection ∇h and the Higgs field φh.
Lemma 2.12. We have
(
F (∇h)−∗∇hφh
)(0,1),η
= % idE and
(
F (∇h)−∗∇hφh
)(1,0),η
= −% idE.
10 T. Mochizuki
Proof. We have ∂E = ∇0,1
h,Q + ∇h,S −
√
−1φhη and ∂E,h = ∇1,0
h,Q + ∇h,S +
√
−1φhη. Because
∂E ◦ ∂E = % id, we obtain ∂E,h ◦ ∂E,h = −% id. Then, we obtain the claim of the lemma by
computations. �
Corollary 2.13. There exists a real 2-form B such that (E, h,∇h, φh) is a B-twisted monopole
if and only if the trace-free part of
(
F (∇h)− ∗∇hφh
)(1,1)
is 0, i.e., there exists a real 2-form $
such that
(
F (∇h)− ∗∇hφh
)(1,1)
=
√
−1$ id. In that case, B = −
√
−1(%− %) +$.
Remark 2.14. If the condition in Corollary 2.13 is satisfied,
(
E, ∂E , h
)
is also called a B-twisted
monopole.
2.3.3 Dirac type singularity
Let Z be a discrete subset of M . Let B be a real 2-form on M . Let (E, h,∇, φ) be a B-twisted
monopole on M \ Z. Let (E, ∂E) be the underlying
√
−1B(0,1),η-twisted mini-holomorphic
bundle.
Definition 2.15. A point P ∈ Z is called Dirac type singularity of (E, h,∇, φ) if the following
conditions are satisfied:
• P is Dirac type singularity of
(
E, ∂E
)
.
• h is an adapted metric of
(
E, ∂E
)
in the sense of Definition 2.6.
We say that (E, h,∇, φ) is a B-twisted monopole with Dirac type singularity on (M ;Z) if
any point P ∈ Z is Dirac type singularity of (E, h,∇, φ).
Lemma 2.16. P is Dirac type singularity of (E, h,∇, φ) if and only if there exists a neighbour-
hood MP of P in M such that |φQ|h = O
(
d(P,Q)−1
)
for Q ∈MP \ {P}.
Proof. IfMP is sufficiently small, there exists a real 1-form AP and an R-valued C∞-function fP
such that B|MP
= dAP − ∗dfP . Then, (Ẽ, h̃) := (E, h)|MP \{P} with ∇̃ := ∇−
√
−1AP idE and
φ̃ := φ−
√
−1fP idE is a monopole on MP \ {P}. If P is Dirac type singularity of (E, h,∇, φ),
then P is Dirac type singularity of (Ẽ, h̃, ∇̃, φ̃). According to [6], it is equivalent to |φ̃Q|h =
O
(
d(P,Q)−1
)
around any point P ∈ Z, which is equivalent to |φQ|h = O
(
d(P,Q)−1
)
around
any point P ∈ Z. �
2.4 Twisted difference modules
Let Γ0 ⊂ C be a lattice. We put T := C/Γ0. Take any a ∈ T , and define the automorphism Φ
of T by Φ(z) = z + a. Let L be a holomorphic line bundle of degree 0 on T .
A parabolic L-twisted difference module V∗ = (V, (τP ,LP )P∈D) on T consists of the following
data:
• A locally free OT -module V equipped with an isomorphism V ⊗OT (∗D) ' (Φ∗)−1(V ) ⊗
L⊗OT (∗D), where D is a finite subset of T .
• A sequence 0 ≤ τP,1 < τP,2 < · · · < τP,m(P ) < 1 for each P ∈ D.
• Lattices LP,i (i = 1, . . . ,m(P ) − 1) of the stalk V (∗D)P at each P ∈ D. We formally set
LP,0 := VP and LP,m(P ) :=
(
(Φ∗)−1(V )⊗ L
)
P
at each P ∈ D.
Triply Periodic Monopoles and Difference Modules on Elliptic Curves 11
The degree of V∗ is defined by the formula (1.1), i.e.,
deg(V∗) := deg(V ) +
∑
P∈D
m(P )∑
i=1
(1− τP,i) deg(LP,i,LP,i−1).
We set µ(V∗) := deg(V∗)/ rank(V ).
For any OT (∗D)-submodule 0 6= V ′ ⊂ V such that V ′(∗D) ' (Φ∗)−1(V ′)(∗D), we obtain
lattices L′P,i of V ′(∗D)P by setting L′P,i := LP,i∩V ′(∗D)P in V (∗D)P , and we obtain a parabolic
L-twisted a-difference module V ′∗ = (V ′, (τP ,L′P )P∈D). Such V ′∗ is called a parabolic a-difference
submodule of V∗.
Definition 2.17. V∗ is called stable (resp. semistable) if
µ(V ′∗) < µ(V∗)
(
resp. µ(V ′∗) ≤ µ(V∗)
)
for any parabolic a-difference submodules V ′ such that 0 < rankV ′ < rankV . It is called
polystable if it is semistable and a direct sum of stable objects.
2.4.1 Example
It is easy to construct examples of parabolic difference modules.
Lemma 2.18. For any holomorphic line bundle L of degree 0, and for any d ∈ R, there exists
a parabolic L-difference module V∗ of rank one such that deg(V∗) = d.
Proof. There exist P1, . . . , Pn ∈ T and `i ∈ Z such that L
(∑n
i=1 `iPi
)
= OT . Note that∑
`i = 0. We take P0 ∈ T \ {P1, . . . , Pn}. We set D := {P0, P1, . . . , Pn}. We set V := OT . By
our choice of D, there exists an isomorphism F : V (∗D) ' (Φ∗)−1(V )⊗L(∗D). We set m(Pi) = 1
and τPi,1 = 0 for i = 1, . . . , n. We set m(P0) = 2, and we choose 0 ≤ τP0,1 < τP0,2 < 1. We set
LP0,1 = OT (`P0)P0 for an integer `. Then, we obtain a parabolic L-twisted difference module
V
(`,τP0,1,τP0,2)
∗ for which deg
(
V
(`,τP0,1,τP0,2)
∗
)
= (τP0,2 − τP0,1)`. Then, the claim is clear. �
3 Equivalences
We shall study equivalences of twisted mini-holomorphic bundles, twisted difference modules,
and twisted monopoles. First, in Section 3.1, we introduce analytically stability condition for
twisted mini-holomorphic bundles in terms of adapted metrics. We also prepare some formulas
for the curvature and the Higgs field of a twisted mini-holomorphic bundle with a Hermitian
metric which are standard in the context of mini-holomorphic bundles as in [4]. In Section 3.2, we
shall explain the equivalence between twisted mini-holomorphic bundles and twisted difference
modules, which preserves the stability conditions. In Section 3.3, we shall explain the equivalence
between polystable twisted mini-holomorphic bundles and twisted monopoles.
3.1 Analytic stability condition for twisted mini-holomorphic bundles
3.1.1 3-dimensional torus with mini-complex structure
We take an oriented base (ai, αi) (i = 1, 2, 3) of the R-vector space R×C. Let Y := R×C with
the Riemannian metric dt dt + dw dw. It is equipped with the mini-complex structure induced
by the mini-complex coordinate system (t, w). We consider the action of Γ := Ze1 ⊕ Ze2 ⊕ Ze3
on Y given by
ei(t, w) = (t, w) + (ai, αi), i = 1, 2, 3.
LetM denote the quotient space of Y by the action of Γ. It is equipped with a naturally induced
mini-complex structure.
12 T. Mochizuki
3.1.2 Contraction of the curvature
Let Z be a finite subset of M. Take % ∈ C∞(M,Ω0,2M). Let (E, ∂E) be a %-twisted mini-
holomorphic bundle on M\ Z. Let h be a Hermitian metric of E. As in [4], we set
G(h) :=
[
∇h,w,∇h,w
]
−
√
−1
2
∇h,tφh.
If we emphasize the dependence on ∂E , we use the notation G(h, ∂E). Note that
G(h) dw dw =
(
F (∇h)− ∗∇hφh
)(1,1)
(3.1)
for the notation in Section 2.3.2.
Let U be an open subset of M \ Z with ν = νt dt + νw dw ∈ C∞
(
U,Ω0,1
)
. On U , we set
∂
ν
E := ∂E − ν idE . Then,
(
E|U , ∂
ν
E
)
is a (%|U − ∂ν)-twisted mini-holomorphic bundle on U . We
obtain the Chern connection ∇νh and the Higgs field φνh.
Lemma 3.1. The following holds:
φνh = φh −
√
−1 Re(νt) idE , ∇νh = ∇h −
√
−1
(
2 Im(νwdw) + Im(νt dt)
)
idE ,
∇νh
(
φνh
)
= ∇h(φh)−
√
−1dRe(νt) idE ,
F
(
∇νh
)
= F (∇h)−
√
−1d
(
2 Im(νwdw) + Im(νt)dt
)
idE ,
G
(
h, ∂
ν
E
)
= G(h, ∂E)−
(
2 Re(∂wνw) + 2−1 Re(∂tνt)
)
idE .
We can check the formulas by direct computations.
Let E′ be any %-twisted mini-holomorphic subbundle of E, i.e., ∂EC
∞(M\Z,E′) ⊂ C∞
(
M\
Z,Ω0,1M⊗E′
)
. We have the natural %-twisted mini-holomorphic structure ∂E′ on E′. Let hE′
be the induced metric of E′. Let pE′ be the orthogonal projection of E onto E′ with respect
to h.
Lemma 3.2. The following Chern–Weil formula holds:
TrG(hE′) = Tr
(
G(hE) · pE′
)
−
∣∣∂E,wpE′∣∣2 − 1
4
∣∣∂E,tpE′∣∣2. (3.2)
Proof. If % = 0, it is proved in [4, Section 2.8.2]. Let us study the general case. It is enough
to prove the equality locally around any point of Q ∈ M \ Z. On a neighbourhood U of Q,
there exists ν ∈ C∞
(
U,Ω0,1
)
such that ∂
ν
E is a mini-holomorphic structure of E|U . Note that
∂νE,wpE′ = ∂E,wpE′ , ∂
ν
E,tpE′ = ∂E,tpE′ . Moreover,
(
E′, ∂
ν
E′
)
is a mini-holomorphic subbundle of(
E, ∂
ν
E
)
, and G
(
hE′ , ∂
ν
E′
)
= G
(
hE′ , ∂E′
)
−
(
2 Re(∂wνw) + 2−1 Re(∂tνt)
)
idE′ . Then, we obtain
the desired formula. �
3.1.3 Analytic stability condition for mini-holomorphic bundles
with a Hermitian metric
Let
(
E, ∂E
)
be a %-twisted mini-holomorphic bundle on M\ Z with a Hermitian metric h.
Definition 3.3. If TrG(h) is expressed as a sum of an L1-function and a non-positive function,
then we set deg
(
E, ∂E , h
)
:=
∫
M\Z TrG(h) dvolM ∈ R ∪ {−∞}. We also set µ
(
E, ∂E , h
)
:=
deg
(
E, ∂E , h
)
/ rank(E).
Suppose that |G(h)|h is L1. By (3.2), deg(E′, hE′) is defined in R ∪ {−∞} for any %-twisted
mini-holomorphic subbundle E′ of E.
Definition 3.4. Suppose that |G(h)|h is L1. Then,
(
E, ∂E , h
)
is called analytically stable
if µ
(
E′, ∂E′ , hE′
)
< µ
(
E, ∂E , h
)
for any %-twisted mini-holomorphic subbundle E′ ⊂ E with
0 < rank(E′) < rank(E).
Triply Periodic Monopoles and Difference Modules on Elliptic Curves 13
3.1.4 Adapted metrics of twisted mini-holomorphic bundles
with Dirac type singularity
Let
(
E, ∂E
)
be a %-twisted mini-holomorphic bundle with Dirac type singularity on (M;Z).
Lemma 3.5. If h is an adapted metric at P , then G(h)Q = O
(
d(P,Q)−1
)
around P , where
d(P,Q) denotes the distance of P and Q. In particular, if h is an adapted metric of
(
E, ∂E
)
,
then |G(h)|h is L1.
Proof. In the case % = 0, it is proved in [4, Lemma 2.35]. The general case follows from
Lemma 3.1. �
Lemma 3.6. Let
(
E, ∂E
)
be a %-twisted mini-holomorphic bundle with Dirac type singularity
on (M;Z). Let E′ 6= 0 be a %-twisted mini-holomorphic subbundle of E. Let h and h′ be adapted
Hermitian metrics of E and E′, respectively. Let hE′ be the metric of E′ induced by h. Then,
deg(E′, hE′) = deg(E′, h′).
Proof. It is enough to study the case rankE′ = 1. We may assume that there exist neighbour-
hoods UP of P ∈ Z such that hE′ = h′ on M \
⋃
P∈Z UP . Then, we have only to prove that∫
UP G(h′) =
∫
UP G(hE′) for any P ∈ Z. By Lemma 3.1, it is enough to study the case % = 0.
It is proved in the proof of [4, Proposition 9.4] (See the argument to compare
∫
G(h0,E1) and∫
G(h2,E1) in the proof of [4, Proposition 9.4].) �
Corollary 3.7. If h1 and h2 are adapted metrics of
(
E, ∂E
)
, deg
(
E, ∂E , h1
)
= deg
(
E, ∂E , h2
)
holds.
Lemma 3.8. Take a small neighbourhood UP of P ∈ Z. The following estimates hold for
Q ∈ UP \ {P}:
|φh,Q|h = O
(
d(P,Q)−1
)
, |(∇φh)Q|h,gM = O
(
d(P,Q)−2
)
,
|F (∇h)Q|h,gM = O
(
d(P,Q)−2
)
.
In particular,
∣∣∇hφh|h and
∣∣F (∇h)
∣∣
h
are L1.
Proof. Suppose that % = 0. The estimates |φh,Q|h = O
(
d(P,Q)−1
)
and |(∇φh)Q|h,gM =
O
(
d(P,Q)−2
)
directly follow from [6, Proposition 1]. Because of Lemma 2.12, Lemma 3.5
and (3.1), we obtain |F (∇h)Q|h,gM = O
(
d(P,Q)−2
)
. We can reduce the case % to the case % = 0
by using Lemma 3.1. �
3.1.5 Analytic stability condition for %-twisted mini-holomorphic bundles
with Dirac type singularity
Let
(
E, ∂E
)
be a %-twisted mini-holomorphic bundle with Dirac type singularity on (M;Z). We
set
degan
(
E, ∂E
)
:= deg
(
E, ∂E , h
)
, µan
(
E, ∂E
)
:= degan
(
E, ∂E
)
/ rank(E)
for an adapted Hermitian metric h of E, which is independent of the choice of h. The numbers
are called the analytic degree and the analytic slope of
(
E, ∂E , h
)
, respectively.
Definition 3.9. We say that
(
E, ∂E
)
is analytically stable if µan
(
E′, ∂E′
)
< µan
(
E, ∂E
)
holds
for any %-twisted mini-holomorphic subbundle E′ ⊂ E with 0 < rank(E′) < rank(E). It is called
polystable if
(
E, ∂E
)
=
⊕(
Ei, ∂Ei
)
, where each
(
Ei, ∂Ei
)
is stable such that µan
(
Ei, ∂Ei
)
=
µan
(
E, ∂E
)
.
14 T. Mochizuki
We obtain the following lemma from Lemma 3.6.
Lemma 3.10. A %-twisted mini-holomorphic bundle with Dirac type singularity
(
E, ∂E
)
on
(M;Z) is analytically stable if and only if
(
E, ∂E , h
)
is analytically stable for an adapted Her-
mitian metric h of E.
3.1.6 Complement on the choice of %
Let H i(M,OM) denote the i-th cohomology group of the complex
(
C∞
(
M,Ω0,iM
)
, ∂M
)
. For
any ν ∈ C∞
(
M,Ω0,1M
)
, %-twisted mini-holomorphic bundles are equivalent to (%−∂ν)-twisted
mini-holomorphic bundles. Hence, the essential ambiguity of the choice of % lives in H2(M,OM).
Lemma 3.11. We have the following isomorphisms:
H0(M,OM) ' C, H1(M,OM) ' C dt⊕ C dw, H2(M,OM) ' C dt ∧ dw.
Hence, for the study of twisted mini-holomorphic bundles, it is essential to study the case % =
αdt dw for some α ∈ C.
Proof. We have the isomorphism Rs×(Rt×Cw) ' Cz×Cw given by (s, t, w) 7−→
(
s+
√
−1t, w
)
.
We consider the action of Z × Γ on R × (R × C) induced by the natural action of Z on R and
the Γ-action on R× C. Let X denote the quotient space. We have the projection ϕ : X −→M
induced by (s, t, w) 7−→ (t, w). We have the natural S1 = R/Z-action on X, and the quotient
space is identified with M. Let ϕ∗ : C∞
(
M,Ω0,iM
)
−→ C∞
(
X,Ω0,i(X)
)
be the map induced
by ϕ∗(dw) = dw, ϕ∗(dt) = ∂z(t) dz =
√
−1
2 dz and the natural pull back ϕ∗ : C∞(M,C) −→
C∞(X,C). Then, it is easy to check that it is a morphism of complexes, and that it induces an
isomorphism between C∞
(
M,Ω0,•M
)
and the S1-invariant part of C∞
(
X,Ω0,•(X)
)
. Therefore,
it induces the isomorphism of H i(M,OM) and the S1-invariant part of H i(X,OX). Then, the
claim of the lemma follows. �
Remark 3.12. Let M =
⋃
λ∈Λ Uλ be an open covering such that the following holds:
• There exist νλ ∈ C∞
(
Uλ,Ω
0,1
Uλ
)
such that %|Uλ = ∂νλ.
• There exist αλ,µ ∈ C∞(Uλ ∩Uµ) such that νλ− νµ = ∂αλ,µ. We assume that αλ,λ = 0 and
αλ,µ = −αµ,λ.
Let Eλ be the OUλ-module obtained as the sheaf of mini-holomorphic sections of
(
EUλ , ∂E−νλ
)
.
We obtain the isomorphism βλ,µ : Eλ|Uλ∩Uµ ' Eµ|Uλ∩Uµ by the multiplication of exp(−αλ,µ). We
obtain the holomorphic functions θλ,µ,κ on Uλ,µ,κ such that βλ,µ ◦ βµ,κ ◦ βκ,λ = θλ,µ,κ id. Such
a tuple ({Eλ}, {βλ,µ}) is called a twisted sheaf. The cohomology class of [θλ,µ,κ] in H2(M,O∗M)
depends only on %, and it is equal to the image of % via the natural map H2(M,OM) −→
H2(M,O∗M).
3.2 Twisted difference modules and twisted mini-holomorphic bundles
We assume that (i) the tuple (ai, αi) (i = 1, 2, 3) is an oriented base of R×C, (ii) α1 and α2 are
linearly independent over R, (iii) the tuple (α1, α2) is an oriented base of C. Let Γ0 ⊂ C be the
lattice generated by α1 and α2.
Let Mcov denote the quotient space of Y by the action of Ze1 ⊕ Ze2. We have the natural
isomorphism Mcov/Ze3 ' M. The projection Y −→ C induces a morphism Mcov −→ T :=
C/Γ0.
Triply Periodic Monopoles and Difference Modules on Elliptic Curves 15
3.2.1 Another mini-complex coordinate system
We introduce another mini-complex coordinate system (s, u) on Y . We set
γ := − a1α2 − a2α1
α1α2 − α2α1
.
We introduce another mini-complex coordinate system (s, u) on the mini-complex manifold Y
as follows:
s := t+ 2 Re(γw) = t+ γw + γw, u := w.
Then, we obtain ei(s, u) = (s, u+αi) for i = 1, 2. We also obtain e3(s, u) = (s+ t, u+ a), where
t := a3 + 2 Re(γα3), a := α3.
Note that t > 0, which follows from that the tuple {(ai, αi)}i=1,2,3 is an oriented base of R×C,
and that {α1, α2} is an oriented base of C. We have the following relations of complex vector
fields:
∂w = ∂u + γ∂s, ∂w = ∂u + γ∂s, ∂t = ∂s.
The product Rs×T is equipped with the natural mini-complex structure. The mini-complex
coordinate system (s, u) induces an isomorphism of mini-complex manifolds Mcov ' Rs × T .
3.2.2 Twisted mini-holomorphic bundles and twisted difference modules
Let Z be a finite subset in M. Let Zcov ⊂ Mcov ' Rs × T denote the pull back of Z. For any
a < b, we set [a, b[:= {a ≤ s < b}. We take ε > 0 such that ([−ε, 0[×T ) ∩ Zcov = ∅. Let D be
the image of Zcov ∩ ([−ε, t[×T ) via the projection Rs×T −→ T . For each P ∈ D, we obtain the
sequence 0 ≤ sP,1 < sP,2 < · · · < sP,m(P ) < t by the condition:
{(sP,i, P ) | i = 1, . . . ,m(P )} = ([0, t[×{P}) ∩ Zcov.
We set τP,i := sP,i/t.
We have the expression % = %0 dt dw = %0 ds du. Let %cov
0 be the function on Mcov = Rs × T
obtained as the pull back of %0 by Mcov −→M. We define ν% = ν%,w dw ∈ C∞(Mcov,Ω0,1M)
by setting
ν%,w(s, u) =
∫ s
0
%cov
0 (σ, u) dσ.
We set ϑ% := ν%|{t}×T . Let L% be the holomorphic line bundle on T given by the product bundle
C× T with ∂T − ϑ%.
Let
(
E, ∂E
)
be a %-twisted mini-holomorphic bundle with Dirac type singularity on (M;Z).
Let us observe that (E, ∂E) induces a parabolic L%-twisted difference module Υ(E, ∂E) over
(T, (τP )P∈D).
Let %cov ∈ C∞
(
M,Ω0,1M
)
be the pull back of %. Let
(
Ecov, ∂Ecov
)
denote the %cov-twisted
mini-holomorphic bundle onMcov obtained as the pull back of
(
E, ∂E
)
. We set
(
Ẽcov, ∂
Ẽcov
)
:=(
Ecov, ∂Ecov − ν%
)
which is a mini-holomorphic bundle on Mcov.
Let V be the locally free OT -module obtained as Ẽcov
|{−ε}×T . It is independent of the choice
of ε as above, up to canonical isomorphisms.
Let Φ: T −→ T be the morphism induced by Φ(u) = u+a. We have the natural isomorphism
Φ∗
(
Ecov
|{t−ε}×T
)
' Ecov
|{−ε}×T .
16 T. Mochizuki
It induces the following isomorphism of holomorphic bundles on T :
Φ∗
((
Ẽcov, ∂
Ẽcov
)
|{t−ε}×T
)
'
(
Ẽcov, ∂
Ẽcov
)
|{−ε}×T ⊗ L%.
The scattering map induces an isomorphism(
Ẽcov, ∂
Ẽcov
)
|{−ε}×T (∗D) '
(
Ẽcov, ∂
Ẽcov
)
|{t−ε}×T (∗D).
Hence, V is equipped with an isomorphism V (∗D) '
(
(Φ∗)−1(V )⊗ L%
)
(∗D).
For each P ∈ D and for i = 1, . . . ,m(P ) − 1, we take sP,i < bP,i < sP,i+1. Let
(
Ẽcov
|{−ε}×T
)
P
denote the OT,P -module obtained as the stalk of the sheaf of holomorphic sections of Ẽcov
|{−ε}×T
at P . Similarly,
(
Ẽcov
|{bP,i}×T
)
P
denote the OT,P -module obtained as the stalk of the sheaf of
holomorphic sections of Ẽcov
|{bP,i}×T at P . The scattering map induces isomorphisms of OT (∗P )P -
modules:(
Ẽcov
|{−ε}×T
)
P
(∗P ) '
(
Ẽcov
|{bP,i}×T
)
P
(∗P ).
Hence,
(
Ecov
|{bP,i}×T
)
P
(i = 1, . . . ,m(P )−1) induce a sequence of lattices LP,i (i = 1, . . . ,m(P )−1)
of V (∗D)P . Thus, we obtain the following parabolic a-difference module on (T, (τP )P∈D):
Υ
(
E, ∂E
)
:=
(
V, (τP ,LP )P∈D
)
.
The following proposition is clear by the construction.
Proposition 3.13. Υ induces an equivalence between %-twisted mini-holomorphic bundles with
Dirac type singularity on (M;Z) and parabolic L%-twisted a-difference modules on (T, (τP )P∈D).
3.2.3 Comparison of stability conditions
Let
(
E, ∂E
)
be a %-mini-holomorphic bundle with Dirac type singularity on (M;Z).
Proposition 3.14. We have µan
(
E, ∂E
)
= tπµ
(
Υ
(
E, ∂E
))
+2
∫
MRe(γ%0). As a result,
(
E, ∂E
)
is analytically (poly)stable if and only if Υ
(
E, ∂E
)
is (poly)stable.
Proof. We consider the real vector field v := 2γ∂w + 2γ∂w −
(
2|γ|2− 1
2
)
∂t onM. Let h be any
Hermitian metric of E. Let ∂E,u denote the operator on E induced by ∂E and ∂u. Let ∂E,h,u
denote the operator on E induced by ∂E,h and ∂u.
Lemma 3.15. G(h) =
[
∂E,h,u, ∂E,u
]
−
√
−1∇h,vφh + 2 Re(γ%0) idE holds.
Proof. Because ∂E,t = ∇h,t −
√
−1φh and ∂E,h,t = ∇h,t +
√
−1φh, the following holds:
∂E,u = ∇h,w − γ
(
∇h,t −
√
−1φh
)
, ∂E,h,u = ∇h,w − γ
(
∇h,t +
√
−1φh
)
.
Hence, we obtain[
∂E,h,u, ∂E,u
]
=
[
∇h,w,∇h,w
]
− γ
[
∇h,w,∇h,t
]
+ γ
√
−1∇h,wφh
+ γ[∇h,w,∇h,t] + γ
√
−1∇h,wφ− 2
√
−1|γ|2∇h,tφh.
According to Lemma 2.12, we have
[
∇h,w,∇h,t
]
−
√
−1∇h,wφ = −%0 idE and
[
∇h,w,∇h,t
]
+√
−1∇h,wφ = %0 idE . Hence, we obtain[
∂E,h,u, ∂E,u
]
=
[
∇h,w,∇h,w
]
+ 2
√
−1γ∇wφ+ 2
√
−1γ∇wφ
− 2
√
−1|γ|2∇tφ− 2 Re(γ%0) idE .
Then, we obtain the claim of the lemma. �
Triply Periodic Monopoles and Difference Modules on Elliptic Curves 17
Let h be an adapted metric of
(
E, ∂E
)
. According to Lemmas 3.5 and 3.8, G(h) and ∇hφh
are L1. Hence, we obtain
degan(E) =
∫
M
TrG(h) =
∫
M
Tr
[
∂E,h,u, ∂E,u
]
−
∫
M
√
−1 Tr∇h,vφh + 2 rank(E)
∫
M
Re(γ%0).
Note that the volume form ofM is equal to
√
−1
2 dt dw dw =
√
−1
2 ds du du. By the Stokes theorem
and the estimate in Lemma 3.8, we obtain that
∫
MTr
(
∇h,vφh
)√−1
2 dt dw dw = 0. By the Fubini
theorem, we obtain that∫
M
Tr
[
∂E,h,u, ∂E,u
]
=
∫ t
0
ds
∫
{s}×T
Tr
[
∂E,h,u, ∂E,u
] √−1
2
du du
=
∫ t
0
ds
∫
{s}×T
πc1
(
Ecov
|{s}×T
)
= tπ deg Υ
(
E, ∂E
)
.
Thus, we obtain the claim of Proposition 3.14. �
3.3 Twisted monopoles and twisted mini-holomorphic bundles
3.3.1 Statements
Let B be a real 2-form on M. We set %B :=
√
−1B(0,1),η and µB := −1
2
∫
MB(1,1) dt. Let
(E, h,∇, φ) be a B-twisted monopole with Dirac type singularity onM\Z. We have the associ-
ated %B-twisted mini-holomorphic bundle
(
E, ∂E
)
. Note that G
(
h, ∂E
)
dw dw =
√
−1B(1,1) idE .
Hence, we obtain
µan
(
E, ∂E
)
=
1
rank(E)
∫
M
Tr
(
G
(
h, ∂E
))√−1
2
dt dw dw = −1
2
∫
M
B(1,1) dt = µB.
We shall prove the following theorem in Sections 3.3.2–3.3.4, which is a variant of the corre-
spondence in [1] on the basis of [7].
Theorem 3.16. The above construction induces an equivalence between B-twisted monopoles
with Dirac type singularity on M \ Z and analytically polystable %B-twisted mini-holomorphic
bundles with Dirac type singularity with slope µB on (M;Z).
More precisely, Theorem 3.16 consists of Propositions 3.18, 3.19, and 3.21 below.
Remark 3.17. According to Lemma 2.9 and Remark 2.10, it is essential to study the case
where
B = c
√
−1
2
dw ∧ dw + αdt ∧ dw + αdt ∧ dw
for (c, α) ∈ R× C. We have %B =
√
−1αdt ∧ dw and µB = −1
2 vol(M)c in this case.
3.3.2 Polystability
Let
(
E, ∂E , h
)
be a B-twisted monopole with Dirac type singularity on M\ Z.
Proposition 3.18.
(
E, ∂E
)
is analytically polystable with degan
(
E, ∂E
)
= rank(E)µB.
18 T. Mochizuki
Proof. Let E′ be a %B-twisted mini-holomorphic subbundle of E. Let hE′ be the metric of E′
induced by h. By the Chern–Weil formula (3.2) and Lemma 3.6, we have
degan
(
E′, ∂E′
)
=
∫
TrG(hE′) = rank(E′)µB −
∫ ∣∣∂E,wpE′∣∣2
− 1
4
∫ ∣∣∂E,tpE′∣∣2 ≤ rank(E′)µB.
If µan
(
E′, ∂E′
)
= µB, we obtain ∂E,wpE′ = ∂E,tpE′ = 0. We obtain that the orthogonal
complement E′⊥ is also a %B-twisted mini-holomorphic subbundle of E. Let hE′⊥ be the
metric of E′⊥ induced by h. Thus, we obtain a decomposition of monopoles
(
E, ∂E , h
)
=(
E′, ∂E′ , hE′
)
⊕
(
E′⊥, ∂E′⊥ , hE′⊥
)
. Hence, we obtain the polystability of
(
E, ∂E
)
by an easy
induction. �
3.3.3 Uniqueness
The uniqueness is also standard.
Proposition 3.19. Let
(
E, ∂E
)
be a %B-twisted mini-holomorphic bundle with Dirac type sin-
gularity on (M;Z). Let h1 and h2 be adapted Hermitian-metrics of E such that G(hi) dw dw =√
−1B(1,1) idE. Then, there exists a decomposition
(
E, ∂E
)
=
⊕(
Ej , ∂Ej
)
such that (i) it
is orthogonal with respect to both h1 and h2, (ii) there exist positive constants aj such that
h2|Ej = ajh1|Ej .
Proof. Let s be the automorphism of E determined by h2 = h1s.
Lemma 3.20. The following inequality holds on M\ Z:
−
(
∂w∂w +
1
4
∂2
t
)
Tr(s) = −
∣∣s−1/2∂E,h1,w(s)
∣∣2
h1
− 1
4
∣∣s−1/2∂E,h1,t(s)
∣∣2
h1
≤ 0.
Proof. In the case %B = 0, it follows from [4, Corollary 2.30]. (Note that ∂E,h1,t is denoted
by ∂′E,h1,t in [4, Corollary 2.30].) Let us study the general case. We have only to check the
inequality locally around any point P of M \ Z. We take a small neighbourhood U of P
and ν = νt dt + νw dw ∈ C∞(U,Ω0,1) such that ∂E − ν id is mini-holomorphic. We obtain
∂νE,h,w = ∂E,h,w + νw id and ∂νE,h,t = ∂E,h,t + νt id. Hence, we obtain ∂νE,h,w(s) = [∂νE,h,w, s] =
[∂E,h,w, s] = ∂E,h,w(s). Similarly, we obtain ∂νE,h,t(s) = ∂E,h,t(s). Hence, the general case can be
reduced to the case %B = 0. �
By the assumption, Tr(s) ≥ 0 is bounded. Then, the inequality holds on M in the sense of
distributions. (See the proof of [7, Proposition 2.2].) Hence, we obtain that Tr(s) is constant,
and ∂E,h1,w(s) = ∂E,h1,t(s) = 0. Because s is self-adjoint with respect to h1, we also obtain
that ∂E,w(s) = ∂E,t(s) = 0. We obtain that the eigenvalues of s are constant, and the eigen
decomposition E =
⊕
Ei is compatible with the mini-holomorphic structure. Then, the claim
of the proposition follows. �
3.3.4 Construction of twisted monopoles
Let
(
E, ∂E
)
be a stable %B-twisted mini-holomorphic bundle with Dirac type singularity on
(M;Z) with µan
(
E, ∂E
)
= µB.
Proposition 3.21. There exists a Hermitian metric h of
(
E, ∂E
)
such that
(
E, ∂E , h
)
is a B-
twisted monopole with Dirac type singularity on M\ Z.
Triply Periodic Monopoles and Difference Modules on Elliptic Curves 19
Proof. As a preliminary, let us consider the rank one case. Note that the stability condition is
trivial in the rank one case.
Lemma 3.22. Assume rankE = 1. Then, there exists a Hermitian metric h of
(
E, ∂E
)
such
that
(
E, ∂E , h
)
is a B-twisted monopole with Dirac type singularity on M\ Z.
Proof. We take a Hermitian metric h0 of E such that the following holds:
• Each P ∈ Z has a neighbourhood UP in M such that (i) G(h0) = 0 on UP \ {P}, (ii) P is
Dirac type singularity of the monopole
(
E, ∂E , h0
)
|UP \{P}
.
Let f be any C∞-function on M. Note that G
(
h0e
f
)
− G(h0) = 4−1∆f , where ∆ denote the
Laplacian of M. (See [4, Section 2.8.4] for the untwisted case. The twisted case can be argued
similarly.) Because∫
M
G(h0)
√
−1
2
dt dw dw = µB = −1
2
∫
M
B(1,1) dt,
there exists an R-valued C∞-function f1 such that (∆f1)dw dw = −4
(
G(h0) dw dw−
√
−1B(1,1)
)
.
Then, the claim of Lemma 3.22 follows. �
Let us study the case where %B = 0, which implies B = B(1,1). On R4 = R × R3, we
use the real coordinate system (s, t, x, y) and the complex coordinate system (z, w) given by
z = s+
√
−1t and w = x+
√
−1y.
Let Γ̃ denote the lattice of R4 = R× (R×C) generated by (1, 0, 0) and (0, ai, αi) (i = 1, 2, 3).
We consider the action of Γ̃ on R4 induced by the natural Z-action on R and the Γ-action on
R×C. Let (X, gX) denote the Kähler manifold obtained as the quotient of (C2, dz dz + dw dw)
by the Γ̃-action. Let p : X −→M denote the naturally defined projection.
We set Ẽ := p−1(E) on X \p−1(Z). It is equipped with the complex structure ∂
Ẽ
determined
by
∂
Ẽ,w
p−1(u) = p−1(∂E,wu), ∂
Ẽ,z
p−1(u) =
1
2
· p−1
(
φ · u+
√
−1∂E,tu
)
for sections u of E. For any adapted Hermitian metric h0 of E, set h̃0 := p−1(h0).
Let F
(
h̃0
)
denote the curvature of the Chern connection of
(
Ẽ, ∂
Ẽ
, h̃0
)
. Let Λ denote the
contraction from (1, 1)-forms to (0, 0)-forms with respect to the Kähler form of (X, gX). Then,√
−1ΛF
(
h̃0
)
= p−1
(
G(h0)
)
holds.
For any saturated coherent OX\p−1(Z)-submodule Ẽ′ ⊂ Ẽ, we have a closed complex analytic
subset W ⊂ X \ p−1(Z) with complex codimension 2 such that Ẽ′ is a subbundle of Ẽ outside
of W . We have the induced metric h̃
0,Ẽ′ of Ẽ′|X\(p−1(Z)∪W ). We define
deg
(
Ẽ′, h̃0
)
:=
√
−1
∫
Tr ΛF
(
h̃
0,Ẽ′
)
dvolX .
Because of the Chern–Weil formula, it is well defined in R ∪ {−∞} as explained in [7]. Then,(
Ẽ, ∂
Ẽ
, h̃0
)
is defined to be analytically stable with respect to the S1-action if
deg
(
Ẽ′, h̃0
)
rank Ẽ′
<
deg
(
Ẽ, h̃0
)
rank Ẽ
holds for any S1-invariant saturated subsheaf Ẽ′ ⊂ Ẽ with 0 < rank Ẽ′ < rank Ẽ. The following
is clear.
20 T. Mochizuki
Lemma 3.23.
(
Ẽ, ∂
Ẽ
, h̃0
)
is analytically stable with respect to the S1-action if and only if(
E, ∂E , h0
)
is analytically stable.
According to Lemma 3.22, there exists a Hermitian metric hdet(E) such that
(
E, ∂E , hdet(E)
)
is a (rankE)B-twisted monopole. We take an adapted Hermitian metric h−1 such that each
P ∈ Z has a neighbourhood UP such that G(h−1)|UP \{P} = 0. An R-valued C∞-function f is
determined by det(h−1) = hdet(E)e
f . We set h0 = h−1e
−f/ rank(E). Then, h0 is an adapted metric
of E. By Lemma 3.23,
(
Ẽ, ∂
Ẽ
, h̃0
)
is analytically stable with respect to the S1-action. We also
have Λ TrF
(
h̃0
)
=
√
−1 rank(E)p−1(B). According to a theorem of Simpson [7, Theorem 1],
there exists an S1-invariant metric h̃ of Ẽ such that (i) det
(
h̃
)
= det
(
h̃0
)
, (ii) ΛF
(
h̃
)
=√
−1p−1(B) id
Ẽ
, (iii) h̃ and h̃0 are mutually bounded. We obtain the corresponding metric h
of E, for which G(h) =
√
−1B idE holds. Because h and h0 are mutually bounded, each P ∈ Z
is a Dirac type singularity of
(
E, ∂E , h
)
which is implied by [6, Theorem 3]. Thus, we obtain
the claim of Proposition 3.21 in the case %B = 0.
Let us study the case where %B is not necessarily 0.
Lemma 3.24. There exist a finite subset Z1 ⊂ M and a %B-twisted mini-holomorphic bundle(
E1, ∂E1
)
with Dirac type singularity of rank one on (M;Z1) such that degan
(
E1, ∂E1
)
= µB.
Proof. It follows from Lemma 2.18 and Proposition 3.13. �
We set
(
E′, ∂E′
)
:=
(
E, ∂E
)
⊗
(
E1, ∂E1
)−1
. Then,
(
E′, ∂E′
)
is a stable mini-holomorphic
bundle with µan
(
E′, ∂E′
)
= 0. According to the claim in the case %B = 0, there exists an
adapted Hermitian metric h′ of
(
E′, ∂E′
)
such that
(
E′, ∂E′ , h
′) is a monopole. According to
Lemma 3.22, there exists a Hermitian metric h1 of E1 such that
(
E1, ∂E1 , h1
)
is a B-twisted
monopole with Dirac type singularity. Let h be the Hermitian metric of E induced by h′ and h1.
Then, h is adapted to
(
E, ∂E
)
, and
(
E, ∂E , h
)
is a B-twisted monopole. Thus the proof of
Proposition 3.21 is completed. �
4 A more sophisticated formulation of the stability condition
We explain that the analytic stability condition (Definition 3.9) is equivalent to the stability
condition introduced by Kontsevich and Soibelman in the case % = 0 (see Section 1.2). This
section is devoted to explain their idea of degree.
4.1 Preliminary
4.1.1 Closed 1-forms and 1-homology
Let A be a 3-dimensional manifold. Let ZiDR(A) denote the space of closed i-form τ on A. Let B
be finite subset of A. Let Hj(A,B) denote the relative j-th homology group with R-coefficient.
Let γ be any element of H1(A,B). We take a representative of γ by a smooth 1-chain γ̃. For
any ω ∈ Z1
DR(A), the number
∫
γ̃ ω is independent of the choice of a representative γ̃. They are
denoted by
∫
γ ω.
Let C∞(A,B) denote the space of C∞-functions f on A such that f(P ) = 0 for any P ∈ B.
Let Z1
DR(A) denote the space of closed 1-forms on A. Let B1
DR(A,B) denote the image of
d : C∞(A,B) −→ Z1
DR(A). Because
∫
γ df = 0 for any f ∈ C∞(A,B), we obtain the well defined
map ∫
γ
: Z1
DR(A)
/
B1
DR(A,B) −→ R.
Triply Periodic Monopoles and Difference Modules on Elliptic Curves 21
4.1.2 Duality
Suppose that A is compact and oriented. Let Hj(A \B) denote the j-th de Rham cohomology
group of A \ B. Let Hj
c (A \ B) denote the j-th de Rham cohomology group with compact
support. We have the non-degenerate pairing between H2(A\B) and H1
c (A\B) induced by the
cup product and the integration. We also have the non-degenerate pairing between H1
c (A \ B)
and H1(A,B) induced by the integration. Hence, we obtain the isomorphism
ΦA,B : H2(A \B) ' H1(A,B).
By definition, for any a ∈ H2(A \B) and b ∈ H1
c (A \B), the following holds:∫
ΦA,B(a)
b =
∫
A
a ∧ b.
Take any Riemannian metric gA of A. For any j-form τ on A\B, let |τ |gA denote the function
on A \B obtained as the norm of τ with respect to gA.
Lemma 4.1. Let τ ∈ Z2
DR(A \B) such that |τ |gA is an L1-function on A. Then, the following
holds for any ρ ∈ Z1
DR(A):∫
ΦA,B([τ ])
ρ =
∫
A
ρ ∧ τ.
Here, [τ ] ∈ H2
DR(A \B) denotes the cohomology class of τ .
Proof. For any point P ∈ Z, we take a small coordinate neighbourhood (AP , xP,1, xP,2, xP,3)
of P such that (i) P corresponds to (0, 0, 0), (ii) AP '
{
(x1, x2, x3) ∈ R3 |
∑
x2
i < 1
}
by the
coordinate system. Set ‖xP ‖ :=
(
x2
P,1 + x2
P,2 + x2
P,3
)1/2
. Then, there exists a C∞-function fP
on AP such that (i) dfP = ρ on {‖xP ‖ < 1/2}, (ii) fP (P ) = 0, (iii) fP (Q) = 0 for Q ∈ {‖xP ‖ >
2/3}. We naturally regard fP as a C∞-function on A. Then, the following holds:∫
ΦA,B([τ ])
ρ =
∫
ΦA,B([τ ])
(
ρ−
∑
P∈B
dfP
)
=
∫
A
(
ρ−
∑
P∈B
dfP
)
∧ τ =
∫
A
ρ ∧ τ −
∑
P
∫
A
d(fP τ).
For each P , we set S2
P (r) :=
{
‖xP ‖ = r
}
with the orientation as the boundary of
{
‖xP ‖ ≤ r
}
.
Then, we obtain the following∫
A
d(fP τ) = − lim
ε→0
∫
S2
P (ε)
fP τ. (4.1)
Note that the limit exists because d(fP τ) = dfP ∧ τ is integrable. Because |τ |gA is L1, we have∫
dr
∫
S2
P (r) |τ |gA < ∞, and hence there exists a sequence ri → 0 such that ri
∫
S2
P (ri)
|τ |gA → 0.
Because |fP | = O(‖xP ‖), we obtain that (4.1) is 0. �
4.2 Relation between degrees of mini-holomorphic bundles
Let M be as in Section 3. We may naturally regard M as a 3-dimensional abelian Lie group.
Let T denote the space of the invariant vector fields on M. Let T∨ denote the space of the
invariant 1-forms on M. We have the natural non-degenerate paring T ⊗ T∨ −→ R. We have
the dual morphism R −→ T∨⊗T. Let σ denote the image of 1. If we take a base ei (i = 1, 2, 3)
22 T. Mochizuki
of T and the dual frame e∨i (i = 1, 2, 3), then σ =
∑
e∨i ⊗ ei. For the mini-complex coordinate
(t, w), we have σ = dt⊗ ∂t + dw ⊗ ∂w + dw ⊗ ∂w.
Let E be a vector bundle on M\ Z. Kontsevich and Soibelman [2] introduced the following
element:∫
ΦM,Z(c1(E))
σ ∈ T.
Proposition 4.2. Let % = %0 dt dw be a 2-form on M. Let
(
E, ∂E
)
be a %-twisted mini-
holomorphic bundle with Dirac type singularity on (M;Z). Then,∫
ΦM,Z(c1(E))
σ =
1
π
degan(E) · ∂t −
rank(E)
π
((∫
M
%0
)
∂w +
(∫
M
%0
)
∂w
)
.
In particular, if % = 0, then the following holds:∫
ΦM,Z(c1(E))
σ =
1
π
degan(E) · ∂t.
Proof. Let h be an adapted metric of
(
E, ∂E
)
. By Lemma 4.1, it is enough to prove the
following equality:
√
−1
2
∫
M
TrF (h) · σ =
∫
M
TrG(h) dvolM ·∂t
− rank(E)
((∫
M
%0
)
∂w +
(∫
M
%0
)
∂w
)
. (4.2)
For κ = t, w,w, we obtain the following by the Stokes formula and the estimate |φh,Q|h =
O
(
d(P,Q)−1
)
:∫
Tr
(
∇h,κφh
)√−1
2
dt dw dw = 0. (4.3)
Note that F (h)tw +
√
−1∇wφ = %0 idE and F (h)tw −
√
−1∇wφ = −%0 idE , according to
Lemma 2.12. We obtain
√
−1
2
∫
TrF (h) dw ⊗ ∂w =
∫
Tr
(
F (h)tw +
√
−1∇wφ
)√−1
2
dt dw dw ⊗ ∂w
= − rank(E)
(∫
M
%0
)
∂w.
Similarly, we obtain
√
−1
2
∫
TrF (h) dw ⊗ ∂w = − rank(E)
(∫
M
%0
)
∂w.
We also obtain the following from (4.3):
√
−1
2
∫
TrF (h)ww dw dw dt⊗ ∂t =
∫
Tr
(
F (h)ww −
√
−1
2
∇h,tφh
) √
−1
2
dw dw dt ⊗ ∂t
=
(∫
M
TrG(h)
)
∂t.
Thus, we obtain (4.2), and the proof of Proposition 4.2 is completed. �
Remark 4.3. As explained in Section 1.2, Kontsevich and Soibelman [2] formulated the stability
condition for mini-holomorphic bundles in terms of the coefficient of ∂t in
∫
ΦZ(c1(E)) σ.
Triply Periodic Monopoles and Difference Modules on Elliptic Curves 23
Acknowledgements
I thanks Maxim Kontsevich and Yan Soibelman for the communications and for sending the
preprint [2]. Indeed, this study grew out of my answer to one of their questions. They also kindly
suggested that there should be a generalization to the twisted case. I hope that this would be
useful for their big project. I owe much to Carlos Simpson whose ideas on the Kobayashi–Hitchin
correspondence are fundamental in this study. I thank Masaki Yoshino for discussions. I thank
the referees for their careful readings and valuable comments.
I am grateful to the organizers of the conference “Integrability, Geometry and Moduli” to ce-
lebrate 60th birthday of Motohico Mulase. The twisted version of the equivalences was explained
in my talk at the conference.
It is my great pleasure to dedicate this paper to Motohico Mulase with appreciation to his
friendly encouragements and suggestions on many occasions.
I am partially supported by the Grant-in-Aid for Scientific Research (S) (No. 17H06127),
the Grant-in-Aid for Scientific Research (S) (No. 16H06335), and the Grant-in-Aid for Scientific
Research (C) (No. 15K04843), Grant-in-Aid for Scientific Research (C) (No. 20K03609), Japan
Society for the Promotion of Science.
References
[1] Charbonneau B., Hurtubise J., Singular Hermitian–Einstein monopoles on the product of a circle and
a Riemann surface, Int. Math. Res. Not. 2011 (2011), 175–216, arXiv:0812.0221.
[2] Kontsevich M., Soibelman Y., Riemann–Hilbert correspondence in dimension one, Fukaya categories and
periodic monopoles, Preprint.
[3] Kronheimer P.B., Monopoles and Taub-NUT metrics, Master Thesis, Oxford, 1986.
[4] Mochizuki T., Periodic monopoles and difference modules, arXiv:1712.08981.
[5] Mochizuki T., Doubly-periodic monopoles and q-difference modules, arXiv:1902.08298.
[6] Mochizuki T., Yoshino M., Some characterizations of Dirac type singularity of monopoles, Comm. Math.
Phys. 356 (2017), 613–625, arXiv:1702.06268.
[7] Simpson C.T., Constructing variations of Hodge structure using Yang–Mills theory and applications to
uniformization, J. Amer. Math. Soc. 1 (1988), 867–918.
[8] Yoshino M., A Kobayashi–Hitchin correspondence between Dirac-type singular mini-holomorphic bundles
and HE-monopoles, arXiv:1902.09995.
https://doi.org/10.1093/imrn/rnq059
https://arxiv.org/abs/0812.0221
https://arxiv.org/abs/1712.08981
https://arxiv.org/abs/1902.08298
https://doi.org/10.1007/s00220-017-2981-z
https://doi.org/10.1007/s00220-017-2981-z
https://arxiv.org/abs/1702.06268
https://doi.org/10.2307/1990994
https://arxiv.org/abs/1902.09995
1 Introduction
1.1 Triply periodic monopoles with Dirac type singularity
1.2 Mini-holomorphic bundles with Dirac type singularity
1.2.1 Mini-complex structure
1.2.2 Mini-holomorphic bundles with Dirac type singularity
1.2.3 Stability condition
1.2.4 Kobayashi–Hitchin correspondence
1.3 Parabolic difference modules on elliptic curves
1.3.1 Parabolic difference modules on elliptic curves and a stability condition
1.3.2 Equivalence
2 Preliminary
2.1 Mini-complex structure on 3-dimensional manifolds
2.1.1 Riemannian case
2.2 Twisted mini-holomorphic bundles
2.2.1 Scattering map
2.2.2 Twisted mini-holomorphic bundles with Dirac type singularity
2.2.3 Chern connections and Higgs fields
2.3 Twisted monopoles in the locally Euclidean case
2.3.1 Twisted monopoles
2.3.2 Twisted monopoles and twisted mini-holomorphic bundles in the locally Euclidean case
2.3.3 Dirac type singularity
2.4 Twisted difference modules
2.4.1 Example
3 Equivalences
3.1 Analytic stability condition for twisted mini-holomorphic bundles
3.1.1 3-dimensional torus with mini-complex structure
3.1.2 Contraction of the curvature
3.1.3 Analytic stability condition for mini-holomorphic bundles with a Hermitian metric
3.1.4 Adapted metrics of twisted mini-holomorphic bundles with Dirac type singularity
3.1.5 Analytic stability condition for -twisted mini-holomorphic bundles with Dirac type singularity
3.1.6 Complement on the choice of
3.2 Twisted difference modules and twisted mini-holomorphic bundles
3.2.1 Another mini-complex coordinate system
3.2.2 Twisted mini-holomorphic bundles and twisted difference modules
3.2.3 Comparison of stability conditions
3.3 Twisted monopoles and twisted mini-holomorphic bundles
3.3.1 Statements
3.3.2 Polystability
3.3.3 Uniqueness
3.3.4 Construction of twisted monopoles
4 A more sophisticated formulation of the stability condition
4.1 Preliminary
4.1.1 Closed 1-forms and 1-homology
4.1.2 Duality
4.2 Relation between degrees of mini-holomorphic bundles
References
|
| id | nasplib_isofts_kiev_ua-123456789-210702 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-17T12:04:31Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Mochizuki, Takuro 2025-12-15T15:24:52Z 2020 Triply Periodic Monopoles and Difference Modules on Elliptic Curves. Takuro Mochizuki. SIGMA 16 (2020), 048, 23 pages 1815-0659 2020 Mathematics Subject Classification: 53C07; 58E15; 14D21; 81T13 arXiv:1903.03264 https://nasplib.isofts.kiev.ua/handle/123456789/210702 https://doi.org/10.3842/SIGMA.2020.048 We explain the correspondences between twisted monopoles with Dirac-type singularity and polystable twisted mini-holomorphic bundles with Dirac-type singularity on a 3-dimensional torus. We also explain that they are equivalent to polystable parabolic twisted difference modules on elliptic curves. I thank Maxim Kontsevich and Yan Soibelman for the communications and for sending the preprint [2]. Indeed, this study grew out of my answer to one of their questions. They also kindly suggested that there should be a generalization to the twisted case. I hope that this will be useful for their big project. I owe much to Carlos Simpson, whose ideas on the Kobayashi-Hitchin correspondence are fundamental in this study. I thank Masaki Yoshino for the discussions. I thank the referees for their careful readings and valuable comments. I am grateful to the organizers of the conference Integrability, Geometry and Moduli to celebrate the 60th birthday of Motohico Mulase. The twisted version of the equivalences was explained in my talk at the conference. It is my great pleasure to dedicate this paper to Motohico Mulase with appreciation for his friendly encouragement and suggestions on many occasions. I am partially supported by the Grant-in-Aid for Scientific Research (S) (No. 17H06127), the Grant-in-Aid for Scientific Research (S) (No. 16H06335), the Grant-in-Aid for Scientific Research (C) (No. 15K04843), the Grant-in-Aid for Scientific Research (C) (No. 20K03609), Japan Society for the Promotion of Science. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Triply Periodic Monopoles and Difference Modules on Elliptic Curves Article published earlier |
| spellingShingle | Triply Periodic Monopoles and Difference Modules on Elliptic Curves Mochizuki, Takuro |
| title | Triply Periodic Monopoles and Difference Modules on Elliptic Curves |
| title_full | Triply Periodic Monopoles and Difference Modules on Elliptic Curves |
| title_fullStr | Triply Periodic Monopoles and Difference Modules on Elliptic Curves |
| title_full_unstemmed | Triply Periodic Monopoles and Difference Modules on Elliptic Curves |
| title_short | Triply Periodic Monopoles and Difference Modules on Elliptic Curves |
| title_sort | triply periodic monopoles and difference modules on elliptic curves |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/210702 |
| work_keys_str_mv | AT mochizukitakuro triplyperiodicmonopolesanddifferencemodulesonellipticcurves |