Triply periodic monopoles and difference modules on elliptic curves

We explain the correspondences between monopoles with Dirac type singularity and polystable mini-holomorphic bundles with Dirac type singularity of degree $0$ on a $3$-dimensional torus. We also explain that they are equivalent to polystable parabolic difference modules of degree $0$ on elliptic curves.


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.

Triply periodic monopoles with Dirac type singularity
Let Y be an oriented 3-dimensional R-vector space with an Euclidean metric g Y . Let Γ be a lattice of Y . We set M := Y /Γ, which is equipped with the induced metric g M . 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. The 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 g M . A point of P ∈ Z is called a Dirac type singularity of the monopole (E, h, ∇, φ) if |φ Q | h = O(d(Q, Z) −1 ) for any Q ∈ M \ Z, where 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 one, according to [6].

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. It was formulated by Kontsevich and Soibelman [2].

Mini-complex structure
We take a mini-complex structure on M \ Z. Namely, we take a linear coordinate system (x 1 , x 2 , x 3 ) on Y compatible with the orientation such that g Y = dx i dx i , and we set t := x 1 and w = x 2 + √ −1x 3 . They induce the complex vector fields ∂ t and ∂ w on M. A C ∞ -function f on an open subset of M is called mini-holomorphic if ∂ t f = ∂ w f = 0. Let O M\Z denote the sheaf of mini-holomorphic functions on M \ Z.

Mini-holomorphic bundles with Dirac type singularity
Let V be a locally free O M\Z -module. Let P be a point of Z. We take a lift (t 0 , w 0 ) ∈ Y . Set B * w0 (δ) := w ∈ C 0 < |w − w 0 | < δ for any δ > 0. For any t ∈]t 0 − δ, t 0 + δ[, the restriction V |{t}×B * w 0 (δ2) is naturally a locally free O B * w 0 (δ2) -module. Because mini-holomorphic functions are constant in the t-direction, we obtain an isomorphism of O B * w 0 (δ2) -modules V |{t0−δ1}×B * w 0 (δ2) ≃ V |{t0+δ1}×B * w 0 (δ2) . If it is meromorphic at w 0 , 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).

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 H j (M \ Z) denote the cohomology group of M \ Z with R-coefficient. Let H j (M, Z) denote the relative j-th homology group of (M, Z) with R-coefficient. Note that there exists the natural isomorphism Let T denote the space of left invariant vector fields on M, 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 For any mini-holomorphic bundle with Dirac type singularity V on (M; Z), we obtain c 1 (V) ∈ H 2 (M \ Z), and hence Φ Z (c 1 (V)) ∈ H 1 (M, Z). Then, we obtain the following: Kontsevich and Soibelman found that ΦZ (c1(V)) σ is a scalar multiplication of ∂ t = ∂ x1 , and they define the degree deg KS (V) for V as follows: They introduced the following stability condition.
. It is called polystable if it is semistable and a direct sum of stable submodules.

Kobayashi-Hitchin correspondence
It is a standard fact that a monopole with Dirac type singularity on M \ Z induces a mini-holomorphic bundle with Dirac type singularity on (M; Z). The following theorem was formulated by Kontsevich and Soibelman [2].
Theorem 1.2 (Theorem 2.7, Proposition 3.2) 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-Soibelman with the analytic degree defined in terms of Hermitian metrics (Proposition 3.2). Then, Theorem 1.2 follows from the fundamental theorem due to Simpson [7] as we shall explain in the proof of Theorem 2.7, 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 S 1 and a compact Riemann surface.

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.

Parabolic difference modules on elliptic curves and a stability condition
Let Γ 0 be a lattice of C. We set T := C/Γ 0 . Let a ∈ C. Let Φ : T −→ T be the morphism induced by Φ(z) = z + a. Let D ⊂ T be a finite subset. A parabolic a-difference module on T consists of the following data V * = V, (τ P , L P ) P ∈D : • A sequence 0 ≤ τ P,1 < τ P,2 < · · · < τ P,m(P ) < 1 for each P ∈ D.
When we fix (τ P ) P ∈D , it is called a parabolic a-difference module on (T, (τ P ) P ∈D ).
The degree of a parabolic a-difference module (V, (τ P , L P ) P ∈D ) is defined as follows: , we obtain lattices L ′ P,i of V ′ ( * D) P by setting L ′ P,i := L P,i ∩ V ′ ( * D) P in V ( * D) P , and we obtain a parabolic a-difference module (V ′ , (τ P , L ′ P ) P ∈D ). Such (V ′ , (τ P , L ′ P ) P ∈D ) is called a parabolic a-difference submodule of (V, (τ P , L P ) P ∈D ).
for any parabolic a-difference submodules such that 0 < rank V ′ < rank V . It is called polystable if it is semistable and a direct sum of stable objects.

Equivalence
We return to the situation in §1.2. We take a generator e i = (a i , α i ) (i = 1, 2, 3) of Γ ⊂ R t × C w = Y , which is compatible with the orientation of Y . We also assume that α 1 and α 2 generate a lattice in C. Let Γ 0 denote the lattice, and we set T := C/Γ 0 . We set It is easy to see that t > 0. We define the isomorphism F : Note that the induced action of Γ on R s × C u is expressed as follows: Let Z Y be the pull back of Z by Y −→ M. Let D denote the image of the composite of the following maps: For any P ∈ D, we take u 0 ∈ C which is mapped to P . We obtain a sequence 0 ≤ s P,1 < s P,2 < · · · < s P,m(P ) < t by the condition: It is independent of the choice of u 0 . We set τ P,i := s P,i /t. Proposition 1.4 (Proposition 4.1, Proposition 4.2) There exists an equivalence between parabolic difference modules on (T, (τ P ) 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 §4.2 for the explicit correspondence. As a consequence of Theorem 1.2 and Proposition 1.4, we obtain the following theorem. Theorem 1. 5 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).
Here, Z and (τ P ) P ∈D are related as above.
Acknowledgement The author thanks Maxim Kontsevich and Yan Soibelman for the communication and for sending the preprint [2]. Indeed, this study grew out of my answer to one of their questions. I hope that this would be useful for their 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. 2 Monopoles and analytically stable mini-holomorphic bundles 2.1 Mini-holomorphic bundles with Dirac type singularity We also consider the mini-complex structure on Y induced by the coordinate system (t, w). We consider the action of Ze 1 ⊕ Ze 2 ⊕ Ze 3 on Y given by Let M denote the quotient space of Y by the action of Ze 1 ⊕ Ze 2 ⊕ Ze 3 . It is equipped with a naturally induced mini-complex structure.

Analytic stability condition for mini-holomorphic bundles with a Hermitian metric
Let Z be a finite subset of M. Let (E, ∂ E ) be a mini-holomorphic bundle on M \ Z. (See [4, §2.2] for the notion of mini-holomorphic bundle.) Let h be a Hermitian metric of E. We obtain the Chern connection ∇ h and the Higgs field φ h as explained in [4, §2.2.3]. Recall that we set is expressed as a sum of an L 1 -function and a non-positive function, then we set

Adapted metrics of mini-holomorphic bundles with Dirac type singularity
Let (E, ∂ E ) be a mini-holomorphic bundle with Dirac type singularity on (M; Z). Let P ∈ Z. Let (t 0 , w 0 ) ∈ Y be a lift of P . We set t P := t − t 0 and w P := w − w 0 . Then, (t P , w P ) induces a mini-complex coordinate system on a neighbourhood U P of P . By using the coordinate system, we may regard U P as an open neighbourhood of (0, 0) in R × C. Let ϕ : Proof It can be proved by the argument in the proof of [4, Proposition 9.4].

Analytic stability condition for mini-holomorphic bundles with Dirac type singularity
Let (E, ∂ E ) be a mini-holomorphic bundle with Dirac type singularity on (M; Z). We set for an adapted Hermitian metric h of E, which is independent of the choice of h.
Definition 2.5 Let (E, ∂ E ) be a mini-holomorphic bundle with Dirac type singularity on (M; Z). We say that • Each point of Z is a Dirac type singularity of the underlying mini-holomorphic bundle (E, ∂ E ).
• h is an adapted metric of (E, ∂ E ) at P .
By definition, a monopole with Dirac type singularity (E, h, ∇, φ) on M \ Z induces a mini-holomorphic bundle with Dirac type singularity (E, ∂ E ) on (M; Z). The following is a variant of the correspondence in [1] on the basis of [7].
Theorem 2.7 The above construction induces an equivalence between monopoles with Dirac type singularity on M \ Z and analytically polystable mini-holomorphic bundles with Dirac type singularity of degree 0 on (M; Z).
More precisely, Theorem 2.7 consists of Proposition 2.8, Proposition 2.9 and Proposition 2.12 below.

Polystability
Let (E, ∂ E , h) is a monopole with Dirac type singularity on M \ Z.
If deg an (E ′ , ∂ E ′ ) = 0, we obtain ∂ E,w p E ′ = ∂ E,t p E ′ = 0. We obtain that the orthogonal complement E ′⊥ is also a mini-holomorphic subbundle of E. Let h E ′⊥ be the metric of E ′⊥ induced by h. Thus, we obtain a decomposition of monopoles (E, . Hence, we obtain the polystability of (E, ∂ E ) by an easy induction.

Construction of monopoles
Let (E, ∂ E ) be a stable mini-holomorphic bundle with Dirac type singularity on (M; Z) of deg an (E, ∂ E ) = 0.

Proposition 2.9
There exists a Hermitian metric h of (E, ∂ E ) such that (E, ∂ E , h) is a monopole with Dirac type singularity on M \ Z.
Proof As a preliminary, let us consider the rank one case. Note that the stability condition is trivial in the rank one case.
Lemma 2.10 Assume that rank E = 1. Then, there exists a Hermitian metric h of (E, ∂ E ) such that (E, ∂ E , h) is a monopole with Dirac type singularity on M \ Z.
Proof We take a Hermitian metric h 0 of E such that the following holds: • Each P ∈ Z has a neighbourhood U P in M such that (i) G(h 0 ) = 0 on U P \ {P }, (ii) P is Dirac type singularity of the monopole (E, ∂ E , h) |UP \{P } . Let us study the general case. On R 4 = R× R 3 , 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 Γ denote the lattice of R 4 = R × (R × C) generated by (1, 0, 0) and (0, a i , α i ) (i = 1, 2, 3). We consider the action of Γ on R 4 induced by the natural Z-action on R and the Γ-action on R × C. Let (X, g X ) denote the Kähler manifold obtained as the quotient of (C 2 , dz dz + dw dw) by the Γ-action. We have the natural projection p : X −→ M.
We set E := p −1 (E) on X \ p −1 (Z). It is equipped with the complex structure ∂ E determined by for sections f of E. For any adapted Hermitian metric h 0 of E, set h 0 := p −1 (h 0 ). Let F ( h 0 ) denote the curvature of the Chern connection of ( E, ∂ E , h 0 ). Let Λ denote the contraction from (1, 1)-forms to (0, 0)-forms with respect to the Kähler form of (X, g X ). Then, Because of the Chern-Weil formula, it is well defined in R ∪ {−∞} as explained in [7]. Then, ( E, ∂ E , h 0 ) is defined to be analytically stable with respect to the S 1 -action if The following is clear. Suppose that (E, ∂ E ) is analytically stable of degree 0. We take an adapted Hermitian metric h 0 such that each P ∈ Z has a neighbourhood U P such that G(h 0 ) |UP \{P } = 0. We may also assume that Tr G(h 0 ) = 0 by the construction in the rank one case in §2.2.3. By Lemma 2.11, ( E, ∂ E , h 0 ) is analytically stable with respect to the S 1 -action. We also have Λ Tr F ( h 0 ) = 0. According to a theorem of Simpson [7, Theorem 1], there exists an We obtain the corresponding metric h of E, for which G(h) = 0 holds. Because h and h 0 are mutually bounded, each P ∈ Z is a Dirac type singularity of (E, ∂ E , h) which is implied by [6, Theorem 3].

Uniqueness
The uniqueness is also standard. Proof Let s be the automorphism of E determined by h 2 = h 1 s. By [4,Corollary 2.30], we have the following inequality on M \ Z: 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 h 1 , we also obtain that We obtain that the eigenvalues of s are constant, and the eigen decomposition E = E i is compatible with the mini-holomorphic structure. Then, the claim of the proposition follows.

A more sophisticated formulation of the stability condition
We explain a different way to define the stability condition of mini-holomorphic bundles introduced by Kontsevich and Soibelman [2], which we already mentioned in §1.2. This section is devoted to explain their idea. Let γ be any element of H 1 (A, B). We take a representative of γ by a smooth 1-chain γ. For any ω ∈ Z 1 DR (A), the number γ ω is independent of the choice of a representative γ. They are denoted by γ ω.

Preliminary
Let

Duality
Suppose that A is compact and oriented. Let By definition, for any a ∈ H 2 (A \ B) and b ∈ H 1 c (A \ B), the following holds: Take any Riemannian metric g A 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 g A .
such that |τ | gA is an L 1 -function on A. Then, the following holds for any ρ ∈ Z 1 DR (A): Here, [τ ] ∈ H 2 DR (A \ B) denotes the cohomology class of τ .
Proof For any point P ∈ Z, we take a small coordinate neighbourhood (A P , x P,1 , x P,2 , x P,3 ) of P such that (i) P corresponds to (0, 0, 0), (ii) A P ≃ {(x 1 , x 2 , x 3 ) ∈ R 3 | x 2 i < 1} by the coordinate system. Set x P := x 2 P,1 + x 2 P,2 + x 2 P,3 1/2 . Then, there exists a C ∞ -function f P on A P such that (i) df P = ρ on { x P < 1/2}, (ii) f P (P ) = 0, (iii) f P (Q) = 0 for Q ∈ { x P > 2/3}. We naturally regard f P as a C ∞ -function on A. Then, the following holds: We set S 2 P (r) := x P = r with the orientation as the boundary of x P ≤ r . Then, we obtain the following (2) Note that the limit exists because d(f P τ ) = df P ∧τ is integrable. Because |τ | gA is L 2 , we have dr S 2 P (r) |τ | gA < ∞, and hence there exists a sequence r i → 0 such that r i S 2 P (ri) |τ | gA → 0. Because |f P | = O( x P ), we obtain that (2) is 0.

Relation between degrees of mini-holomorphic bundles
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 e i (i = 1, 2, 3) of T and the dual frame e ∨ i (i = 1, 2, 3), then σ = e ∨ i ⊗ e i . Let E be a vector bundle on M \ Z. Kontsevich and Soibelman [2] introduced the following element: Proof Let h be an adapted metric of (E, ∂ E ). For each P ∈ Z, we take any small neighbourhood U P of P in M. Let d(P, Q) denote the distance of P and Q ∈ U P . Then, the following holds for Q ∈ U P \ {P } (for example, see [6, §5]): By Lemma 3.1, it is enough to prove the following equality: For κ = t, x, y, we obtain the following by the Stokes formula and |φ h,Q | h = O d(P, Q) −1 : Note that F (h) tx = ∇ h,y φ h and F (h) yt = ∇ h,x φ h holds. Hence, we obtain Tr F (h) tx dt dx dy = Tr F (h) yt dt dx dy = 0.
We also obtain the following from (4): Tr F (h) xy dt dx dy = Tr F (h) xy − ∇ h,t φ h dt dx dy.
Then, we obtain (3), and the proof of Proposition 3.2 is completed. [2] formulated the stability condition for mini-holomorphic bundles in terms of the coefficient of ∂ t in ΦZ (c1(E)) σ, as explained in §1.2.

Parabolic difference modules on elliptic curves
We use the notation in §2.1.1. We assume that (i) the tuple (a i , α i ) (i = 1, 2, 3) is an oriented base of R × C, (ii) α 1 and α 2 are linearly independent. Let Γ 0 ⊂ C be the lattice generated by α 1 and α 2 . The projection Y −→ C induces a morphism M −→ T := C/Γ 0 . Let M cov denote the quotient space of Y by the action of Ze 1 ⊕ Ze 2 . We have the natural isomorphism M cov /Ze 3 ≃ M.

Another mini-complex coordinate system
We introduce another mini-complex coordinate system (s, u) on Y . We set 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 have e i (s, u) = (s, u + α i ) for i = 1, 2. We also have e 3 (s, u) = (s + t, u + a): Note that t > 0 because the tuple {(a i , α i )} i=1,2,3 is an oriented base of R × C. We have the following relations of complex vector fields: The product R s × T is equipped with the natural mini-complex structure. The mini-complex coordinate system (s, u) induces an isomorphism of mini-complex manifolds M cov ≃ R s × T .

Mini-holomorphic bundles and difference modules
Let Z be a finite subset in M. Let Z cov ⊂ M cov ≃ R s × T denote the pull back of Z. We take ǫ > 0 such that ([−ǫ, 0[×T ) ∩ Z cov = ∅. Let D be the image of Z cov ∩ ([−ǫ, t[×T ) via the projection R s × T −→ T . For each P ∈ D, we obtain the sequence 0 ≤ s P,1 < s P,2 < · · · < s P,m(P ) ≤ 1 by the condition: We set τ P,i := s P,i /t.
Let (E, ∂ E ) be a mini-holomorphic bundle on M \ Z with Dirac type singularity at Z. Let us observe that (E, ∂ E ) induces a parabolic a-difference module Υ(E, ∂ E ) over (T, (τ P ) P ∈D ).
Let V be the locally free O T -module obtained as E cov |{−ǫ}×T . It is independent of the choice of ǫ as above, up to canonical isomorphisms.
Let Φ : T −→ T be the morphism induced by Φ(u) = u + a. We have the natural isomorphism The scattering map induces an isomorphism Hence, V is equipped with an isomorphism V ( * D) ≃ (Φ * ) −1 (V )( * D).
For each P ∈ D and for i = 1, . . . , m(P ) − 1, we take s P,i < b P,i < s P,i+1 . Let (E cov |{−ǫ}×T ) P denote the O T,Pmodule obtained as the stalk of the sheaf of holomorphic sections of E cov |{−ǫ}×T at P . Similarly, (E cov |{bP,i }×T ) P denote the O T,P -module obtained as the stalk of the sheaf of holomorphic sections of E cov |{bP,i}×T at P . The scattering map induces isomorphisms of O T ( * P ) P -modules: (E cov |{−ǫ}×T ) P ( * P ) ≃ (E cov |{bP,i }×T ) P ( * P ).
The following proposition is easy to see.

Comparison of stability conditions
Let (E, ∂ E ) be a mini-holomorphic bundle with Dirac type singularity on (M; Z). Proof We consider the real vector field v := 2γ∂ w + 2γ∂ w − 2|γ| 2 − 1 2 ∂ t on M. Let h be any Hermitian metric of E.
Proof The following holds: Then, we obtain the claim of the lemma.
Let h be an adapted metric of (E, ∂ E ). Then, G(h), F (h) and ∇ h φ h are L 1 . Hence, we obtain