arXiv:1608.08992v2 [math.SG] 29 Aug 2018

arXiv:1608.08992v1 [math.SG] 31 Aug 2016

Associative Yang-Baxter equation and Fukaya categories of square-tiled surfaces YANKI LEKILI ALEXANDER POLISHCHUK

We show that all strongly non-degenerate trigonometric solutions of the associative Yang-Baxter equation (AYBE) can be obtained from triple Massey products in the Fukaya category of square-tiled surfaces. Along the way, we give a classification result for cyclic A∞ -algebra structures on a certain Frobenius algebra associated with a pair of spherical objects in terms of the equivalence classes of the corresponding solutions of the AYBE. As an application, combining our results with homological mirror symmetry for punctured tori (cf. [15]), we prove that any two simple vector bundles on a cycle of projective lines are related by a sequence of spherical twists.

Introduction The associative Yang-Baxter equation (AYBE) is the equation (0–1)

r12 (−u0 , v)r13 (u + u0 , v + v0 ) − r23 (u + u0 , v0 )r12 (u, v) + r13 (u, v + v0 )r23 (u0 , v0 ) = 0,

where r : C2 → Matn (C) ⊗ Matn (C) is a meromorphic function of two complex variables (u, v) in a neighborhood of (0, 0) taking values in Matn (C) ⊗ Matn (C), and Matn (C) is the matrix algebra. Here we use the notation r12 = r ⊗ 1 ∈ Matn (C) ⊗ Matn (C) ⊗ Matn (C), etc. The equation (0–1) is usually coupled with the skew-symmetry (also called unitarity) condition (0–2)

r21 (−u, −v) = −r(u, v)

where r21 is obtained from r by the transposition of tensor factors a2 ⊗ a1 → a1 ⊗ a2 . Note that the constant version of the AYBE was studied by Aguiar [2] in connection with infinitesimal Hopf algebras. The AYBE is an analog in the world of associative algebras of the well-known classical Yang-Baxter equation (CYBE), [r12 (v), r13 (v + v0 )] + [r12 (v), r23 (v0 )] + [r13 (v + v0 ), r23 (v0 )] = 0, where r(v) is a meromorphic function in a neighborhood of 0 taking values in a Lie algebra g ⊗ g. Solutions of CYBE lead to Poisson-Lie groups and classical integrable systems (see for ex. [9], [7]). There is a direct relation between the two equations in the case g = sln (C): if r(u, v) is a solution of the AYBE such that the limit r(v) = (pr ⊗ pr)r(u, v)|u=0 exists (where pr is the projection away from the identity to traceless matrices), then r(v) is a solution of the CYBE for g = sln (C). 1

It was discovered in [16] that solutions of the AYBE often arise from 1-Calabi-Yau A∞ -categories. More precisely, assume we have such an A∞ -category C and two sets of isomorphism classes of 1-spherical objects in C , X and Y , such that for every pair of distinct objects x1 , x2 ∈ X (resp. y1 , y2 ∈ Y ), Hom∗ (x1 , x2 ) = 0 (resp. Hom∗ (y1 , y2 ) = 0). Furthermore, assume Hom6=0 (x, y) = 0 (and so Hom6=1 (y, x) = 0) for x ∈ X and y ∈ Y . Then dualising the triple product 1 m3 : Hom0 (x2 , y2 ) ⊗ Hom1 (y1 , x2 ) ⊗ Hom0 (x1 , y1 ) → Hom0 (x1 , y2 ), where x1 , x2 ∈ X , y1 , y2 ∈ Y , using the Calabi-Yau pairing, we get a tensor ,x2 ryx11,y : Hom0 (x2 , y2 ) ⊗ Hom0 (x1 , y1 ) → Hom0 (x1 , y2 ) ⊗ Hom0 (x2 , y1 ). 2

defined by ,x2 hryx11,y (f ⊗ f11 ), g21 ⊗ g12 i = hm3 (f22 , g12 , f11 ), g21 i. 2 22

(0–3)

where fii ∈ Hom0 (xi , yi ), gij ∈ Hom1 (yi , xj ). Now, let x1 , x2 , x3 (resp., y1 , y2 , y3 ) be distinct elements of X (resp., Y ). Then taking into account the assumptions on X , Y the relevant A∞ -relation involving m3 takes the form m3 (m3 (f33 , g23 , f22 ), g12 , f11 ) + m3 (f33 , m3 (g23 , f22 , g12 ), f11 ) − m3 (f33 , g23 , m3 (f22 , g12 , f11 )) = 0, where fii ∈ Hom0 (xi , yi ), gij ∈ Hom1 (yi , xj ), Rewritten in terms of the corresponding dual tensor ryx11yx22 this equation takes form (ryx22yx31 )12 (ryx11yx33 )13 − (ryx11yx23 )23 (ryx22yx33 )12 + (ryx12yx33 )13 (ryx11yx22 )23 = 0,

(0–4)

defined as an equation on Hom0 (x3 , y3 ) ⊗ Hom0 (x2 , y2 ) ⊗ Hom0 (x1 , y1 ) → Hom0 (x2 , y3 ) ⊗ Hom0 (x1 , y2 ) ⊗ Hom0 (x3 , y1 ) (see [16, Thm. 1]2 ). In addition, because of the cyclicity of the A∞ -structure, this tensor satisfies the following skew-symmetry condition: (ryx11yx22 )21 = −ryx22yx11 .

(0–5)

We will call the equation (0–4) the set-theoretic AYBE. It was further shown in [16] that in the case when C is the derived category of coherent sheaves on an elliptic curve (or some of its degenerations) then there exist natural choices of X and Y as above, so that all the spaces Hom0 (x, y), x ∈ X , y ∈ Y , can be identified with the fixed finite-dimensional vector space V . Furthermore, in this case X and Y have 1

Our convention is that in an A∞ -category, we read the compositions from right-to-left (as in [20]). This affects certain signs in computations. In particular, the A∞ -relations are given by: X (−1)|a1 |+...+|an |−n md−m+1 (ad , . . . , an+m+1 , mm (an+m , . . . , an+1 ), an , . . . a1 ) = 0 m,n 2

Our equation differs from [16, Eq. (1.2)] due to different conventions. The two equations become equivalent xx0 x0 x if we replace ryy 0 by ry0 y .

2

,x2 ⊗2 → V ⊗2 depend only on the differences abelian group structures, and the obtained tensors ryx11,y 2 : V u = x2 − x1 , v = y2 − y1 , which leads to the equation (0–1).

Note that different choices of trivialization of the Hom-spaces in the above construction correspond to the natural equivalence relation on solutions of the set-theoretic AYBE introduced in [16]. Namely, given a function ϕxy with values in GLn (C), we can transform a solution ryx11yx22 of (0–4) to the new solution eryx11yx22 = (ϕxy21 ⊗ ϕxy12 )ryx11yx22 (ϕxy11 ⊗ ϕxy22 )−1 .

(0–6)

Our first result is that an analog of the above construction gives a bijection between formal solutions of the set-theoretic AYBE and a class of A∞ -structures. Namely, we will consider deformations of the formal A∞ -category A = An defined as follows. Definition 0.0.1 The A∞ -category A = An has two objects X and Y , and the Hom-spaces Hom(X, Y) = Hom0 (X, Y) = Zθ1 ⊕ . . . ⊕ Zθn , Hom(Y, X) = Hom1 (Y, X) = Zη1 ⊕ . . . ⊕ Zηn , Hom0 (X, X) = Z idX , Hom0 (Y, Y) = Z idY , Hom1 (X, X) = ZξX , Hom1 (Y, Y) = ZξY . The elements idX and idY act as strict units in the sense that m2 (a, idX ) = a , m2 (idX , a) = (−1)|a| a , m2 (a, idY ) = a , m2 (idY , a) = (−1)|a| a, whenever composition with a ∈ A is non-zero, where |a| is the degree of a, and the other compositions are given by m2 (ηα , θβ ) = δαβ ξX , m2 (θα , ηβ ) = −δαβ ξY . Note that one can view A as a graded category by defining the composition as: a2 · a1 = (−1)|a1 | m2 (a2 , a1 ) We also define the symmetric perfect pairing on the Hom-spaces of A by hηα , θβ i = −hθb , ηa i = δαβ , hξX , idX i = −hidX , ξX i = hξY , idY i = −hidY , ξY i = 1. Let k be a field. We are going to consider A∞ -structures on A ⊗ k, with given m2 , which are cyclic with respect to this pairing. Recall that a strictly cyclic A∞ -category of dimension 1 is a strictly unital, proper A∞ -category together with nondegenerate pairings h , i : hom∗ (X, Y) ⊗ hom1−∗ (Y, X) → k satisfying ha1 , a2 i = (−1)(|a1 |−1)(|a2 |−1)+1 ha2 , a1 i and the cyclic symmetry condition: hak+1 , mk (ak , ak−1 , . . . , a1 )i = (−1)(|ak+1 |−1|)(|a1 |+|a2 |+...+|ak |−k) hak , mk (ak−1 , ak−2 . . . , a1 , ak+1 )i An A∞ -functor f = (fn )n≥1 : A → B between cyclic A∞ categories is said to be cyclic if the following hold: hf1 (a2 ), f1 (a1 )i = ha2 , a1 i 3

for any a2 , a1 and X

hfl (an , . . . , ak+1 ), fk (ak , . . . a1 )i = 0

k+l=n

for any sequence of composable morphisms an , . . . , a1 with n ≥ 3. For a commutative k-algebra R we denote by M∞ (A ⊗ R) the set of cyclic, strictly unital, minimal A∞ -structures on A ⊗ R, up to a strict cyclic A∞ -equivalence (i.e., the one with f1 = id). Let us set X (0–7) P := eij ⊗ eji ∈ Matn (k) ⊗ Matn (k), i,j

where (eij ) is the standard basis of Matn (k) defined by eij (ek ) = δjk ei if (ei )ni=1 is a basis of kn . In other words, P is the transposition operator given by: P(x ⊗ y) = y ⊗ x. Theorem A. There is a natural explicit bijection between M∞ (A ⊗ R) and the equivalence classes of formal skew-symmetric solutions ryx11yx22 of the set-theoretic AYBE of the following type. We let x1 , x2 , y1 , y2 to be formal variables and consider ryx11yx22 ∈ Matn (k) ⊗ Matn (k) ⊗ R[[x1 , x2 , y1 , y2 ]][(x2 − x1 )−1 (y2 − y1 )−1 ] of the form (0–8)

ryx11yx22 ≡

P id ⊗ id + {mod Matn (k) ⊗ Matn (k) ⊗ R[[x1 , x2 , y1 , y2 ]]}, x2 − x1 y1 − y2

Q such that (0–4) is satisfied in Matn (k) ⊗ Matn (k) ⊗ R[[x1 , x2 , x3 , y1 , y2 , y3 ]][∆−1 ], where ∆ = i 1 is much more complicated: in this case one can still classify all simple vector bundles on C (see [5]) but the relevant combinatorics is quite involved. The idea of the proof of Theorem C is to consider the solution of the formal set-theoretic AYBE associated with the pair (O, V) (where V is sufficiently positive), and to use Theorem A which states that the subcategory generated by O and V is determined by this solution. The point is that we know this solution of the AYBE to be the same as for the Lagrangians L1 , L2 in the symplectic surface Σ of genus 1 associated with some associative Belavin-Drinfeld structure. We prove that in this situation the pair (O, V) (resp. (L1 , L2 )) split generates the perfect derived category of C (resp., the Fukaya category of 6

Σ). Thus, we reduce the problem to a similar question about Lagrangians in the Fukaya category, where we can use the action of the mapping class group. The paper is organized as follows. In Section 1 we study the relation between formal solutions of the set-theoretic AYBE and A∞ -structures, in particular, proving Theorem A in 1.3. In addition, in Sec. 1.4 we discuss the natural involution on solutions of the AYBE, which allows us to deduce the pole conditions imposed in [17] for strongly nondegenerate solutions of the AYBE (see Prop. 1.4.4). Also, in Sec. 1.5 we explain, basing on ideas of [21], the connection between set-theoretical solutions of the AYBE coming from algebraic (or analytic) families of 1-spherical objects (see (0–3)) and the corresponding formal solutions from Theorem A. Section 2 is devoted to the construction of trigonometric solutions of the AYBE from Fukaya products on the square-tiled surfaces associated with Belavin-Drinfeld structures. In Section 3 we consider two applications of Theorems A and B to vector bundles over a (nodal) cycle C of projective lines. One is a criterion, in terms of some combinatorial data, for a pair of simple vector bundles on C to be related by a Fourier-Mukai autoequivalence (see Theorem 3.1.4). Another is Theorem C, proved in Sec. 3.2. Acknowledgments. Y.L. is supported in part by the Royal Society and the NSF grant DMS-1509141, and would like to thank Denis Auroux for a helpful correspondence. A.P. is supported in part by the NSF grant DMS-1400390, and would like to thank SISSA, where part of this work was done, for hospitality and excellent working conditions.

1

1.1

A class of cyclic A∞ -structures and formal solutions of the set-theoretic AYBE Twisted objects over complete rings

Let us quickly review the definition of the A∞ -category of twisted objects with coefficients in a complete ring (see [11], [10], [14]), mostly following [20, Ch. 1] and [13, Sec. 7.6]. Let C be a topological A∞ -category over a complete ring R. We assume that R is topologized by a decreasing filtration (Rn ), such that Rm Rn ⊂ Rm+n , and that the Hom-spaces in C are complete. We will only consider twisted objects of the following kind: (X, δX ), where X is an object of C and δX ∈ R1 Hom1 (X, X) is an element satisfying the Maurer-Cartan equation X mn (δXn ) = 0. n≥1

Note that here the left-hand side converges in Hom2 (X, X). The Hom-space between two such objects (X, δX ) and (Y, δY ) is simply Hom(X, Y). There are natural A∞ -products (mtn ) for the twisted objects, which are obtained by inserting the twisting elements δ in any number wherever possible. More precisely, mtd is given by X i mtd (ad , ad−1 , . . . , a1 ) = md+i0 +...+id (δXidd , ad , δXd−1 , ad−1 , . . . , a1 , δXi00 ) d−1 i0 ,...,id ≥0

7

(We follow the sign conventions of [20, Ch.1]). An A∞ -functor f = (fn )n≥1 : C → C 0 between A∞ -categories over R, as above, leads to an A∞ -functor F = (F n )n≥1 between their categories of twisted objects. Namely, (X, δX ) maps to (F(X), F(δX )), where X F(X) = f(X) , F(δX ) = fn (δXn ). The maps F d on Hom-spaces between twisted objects is given by X i F d (ad , ad−1 , . . . , a1 ) = fd+i0 +...+id (δXidd , ad , δXd−1 , ad−1 , . . . , a1 , δXi00 ) d−1 i0 ,...,id ≥0

1.2

Triple Massey products and a construction of formal solutions of the set-theoretic AYBE

First, let us recall the general definition of the triple Massey product in an A∞ -category C following [16], but with different sign conventions. For a triple of m1 -closed composable morphisms X0

a1-

X1

a2-

X2

a3-

X3

such that m2 (a2 , a1 ) = m1 (h1 ), m2 (a3 , a2 ) = m1 (h2 ), one sets (1–1)

MP(a3 , a2 , a1 ) = m3 (a3 , a2 , a1 ) − m2 (h2 , a1 ) − m2 (a3 , h1 )

{mod Im(m1 )},

which is well-defined as a coset for the image of H |a1 |+|a2 |−1 Hom(X0 , X2 ) ⊕ H |a2 |+|a3 |−1 Hom(X1 , X3 )

(m2 (a3 ,?),m2 (?,a1 ))

H |a1 |+|a2 |+|a3 |−1 Hom(X0 , X3 ).

The main result about such triple Massey products is that they are preserved under A∞ -functors: if f : C → C 0 is an A∞ -functor then f1 (MP(a3 , a2 , a1 )) = MP(f1 (a3 ), f1 (a2 ), f1 (a1 )) in the appropriate quotient-group (see [16, Prop. 1.1]). Next, given a minimal cyclic A∞ -structure with the given m2 on the category A ⊗ R, where R is a commutative ring (see Def. 0.0.1), we extend the coefficients to e := R[[x1 , x2 , y1 , y2 ]][(x2 − x1 )−1 (y2 − y1 )−1 ] R e and consider the following twisted objects of A ⊗ R: (1–2)

Xi = (X, xi ξX ), Yi = (Y, yi ξY ), for i = 1, 2.

Note that the Hom-spaces Hom(Xi , Yj ) and Hom(Yj , Xi ) are still concentrated in one degree and so have trivial mt1 . We denote by θαij (resp., ηαji ) the basis elements in Hom(X, Y) (resp, Hom(Y, X)) viewed as elements of Hom(Xi , Yj ) (resp., Hom(Yj , Xi )). On the other hand, Hom(X1 , X2 ) and Hom(Y1 , Y2 ) now have a nontrivial differential: mt1 (idX ) = (x2 − x1 )ξX , mt1 (idY ) = (y2 − y1 )ξY , 8

e so the corresponding cohomology vanishes (due to the localization in the definition of R). We consider the triple Massey product corresponding to the composable morphisms X1

11 θα

- Y1

ηβ12

- X2

22 θα 0

- Y2 .

We claim that this Massey product is well-defined and univalued. Indeed, we have mt2 (ηβ12 , θα11 ) = δαβ ξX =

δαβ · mt1 (idX ), x2 − x1

mt2 (θα220 , ηβ12 ) = −δα0 β ξY =

δ α0 β · mt1 (idY ), y1 − y2

hence it is well-defined. The fact that it is univalued follows immediately from the vanishing of H 0 Hom(X1 , X2 ) and H 0 Hom(Y1 , Y2 ). According to the formula (1–1) we have δα0 β δαβ · mt2 (θα220 , idX ) − · mt2 (idY , θα11 ) x2 − x1 y1 − y2 δα0 β δαβ · θα220 − · θ11 , = mt3 (θα220 , ηβ12 , θα11 ) − x2 − x1 y1 − y2 α

MP(θα220 , ηβ12 , θα11 ) = mt3 (θα220 , ηβ12 , θα11 ) −

where the last equality follows since our A∞ -structure on A ⊗ R is strictly unital, so the products mt2 involving the identity remain the same as m2 . Therefore, we have: hMP(θα220 , ηβ12 , θα11 ), ηβ210 i = hmt3 (θα220 , ηβ12 , θα11 ), ηβ210 i +

(1–3)

1.3

δαβ δα0 β 0 δα0 β δαβ 0 + . x2 − x1 y1 − y2

Proof of Theorem A

e To a minimal cyclic A∞ -structure on A ⊗ R we associate the element ryx11yx22 ∈ Matn (k) ⊗ Matn (k) ⊗ R obtained from MP(θα220 , ηβ12 , θα11 ) by dualization. In other words, X (1–4) ryx11yx22 = hMP(θα220 , ηβ12 , θα11 ), ηβ210 i · eβ 0 α0 ⊗ eβα . α,α0 ,β,β 0

Let us set fyx11yx22 :=

X

hmt3 (θα220 , ηβ12 , θα11 ), ηβ210 i · eβ 0 α0 ⊗ eβα ,

α,α0 ,β,β 0

which is an element of Matn (k) ⊗ Matn (k) ⊗ R[[x1 , x2 , y1 , y2 ]]. We can rewrite (1–3) as X X 1 1 ryx11yx22 = fyx11yx22 + eαα ⊗ eα0 α0 + eα0 α ⊗ eαα0 = · · x2 − x1 y1 − y2 0 0 α,α

fyx11yx22

α,α

id ⊗ id P + + , x2 − x1 y1 − y2

In particular, the singular part of ryx11yx22 has the required form. Also, the skew-symmetry equation (0–5) is equivalent to (fyx11yx22 )21 = −fyx22yx11 , 9

which can be deduced from the cyclic symmetry equation as follows: X (fyx11yx22 )21 = hmt3 (θα220 , ηβ12 , θα11 ), ηβ210 i · eβα ⊗ eβ 0 α0 α,α0 ,β,β 0

=−

X

hmt3 (θα22 , ηβ120 , θα110 ), ηβ21 i · eβα ⊗ eβ 0 α0

α,α0 ,β,β 0

=−

X

hmt3 (θα220 , ηβ12 , θα11 ), ηβ210 i · eβ 0 α0 ⊗ eβα = −fyx22yx11 .

α,α0 ,β,β 0

We claim that the set-theoretic AYBE for ryx11yx22 is equivalent to the A∞ -constraints in A ⊗ R applied to all possible strings of composable elements ξYf 1 , ηα313 , ξXe 3 , θβ332 , ξYd3 , ηα232 , ξXc 2 , θβ221 , ξYb2 , ηα121 , ξXa 1 .

(1–5)

Note that because of the cyclic symmetry these constraints are equivalent to the full A∞ -constraints. Recall that the set-theoretic AYBE takes place over the ring R[[x1 , x2 , x3 , y1 , y2 , y3 ]][∆−1 ]. Over this ring we can define twisted objects (Xi , Yi ) for i = 1, 2, 3 as in (1–2). We extend the notation θαij and ηβji for basis elements in the Hom-spaces to this case. Let us set for brevity P id ⊗ id , µij = . λij = xj − xi yi − yj Thus, the set-theoretic AYBE takes the following form: (fyx22yx31 + λ21 + µ23 )12 (fyx11yx33 + λ13 + µ13 )13 − (fyx11yx23 + λ13 + µ12 )23 (fyx22yx33 + λ23 + µ23 )12 + (fyx12yx33 + λ23 + µ13 )13 (fyx11yx22 + λ12 + µ12 )23 = 0. A straightforward calculation shows that the terms depending quadratically on (λij , µij ) cancel out, so this is equivalent to an equation of the form   (1–6) AYBE[f ] + (fyx22yx31 )12 (λ13 )13 − (λ13 )23 (fyx22yx33 )12 + . . . = 0, where AYBE[f ] is the left-hand side of (0–4) with f instead of r, and the remaining terms similarly combine the terms linear in λij and µij . Now we claim that looking at the coefficients of the expansion of (1–6) in x1 , x2 , x3 , y1 , y2 , y3 we get precisely the A∞ -constraints for the strings (1–5). These constraints have the form X − m∗ (ξYf21 , m∗ (ξYf11 , ηα3 , ξXe 3 , θβ2 , ξYd3 , ηα2 , ξXc22 ), ξXc12 , θβ1 , ξYb2 , ηα1 , ξXa 1 ) f =f2 +f1 ;c=c2 +c1

−

X

m∗ (ξYf 1 , ηα3 , ξXe23 , m∗ (ξXe13 , θβ2 , ξYd3 , ηα2 , ξXc 2 , θβ1 , ξYb22 ), ξYb21 , ηα1 , ξXa 1 )

e=e2 +e1 ;b=b2 +b1

+

X

m∗ (ξYf 1 , ηα3 , ξXe 3 , θβ2 , ξYd32 , m∗ (ξYd31 , ηα2 , ξXc 2 , θβ1 , ξYb2 , ηα1 , ξXa21 ), ξXa11 )

d=d2 +d1 ;a=a2 +a1

+ . . . = 0, where the additional terms appear when one of a, b, c, d, e, f is zero, and have the form either m∗ (. . . , m2 , . . .) or m2 (. . . , m∗ , . . .). Using cyclic symmetry, one can immediately check that the coefficients of x1a yb2 x2c yd3 x3e yf1 in the three terms in −AYBE[f ] match the first three terms in the A∞ -constraint above. 10

Now let us show how the second term in (1–6) matches the terms with f = 0 in the A∞ -constraint. The matching of the other terms is done similarly. First, we observe that (fyx22yx31 )12 (λ13 )13 − (λ13 )23 (fyx22yx33 )12 =

1 (f x2 x1 − fyx22yx33 )12 . x3 − x1 y2 y3

If we expand fyx22yx31 in powers of x1 , fyx22yx31 =

X

fyx22y,n3 · x1n ,

n≥0

then the above expression becomes X xn − xn 1

n≥0

3

x3 − x1

· (fyx22y,n3 )12 = −

X

x1a x3e · (fyx22y,a+e+1 )12 . 3

a,e≥0

Now one can easily check that this matches the contribution of the terms of the form m2 (ηα3 , m∗ (ξXe 3 , θβ2 , ξYd3 , ηα2 , ξXc 2 , θβ1 , ξYb2 , ηα1 , ξXa 1 )) in the A∞ -constraint. Next, we claim that changing an A∞ -structure by a cyclic homotopy transforms the corresponding solution of set-theoretic AYBE to an equivalent one, as in (0–6), and that all equivalences appear in this way. Namely, a cyclic homotopy (fn ) gives rise to the formal series X ϕxy := hfa+b+1 (ξYb , ηα , ξXa ), θβ i · xa yb · eβα a,b≥0

in Matn (R)[[x, y]]. Note that since f1 is the identity, the constant term of ϕxy is equal to id ∈ Matn (R). One can easily check that if (m0n ) is obtained from (mn ) by a cyclic homotopy (fn ) then the corresponding solutions of the set-theoretic AYBE are related by (0–6). Finally, we claim that a cyclic A∞ -equivalence (fn ) is uniquely determined by the corresponding formal series ϕxy , which could be an arbitrary series with the constant term equal to the identity. Indeed, recall that the condition for a strict A∞ -functor to be cyclic is that X (1–7) hfl (an , . . . , ak+1 ), fk (ak , . . . , a1 )i = 0 k+l=n

for any sequence of composable morphisms a1 , . . . , an , where n ≥ 3. For our A∞ -category, the only potentially non-trivial values of f∗ are f∗ (ξYn , ηα , ξXm ), f∗ (ξXn , θβ , ξYm ), f∗ (ξYp , ηα , ξXn , θβ , ξYm ), f∗ (ξXp , θβ , ξYn , ηα , ξXm ). The constraints between them are given by (1–7) applied to the following two kinds of composable strings (ξXc , θβ , ξYb , ηα , ξXa ) and (ξYc , ηα , ξXb , θβ , ξYa ), where a + b + c > 0. The first of these strings gives the identity hξ, f∗ (ξXc−1 , θβ , ξYb , ηα , ξXa )i + hf∗ (ξXc , θβ , ξYb , ηα , ξXa−1 ), ξi X − hf∗ (ξXc , θβ , ξYb2 ), ηγ i · hf ∗ (ξYb1 , ηα , ξXa ), θγ i = 0, b1 +b2 =b,γ

11

where the first (resp., second) term appears only for a ≥ 1 (resp., c ≥ 1). We can rewrite these identities in terms of the generating series X ϕ exy := hfb+c+1 (ξXc , θβ , ξYb ), ηα i · xc yb · eβα , b,c≥0

ψyx1 ,x2 :=

X

hfa+b+c+2 (ξXc , θβ , ξYb , ηα , ξXa ), ξX i · x1a yb x2c · eβα ,

a,b,c≥0

as follows: (x1 − x2 )ψyx1 ,x2 − ϕ exy2 · ϕxy1 − Id = 0. Note that subtracting Id here corresponds to avoiding the case a = b = c = 0 in cyclic homotopy equation. Setting x2 = x1 we deduce from this that (1–8)

ϕ exy = −(ϕxy )−1 ,

and so the above identity can be solved for ψyx1 ,x2 : Id − ϕxy1 (ϕxy2 )−1 . x1 − x2 Similarly, the constraints associated with the strings (ξYc , ηα , ξXb , θβ , ξYa ) boil down to (1–8) and to an equation expressing all f∗ (ξYc , ηα , ξXb , θβ , ξYa ) in terms of the coefficients of ϕxy . ψyx1 ,x2 =

Remark 1.3.1 A more natural proof of the fact that (1–4) satisfies the set-theoretic AYBE can be obtained using the A∞ -constraints (several times) in the category of twisted objects of A⊗R[[x1 , x2 , x3 , y1 , y2 , y3 ]][∆−1 ]. We leave this as an exercise to the reader. However, this proof does not show that conversely, the settheoretic AYBE for ryx11yx22 , associated with (mn ) as above, implies that (mn ) defines a cyclic A∞ -structure on A ⊗ R.

1.4

Involution on skew-symmetric solutions of the AYBE

In the following Proposition we define a natural involution on the skew-symmetric solutions of the settheoretic AYBE (0–4), where the variables xi and yi could be either distinct elements of some sets X and Y , or formal variables, as in Theorem A. Proposition 1.4.1 Let ryx11yx22 be a solution of the set-theoretic AYBE with values in Matn (k)⊗Matn (k)⊗R, where R is a commutative k-algebra, satisfying the skew-symmetry condition (0–5). Set rˆyxˆˆ11yxˆˆ22 := (rxyˆˆ21xyˆˆ12 )t · P, where (a1 ⊗ a2 )t = at1 ⊗ at2 (here at is the transpose of a matrix a) and P is given by (0–7), and in the non-formal case the arguments xˆ i (resp., yˆ i ) take values in Y (resp., X ). Then (i) rˆyxˆˆ11yxˆˆ22 is again a solution of the AYBE satisfying the skew-symmetry condition. Furthermore, one has rˆˆ = r. If xˆ i are yˆ i are formal variables, and r has an expansion of the form (0–8) then so does −ˆrxˆ 1 xˆ 2 . −ˆy1 ,−ˆy2

(ii) In the context of Theorem A, assume that ryx11yx22 corresponds to an A∞ -structure on A ⊗ R. Formally ˆ setting Xˆ = Y and Yˆ = X[1], θˆα = ηα , ηˆβ = θβ , etc., we get a new A∞ -structure on A ⊗ R. Then the corresponding formal solution of the AYBE is precisely rˆ . 12

Proof. (i) Note that the map Matn (k) ⊗ Matn (k) → Matn (k) ⊗ Matn (k) : x 7→ P · x · P is just the permutation of factors. Thus, the skew-symmetry condition can be rewritten as Pryx11yx22 P = −ryx22yx11 . Passing to the transposed matrices and making the substitution xi = yˆ i , y2 = xˆ 1 , y1 = xˆ 2 , we derive that rˆ also satisfies (0–5). The fact that rˆˆ = r follows from the identity P · P = 1 ⊗ 1. The AYBE equation for rˆ is the following equation for r: (ryx12yx23 )12,t P12 (ryx31yx13 )13,t P13 − (ryx31yx12 )23,t P23 (ryx32yx23 )12,t P12 + (ryx31yx23 )13,t P13 (ryx21yx12 )23,t P23 = 0, where we set xi = yˆ i , yi = xˆ i . Using the skew-symmetry of r (and the fact that Pt = P) we can rewrite this as −P12 (ryx23yx12 )12,t (ryx31yx13 )13,t P13 + P23 (ryx12yx31 )23,t (ryx32yx23 )12,t P12 − P13 (ryx23yx31 )13,t (ryx21yx12 )23,t P23 = 0. or passing to the transpose, −P13 (ryx31yx13 )13 (ryx23yx12 )12 P12 + P12 (ryx32yx23 )12 (ryx12yx31 )23 P23 − P23 (ryx21yx12 )23 (ryx23yx31 )13 P13 = 0. Using the fact that P · P = 1, we can rewrite this as −(ryx31yx13 )13 (ryx23yx12 )12 + P13 P12 (ryx32yx23 )12 (ryx12yx31 )23 P23 P12 − P13 P23 (ryx21yx12 )23 (ryx23yx31 )13 P13 P12 = 0. Now we use the identities P13 P12 = P12 P23 , P13 P23 = P12 P13 and the fact that x 7→ Pij xPij acts as a transposition (ij), to rewrite this as −(ryx31yx13 )13 (ryx23yx12 )12 + (ryx32yx23 )23 (ryx12yx31 )31 − (ryx21yx12 )12 (ryx23yx31 )32 = 0. Swapping 2 and 3 we get the equation −(ryx31yx13 )12 (ryx23yx12 )13 + (ryx32yx23 )32 (ryx12yx31 )21 − (ryx21yx12 )13 (ryx23yx31 )23 = 0. Using the skew-symmetry to rewrite the middle summand we get the equation obtained from the AYBE by the change of variables (x1 , x2 , x3 ; y1 , y2 , y3 ) 7→ (x3 , x1 , x2 ; y2 , y3 , y1 ). (ii) This follows from the cyclic symmetry. Namely, if we set y1 = xˆ 2 , y2 = xˆ 1 , xi = yˆ i , for i = 1, 2, and use the corresponding identification between the twisted objects, we see that the new solution comes from the Massey products for the composable arrows Y2 → X1 → Y1 → X2 . Furthermore, it is given by X hmt3 (ηβ12 , θα11 , ηβ210 ), θα220 i · eα0 β ⊗ eαβ 0 + . . . , α,α0 ,β,β 0

where the other terms are standard singular parts. Using the cyclic symmetry we can rewrite this as X hmt3 (θα220 , ηβ12 , θα11 ), ηβ210 i · eα0 β ⊗ eαβ 0 + . . . . This matches the formula

α,α0 ,β,β 0 for (ryx11yx22 )t

· P due to the identity

(eβ 0 α0 ⊗ eβα )t · P = eα0 β ⊗ eαβ 0 .

13

Corollary 1.4.2 Let r(u, v) be a solution of the AYBE with values in Matn (k) ⊗ Matn (k), satisfying the skew-symmetry condition (0–2). Then r(v, u)t · P is again a solution of the AYBE satisfying the skew-symmetry condition. Recall that the nondegeneracy condition on solutions of the AYBE imposed in [17] is that the tensor r(u, v) ∈ Matn (k) ⊗ Matn (k) is nondegenerate for generic (u, v). Now we are going to use the above involution to show that the pole conditions for r(u, v), imposed in the classification result of [17], are implied by the following stronger nondegeneracy condition, involving r(u, v) and r(u, v)t · P. Definition 1.4.3 Let us say that an Matn (k) ⊗ Matn (k)-valued function r(u, v), meromorphic in a neighborhood of (0, 0), is strongly nondegenerate if the tensors r(u, v) and r(u, v)t · P are nondegenerate for generic (u, v). Proposition 1.4.4 Assume N > 1. Let r(u, v) be a strongly nondegenerate skew-symmetric solution of the AYBE (meromorphic in a neighborhood of (0, 0)). Then r(u, v) has a simple pole at u = 0 (resp., 0 P 0 v = 0) with the polar term c · 1⊗1 u (resp., c · v ), where c and c are nonzero constants. Proof. First, we claim that the involution r(u, v) 7→ rˆ (u, v) := r(v, u)t · P on skew-symmetric solutions of the AYBE preserves the notion of strong nondegeneracy. Indeed, this immediately follows from the observation that P · r · P = r21 , so it is nondegenerate if and only if r is nondegenerate. Next, given r(u, v), a strongly nondegenerate skew-symmetric solution of the AYBE, we claim that r(u, v) necessarily has a pole at u = 0. Indeed, assume r(u, v) is regular at u = 0. Then by [17, Lem. 1.2], we have an expansion r(u, v) = r0 (v) + ur1 (v) + . . ., where r0 (v) = r(u, 0) is a nondegenerate skew-symmetric solution of the AYBE. Then by [17, Thm. 0.2], r0 (v) has a pole at v = 0, hence, r(u, v) has a pole at v = 0. By [17, Lem. 1.3], this implies that r(u, v) has a simple pole at v = 0 with the polar part c · Pv . Hence, rˆ (u, v) has a simple pole at u = 0 with the polar part c · 1⊗1 u . Since rˆ (u, v) is still a nondegenerate skew-symmetric solution of the AYBE, by [17, Lem. 1.5], rˆ (u, v) has a simple pole at v = 0. Equivalently, r(u, v) has a simple pole at u = 0, which is a contradiction. Thus, we know that r(u, v) has a pole at u = 0, or equivalently, rˆ (u, v) has a pole at v = 0. By [17, Lem. 1.3], this implies that rˆ (u, v) has a simple pole at v = 0 with the polar part c · Pv . Hence, r(u, v) has a simple at u = 0 with the polar part c · 1⊗1 u . Now the assertion follows from [17, Lem. 1.5].

1.5

From algebraic/analytic to formal solutions of the set-theoretic AYBE

In this section we want to consider the solutions of the set-theoretic AYBE arising, as described in Introduction, from two algebraic families of objects X and Y . We want to show how to pass from these solutions to the corresponding formal solutions associated to picking one object in each family. We will use the formalism from [21, Sec. 1] concerning families of objects in A∞ -categories. Let A be an A∞ -category over k, and let X and Y be smooth affine curves over k, such that we have perfect families of A -modules M and N parametrized by X and Y . As in the introduction, we assume 14

that for x 6= x0 (resp., y 6= y0 ) one has Hom∗ (Mx , Mx0 ) = 0 (resp., Hom∗ (Ny , Ny0 ) = 0), that each Mx (resp., Ny ) is 1-spherical, and that Hom(Mx , Ny ) are concentrated in degree 0. Furthermore, we assume that Hom∗ (p∗X M, p∗Y N ) is a vector bundle over X × Y . Recall (see [21, (1h)]) that one can associate with the families M and N the deformation classes Def (M) ∈ Ω1X ⊗ Hom1 (M, M), Def (N ) ∈ Ω1Y ⊗ Hom1 (M, N ). Let U ⊂ X 2 × Y 2 be the complement to the diagonals ∆X × Y 2 ∪ X 2 × ∆Y . Then over U we have the induced families M(x1 ), M(x2 ), N (y1 ) and N (y2 ) (pull-backs from the families over X and Y ), such that Hom∗ (M(x1 ), M(x2 )) = Hom∗ (N (y1 ), N (y2 )) = 0. Thus, we have a well defined triple Massey product MP(x1 , x2 ; y1 , y2 ) : Hom0 (M(x1 ), N (y1 )) ⊗ Hom1 (N (y1 ), M(x2 )) ⊗ Hom0 (M(x2 ), N (y1 )) → Hom0 (M(x1 ), N (y2 )), which is a morphism of vector bundles over U . On the other hand, let us fix points x0 ∈ X and y0 ∈ Y , and let us fix generators ξx0 ∈ Hom1 (Mx0 , Mx0 ) and ξy0 ∈ Hom1 (Ny0 , My0 ). Assume that Def (M)|x0 ∈ Ω1X |x0 ⊗ Hom1 (Mx0 , Mx0 ) ' Ω1X |x0 is nonzero, and similarly Def (N )|y0 ∈ Ω1Y |y0 is nonzero. Then we can choose a formal parameter t near x0 on X (resp., parameter s near y0 on Y ) such that Def (M) = dt in a formal neighborhood of x0 (resp., Def (N ) = ds near y0 ). ˆ x be the family over k[[t]] obtained from M by restriction to a formal disk around x0 , and let Nˆ y Let M 0 0 be the similar family over k[[s]]. On the other hand, we have twisted objects (Mx0 , tξx0 ) and (Ny0 , sξy0 ), over k[[t]] and k[[s]], respectively. These twisted objects produce the same deformation classes dt and ds, so by the proof of [21, Prop. 1.21], we derive the existence of quasi-isomorphisms of families ˆ x ' (Mx , tξx ), Nˆ y ' (Ny , sξy ). M 0 0 0 0 0 0 By the functoriality of Massey products, this implies that the formal expansion of the Massey product MP(x1 , x2 , y1 , y2 ) near x0 and y0 , is equal to the triple Massey product considered in the proof of Theorem A. Note that Def (M)|x0 can be identified with the usual class of the first-order deformation of Mx0 , associated with M. In particular, in the situation when M is a universal deformation of x0 then Def (M)|x0 is nonzero. This is the situation that occurs when we consider families of simple vector bundles (or structure sheaves of points) on elliptic curves and their degenerations, as in [16], [17]. In the case of families of Lagrangians in Fukaya category, we have a similar picture, with algebraic families replaced by analytic. 15

2 2.1

Trigonometric solutions of the AYBE from symplectic geometry A square-tiled surface from Belavin-Drinfeld structures

In this section, starting from an associative Belavin-Drinfeld structure (S, C1 , C2 , A), we construct a punctured Riemann surface Σ together with a non-vanishing holomorphic one-form α ∈ Γ(C, ΩC1,0 ). Let us begin with a finite set S of n elements, and two arbitrary permutations C1 , C2 ∈ Aut(S) ∼ = Sn . Let T be the square torus C/(Z ⊕ iZ) and let T0 = T \ {0} be the punctured torus. Let us also consider the oriented curves, l1 , l2 : [0, 1] → T defined by: 1 − it t+i l1 (t) = , l2 (t) = 2 2 Let p0 ∈ T0 be the point (1 + i)/2, which is the unique point in l1 ∩ l2 . Consider the n-fold (unramified) covering: π : Σ0 → T0 corresponding to the subgroup H ⊂ π1 (T0 , l1 ∩ l2 ) = hl1 , l2 i ∼ = F2 given as the preimage of the stabilizer of a point s0 ∈ S under the homomorphism ρ : π1 (T0 ) → Aut(S) defined by ρ(l1 ) = C1 , ρ(l2 ) = C2 . Thus, we identify the set S with the fiber π −1 (p0 ), so that the action of the generators l1 and l2 of the fundamental group π1 (T0 , p0 ) on the fibre S is given by the permutations C1 , C2 : S → S. We let Li = π −1 (li ) to be the multi-curves in Σ0 covering the circles li , so that L1 ∩ L2 = π −1 (p0 ) = S. The surface Σ0 is connected if and only if the subgroup hC1 , C2 i ⊂ Aut(S) is transitive. The curves L1 and L2 are connected if and only if the permutations C1 , C2 are individually transitive (cyclic) permutations. We will always require that Σ0 is connected. If (S, C1 , C2 ) comes from an associative Belavin-Drinfeld data, then C1 , C2 are required to be cyclic permutations, however this condition is not strictly necessary in what follows. One can lift the flat metric on T0 to Σ0 . To visualise this metric on Σ0 , let us now give a more geometric construction of the covering map π : Σ0 → T0 . Recall that T is obtained from the unit square in [0, 1] × [0, 1] ⊂ R2 by identifying the opposite sides. T0 is obtained from this by removing the corner point. Now let us take n copies of the unit square (with corners removed) in bijection with the set S = {1, . . . , n}. Given automorphism C1 , C2 ⊂ Sn ∼ = Aut(S), construct a surface Σ0 as follows: 1) identify the right edge of the ith square with the left edge of jth square if C1 (i) = j; 2) identify the bottom edge of the ith square with the top edge of the jth square if C2 (i) = j. It is because of this construction Σ0 is called a square-tiled surface. The name was first suggested to Anton Zorich by Alex Eskin [24]. b0 By the Riemann existence theorem ([8, Sec. 4.2.2]), the surface Σ0 can be completed to a surface Σ and the covering map extends to a branched covering map: b0 → T π b:Σ ramified along the origin (0, 0) ∈ T. The ramification locus {p1 , p2 , . . . pb } = π b−1 (0) 16

consists of a number of points, which is equal to the number of cycles in the cycle decomposition of the commutator [C1 , C2 ] into disjoint union of cycles of varying lengths (from 1 to n). Indeed, the curves L1 and L2 divide the surface Σ into polygons, such that the point pi is contained in a (2e(pi ) + 2)-gon, where e(pi ) is the ramification index of the point pi . P We let bk denote the number of k-cycles, so that we have n = nk=1 kbk . We record the following elementary computation, which follows from the above explicit description of Σ as a union of bpolygonal regions, or also by the Riemann-Hurwitz formula. c0 \ Σ0 is equal to b = Pn bk . The Euler characteristic Proposition 2.1.1 The number of points in Σ k=1 of Σ0 is χ(Σ0 ) = −n. Consequently, the genus g is determined by the formula χ(Σ0 ) = 2 − 2g − b = −n. In particular, g = 1 if and only if C1 and C2 commute. Finally, we need to incorporate the proper subset A ⊂ S that appears in an associative Belavin-Drinfeld structure (S, C1 , C2 , A). This data enters in determining a partial compactification of Σ0 . Namely, recall that A is, by definition, a subset of the set of fixed points of the action of the commutator b 0 → T, the set of fixed points of the [C1 , C2 ] on S. In terms of the branched covering map π b : Σ commutator [C1 , C2 ] can be identified with the set of points pi in the preimage π b−1 (0) which have ramification index e(pi ) = 1. Using this we can identify A with a subset of points pi where the map π b is unramified. To be precise, an element a ∈ A gives a square with corners {a, C2 (a), C1 C2 (a), C2−1 C1 C2 (a)}, which determines a point pa ∈ π b−1 (0) of ramification index 1 contained in this square. We define Σ = ΣA to be the partial compactification Σ0 ∪ {pa |a ∈ A}. Note that the covering map π : Σ0 → T0 extends to a local diffeomorphism: π:Σ→T Hence, the flat metric on T lifts to a flat metric on Σ so as to make π into a local isometry. From now on, we will consider the square-tiled surface Σ equipped with this flat metric. Note that, for convenience, we always normalize the metric on T so that the length of the curves l1 and l2 are 1. Example 2.1.2 Figure 1 shows an example of this construction corresponding to S = {1, 2, 3, 4}, C1 = (1, 4, 2, 3), C2 = (1, 2, 3, 4). The red curve L1 ⊂ Σ0 and the blue curve L2 ⊂ Σ0 depict the preimages of the curves l1 , l2 ⊂ T0 . The flat metric can be extended over the black labelled point without any singularities. Thus, we can choose A to be either empty or include the unique black labelled point, which corresponds to {3} - the unique fixed point of [C1 , C2 ]. If A = {3}, then correspondingly, we compactify Σ0 by filling in the puncture labelled black. Remark 2.1.3 Topologically the punctured torus T0 can be seen as the plumbing of two cotangent bundles T ∗ li ∼ = T ∗ S1 at one point l1 ∩ l2 . Similarly, one can construct the surface Σ0 as the plumbing ∗ of T L1 and T ∗ L2 at n-points corresponding to L1 ∩ L2 . The Figure 2 illustrates this construction in the case of S = {1, 2, 3, 4}, C1 = (1, 2, 3, 4), C2 = (1, 3, 2, 4). 17

l1 l2

Figure 1: A square-tiled surface

Figure 2: S = {1, 2, 3, 4}, C1 = (1, 2, 3, 4), C2 = (1, 3, 2, 4)

2.2

A Fukaya category from Belavin-Drinfeld structures

Let (C, α) be the square-tiled surface obtained from an associative Belavin-Drinfeld structure as above. ⊗2 Let Σ be the topological surface underlying C. The square Ω = α ⊗ α ∈ Γ(C, (Ω1,0 C ) ) determines a non-vanishing quadratic form, which gives a flat Riemannian metric |Ω| on Σ and a horizontal foliation 18

of tangent vectors v with Ω(v, v) > 0. The Riemannian metric determines an area form ω 4 and the horizontal foliation determines a grading structure on Σ, i.e a section of the projectivized tangent bundle of S, which we view as an unoriented line field l ⊂ T(Σ). We note that such line fields form a torsor for C∞ (Σ, RP1 ) and the connected components can be identified with H 1 (Σ; Z). To work over C, one works with exact Fukaya categories as in [20]. Thus, we will need to choose a primitive θ for ω , which exists since Σ is non-compact. We choose this so that the Lagrangians L1 and L2 are exact. One can arrange this as follows: Choose any primitive θ0 for ω ; find a closed 1-form R σ0 such that σ0 ([Li ]) = Li θ0 , which exists since Li give independent non-trivial homology classes in H1 (Σ); and let θ = θ0 − σ0 . We also normalize the area form so that the geodesics L1 and L2 have length 1. We can now form a Z-graded triangulated 1-Calabi-Yau A∞ category - Fukaya category F(Σ) of (Σ, dθ, Ω) - linear over C. The objects of F(Σ) are closed, exact, oriented curves L equipped with grading structures and a C× -local system ξ → L whose fibre is C and holonomy is in C× . Recall that a grading structure on a curve L means a choice of a homotopy class of a path from TL to the line field l|L . If x ∈ L ∩ L0 is a transverse intersection point, the grading |x| is given by bα/πc + 1 where α is the net rotation from Tx L → lx → Tx L0 . This lifts the Z2 -grading on the intersection points given by bL ·x L0 /πc + 1 where L ·x L0 is the local algebraic intersection number of L and L0 at x associated to orientations of L and L0 . Note also that the C× -local system on a curve implicitly determines a Spin structure corresponding to the subgroup {±1} ⊂ C× , i.e. changing the Spin structure by the action of H 1 (S1 ; Z2 ) = Z2 is equivalent to modifying the monodromy of the local system by the action of {±1} ⊂ C× . Spin structures enter in defining orientations of various moduli spaces of holomorphic curves and they play a role in determining the signs in various counts. In the case of Fukaya categories of 2-dimensional surfaces, which is the only situation considered in this paper, there is a combinatorial method given in [19, Sec. 7] that allows us to compute these signs. Throughout, in our explicit computations, we follow this method to determine the signs without giving further explanation. The morphisms spaces in the Fukaya category is given by Floer cochain complexes: M CF ∗ ((L1 , ξ1 ), (L2 , ξ2 )) = homC (ξ1 |x , ξ2 |x ) x∈L1 ∩L2

For brevity, we often suppress the local systems ξi from the notation. The A∞ -structure comprises a collection of maps: mk : CF(Lk−1 , Lk ) ⊗ . . . ⊗ CF(L0 , L1 ) → CF(L0 , Lk )[2 − k] For pi ∈ Li−1 ∩ Li and p0 ∈ L0 ∩ Lk , the components of these maps involving homC (ξi−1 |pi , ξi |pi ) are defined by counting holomorphic disks with (k + 1)-boundary punctures such that the boundary 4

Note that our symplectic form ω is not convex at infinity. This is usually required in setting up Floer theory in order to ensure that a maximum principle holds which guarantees that pseudo-holomorphic disks remain in a compact region. However, in dimension 2, this holds essentially for topological reasons. Alternatively, one could modify ω near infinity to make it convex. Either way, the outcome is unchanged and we will simply use the area form ω .

19

components are mapped to (L0 , L1 , . . . Lk ) in the cyclic order. Let us denote the moduli space of such pseudoholomorphic disks u in the homotopy class [u] by M(pk , pk−1 , . . . , p1 , p0 ; [u]). If the index of [u] is fixed to be 2 − k and the regularity is arranged then Gromov-Floer compactness ensures that this moduli space is a finite set of points, which we can then count (with signs). For ρi ∈ homC (ξi−1 |pi , ξi |pi ), we set X (2–1) mk (ρk , . . . , ρ1 ) = #M(pk , pk−1 , . . . , p1 , p0 ; [u]) · hol∂u ∈ homC (ξ0 |p0 , ξk |p0 ), [u]:ind([u])=2−k

where the term hol∂u is defined as follows. The boundary component of u mapping to Li gives isomorphisms ξi |pi → ξi |pi+1 . Therefore, given elements ρi ∈ homC (ξi−1 |pi , ξi |pi ), using the isomorphisms provided by u, we can construct the composition: hol∂u = ρk ◦ . . . ◦ ρ1 ∈ homC (ξ0 |p0 , ξk |p0 ). Note that exactness ensures that the sum is finite, hence is well-defined. In practice, when we do explicit computations, we will mark points by ? on each Lagrangian circle, and the contribution of a holomorphic disk will be weighted by the holonomy factor each time the boundary of the disk passes through the marked point.

2.3

Constructing solutions to the AYBE via Massey products in F(Σ)

As was shown in [17], with every associative Belavin-Drinfeld structure (S, C1 , C2 , A) one can associate a trigonometric solution of the AYBE. A slightly different looking definition of an associative BelavinDrinfeld structure was used in [17]. In the next lemma we show the equivalence of the Definition (0.0.2) with the definition of the associative Belavin-Drinfeld structure in [17]. Let S be a finite set of n elements. We denote a transitive cyclic permutation as a map C : S → S and we write ΓC := {(s, C(s))|s ∈ S} ⊂ S × S for its graph. Lemma 2.3.1 Let S be a set equipped with a pair of transitive cyclic permutations C1 , C2 : S → S. Then to give a proper subset A ⊂ S, such that (S, C1 , C2 , A) is an associative Belavin-Drinfeld structure, is equivalent to giving a pair of proper subsets Γ1 , Γ2 ⊂ ΓC1 such that (C2 × C2 )Γ1 = Γ2 . Proof. We set Γ1 = {(a, C1 (a))|a ∈ A}, Γ2 = {(C2 (a), C1 (C2 (a))|a ∈ A.} One immediately sees that the condition (C2 × C2 )(Γ1 ) = Γ2 is equivalent to C1 C2 (a) = C2 C1 (a) for every a ∈ A. We prefer the form given in Definition 0.0.2 as it makes the symmetry with respect to switching C1 and C2 more clear (cf. Proposition 1.4.1). In examples, it may be convenient to identify S = {1, . . . , n} such that C1 (i) = i + 1. One then thinks of C2 as an n-cycle in the symmetric group Sn 20

The commutator [C1 , C2 ] ∈ An ⊂ Sn plays a special role in the definition as the elements of the set A correspond to a subset of the set of fixed points of the commutator [C1 , C2 ]. We remark that it can be proven by induction that any element of the alternating group An arises as the commutator [C1 , C2 ] of two n-cycles in Sn (see Prop. 4 [12]). Let AS denote the algebra of endomorphisms of the C-vector space with the basis (ei )i∈S , so that AS ' Matn (C), where n = |S|. We denote by eij ∈ AS the endomorphism defined by eij (ek ) = δjk ei . The solution of the AYBE associated with (S, C1 , C2 , A) is given by (2–2) X X 1 1 r(u, v) = eii ⊗ eii + eii ⊗ eii exp(u) − 1 i 1 − exp(−v) i X X 1 ku mv 1 + exp( )eCk (i),Ck (i) ⊗ eii + exp( )ei,C2m (i) ⊗ eC2m (i),i 1 1 exp(u) − 1 n exp(v) − 1 n 0