Arithmetic mirror symmetry for genus 1 curves with n marked points

Advances in Mathematics

2019

Self Cite

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 categories 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 1-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. [17]), we prove that any two simple vector bundles on a cycle of projective lines are related by a sequence of 1-spherical twists and their inverses. arXiv:1608.08992v2 [math.SG] 29 Aug 2018 1 Our convention is that in an A ∞ -category, we read the compositions from right-to-left (as in [24]). This affects certain signs in computations. In particular, the A ∞ -relations are given by: m,n (−1) |a 1 |+...+|an|−n m d−m+1 (a d , . . . , a n+m+1 , m m (a n+m , . . . , a n+1 ), a n , . . . a 1 ) = 0

Section: Simple Vector Bundles On Cycles Of Projective Linesmentioning

confidence: 87%

Section: Introductionmentioning

confidence: 99%

Associative Yang–Baxter equation and Fukaya categories of square-tiled surfaces

Lekili

Advances in Mathematics

2019

Self Cite

“…In a previous work (cf. ), the authors studied these categories for

X = T_{d}

d

‐holed torus, and

C = {scriptC}_{d}

the standard (Néron)

d

‐gon (For

d = 1

C_{1}

is the nodal projective cubic

{y^{2} z + x y z = x^{3}}

P_{Z}^{2}

), and proved the homological mirror symmetry statement that identifies the following triangulated categories, all defined over

double-struckZ

F false(T_{d} false) ≃ Perf false(C_{d} false)

W false(T_{d} false) ≃ D^{b} false(prefixCoh C_{d} false) .

…”

Section: Introductionmentioning

confidence: 92%

“…As an application of Theorem A, we deduce an equivalence of the perfect derived category of

scriptC

(respectively, the derived category of coherent sheaves on

scriptC

) with the appropriate infinitesimally wrapped Fukaya category (respectively, wrapped Fukaya category). In particular, we obtain simpler proofs of mirror symmetry for punctured surfaces of genus 0 and 1 (see [; , Theorem B]), and we also get a homological mirror symmetry result for all surfaces of genus

g > 1

with at least one puncture.…”

Section: Introductionmentioning

confidence: 99%

Auslander orders over nodal stacky curves and partially wrapped Fukaya categories

Lekili

2018

Journal of Topology

Self Cite

It follows from the work of Burban and Drozd [Math. Ann. 351 (2011) 665-709] that for nodal curves C, the derived category of modules over the Auslander order AC provides a categorical (smooth and proper) resolution of the category of perfect complexes Perf(C). On the A-side, it follows from the work of Haiden-Katzarkov-Kontsevich [Publ. Math. Inst. HautesÉtudes Sci. 126 (2017) 247-318] that for punctured surfaces X with stops Λ at their boundary, the partially wrapped Fukaya category W(X, Λ) provides a categorical (smooth and proper) resolution of the compact Fukaya category F (X). Inspired by this analogy, we establish an equivalence between the derived category of modules over the Auslander orders over certain nodal stacky curves and partially wrapped Fukaya categories associated to punctured surfaces of arbitrary genus equipped with stops at their boundary. As an application, we deduce equivalences between derived categories of coherent sheaves (respectively perfect complexes) on such nodal stacky curves and the wrapped (respectively compact) Fukaya categories of punctured surfaces of arbitrary genus.

“…In [25,26] we started to develop a systematic approach to the moduli spaces of minimal A ∞ -structures on a given graded vector space. In [25,13,26] we related certain moduli spaces of A ∞ -stuctures to appropriate moduli spaces of curves, and in [14] this was applied to proving an arithmetic version of homological mirror symmetry for n-punctured tori.…”

mentioning

confidence: 99%

𝐴_{∞}-structures associated with pairs of 1-spherical objects and noncommutative orders over curves

Trans. Amer. Math. Soc.

2020

Self Cite

We show that pairs (X, Y ) of 1-spherical objects in A ∞ -categories, such that the morphism space Hom(X, Y ) is concentrated in degree 0, can be described by certain noncommutative orders over (possibly stacky) curves. In fact, we establish a more precise correspondence at the level of isomorphism of moduli spaces which we show to be affine schemes of finite type over Z.Thus, up to an isomorphism, the graded associative algebra Hom(X ⊕ Y, X ⊕ Y ) is determined by a linear automorphism g of the finite-dimensional space Hom 0 (X, Y ), which measures the difference between the above two pairings. More precisely, fixing trivializations of Hom 1 (X, X) and Hom 1 (Y, Y ) and a basis α 1 , . . . , α n in Hom 0 (X, Y ), we get an identification of graded associative algebraswhere S(g) = S(k n , g) is a certain (2n + 4)-dimensional algebra depending on g ∈ GL n (k) (see Sec. 1.1). Furthermore, it is easy to see that replacing g by λ · g, where λ ∈ k * , leads to an isomorphic algebra.Since X and Y were objects of a minimal A ∞ -category, we get a minimal A ∞ -structure on S(g) extending the given m 2 . Thus, the problem of describing pairs of 1-spherical objects (X, Y ) as above with Hom 0 (X, Y ) of dimension n (in this case we refer to (X, Y ) as an n-pair) fits into the framework of [26, Sec. 2.2]. As in [26] we consider the moduli space of all minimal A ∞ -structures on the family of algebras S(·) over PGL n (extending the given m 2 ). The corresponding moduli functor M ∞ (sph, n) associates to a commutative ring R the set of gauge equivalence classes of minimal R-linear A ∞ -structures on an algebra of the form S(R n , g), where g ∈ PGL n (R) (see Sec. 1.1 for the precise definition).Note that by definition, we have a natural projectionwhere we identify the affine scheme PGL n with the corresponding functor on commutative rings.In the case n = 2 we need to restrict possible elements g, so we consider the principal open subscheme PGL 2 [tr −1 ] ⊂ PGL 2 where tr(g) is invertible. We denote by M ∞ (sph, 2)[tr −1 ] ⊂ M ∞ (sph, 2) the corresponding subfunctor.Our first main result, Theorem A below, relates A ∞ -structures in M ∞ (sph, n) for n ≥ 3 (resp., M ∞ (sph, 2)[tr −1 ] for n = 2) to certain noncommutative projective schemes in the sense of [2]. Recall that for a Noetherian graded algebra R one considers the quotient-category qgr R = grmod −R/ tors of finitely generated graded R-modules by the subcategory of torsion modules (it should be viewed as the category of coherent sheaves on the corresponding noncommutative scheme).Definition 0.1.1. Let R be a commutative ring, V a finitely generated projective Rmodule, and L an invertible R-module. For g ∈ End(V ) ⊗ L we denote by End g (V ) ⊂ End(V ) the R-submodule of transformations a such that tr(ga) = 0. We define the subalgebra in End(V )[z] byWe view E(V, g) as a graded R-algebra, where deg(z) = 1.Theorem A. For n ≥ 2, let us consider the functor M f ilt (n) associating with a commutative ring R the following data: a morphism g : Spec(R) → PGL n and an isomorphism 1...