Fukaya categories of plumbings and multiplicative preprojective algebras

Etgü, Tolga; Lekili, Yankı

doi:10.4171/qt/131

Cited by 15 publications

(14 citation statements)

References 25 publications

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“…and extended to the whole path algebra by linearity and the Leibniz rule. Etgü and Lekili [EL19] computed the Chekanov-Eliashberg dg-algebra of the Legendrian attaching spheres of a plumbing of copies of T * S n and showed by direct computation that it is quasi-isomorphic to G n Q for n = 2 where Q is a Dynkin quiver of type A and D. The result for the remaining Dynkin cases Q = E 6 , E 7 , E 8 was proven by Lekili and Ueda [LU20, Section 5].…”

Section: Applicationsmentioning

confidence: 98%

Simplicial descent for Chekanov-Eliashberg dg-algebras

Asplund¹

2021

Preprint

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We introduce a type of surgery decomposition of Weinstein manifolds we call simplicial decompositions. The main result of this paper is that the Chekanov-Eliashberg dg-algebra of the attaching spheres of a Weinstein manifold satisfies a descent (cosheaf) property with respect to a simplicial decomposition. Simplicial decompositions generalize the notion of Weinstein connected sum and we show that there is a one-to-one correspondence (up to Weinstein homotopy) between simplicial decompositions and so-called good sectorial covers. As an application we explicitly compute the Chekanov-Eliashberg dg-algebra of the Legendrian attaching spheres of a plumbing of copies of cotangent bundles of spheres of dimension at least three according to any plumbing quiver. We show by explicit computation that this Chekanov-Eliashberg dg-algebra is quasi-isomorphic to the Ginzburg dg-algebra of the plumbing quiver. Contents 19 3.1. Simplicial decompositions and good sectorial covers 19 3.2. Relation to sectorial descent 26 4. Applications 27 4.1. Legendrian submanifolds-with-boundary-and-corners 27 4.2. The Chekanov-Eliashberg cosheaf 30 4.3. Plumbing of cotangent bundles of spheres 31 References 36

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Section: Applicationsmentioning

confidence: 98%

Simplicial descent for Chekanov-Eliashberg dg-algebras

Asplund¹

2021

Preprint

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“…For related results which identify certain (partially) wrapped Fukaya categories with relative Calabi–Yau completions (see ).…”

Section: Combinatorial Computationsmentioning

confidence: 99%

“…However, we should remark that the Liouville manifolds which satisfy the Koszul duality form only a very restrictive class. Known counterexamples include cotangent bundles of most of the non‐simply connected manifolds and their plumbings, and plumbings of

T^{*} S^{2}

according to a non‐Dynkin tree . The common feature of these counterexamples is that the degree 0 part of the symplectic cohomology

{italicSH}^{*} false(M false)

is infinite‐dimensional.…”

Section: Introductionmentioning

confidence: 99%

Koszul duality via suspending Lefschetz fibrations

2019

Journal of Topology

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Let M be a Liouville 6‐manifold which is the smooth fiber of a Lefschetz fibration on C4 constructed by suspending a Lefschetz fibration on C3. We prove that for many examples including stabilizations of Milnor fibers of hypersurface cusp singularities, the compact Fukaya category F(M) and the wrapped Fukaya category W(M) are related through A∞‐Koszul duality, by identifying them with cyclic and Calabi–Yau completions of the same quiver algebra. This implies the split‐generation of the compact Fukaya category F(M) by vanishing cycles. Moreover, new examples of Liouville manifolds which admit quasi‐dilations in the sense of Seidel–Solomon are obtained.

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“…Subsequently they have been shown to arise as moduli spaces of irregular connections in the work [3,4] of Boalch and Yamakawa (indeed Boalch's work [3] lead him to define an even more general notion of multiplicative quiver variety than that considered here). Bezrukavnikov and Kapranov [2] realise them as moduli of microlocal sheaves on nodal curves (see also the work of Crawley-Boevey [11]), while in symplectic topology the work of Etgü and Lekili [15] shows that the Fukaya categories of certain symplectic four-manifolds, which are built from quiver-type data, are controlled by a derived version of the associated multiplicative preprojective algebra. Moreover, results of Chalykh and Fairon [7] and Braverman-Etingof-Finkelberg [5] reveal exciting new connections between multiplicative quiver varieties and new families of integrable systems which have also been constructed using double affine Hecke algebras.…”

Section: Introductionmentioning

confidence: 99%

The pure cohomology of multiplicative quiver varieties

McGerty

Nevins

2021

Sel. Math. New Ser.

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To a quiver Q and choices of nonzero scalars $$q_i$$ q i , non-negative integers $$\alpha _i$$ α i , and integers $$\theta _i$$ θ i labeling each vertex i, Crawley-Boevey–Shaw associate a multiplicative quiver variety$${\mathcal {M}}_\theta ^q(\alpha )$$ M θ q ( α ) , a trigonometric analogue of the Nakajima quiver variety associated to Q, $$\alpha $$ α , and $$\theta $$ θ . We prove that the pure cohomology, in the Hodge-theoretic sense, of the stable locus $${\mathcal {M}}_\theta ^q(\alpha )^{{\text {s}}}$$ M θ q ( α ) s is generated as a $${\mathbb {Q}}$$ Q -algebra by the tautological characteristic classes. In particular, the pure cohomology of genus g twisted character varieties of $$GL_n$$ G L n is generated by tautological classes.

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Fukaya categories of plumbings and multiplicative preprojective algebras

Cited by 15 publications

References 25 publications

Simplicial descent for Chekanov-Eliashberg dg-algebras

Simplicial descent for Chekanov-Eliashberg dg-algebras

Koszul duality via suspending Lefschetz fibrations

The pure cohomology of multiplicative quiver varieties

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