Atiyah-Floer Conjecture: a Formulation, a Strategy to Prove and Generalizations

Daemi, Aliakbar; Fukaya, Kenji

doi:10.48550/arxiv.1707.03924

Cited by 9 publications

(11 citation statements)

References 29 publications

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“…More ambitiously, one could try and work with or around the symplectic singularities; see [38,12] for some ideas in this direction in a similar setting.…”

Section: Remark the Map Shmentioning

confidence: 99%

Microsheaves from Hitchin fibers via Floer theory

Shende¹

2021

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Consider a smooth component of the moduli space of stable Higgs bundles on which the Hitchin fibration is proper. We record in this note the following corollaries to existing results on Floer theory and Fukaya categories. (1) Any any smooth Hitchin fiber determines a microsheaf on the global nilpotent cone. (2) Distinct fibers give rise to orthogonal microsheaves. (3) The endomorphisms of the microsheaf is isomorphic to the cohomology of the Hitchin fiber.Any sheaf on the moduli stack of bundles which is microsupported in the nilpotent cone restricts (by definition) to a microsheaf on the locus of stable and nilpotent Higgs bundles. Conversely, such microsheaves can be restricted to the moduli of stable bundles, and then extended to the stack of all bundles. We expect (but do not prove) that our microsheaves should restrict from, and extend to, the Hecke eigensheaves of the geometric Langlands programme. HITCHIN FIBERS AND HECKE EIGENSHEAVESLet us recall what Hecke eigensheaves are, and why one might hope to get them from fibers of the Hitchin system. More detailed discussions can be found in e.g. [7,32,15]. This section is purely motivational and logically unrelated to the results presented in the remainder of the article.Let C be a smooth compact complex curve, G a reductive group, and Bun G (C) the moduli of G-bundles on C. We write [X] for a category of sheaves on X, whose precise nature is irrelevant here. There are 'Hecke operators'given by convolution with respect to a correspondence parameterizing pairs of bundles which are isomorphic away from a single point c ∈ C, and whose difference at this point is controlled by a cocharacter µ of G.The geometric Langlands conjecture asserts, in particular, the existence of certain F ∈ [Bun G (C)] which are 'Hecke eigensheaves' in the sense that H µ (F ) = F ρ µ (χ), where χ is a G ∨ local system on C, and ρ µ is the representation corresponding to the character µ of G ∨ .The Hecke action transforms microsupports (or characteristic cycles) by setwise convolution with the conic Lagrangian ss(H µ ) ⊂ T * Bun G (C) × T * Bun G (C) × T * C; which is roughly the conormal to the image of the Hecke correspondence. This conormal respects the fibers of Hitchin's integrable system h :is the spectral cover corresponding to the point b ∈ B and representation µ.The above geometry suggests that any sufficiently functorial procedure relating Hitchin fibers to sheaves on Bun G (C) -and spectral curves to sheaves on C -may be expected to yield Hecke eigensheaves. Quantization is one such procedure: the semiclassical limit of a quantum state on M determines a Lagrangian submanifold of T * M (see e.g. [6]). Taking M = Bun G (C), and the corresponding Lagrangians to be the Hitchin fibers, quantizing the Hitchin system may be expected to yield Hecke eigensheaves, and indeed does [7]. Related approaches include [5,14,32,9,15]. This note explains how to use Floer theory to produce sheaves on Bun G (C) from Hitchin fibers.

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“…More ambitiously, one could try and work with or around the symplectic singularities; see [38,12] for some ideas in this direction in a similar setting.…”

Section: Remark the Map Shmentioning

confidence: 99%

Microsheaves from Hitchin fibers via Floer theory

Shende¹

2021

Preprint

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“…Remark 3.8.3 (A way of proving Conjecture 3.8.1). In [DF17], Daemi and Fukaya have proposed a proof for the Atiyah-Floer conjecture. In particular, they construct a different version of Lagrangian Floer homology and translate the Atiyah-Floer conjecture to the equivalent conjecture which states that this new version is actually isomorphic to H inst F • .…”

Section: 3mentioning

confidence: 99%

4-Manifold Topology, Donaldson-Witten Theory, Floer Homology and Higher Gauge Theory Methods in the BV-BFV Formalism

Moshayedi¹

2021

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We study the behavior of Donaldson's invariants of 4-manifolds based on the moduli space of anti self-dual connections (instantons) in the perturbative field theory setting where the underlying source manifold has boundary. It is well-known that these invariants take values in the instanton Floer homology groups of the boundary 3-manifold. Gluing formulae for these constructions lead to a functorial topological field theory description according to a system of axioms developed by Atiyah, which can be also regarded in the setting of perturbative quantum field theory, as it was shown by Witten, using a version of supersymmetric Yang-Mills theory, known today as Donaldson-Witten theory. One can actually formulate an AKSZ model which recovers this theory for a certain gauge-fixing. We consider these constructions in a perturbative quantum gauge formalism for manifolds with boundary that is compatible with cutting and gluing, called the BV-BFV formalism, which was recently developed by Cattaneo, Mnev and Reshetikhin. We prove that this theory satisfies a modified Quantum Master Equation and extend the result to a global picture when perturbing around constant background fields. These methods are expected to extend to higher codimensions and thus might help getting a better understanding for fully extendable n-dimensional field theories (in the sense of Baez-Dolan and Lurie) in the perturbative setting, especially when n ≤ 4. Additionally, we relate these constructions to Nekrasov's partition function by treating an equivariant version of Donaldson-Witten theory in the BV formalism. Moreover, we discuss the extension, as well as the relation, to higher gauge theory and enumerative geometry methods, such as Gromov-Witten and Donaldson-Thomas theory and recall their correspondence conjecture for general Calabi-Yau 3-folds. In particular, we discuss the corresponding (relative) partition functions, defined as the generating function for the given invariants, and gluing phenomena. 3.2. The (holomorphic) Chern-Simons action functional 3.3. A 4D-3D bulk-boundary correspondence on (infinite) cylinders 3.4. Instanton Floer homology 3.5. Relation to Donaldson polynomials 3.6. Field theory approach to instanton Floer homology 3.7. Lagrangian Floer homology 3.8. The Atiyah-Floer conjecture 4. The BV-BFV formalism 4.1. BV formalism 4.1.1. Quantization 4.2. BV algebras 4.3. BFV formalism 4.4. BV-BFV formalism 4.4.1. Quantization 4.5. Gluing of BV-BFV partition functions 4.6. Example: abelian BF theory 4.7. BV-BF k V extension and shifted symplectic structures 5. AKSZ formulation of DW theory 5.1.

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“…In [HLS16a,HLS16b], Hendricks, Lipshitz, and Sarkar used a homotopy theoretic method to define G-equivariant Lagrangian Floer homology for G a compact Lie group. More recently, in [DF17], Daemi and Fukaya defined an equivariant de Rham model using G-equivariant Kuranishi structures developed in [FOOO09b,Fuk17].…”

Section: A Morse Model For Equivariant Lagrangian Floer Theorymentioning

confidence: 99%

“…Hendricks-Lipshitz-Sarkar [HLS16a,HLS16b] developed a homotopy coherent method to build up a G-equivariant Floer theory. Fukaya [Fuk17] and Daemi-Fukaya [DF17] used G-equivariant Kuranishi structure to tackle the G-equivariant transversality problem, and make a formulation using differential forms. There are other interesting works related to this subject [Sei15,LP16].…”

Section: Introductionmentioning

confidence: 99%

$T$-equivariant disc potential and SYZ mirror construction

Kim,

Lau,

Zheng

2019

Preprint

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We develop a G-equivariant Lagrangian Floer theory by counting pearly trees in the Borel construction L G . We apply the construction to smooth moment-map fibers of toric semi-Fano varieties and obtain the T-equivariant Landau-Ginzburg mirrors. We also apply this to the typical S 1 -invariant SYZ singular fiber, which is the single-pinched torus, and compute its S 1 -equivariant disc potential. Contents 1. Introduction 1 2. A Morse model for equivariant Lagrangian Floer theory 4 2.1. The non-equivariant singular chain model 5 2.2. The non-equivariant Morse model with a strict unit 8 2.3. The equivariant Morse model 15 2.4. Homotopy partial units and the equivariant disc potentials 20 2.5. Equivariant Floer theory for Lagrangian immersions 25 3. The T-equivariant disc potentials of toric manifolds 26 3.1. Morse theory on the approximation spaces 27 3.2. Computing the T-equivariant disc potentials 31 4. S 1 -equivariant disc potential for the immersed 2-sphere 36 References 40

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Atiyah-Floer Conjecture: a Formulation, a Strategy to Prove and Generalizations

Cited by 9 publications

References 29 publications

Microsheaves from Hitchin fibers via Floer theory

Microsheaves from Hitchin fibers via Floer theory

4-Manifold Topology, Donaldson-Witten Theory, Floer Homology and Higher Gauge Theory Methods in the BV-BFV Formalism

$T$-equivariant disc potential and SYZ mirror construction

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