Sheaf quantization from exact WKB analysis

Kuwagaki, Tatsuki

doi:10.48550/arxiv.2006.14872

Cited by 6 publications

(8 citation statements)

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“…As is mentioned in [Br1,§7], the BPS Riemann-Hilbert problem is also closely related to the Stokes structure of the Voros symbols in the theory of exact WKB analysis of a Schrödinger-type ODE discussed in [IN1,IN2]. See [I1,Al1,Ku] for a relation between Fock-Goncharov coordinates and Voros symbols, and see also [Al2] for the further development in this direction. In our context, the BPS structure for the BPS Riemann-Hilbert problem arises from the meromorphic quadratic differential which appears in the classical limit of the ODE.…”

Section: Bps Structures and Spectral Networkmentioning

confidence: 97%

Topological recursion and uncoupled BPS structures I: BPS spectrum and free energies

Iwaki¹,

Kidwai²

2020

Preprint

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For the hypergeometric spectral curve and its confluent degenerations, we obtain a simple formula expressing the topological recursion free energies as a sum over BPS states (degenerate spectral networks) for an appropriate quadratic differential. In doing so, we provide a complete description of the corresponding BPS structures over a generic locus in the parameter space. In particular, we count degenerations of spectral networks occurring for the nine examples considered. We conjecture that a similar relation should hold more generally whenever the corresponding BPS structure is uncoupled.

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Section: Bps Structures and Spectral Networkmentioning

confidence: 97%

Topological recursion and uncoupled BPS structures I: BPS spectrum and free energies

Iwaki¹,

Kidwai²

2020

Preprint

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“…describe the mutation of the Fock-Goncharov cluster coordinates (on the moduli space of flat SL 2 (C)-connections) caused by the degeneration of the spectral network. On the other hand, as is shown in [I1,All1,Ku], the Fock-Goncharov cluster coordinates can be identified (on the oper locus, where this makes sense) with the Borel-resummed Voros symbols (exponential of Voros coefficients) in the exact WKB analysis of a certain Schrödinger-type equation with the spectral curve as its classical limit (see also [IN1,IN2]), where the mutation formula (BPS automorphism) is a consequence of the Stokes phenomenon for the Voros symbols.…”

Section: Introductionmentioning

confidence: 82%

“…In fact, it is easy to see this jump formula is in fact nothing but the required property (2.10) in the BPS Riemann-Hilbert problem. The purpose of this subsection is to solve the BPS Riemann-Hilbert problem, and thus show that our definition (3.24) of BPS indices proposed in [IK] is consistent with this point of view if we identify the Fock-Goncharov coordinates with the Borel-resummed Voros symbols (c.f., [I1,All2,Ku]).…”

Section: Solution To Bps Riemann-hilbert Problem and The τ -Functionmentioning

confidence: 99%

Topological recursion and uncoupled BPS structures II: Voros symbols and the $τ$-function

Iwaki¹,

Kidwai²

2021

Preprint

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We continue our study of the correspondence between BPS structures and topological recursion in the uncoupled case, this time from the viewpoint of quantum curves. For spectral curves of hypergeometric type, we show the Borel-resummed Voros symbols of the corresponding quantum curves solve Bridgeland's "BPS Riemann-Hilbert problem". In particular, they satisfy the required jump property in agreement with the generalized definition of BPS indices Ω in our previous work. Furthermore, we observe the Voros coefficients define a closed one-form on the parameter space, and show that (log of) Bridgeland's τ -function encoding the solution is none other than the corresponding potential, up to a constant. When the quantization parameter is set to a special value, this agrees with the Borel sum of the topological recursion partition function Z TR , up to a simple factor. KOHEI IWAKI AND OMAR KIDWAI 4.7. Stokes phenomenon for Voros symbols 35 4.8. The Voros potential 36 5. Solution to BPS Riemann-Hilbert problem and the τ -function 37 5.1. BPS indices and the Stokes structure of the Voros coefficients 37 5.2. Relation to the minimal and holomorphic solutions 39 5.3. BPS τ -function and the Voros potential 41 5.4. Borel-resummed TR partition function as a BPS τ -function 43 5.5. Higher degree spectral curves 44 Appendix A. Explicit expression for Voros coefficients 46 References 47

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“…Another point of view is provided by the 'holomorphic Floer theory' conjectures-in-progress of Kontsevich and Soibelman [34]; my vague understanding of which is that one finds, in examples, an identification between the flow tree limit of Floer theory with Stokes rays associated to corresponding D-modules. See [36] for another point of view on this conjectural correspondence.…”

Section: Floer Theory In Hyperk äHler Manifoldsmentioning

confidence: 99%

Microsheaves from Hitchin fibers via Floer theory

Shende¹

2021

Preprint

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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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Sheaf quantization from exact WKB analysis

Cited by 6 publications

References 0 publications

Topological recursion and uncoupled BPS structures I: BPS spectrum and free energies

Topological recursion and uncoupled BPS structures I: BPS spectrum and free energies

Topological recursion and uncoupled BPS structures II: Voros symbols and the $τ$-function

Microsheaves from Hitchin fibers via Floer theory

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