A-branes, Foliations and Localization

Banerjee, Sibasish; Longhi, Pietro; Romo, Mauricio

doi:10.1007/s00023-022-01231-8

Cited by 5 publications

(4 citation statements)

References 74 publications

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“…with X γ as defined in (51). Below we will provide supporting evidence for this relation in a few examples.…”

Section: Introductionmentioning

confidence: 59%

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The threefold way to quantum periods: WKB, TBA equations and q-Painlevé

Del Monte,

Longhi

2023

SciPost Phys.

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We show that TBA equations defined by the BPS spectrum of 5d5d\mathcal{N}=1𝒩=1SU(2)SU(2) Yang-Mills on S^1\times \mathbb{R}^4S1×ℝ4 encode the q-Painlevé III_33 equation. We find a fine-tuned stratum in the physical moduli space of the theory where solutions to TBA equations can be obtained exactly, and verify that they agree with the algebraic solutions to q-Painlevé. Switching from the physical moduli space to that of stability conditions, we identify two one-parameter deformations of the fine-tuned stratum, where the general solution of the q-Painlevé equation in terms of dual instanton partition functions continues to provide explicit TBA solutions. Motivated by these observations, we propose a further extensions of the range of validity of this correspondence, under a suitable identification of moduli. As further checks of our proposal, we study the behavior of exact WKB quantum periods for the quantum curve of local \mathbb{P}^1\times\mathbb{P}^1ℙ1×ℙ1.

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“…with X γ as defined in (51). Below we will provide supporting evidence for this relation in a few examples.…”

Section: Introductionmentioning

confidence: 59%

“…as explained in [42, Section 3]. Readers are referred to [42][43][44][45][46][47][48][49][50][51] and references therein for background and for more details on the case at hand. The intersection pairing of the four basis cycles is…”

Section: Remarkmentioning

confidence: 99%

The threefold way to quantum periods: WKB, TBA equations and q-Painlevé

Del Monte,

Longhi

2023

SciPost Phys.

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“…Although our guiding example in this paper is the resolved conifold and its mirror, we are certain that many aspects of our work can be generalized to a larger class of geometries, and in particular to toric Calabi-Yau threefolds. 15 In this case, the mirror curves are available and the corresponding quantum curves have been intensively studied. While the closed quantum curve in our work was obtained from the explicit knowledge of the asymptotic series of the topological string free energy in this case, we think that the path to generalization is through understanding the more general link between the open and the closed quantum curve.…”

Section: Discussion and Outlookmentioning

confidence: 99%

“…In the dual 5d gauge theory these topology changes correspond to 5d BPS particles. Although the spectrum for more general toric geometries may be extracted from the corresponding Stokes graphs (see [14] for the example of the local P 1 × P 1 geometry), it is not clear whether the corresponding phases ϑ have a similar interpretation in terms of the Borel sum of the corresponding NS free energy (some aspects of this Borel resummation have been studied numerically in [54]) and which type of invariants these BPS states correspond to in terms of the Calabi-Yau geometry (see [15] for recent progress on this).…”

Section: Discussion and Outlookmentioning

confidence: 99%

Quantum Curves, Resurgence and Exact WKB

Alim¹,

Hollands²,

Tulli³

2023

SIGMA

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We study the non-perturbative quantum geometry of the open and closed topological string on the resolved conifold and its mirror. Our tools are finite difference equations in the open and closed string moduli and the resurgence analysis of their formal power series solutions. In the closed setting, we derive new finite difference equations for the refined partition function as well as its Nekrasov-Shatashvili (NS) limit. We write down a distinguished analytic solution for the refined difference equation that reproduces the expected non-perturbative content of the refined topological string. We compare this solution to the Borel analysis of the free energy in the NS limit. We find that the singularities of the Borel transform lie on infinitely many rays in the Borel plane and that the Stokes jumps across these rays encode the associated Donaldson-Thomas invariants of the underlying Calabi-Yau geometry. In the open setting, the finite difference equation corresponds to a canonical quantization of the mirror curve. We analyze this difference equation using Borel analysis and exact WKB techniques and identify the 5d BPS states in the corresponding exponential spectral networks. We furthermore relate the resurgence analysis in the open and closed setting. This guides us to a five-dimensional extension of the Nekrasov-Rosly-Shatashvili proposal, in which the NS free energy is computed as a generating function of q-difference opers in terms of a special set of spectral coordinates. Finally, we examine two spectral problems describing the corresponding quantum integrable system.

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Hadamard products and BPS networks

Elmi

2024

J. High Energ. Phys.

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We study examples of fourth-order Picard-Fuchs operators that are Hadamard products of two second-order Picard-Fuchs operators. Each second-order Picard-Fuchs operator is associated with a family of elliptic curves, and the Hadamard product computes period integrals on the fibred product of the two elliptic surfaces. We construct 3-cycles on this geometry as the union of 2-cycles in the fibre over contours on the base. We then use the special Lagrangian condition to constrain the contours on the base. This leads to a construction that is reminiscent of spectral networks and exponential networks that have previously appeared in string theory literature.

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A-branes, Foliations and Localization

Cited by 5 publications

References 74 publications

The threefold way to quantum periods: WKB, TBA equations and q-Painlevé

The threefold way to quantum periods: WKB, TBA equations and q-Painlevé

Quantum Curves, Resurgence and Exact WKB

Hadamard products and BPS networks

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