Direct Integration for Mirror Curves of Genus Two and an Almost Meromorphic Siegel Modular Form

Klemm, Albrecht; Poretschkin, Maximilian; Schimannek, Thorsten; Raum, Martin

doi:10.48550/arxiv.1502.00557

Cited by 19 publications

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“…The definition of the matrix C is given in Sec. 2.1 and it is completely determined by the toric data of X Σ as explained in [65]. The g Σ -dimensional integral vector r is an explicit "representation" of the B-field satisfying…”

Section: Quantization Conditions and Topological Stringsmentioning

confidence: 99%

BPS relations from spectral problems and blowup equations

Grassi

2019

Lett Math Phys

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Recently an exact duality between topological string and the spectral theory of operators constructed from mirror curves to toric Calabi-Yau threefolds has been proposed. At the same time an exact quantization condition for the cluster integrable systems associated to these geometries has been conjectured. The consistency between the two approaches leads to an infinite set of constraints for the refined BPS invariants of the toric Calabi-Yau threefolds. We show that, for the Y N,m geometries, these constraints can be derived from the K-theoretic blowup equations for SU (N ) SYM with generic Chern-Simons invariant m. In particular this furnishes a proof of the BPS constraints in the case of m = 0.

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Section: Quantization Conditions and Topological Stringsmentioning

confidence: 99%

BPS relations from spectral problems and blowup equations

Grassi

2019

Lett Math Phys

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“…Solving the anomaly equation can be done with the direct integration procedure [18,20,25]. Here we will use the direct integration method from [21], which is formulated in terms of the propagator S ij , the latter being defined by…”

Section: Direct Integration Procedures In Terms Of Propagatorsmentioning

confidence: 99%

“…From special geometry F (0,0) can be determined. For F (1,0) and F (0,1) , respectively, we take [25] F (1,0) = k 1 24 log (∆ ξ α ) and…”

Section: Direct Integration Procedures In Terms Of Propagatorsmentioning

confidence: 99%

“…Despite the fact that part of the original motivation of this work was to use the extension of the SL(2, Z) modular approach of the g = 1 case to the one of Siegel modular forms in the higher genus case in order to solve the holomorphic anomaly equation as in [25], it has turned out to be much easier to extend the D 2n to a system of multi-parameter operators that yield the higher correction terms from the classical periods. We develop this formalism, which applies to arbitrary genus, and exemplify it with the quintic potential with additional deformation parameters turned on.…”

Section: Introductionmentioning

confidence: 99%

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WKB method and quantum periods beyond genus one

Fischbach¹,

Klemm

Nega³

2019

J. Phys. A: Math. Theor.

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We extend topological string methods in order to perform WKB approximations for quantum mechanical problems with higher order potentials efficiently. This requires techniques for the evaluation of the relevant quantum periods for Riemann surfaces beyond genus one. The basis of these quantum periods is fixed using the leading behaviour of the classical periods. The full expansion of the quantum periods is obtained using a system of Picard-Fuchs like operators for a sequence of integrals of meromorphic forms of the second kind. Discrete automorphisms of simple higher order potentials allow to view the corresponding higher genus curves as covering of a genus one curve. In this case the quantum periods can be alternatively obtained using the holomorphic anomaly solved in the holomorphic limit within the ring of quasi modular forms of a congruent subgroup of SL(2, Z) as we check for a symmetric sextic potential. 1

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“…Most of classical computations have already done in e.g. [36,51,41,42,22], and we gather the results to make the paper self-contained. In the following computations, we may omit some arguments in the functions for short notation.…”

Section: Examplesmentioning

confidence: 99%

Quantum Periods and Spectra in Dimer Models and Calabi-Yau Geometries

Huang,

Sugimoto,

Wang

2020

Preprint

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We study a class of quantum integrable systems derived from dimer graphs and also described by local toric Calabi-Yau geometries with higher genus mirror curves, generalizing some previous works on genus one mirror curves. We compute the spectra of the quantum systems both by standard perturbation method and by Bohr-Sommerfeld method with quantum periods as the phase volumes. In this way, we obtain some exact analytic results for the classical and quantum periods of the Calabi-Yau geometries. We also determine the differential operators of the quantum periods and compute the topological string free energy in Nekrasov-Shatashvili (NS) limit. The results agree with calculations from other methods such as the topological vertex.

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Direct Integration for Mirror Curves of Genus Two and an Almost Meromorphic Siegel Modular Form

Cited by 19 publications

References 0 publications

BPS relations from spectral problems and blowup equations

BPS relations from spectral problems and blowup equations

WKB method and quantum periods beyond genus one

Quantum Periods and Spectra in Dimer Models and Calabi-Yau Geometries

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