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

Abstract

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.

Brane Transitions from Exceptional Groups

“…In section 4.4, we have considered more general setups to discuss the implications of the proposed brane transitions and find symmetries of infinite dimensions. To discuss the infinite-dimensional symmetries for these setups, it is important to extend the studies of spectral determinants to higher genus in [26,29,41] or [42][43][44].…”

Quantum Periods and TBA-like Equations for a Class of Calabi-Yau Geometries

“…Certainly, it would be interesting to further generalize the results to more Calabi-Yau geometries and consider a bigger moduli space instead of the one-parameter space in this paper. For example, in the case of local Calabi-Yau geometries with multiple A-periods, it is found that their quantum corrections are described by the same differential operators [10]. It seems that the general formalism here would need to be much improved to find all the TBA-like equations for the quantum periods of the different A-cycles of a Calabi-Yau geometry.…”

“…[6,7]. These developments in quantum periods lead to exciting results such as the calculations of topological string free energy in the NS limit [8], quantum spectra [9,10], exact quantizations including non-perturbative effects [11,12]. The quantum mirror maps for a class of del Pezzo Calabi-Yau geometries with exceptional symmetry are recently studied in [13].…”

Quantum Representation of Affine Weyl Groups and Associated Quantum Curves

“…• In relation to the spectral problem mentioned above, computation of the quantum period integrals is also an interesting problem [37,1,2,21,22,23]. Even in genus one cases they are technical challenges in particular for the fully massive E 6 , E 7 , E 8 cases.…”