Exact solutions to quantum spectral curves by topological string theory

Tuesday, 30 June, 2015

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We generalize the conjectured connection between quantum spectral problems and topological strings to many local almost del Pezzo surfaces with arbitrary mass parameters. The conjecture uses perturbative information of the topological string in the unrefined and the Nekrasov-Shatashvili limit to solve non-perturbatively the quantum spectral problem. We consider the quantum spectral curves for the local almost del Pezzo surfaces of F2, F1, the blowup of P2 in two points and a mass deformation of the E8 del Pezzo corresponding to different deformations of the three-term operators O1,1, O1,2 and O2,3. To check the conjecture, we compare the predictions for the spectrum of these operators with numerical results for the eigenvalues. We also compute the first few fermionic spectral traces from the conjectural spectral determinant, and we compare them to analytic and numerical results in spectral theory. In all these comparisons, we find that the conjecture is fully validated with high numerical precision. For local F2 we expand the spectral determinant around the orbifold point and find intriguing relations for Jacobi theta functions. We also give an explicit map between the geometries of F0 and F2 as well as a systematic way to derive the operators Om,n from toric geometries.


Jie Gu
Albrecht Klemm
Marcos Marino
Jonas Reuter