Non-perturbative approaches to the quantum Seiberg-Witten curve

Abstract

We study various non-perturbative approaches to the quantization of the Seiberg-Witten curve of $\mathcal{N}=2$, $SU(2)$ super Yang-Mills theory, which is closely related to the modified Mathieu operator. The first approach is based on the quantum WKB periods and their resurgent properties. We show that these properties are encoded in the TBA equations of Gaiotto-Moore-Neitzke determined by the BPS spectrum of the theory, and we relate the Borel-resummed quantum periods to instanton calculus. In addition, we use the TS/ST correspondence to obtain a closed formula for the Fredholm determinant of the modified Mathieu operator. Finally, by using blowup equations, we explain the connection between this operator and the $\tau$ function of Painleve $\mathrm{III}$.

46 pages, 4 figures, 12 tables; new references added, typos corrected

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• The resurgent structure of quantum knot invariants
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• WKB periods for higher order ODE and TBA equations
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• Quantum spectral problems and isomonodromic deformations
• Exact WKB analysis and TBA equations for the Mathieu equation
• Wall-crossing of TBA equations and WKB periods for the third order ODE
• Resurgence in the Bi-Yang-Baxter Model
• Quantization of Harer-Zagier formulas
• Conformal TBA for resolved conifolds
• Recursion relation for instanton counting for $SU(2)$ $\mathcal{N}=2$ SYM in NS limit of $\Omega$ background
• Resurgence in Lorentzian quantum cosmology: No-boundary saddles and resummation of quantum gravity corrections around tunneling saddle points
• Quantum phase transition and Resurgence: Lessons from 3d $\mathcal{N}=4$ SQED
• Quantum Seiberg-Witten periods for $\mathcal{N}=2$ $SU(N_c)$ SQCD around the superconformal point
• Quantum periods and TBA equations for $\mathcal{N}=2SU(2)N_f=2$ SQCD with flavor symmetry
• Irregular conformal blocks, Painlevé III and the blow-up equations
• TBA equations and exact WKB analysis in deformed supersymmetric quantum mechanics
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• WKB analysis of the linear problem for modified affine Toda field equations
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• Painlevé Kernels and Surface Defects at Strong Coupling
• A short review on TBA equation and scattering amplitude/Wilson loop duality