Quantum periods and TBA equations for $\mathcal{N}=2\ SU(2)\ N_f=2$ SQCD with flavor symmetry

Abstract

We apply the exact WKB analysis to the quantum Seiberg-Witten curve for 4-dimensional N = 2 SU (2) N f = 2 SQCD with the flavor symmetry. The discontinuity and the asymptotic behavior of the quantum periods define a Riemann-Hilbert problem. We derive the thermodynamic Bethe ansatz (TBA) equations as a solution to this problem. We also compute the effective central charge of the underlying CFT, which is shown to be proportional to the one-loop beta function of the SQCD.

A new method for exact results on Quasinormal Modes of Black Holes

“…We have to numerically input l in the TBA with the boundary condition at θ → −∞: ε±,±(θ) ≃ 4P θ ≃ 4(l + 1/2)θ, also following from the asymptotic of the ODE (25) (the precision improves by adding also the constant at the subleading order). This TBA is a generalization of those found in [29,31] for the Perturbed Hairpin IM and the N f = 2 gauge theory with equal masses m 1 = m 2 respectively, although we shall pay particular attention to the change of variables from gravity or gauge to integrability: this results in n l TBA Leaver WKB 0 1 0.86814 − 0.36917i 0.86893 − 0.37286i 0.89642 − 0.36596i 0 2 1.4755 − 0.36411i 1.4778 − 0.3682i 1.4940 − 0.36596i 0 3 2.0767 − 0.36178i 2.0801 − 0.3671i 2.0916 − 0.36596i 0 4 2.6758 − 0.36007i 2.6803 − 0.3666i 2.6893 − 0.36596i (33), (through (34) with n ′ = 0) and WKB approximation (Σ1 = 0.1, Σ2 = 0.2, Σ3 = 0.3, Σ4 = 1). Since Σ1 = Σ3 the Leaver method seems not applicable (N.A.…”

Wall-crossing of TBA equations and WKB periods for the third order ODE

“…See [70,71] for the case related to the second order ODE. It is also interesting to include more irregular/regular singular points in the potential, which will help us to study the four dimensional N = 2 super Yang-Mills theory [17,18,20,22,23].…”

“…One can determine the quantum periods directly by the Thermodynamic Bethe ansatz (TBA) equations [15] satisfied by the Y-functions. See also [16][17][18][19][20][21][22][23][24], for more general potential.…”