We derive the Thermodynamic Bethe Ansatz (TBA) equations for the exact WKB periods of the Mathieu equation in the weak coupling region. We will use the TBA equations to calculate the quantum corrections to the WKB periods, which are regarded the quantum periods of $\mathcal{N} = 2$ $SU(2)$ super Yang-Mills theory at strong coupling. We calculate the effective central charge of the TBA equations, which is found to be proportional to the coefficient of the one-loop beta function of the 4d theory. We also study the spectral problem for the Mathieu equation based on the TBA equations numerically.
TBA equations for $SU(r+1)$ quantum Seiberg-Witten curve: higher-order Mathieu equation
Institute for Astrophysics, School of Physics, Zhengzhou University（郑州大学物理学院天体物理研究所）；Beijing Institute of Mathematical Sciences and Applications（北京数学与应用研究院）；Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）
In this paper, based on lectures by the authors at the May 2015 workshop {\it Resurgence, Physics and Numbers}, at the Centro di Ricerca Matematica Ennio De Giorgio of the Scuola Normale Superiore in Pisa, we explain the origin of resurgent trans-series in the Mathieu equation spectral problem, using uniform WKB and all-orders (exact) WKB. Exact quantization conditions naturally arise, and their expansion in the weak coupling regime produces resurgent trans-series expressions which exhibit precise relations between different instanton sectors. Indeed, the perturbative expansion encodes all ord…
We study the WKB periods for the third order ordinary differential equation (ODE) with polynomial potential, which is obtained by the Nekrasov-Shatashvili limit of ($A_2,A_N$) Argyres-Douglas theory in the Omega background. In the minimal chamber of the moduli space, we derive the Y-system and the thermodynamic Bethe ansatz (TBA) equations by using the ODE/IM correspondence. The exact WKB periods are identified with the Y-functions. Varying the moduli parameters of the potential, the wall-crossing of the TBA equations occurs. We study the process of the wall-crossing from the minimal chamber t…
We study the WKB periods for the $(r+1)$-th order ordinary differential equation (ODE) which is obtained by the conformal limit of the linear problem associated with the $A_r^{(1)}$ affine Toda field equation. We compute the quantum corrections by using the Picard-Fuchs operators. The ODE/IM correspondence provides a relation between the Wronskians of the solutions and the Y-functions which satisfy the thermodynamic Bethe ansatz (TBA) equation related to the Lie algebra $A_r$. For the quadratic potential, we propose a formula to show the equivalence between the logarithm of the Y-function and …
It has been recently realized that, in the case of polynomial potentials, the exact WKB method can be reformulated in terms of a system of TBA equations. In this paper we study this method in various examples. We develop a graphical procedure due to Toledo, which provides a fast and simple way to study the wall-crossing behavior of the TBA equations. When complemented with exact quantization conditions, the TBA equations can be used to solve spectral problems exactly in Quantum Mechanics. We compute the quantum corrections to the all-order WKB periods in many examples, as well as the exact spe…
We show that TBA equations defined by the BPS spectrum of $5d$ $\mathcal{N}=1$ $SU(2)$ Yang-Mills on $S^1\times \mathbb{R}^4$ encode the q-Painlevé III$_3$ equation. We find a fine-tuned stratum in the physical moduli space of the theory where solutions to TBA equations can be obtained exactly, and verify that they agree with the algebraic solutions to q-Painlevé. Switching from the physical moduli space to that of stability conditions, we identify a one-parameter deformation of the fine-tuned stratum, where the general solution of the q-Painlevé equation in terms of dual instanton partition f…
We study in detail the Schrödinger equation corresponding to the four dimensional $SU(2)$ $\mathcal{N}=2$ SQCD theory with one flavour. We calculate the Voros symbols, or quantum periods, in four different ways: Borel summation of the WKB series, direct computation of Wronskians of exponentially decaying solutions, the TBA equations of Gaiotto-Moore-Neitzke/Gaiotto, and instanton counting. We make computations by all of these methods, finding good agreement. We also study the exact quantization condition for the spectrum, and we compute the Fredholm determinant of the inverse of the Schrödinge…
We present a relation between the Mathieu equation and a particular elliptic curve. We find that the Floquet exponent of the Mathieu equation, for both $q<<1$ and $q>>1$, can be obtained from the integral of a differential one form along the two homology cycles of the elliptic curve. Certain higher order differential operators are needed to generate the WKB expansion. We provide a fifth order proof.
We derive a system of TBA equations governing the exact WKB periods in one-dimensional Quantum Mechanics with arbitrary polynomial potentials. These equations provide a generalization of the ODE/IM correspondence, and they can be regarded as the solution of a Riemann-Hilbert problem in resurgent Quantum Mechanics formulated by Voros. Our derivation builds upon the solution of similar Riemann-Hilbert problems in the study of BPS spectra in $\mathcal{N}=2$ gauge theories and of minimal surfaces in AdS. We also show that our TBA equations, combined with exact quantization conditions, provide a po…
We apply exact WKB methods to the study of the partition function of pure N=2 epsilon_i-deformed gauge theory in four dimensions in the context of the 2d/4d correspondence. We study the partition function at leading order in epsilon_2/epsilon_1 (i.e. at large central charge) and in an expansion in epsilon_1. We find corrections of the form ~ exp[- SW periods / epsilon_1] to this expansion. We attribute these to the exchange of the order of summation over gauge instanton number and over powers of epsilon_1 when passing from the Nekrasov form of the partition function to the topological string t…
The partition function on the three-sphere of ABJM theory contains non-perturbative corrections which correspond to membrane instantons in M-theory. These corrections can be studied in the Fermi gas approach to the partition function, and they are encoded in a system of integral equations of the TBA type. We study a semiclassical or WKB expansion of this TBA system in the ABJM coupling k, which corresponds to the strong coupling expansion of the type IIA string. This allows us to study membrane instanton corrections in M-theory at high order in the WKB expansion. Using these WKB results, we ve…
Based on some recent progress on a relation between four dimensional super Yang-Mills gauge theory and quantum integrable system, we study the asymptotic spectrum of the quantum mechanical problems described by the Mathieu equation and the Lamé equation. The large momentum asymptotic expansion of the eigenvalue is related to the instanton partition function of supersymmetric gauge theories which can be evaluated by a combinatorial method. The electro-magnetic duality of gauge theory indicates that in the parameter space there are three asymptotic expansions for the eigenvalue, we confirm this …
We derive the Thermodynamic Bethe Ansatz (TBA) equations for the Schrödinger equation with an arbitrary polynomial potential and a regular singular (simple and double pole) term. The TBA equations provide a non-trivial generalization of the ODE/IM correspondence and also give a solution for the Riemann-Hilbert problem in the exact WKB method. We study the TBA equations in detail for the linear and the harmonic oscillator potentials together with inverse and centrifugal terms. As an application, we also compute numerically the Voros spectrum for these potentials using the Bohr-Sommerfeld quanti…
We apply the exact WKB analysis to the quantum Seiberg-Witten curve for 4-dimensional $\mathcal{N} = 2\ SU(2)\ N_f=2$ 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.
We study various non-perturbative approaches to the quantization of the Seiberg-Witten curve of ${\cal N}=2$, $SU(2)$ 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 t…
We describe the exact WKB method from the point of view of abelianization, both for Schrödinger operators and for their higher-order analogues (opers). The main new example which we consider is the "$T_3$ equation," an order $3$ equation on the thrice-punctured sphere, with regular singularities at the punctures. In this case the exact WKB analysis leads to consideration of a new sort of Darboux coordinate system on a moduli space of flat $\mathrm{SL}(3)$-connections. We give the simplest example of such a coordinate system, and verify numerically that in these coordinates the monodromy of the…
The Nekrasov-Shatashvili limit for the low-energy behavior of N=2 and N=2* supersymmetric SU(2) gauge theories is encoded in the spectrum of the Mathieu and Lam'e equations, respectively. This correspondence is usually expressed via an all-orders Bohr-Sommerfeld relation, but this neglects non-perturbative effects, the nature of which is very different in the electric, magnetic and dyonic regions. In the gauge theory dyonic region the spectral expansions are divergent, and indeed are not Borel-summable, so they are more properly described by resurgent trans-series in which perturbative and non…
This work develops a new method to calculate non-perturbative corrections in one-dimensional Quantum Mechanics, based on trans-series solutions to the refined holomorphic anomaly equations of topological string theory. The method can be applied to traditional spectral problems governed by the Schrödinger equation, where it both reproduces and extends the results of well-established approaches, such as the exact WKB method. It can be also applied to spectral problems based on the quantization of mirror curves, where it leads to new results on the trans-series structure of the spectrum. Various …
