WKB periods for higher order ODE and TBA equations

Abstract

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(1) r 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 the WKB period, which is confirmed by solving the TBA equation numerically.

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

“…In the case of Schrödinger equations, the Y-functions are identified with the WKB periods in the ODE. A similar identification is also expected for the third order ODE [46]. In this section, we first introduce the Y-functions for the (A 2 , A N )-type ODE.…”

“…which is different from the one in the previous paper [46]. The mass parameters are defined through the classical periods and are computed by (2.41) as…”

“…We also refer it as the (A 2 , A N )-type ODE. In a previous paper [46], we have studied the N = 1 case. We consider the WKB solution of the (2.1) of the form ψ(x) = exp 1 x P (x )dx , P (x) = ∞ n=0 n p n (x).…”

“…Let us consider (A 2 , A 2 )-type ODE in the minimal chamber studied in a previous paper [46]. The Y-system (3.16) becomes…”

“…For recent developments, see [40][41][42][43][44][45] 5 . In a previous paper [46], we studied the WKB expansion of the higher order ODE and found the relation between the WKB periods and the Y-functions associated with the TBA equations. In particular, we considered the case of the quadratic potential, which is found to correspond to the (A r , A 1 )-type TBA system.…”