4D limit of melting crystal model and its integrable structure

This paper addresses the problems of quantum spectral curves and 4D limit for the melting crystal model of 5D SUSY U(1) Yang-Mills theory on R-4 x S-1. The partition function Z(t) deformed by an infinite number of external potentials is a tau function of the KP hierarchy with respect to the coupling constants t = (t(1), t(2), ...). A single-variate specialization Z(x) of Z(t) satisfies a q-difference equation representing the quantum spectral curve of the melting crystal model. In the limit as the radius R of S-1 in R-4 x S-1 tends to 0, it turns into a difference equation for a 4D counterpart Z(4D)(X) of Z(x). This difference equation reproduces the quantum spectral curve of Gromov-Witten theory of CP1. Z(4D)(X) is obtained from Z(x) by letting R -> 0 under an R-dependent transformation x = x(X, R) of x to X. A similar prescription of 4D limit can be formulated for Z(t) with an R-dependent transformation t = t(T, R) of t to T = (T-1, T-2,...). This yields a 4D counterpart Z(4D)(T) of Z(t). Z(4D)(T) agrees with a generating function of all-genus Gromov-Witten invariants of CP1. Fay-type bilinear equations for Z(4D)(T) can be derived from similar equations satisfied by Z(t). The bilinear equations imply that Z(4D)(T), too, is a tau function of the KP hierarchy. These results are further extended to deformations Z(t, s) and Z(4D)(T, s) by a discrete variable s is an element of Z, which are shown to be tau functions of the 1D Toda hierarchy. (C) 2018 Elsevier B.V. All rights reserved.