4D limit of melting crystal model and its integrable structure

Abstract 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 × 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 × 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 CP 1 . 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 CP 1 . 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 ∈ Z , which are shown to be tau functions of the 1D Toda hierarchy.