Punctures and p-Spin Curves from Matrix Models

Abstract

AbstractThis article investigates the intersection numbers of the moduli space of p-spin curves with the help of matrix models. The explicit integral representations that are derived for the generating functions of these intersection numbers exhibit p Stokes domains, labelled by a “spin”-component l taking values $$l = -1, 0, 1, 2,...,p-2$$ l = - 1 , 0 , 1 , 2 , . . . , p - 2 . Earlier studies concerned integer values of p, but the present formalism allows one to extend our study to half-integer or negative values of p, which turn out to describe new types of punctures or marked points on the Riemann surface. They fall into two classes: Ramond $$(l=-1)$$ ( l = - 1 ) , absent for positive integer p, and Neveu–Schwarz $$(l\ne -1)$$ ( l ≠ - 1 ) . The intersection numbers of both types are computed from the integral representation of the n-point correlation functions in a large N scaling limit. We also consider a supersymmetric extension of the random matrix formalism to show that it leads naturally to an additional logarithmic potential. Open boundaries on the surface, or admixtures of R and NS punctures, may be handled by this extension.