Topological σ-models and the large-N matrix integrals

Abstract

In this paper we describe in some detail the representation of the topological CP 1 model in terms of a matrix integral which we have introduced in a previous article. We first discuss the integrable structure of the CP 1 model and show that it is governed by an extension of the 1-dimensional Toda hierarchy. We then introduce a matrix model which reproduces the sum over holomorphic maps from arbitrary Riemann surfaces onto CP 1 . We compute intersection numbers on the moduli space of curves using geometrical method and show that the results agree with those predicted by the matrix model. We also develop a Landau-Ginzburg (LG) description of the CP 1 model using a superpotential e X +e t0,Q e −X given by the Lax operator of the Toda hierarchy (X is the LG field and t 0,Q is the coupling constant of the Kähler class). The form of the superpotential indicates the close connection between CP 1 and N = 2 supersymmetric sine-Gordon theory which was noted some time ago by several authors. We also discuss possible generalizations of our construction to other manifolds and present a LG formulation of the topological CP 2 model.

The algebraic and Hamiltonian structure of the dispersionless Benney and Toda hierarchies

“…where the Q (n) are the charges defined by (28) and the F (n) are the corresponding fluxes with respect to the t 1 flow, Explicitly Calculations of this type were first performed (for N = 2 and small values of n) in [15,16] and then (for N = 3 and small values of n) in [17].…”

“…Calculations of this type were first performed (for N = 2 and small values of n) in [15,16] and then (for N = 3 and small values of n) in [17].…”

Mirror Symmetry in Two Steps: A–I–B

“…The Landau-Ginzburg model with the (twisted) superpotential W , where W is as above, and its complex conjugate W has been considered by K. Hori and C. Vafa [18] (see also [13,5,7,16]). They showed that its correlation functions in the BPS sector are related to those of the twisted sigma model of P 1 , which is the sense in which the two theories are mirror dual to each other.…”