Topological-Antitopological Fusion and the Large N CP^N Model

TL;DR

This paper explores the large N limit of supersymmetric CP^N models using the tt* formalism, connecting it to the 2D Toda lattice and self-dual gravity, and providing analytical solutions through finite temperature and large N techniques.

Contribution

It demonstrates how the tt* equations relate to the 2D Toda lattice in the large N limit of supersymmetric CP^N models, linking field theory with integrable systems and gravity.

Findings

The tt* equation reduces to the 2D Toda lattice in the large N limit.

Finite temperature and large N techniques effectively determine solutions.

Analytical verification of solutions using Legendre transforms from self-dual gravity.

Abstract

We discuss the large $N$ limit of the supersymmetric $CP^N$ models as an illustration of Cecotti and Vafa's $tt^*$ formalism. In this limit the `$tt^*$ equation' becomes the long wavelength limit of the $2D$ Toda lattice, an equation first studied in the context of self-dual gravity. We show how simple finite temperature and large $N$ techniques determine the relevant solution, and verify analytically that it solves the $tt^*$ equation, using Legendre transform techniques from self-dual gravity.