Exact solutions for self-dual Yang-Mills and self-dual tensor multiplets on gravitational instanton background

Abstract

We give exact solutions for recently developed N = 1 locally supersymmetric self-dual gauge theories in 2 + 2 dimensions. We give the exact solutions for an SL(2) self-dual Yang-Mills multiplet and what we call “self-dual tensor multiplet” on the gravitational instanton background by Eguchi and Hanson. We use a general method to get an SL(2) self-dual Yang-Mills solution from any known self-dual gravity solutions. Our result is the first example of exact solutions for the coupled system of these N = 1 locally supersymmetric self-dual multiplets in 2 + 2 dimensions, which is supposed to have strong significance for integrable models in lower dimensions upon appropriate dimensional reduction or truncation. We also inspect the consistency of our exact solutions as a background for the N = 2 superstring coupled to the Wess-Zumino-Witten term in σ-model formulation.