Compact Conformal Manifolds

TL;DR

This paper systematically studies compact conformal manifolds of 4D SCFTs, showing their Kahler structure, realizing complex projective spaces, and providing methods to construct and analyze such spaces via deformations and chiral rings.

Contribution

It introduces a framework for constructing and understanding compact conformal manifolds in 4D SCFTs, including explicit realizations and algorithms for more general spaces.

Findings

Compact conformal manifolds are Kahler and can be complex projective spaces.

Explicit construction of N-dimensional projective space conformal manifolds from N=2 SCFTs.

An algorithm for building more general compact SCFT spaces.

Abstract

In this note we begin a systematic study of compact conformal manifolds of SCFTs in four dimensions (our notion of compactness is with respect to the topology induced by the Zamolodchikov metric). Supersymmetry guarantees that such manifolds are Kahler, and so the simplest possible non-trivial compact conformal manifold in this set of geometries is a complex one-dimensional projective space. We show that such a manifold is indeed realized and give a general prescription for constructing complex N-dimensional projective space conformal manifolds as certain small N=2->N=1 breaking deformations of strongly interacting N=2 SCFTs. In many cases, our prescription reduces the construction of such spaces to a study of the N=2 chiral ring. We also give an algorithm for constructing more general compact spaces of SCFTs.