Conformal Manifolds in Four Dimensions and Chiral Algebras

TL;DR

This paper explores the structure of chiral algebras associated with four-dimensional N=2 superconformal field theories, especially those with conformal manifolds, revealing minimal generator counts and specific algebra realizations.

Contribution

It demonstrates that chiral algebras of N=2 SCFTs with conformal manifolds have at least three generators and identifies explicit algebra realizations at special points.

Findings

Chiral algebras have at least three generators on conformal manifolds.

Explicit example of a chiral algebra as the A(6) theory from a specific hypersurface singularity.

Isolated theories can be conformally gauged only with SUSY moduli spaces.

Abstract

Any N=2 superconformal field theory (SCFT) in four dimensions has a sector of operators related to a two-dimensional chiral algebra containing a Virasoro sub-algebra. Moreover, there are well-known examples of isolated SCFTs whose chiral algebra is a Virasoro algebra. In this note, we consider the chiral algebras associated with interacting N=2 SCFTs possessing an exactly marginal deformation that can be interpreted as a gauge coupling (i.e., at special points on the resulting conformal manifolds, free gauge fields appear that decouple from isolated SCFT building blocks). At any point on these conformal manifolds, we argue that the associated chiral algebras possess at least three generators. In addition, we show that there are examples of SCFTs realizing such a minimal chiral algebra: they are certain points on the conformal manifold obtained by considering the low-energy limit of type IIB string theory on the three complex-dimensional hypersurface singularity x_1^3+x_2^3+x_3^3+A x_1x_2x_3+w^2=0. The associated chiral algebra is the A(6) theory of Feigin, Feigin, and Tipunin. As byproducts of our work, we argue that (i) a collection of isolated theories can be conformally gauged only if there is a SUSY moduli space associated with the corresponding symmetry current moment maps in each sector, and (ii) N=2 SCFTs with a>=c have hidden fermionic symmetries (in the sense of fermionic chiral algebra generators).