Sphere Partition Functions and the Zamolodchikov Metric

TL;DR

This paper explores how sphere partition functions in conformal field theories relate to the Zamolodchikov metric, revealing conditions under which these functions are physical and how they encode geometric data of the theory.

Contribution

It demonstrates that in supersymmetric CFTs, sphere partition functions compute the Kahler potential on conformal manifolds, providing new proofs and extending results to four dimensions.

Findings

In 2D N=(2,2) theories, the partition function computes the Kahler potential.

In 4D N=2 theories, the S^4 partition function also computes the Kahler potential.

Without sufficient supersymmetry, the partition functions are not well-defined functions of marginal couplings.

Abstract

We study the finite part of the sphere partition function of d-dimensional Conformal Field Theories (CFTs) as a function of exactly marginal couplings. In odd dimensions, this quantity is physical and independent of the exactly marginal couplings. In even dimensions, this object is generally regularization scheme dependent and thus unphysical. However, in the presence of additional symmetries, the partition function of even-dimensional CFTs can become physical. For two-dimensional N=(2,2) supersymmetric CFTs, the continuum partition function exists and computes the Kahler potential on the chiral and twisted chiral superconformal manifolds. We provide a new elementary proof of this result using Ward identities on the sphere. The Kahler transformation ambiguity is identified with a local term in the corresponding N=(2,2) supergravity theory. We derive an analogous, new, result in the case of four-dimensional N=2 supersymmetric CFTs: the S^4 partition function computes the Kahler potential on the superconformal manifold. Finally, we show that N=1 supersymmetry in four dimensions and N=(1,1) supersymmetry in two dimensions are not sufficient to make the corresponding sphere partition functions well-defined functions of the exactly marginal parameters.