Anomalies, Conformal Manifolds, and Spheres

TL;DR

This paper explores the universal trace anomaly contributions from exactly marginal operators in conformal field theories, interpreting them through sigma models on conformal manifolds, especially in supersymmetric contexts, revealing geometric properties and relations to partition functions.

Contribution

It provides new insights into the geometry of conformal manifolds in supersymmetric theories, showing they are Kahler-Hodge with vanishing Kahler class, and connects sphere partition functions to Kahler potentials.

Findings

Conformal manifolds are Kahler-Hodge and have vanishing Kahler class.

Sphere partition functions relate directly to Kahler potentials in supersymmetric theories.

Identification of potential trace anomalies that are consistent with Wess-Zumino conditions but can be excluded upon detailed analysis.

Abstract

The two-point function of exactly marginal operators leads to a universal contribution to the trace anomaly in even dimensions. We study aspects of this trace anomaly, emphasizing its interpretation as a sigma model, whose target space M is the space of conformal field theories (a.k.a. the conformal manifold). When the underlying quantum field theory is supersymmetric, this sigma model has to be appropriately supersymmetrized. As examples, we consider in some detail N=(2,2) and N=(0,2) supersymmetric theories in d=2 and N=2 supersymmetric theories in d=4. This reasoning leads to new information about the conformal manifolds of these theories, for example, we show that the manifold is Kahler-Hodge and we further argue that it has vanishing Kahler class. For N=(2,2) theories in d=2 and N=2 theories in d=4 we also show that the relation between the sphere partition function and the Kahler potential of M follows immediately from the appropriate sigma models that we construct. Along the way we find several examples of potential trace anomalies that obey the Wess-Zumino consistency conditions, but can be ruled out by a more detailed analysis.