Structures on the Conformal Manifold in Six Dimensional Theories

TL;DR

This paper analyzes the geometric structures on the conformal manifold of six-dimensional conformal field theories, identifying specific tensor properties and their positivity conditions, with applications to free and interacting theories.

Contribution

It provides a detailed analysis of the tensors on the conformal manifold in six dimensions, including positivity conditions and applications to free and interacting theories.

Findings

Only one tensor satisfies positivity conditions.

All three tensors contribute in the scalar $^3$ theory.

Results apply to both free and interacting six-dimensional CFTs.

Abstract

The tensors which may be defined on the conformal manifold for six dimensional CFTs with exactly marginal operators are analysed by considering the response to a Weyl rescaling of the metric in the presence of local couplings. It is shown that there are three symmetric two index tensors only one of which satisfies any positivity conditions. The general results are specialised to the six dimensional conformal theory defined by free two-forms and also to the interacting scalar $\phi^3$ theory at two loops which preserves conformal invariance to this order. All three two index tensor contributions are present.