Partition functions of higher derivative conformal fields on conformally related spaces

TL;DR

This paper explores the relationships between partition functions of higher derivative conformal fields on conformally related spaces, revealing their equivalences and non-unitarity through character integral representations and geometric mappings.

Contribution

It demonstrates the equivalence of partition functions for higher derivative conformal fields on different conformally related spaces and analyzes their non-unitarity using character integral methods.

Findings

Partition functions of conformally coupled scalars and fermions are identical on $S^a\times AdS_b$ and $S^{a+b}$.

Higher derivative conformal fields have equivalent partition functions on hyperbolic cylinders and branched spheres.

The study shows these theories are non-unitary based on relationships between Hofman-Maldacena variables.

Abstract

The character integral representation of one loop partition functions is useful to establish the relation between partition functions of conformal fields on Weyl equivalent spaces. The Euclidean space $S^a\times AdS_b$ can be mapped to $S^{a+b}$ provided $S^a$ and $AdS_b$ are of the same radius. As an example, to begin with, we show that the partition function in the character integral representation of conformally coupled free scalars and fermions are identical on $S^a\times AdS_b$ and $S^{a+b}$. We then demonstrate that the partition function of higher derivative conformal scalars and fermions are also the same on hyperbolic cylinders and branched spheres. The partition function of the four-derivative conformal vector gauge field on the branched sphere in $d=6$ dimension can be expressed as an integral over `naive' bulk and `naive' edge characters. However, the partition function of the conformal vector gauge field on $S^1_q\times AdS_5$ contains only the `naive' bulk part of the partition function. This follows the same pattern which was observed for the partition of conformal $p$-form fields on hyperbolic cylinders. We use the partition function of higher derivative conformal fields on hyperbolic cylinders to obtain a linear relationship between the Hofman-Maldacena variables which enables us to show that these theories are non-unitary.