A Compendium of Sphere Path Integrals

TL;DR

This paper systematically derives covariant 1-loop path integrals for various symmetric tensor fields of arbitrary spin on spheres, extending known results to higher spins and different mass regimes with explicit formulas and prescriptions.

Contribution

It provides a comprehensive derivation of covariant path integrals for arbitrary spin fields on spheres, including new higher spin analogs of Polchinski's phase and general prescriptions for diverse field types.

Findings

Derived path integrals for massless fields of arbitrary spin.

Extended the phase analysis to higher spins, generalizing Polchinski's phase.

Provided explicit formulas and prescriptions for massive, shift-symmetric, and partially massless fields.

Abstract

We study the manifestly covariant and local 1-loop path integrals on $S^{d+1}$ for general massive, shift-symmetric and (partially) massless totally symmetric tensor fields of arbitrary spin $s\geq 0$ in any dimensions $d\geq 2$. After reviewing the cases of massless fields with spin $s=1,2$, we provide a detailed derivation for path integrals of massless fields of arbitrary integer spins $s\geq 1$. Following the standard procedure of Wick-rotating the negative conformal modes, we find a higher spin analog of Polchinski's phase for any integer spin $s\geq 2$. The derivations for low-spin ($s=0,1,2$) massive, shift-symmetric and partially massless fields are also carried out explicitly. Finally, we provide general prescriptions for general massive and shift-symmetric fields of arbitrary integer spins and partially massless fields of arbitrary integer spins and depths.