On the simplified path integral on spheres

TL;DR

This paper develops a simplified path integral approach for particles on spheres and symmetric spaces, using Riemann normal coordinates to incorporate curvature effects via a scalar potential, and applies it to compute trace anomalies in high dimensions.

Contribution

It introduces a quadratic kinetic term in the path integral for particles on spheres, enabling perturbative calculations of trace anomalies in dimensions 14 and 16.

Findings

Successfully extended perturbative evaluation to high orders

Computed type-A trace anomalies in dimensions 14 and 16

Validated the simplified path integral approach for curved spaces

Abstract

We have recently studied a simplified version of the path integral for a particle on a sphere, and more generally on maximally symmetric spaces, and proved that Riemann normal coordinates allow the use of a quadratic kinetic term in the particle action. The emerging linear sigma model contains a scalar effective potential that reproduces the effects of the curvature. We present here further details on the construction, and extend its perturbative evaluation to orders high enough to read off the type-A trace anomalies of a conformal scalar in dimensions d = 14 and d = 16.