Worldline approach to vector and antisymmetric tensor fields

TL;DR

This paper develops a worldline path integral approach to quantize antisymmetric tensor fields, simplifying calculations of effective actions, anomalies, and dualities in arbitrary dimensions, especially when coupled to gravity.

Contribution

It introduces a novel worldline quantization method for antisymmetric tensor fields that simplifies heat-kernel calculations and reveals duality relations.

Findings

Calculated Seeley-DeWitt coefficients for arbitrary rank tensors

Obtained the trace anomaly for a spin 1 particle in 4D

Established duality relations between differential form gauge fields

Abstract

The N=2 spinning particle action describes the propagation of antisymmetric tensor fields, including vector fields as a special case. In this paper we study the path integral quantization on a one-dimensional torus of the N=2 spinning particle coupled to spacetime gravity. The action has a local N=2 worldline supersymmetry with a gauged U(1) symmetry that includes a Chern-Simons coupling. Its quantization on the torus produces the one-loop effective action for a single antisymmetric tensor. We use this worldline representation to calculate the first few Seeley-DeWitt coefficients for antisymmetric tensor fields of arbitrary rank in arbitrary dimensions. As side results we obtain the correct trace anomaly of a spin 1 particle in four dimensions as well as exact duality relations between differential form gauge fields. This approach yields a drastic simplification over standard heat-kernel methods. It contains on top of the usual proper time a new modular parameter implementing the reduction to a single tensor field. Worldline methods are generically simpler and more efficient in perturbative computations then standard QFT Feynman rules. This is particularly evident when the coupling to gravity is considered.