Worldline approach to vector and antisymmetric tensor fields

TL;DR

The paper develops a worldline quantization framework using the N=2 spinning particle in a gravitational background to compute the one-loop effective action for antisymmetric tensor fields, including vectors. By quantizing on a torus and introducing a U(1) modulus, it projects onto a single $p$-form sector and performs a proper-time (Seeley–DeWitt) expansion to obtain $a_0$, $a_1$, and $a_2$ coefficients for arbitrary rank in arbitrary dimension. It validates the approach by reproducing known results for low-rank forms, derives the spin-1 trace anomaly in four dimensions, and yields new coefficients for higher forms such as $F_5$ and $F_6$, along with duality relations among differential forms. The method offers a significant simplification over traditional heat-kernel methods, especially in curved backgrounds, and clarifies the role of the modular parameter in form-duality and topological aspects.

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.