Detours and Paths: BRST Complexes and Worldline Formalism

TL;DR

Detours and Paths develops a BRST-based worldline formalism to construct detour complexes that encode gauge field equations and Bianchi identities for tensor fields of arbitrary symmetry. By coupling N=2 and exhausted sp(2r) spinning particle models to gravity and gauging their worldline symmetries, the authors derive self-adjoint long operators that govern gauge-invariant actions for Maxwell theory and for higher-spin, mixed-symmetry fields through Lie algebra cohomology. The formalism yields explicit degrees-of-freedom counts via path integrals and confirms consistency with lightcone analyses, providing a robust first-quantized route to second-quantized gauge theories. The framework holds promise for controlled higher-spin interactions and background couplings, while clarifying obstructions and geometric invariants relevant to these theories in curved spaces.

Abstract

We construct detour complexes from the BRST quantization of worldline diffeomorphism invariant systems. This yields a method to efficiently extract physical quantum field theories from particle models with first class constraint algebras. As an example, we show how to obtain the Maxwell detour complex by gauging N=2 supersymmetric quantum mechanics in curved space. Then we concentrate on first class algebras belonging to a class of recently introduced orthosymplectic quantum mechanical models and give generating functions for detour complexes describing higher spins of arbitrary symmetry types. The first quantized approach facilitates quantum calculations and we employ it to compute the number of physical degrees of freedom associated to the second quantized, field theoretical actions.