Quantum theories of (p,q)-forms

TL;DR

This paper develops quantum field theories for massless (p,q)-forms on Kähler spaces, exploring gauge and non-gauge models, their derivation from spinning particles, and analyzing their one-loop effective actions and dualities.

Contribution

It introduces four quantum theories for (p,q)-forms, linking them to spinning particle models with U(2) supersymmetry, and investigates their dualities and topological features.

Findings

Derived equations of motion from spinning particle quantization.

Computed heat kernel coefficients and one-loop effective actions.

Identified duality relations and topological index mismatches.

Abstract

We describe quantum theories for massless (p,q)-forms living on Kaehler spaces. In particular we consider four different types of quantum theories: two types involve gauge symmetries and two types are simpler theories without gauge invariances. The latter can be seen as building blocks of the former. Their equations of motion can be obtained in a natural way by first-quantizing a spinning particle with a U(2)-extended supersymmetry on the worldline. The particle system contains four supersymmetric charges, represented quantum mechanically by the Dolbeault operators and their hermitian conjugates. After studying how the (p,q)-form field theories emerge from the particle system, we investigate their one loop effective actions, identify corresponding heat kernel coefficients, and derive exact duality relations. The dualities are seen to include mismatches related to topological indices and analytic torsions, which are computed as Tr(-1)^F and Tr[(-1)^F F] in the first quantized supersymmetric nonlinear sigma model for a suitable fermion number operator F.