Quantum theory of massless (p,0)-forms

TL;DR

This paper develops a quantum framework for massless (p,0)-forms on Kähler spaces, deriving their equations from a spinning particle model, and computes related heat kernel coefficients and duality relations.

Contribution

It introduces a worldline approach to quantize massless (p,0)-forms on Kähler manifolds and derives their effective action and duality properties.

Findings

Derived holomorphic Maxwell equations for (p,0)-forms on Kähler spaces.

Provided a worldline representation of the one-loop effective action.

Established duality relations including a topological mismatch.

Abstract

We describe the quantum theory of massless (p,0)-forms that satisfy a suitable holomorphic generalization of the free Maxwell equations on Kaehler spaces. These equations arise by first-quantizing a spinning particle with a U(1)-extended local supersymmetry on the worldline. Dirac quantization of the spinning particle produces a physical Hilbert space made up of (p,0)-forms that satisfy holomorphic Maxwell equations coupled to the background Kaehler geometry, containing in particular a charge that measures the amount of coupling to the U(1) part of the U(d) holonomy group of the d-dimensional Kaehler space. The relevant differential operators appearing in these equations are a twisted exterior holomorphic derivative and its hermitian conjugate (twisted Dolbeault operators with charge q). The particle model is used to obtain a worldline representation of the one-loop effective action of the (p,0)-forms. This representation allows to compute the first few heat kernel coefficients contained in the local expansion of the effective action and to derive duality relations between (p,0) and (d-p-2,0)-forms that include a topological mismatch appearing at one-loop.