Quantum theory of massless (p, 0)-forms

• Published in: The journal of high energy physics Vol. 2011; no. 9
• Main Authors: Bastianelli, Fiorenzo, Bonezzi, Roberto
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01-09-2011; Springer Nature B.V
• Subjects: Classical and Quantum Gravitation; Counting; Differential equations; Elementary Particles; High energy physics; Hilbert space; Maxwell's equations; Operators (mathematics); Physics; Physics and Astronomy; Quantum Field Theories; Quantum Field Theory; Quantum Physics; Quantum theory; Relativity Theory; Representations; String Theory; Supersymmetry; Thermal expansion; Duality in Gauge Field Theories; Sigma Models
• ISSN: 1029-8479; 1029-8479
• DOI: 10.1007/JHEP09(2011)018

We describe the quantum theory of massless (
p
, 0)-forms that satisfy a suitable holomorphic generalization of the free Maxwell equations on Kähler 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 Kähler 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 Kähler space. The relevant differential operators appearing in these equations are a twisted exterior holomorphic derivative
∂
q
and its hermitian conjugate
∂
q
†
(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.