Interacting 2D Field-Theoretic Model for Hodge Theory

TL;DR

This paper demonstrates that a specific 2D interacting field theory models Hodge theory through its symmetries, showing it remains consistent and unitary despite quantum anomalies, and supports particle interpretation.

Contribution

It provides a tractable 2D field-theoretic example realizing Hodge theory via symmetry operators, with implications for anomaly triviality and unitarity.

Findings

Symmetries realize de Rham cohomological operators.

Quantum anomaly does not compromise theory’s consistency.

Theory supports particle interpretation despite chiral symmetry.

Abstract

We take up the St${\ddot u}$ckelberg-modified version of the two (1+1)-dimensional (2D) Proca theory, in interaction with the Dirac fields, to study its various continuous and discrete symmetry transformations and show that this specific interacting 2D field-theoretic model provides a tractable example for the Hodge theory because its symmetries (and corresponding conserved charges) provide the physical realizations of the de Rham cohomological operators of differential geometry at the algebraic level. The physical state of this theory is chosen to be the harmonic state (of the Hodge decomposed state) in the quantum Hilbert space which is annihilated by the conserved and nilpotent (anti-)BRST as well as (anti-)co-BRST charges. A physical consequence of this study is an observation that the 2D anomaly, at the quantum level, does not lead to any problem as far as the consistency and unitarity of our present 2D theory is concerned. In other words, our present 2D field-theoretic model is amenable to particle interpretation despite the presence of the local chiral symmetry (which is associated with the nilpotent (anti-)co-BRST symmetry transformations) besides the presence of the nilpotent (anti-)BRST symmetries (which are connected with the local gauge symmetry). The physicality condition with the (anti-)co-BRST charges implies that the 2D anomaly term is trivial in our present theory. Hence, our 2D theory is consistent, unitary and amenable to particle interpretation.