q$-Deformed Chern Class, Chern-Simons and Cocycle Hierarchy

TL;DR

This paper develops a $q$-deformed gauge theory framework for $SU_q(2)$, introducing new $q$-deformed invariants, hierarchy relations, and a $q$-deformed Yang-Mills equation, extending classical gauge concepts into the quantum group setting.

Contribution

It introduces a $q$-deformed Chern class, homotopy operator, and cocycle hierarchy, providing a novel approach to quantum group gauge theories and their invariants.

Findings

Defined $q$-deformed Killing form and Chern class for $SU_q(2)

Derived recursive relations for $q$-deformed cocycle hierarchy

Constructed $q$-deformed Yang-Mills equations and Lagrangian

Abstract

In this paper, from the $q$-gauge covariant condition we define the $q$-deformed Killing form and the second $q$-deformed Chern class for the quantum group $SU_{q}(2)$. Developing Zumino's method we introduce a $q$-deformed homotopy operator to compute the $q$-deformed Chern-Simons and the $q$-deformed cocycle hierarchy. Some recursive relations related to the generalized $q$-deformed Killing forms are derived to prove the cocycle hierarchy formulas directly. At last, we construct the $q$-gauge covariant Lagrangian and derive the $q$-deformed Yang-Mills equation. We find that the components of the singlet and the adjoint representation are separated in the $q$-deformed Chern class, $q$-deformed cocycle hierarchy and the $q$-deformed Lagrangian, although they are mixed in the commutative relations of BRST algebra.