Functoriality of Quantum Principal Bundles and Quantum Connections
Gustavo Amilcar Saldaña Moncada
TL;DR
This work generalizes the classical association between principal bundles and gauge theories to the noncommutative setting by developing a quantum association functor $\mathrm{Ass}^{\omega}_{\zeta}$ that pairs representations of a compact quantum group $\mathcal{G}$ with quantum vector bundles carrying linear connections. Built on Durdevich's quantum principal bundles, Woronowicz's corepresentation theory, and Connes–Dubois-Violette quantum vector bundles, the authors construct the contravariant functor from $\mathbf{Rep}_{\mathcal{G}}$ to $\mathbf{qVB}^{\nabla}_{B}$ and prove a categorical equivalence between the quantum bundle category $\mathbf{qPB}^{\omega}_{B}$ and the quantum gauge theory sectors $\mathbf{qGTS}^{\nabla}_{B}$. The key contributions include an explicit realization of $E^{V}=\mathrm{Mor}^{0}_{\mathbf{Rep}^{\infty}_{\mathcal{G}}}(\delta^{V},\Delta_P)$ as a finitely generated projective $B$-module, the induced connections $\nabla^{\omega}_{V}$, and the demonstration that all such structures can be reconstructed from the associated functor, establishing a NC analogue of the classical categorical equivalence. This framework provides a robust noncommutative geometric foundation for gauge theories, enabling NC Yang–Mills constructions and unifying multiple strands of NC geometry under a common functorial perspective.
Abstract
In the framework of Category Theory, we study the association between finite--dimensional representations of a compact quantum group and quantum vector bundles with linear connections for a given quantum principal bundle with a principal connection. In particular, we present a categorical equivalence between this kind of {\it quantum association functors} and quantum principal bundles with principal connections.