Abelian color cycle and abelian color flux dualization methods for non-abelian lattice field theories

TL;DR

This thesis develops dualization methods using abelian color cycles and fluxes for non-abelian lattice models, enabling a reformulation that simplifies the analysis of gauge and fermionic systems.

Contribution

It introduces the abelian color cycle and flux methods for dualizing non-abelian lattice theories, facilitating their analysis in a dual framework.

Findings

Dual variables represent worldsheets and worldlines for gauge and matter fields.

The dual partition function is expressed as a strong coupling series in closed form.

Net-particle number corresponds to the winding number of worldlines.

Abstract

In this thesis we present the application of the \textit{abelian color cycle} (ACC) and the \textit{abelian color flux} (ACF) methods to several models: the SU(2) principal chiral model, the SU(2) gauge theory with staggered fermions and QCD with staggered fermions. The key step of our approaches consists in decomposing the action of the model one is considering into its minimal units. For gauge theories those minimal terms are complex numbers, which we refer to as abelian color cycles, while for fermions the action is decomposed into Grassmann bilinears, which we called abelian color fluxes. As a result of these decompositions the actions are sums of commuting terms, and thus one can proceed with the dualization of the theory as in the abelian case, by factorizing and expanding the Boltzmann weight. The expansion indices, so-called dual variables, become the new degrees of freedom for the description of the system once the conventional fields are integrated out. The integration over the conventional fields results into weight factors and constraints. The constraints implement the symmetries of the theory in the dual form and imply that the dual configurations which contribute to the long range physics are worldsheets for the gauge degrees of freedom and worldlines for matter fields. On the other hand, the weight factors allow one to organize the dual partition function into a strong coupling series of which all terms are known in closed form. Moreover, the form of the dependence on the chemical potential allows one to identify the net-particle number as the total net temporal winding number of the worldlines in the dual representation.