| The color-flavor transformation is an integral identity which first appeared in the study of disordered systems in condensed matter physics. Since then it has been successfully applied to many fields of physics. In this thesis, we study its applications in lattice quantum chromodynamics (QCD), the fundamental theory to study the non-perturbative properties of strongly interacting particles. The advantage of this transformation is that it can simplify the numerical simulations as well as provide analytical insights into lattice gauge theory.; We begin with an outline of the background and the motivation for this thesis. Then we briefly introduce a few general concepts of lattice gauge theory. Next we review the fermionic color-flavor transformation for SU( Nc), where Nc is the number of color degrees of freedom, and its applications in fermion-induced QCD. By studying the resulting baryon loop expansion, we recognize both the advantages of this transformation and the difficulties associated with fermion-induced QCD. We then introduce an alternative scheme, i.e., boson-induced Yang-Mills theory, in which the bosonic color-flavor transformation plays the role of a duality transformation. We proceed by explicitly deriving the bosonic color-flavor transformation for the gauge group SU(Nc), which has not been studied so far. However, the method we use, which has also been used by other authors, leads to a result with a limited range of applicability. We then introduce a new method which can extend the transformation, for both the U(Nc) and the SU(Nc) gauge groups, to a larger regime of applicability. Finally, we study the graded color-flavor transformation for the special unitary group, which is of interest in boson-induced QCD. An open problem in our result on the graded transformations is raised and discussed at the end. |
