Lectures on Two-Dimensional Noncommutative Gauge Theory Quantization

Abstract

These notes comprise the second part of two articles devoted to the construction of exact solutions of noncommutative gauge theory in two spacetime dimensions. Here we shall deal with the quantum field theory. Topics covered include an investigation of the symmetries of quantum gauge theory on the noncommutative torus within the path integral formalism, the derivation of the exact expression for the vacuum amplitude, and the classification of instanton contributions. A section dealing with a new, exact combinatorial solution of gauge theory on a two-dimensional fuzzy torus is also included.Based on invited lectures given by the second author at the

Matrix Quantum Mechanics and Soliton Regularization of Noncommutative Field Theory

“…While we will not completely solve the problem at the level of the matrix quantum mechanics, our analysis will illustrate in a straightforward manner what properties to seek in the search for exactly solvable noncommutative field theories. Analogous computations for the lattice regularizations of noncommutative gauge theory in two dimensions can be found in [63,32].…”

Instantons, fluxons, and open gauge string theory

“…An explicit realization of two-dimensional noncommutative gauge theory as a matrix model has also been constructed in [14,15] by exploiting the general relation [16] between its lattice regularization and the twisted Eguchi-Kawai (TEK) model [17]- [19]. According to the general paradigm of reduced models, ordinary QCD 2 at large N should be recovered from the TEK model as well.…”

Localization for Yang-Mills Theory on the Fuzzy Sphere

“…Our model fills this gap, providing a gauge theory on the fuzzy sphere whose exact solution is on a unified footing with that of gauge theory on the noncommutative torus, in the same way that all two-dimensional gauge theories admit universal solutions. This is even apparent from the strong coupling expansions of the two noncommutative gauge theories [33,34], which exhibit the same degrees of complexity.…”

“…Its downfall is that it leads to somewhat cumbersome expressions for the partition function which arise from a rather different sort of localization. This is analogous to the case of gauge theory on the two-dimensional noncommutative torus whose strong-coupling expansion involves the addition of infinitely many higher Casimir operators to the usual Migdal formula [34], or its matrix model regularization which is given by a complicated combinatorial formula [33]. This complexity makes it difficult to explicitly extract the contributions from fuzzy sphere instantons, and we will examine this problem more thoroughly in the next section.…”

“…One would therefore like to have a similar picture in the noncommutative case. The instanton expansion can be readily generalized to provide the exact solution for gauge theory on a two-dimensional noncommutative torus [32,33]. However, in previous formulations of gauge theory on the fuzzy sphere this is not possible, either because extra scalar degrees of freedom not normally present in commutative Yang-Mills theory destroy the topological nature of the gauge theory and hence its exact solvability, or else because the exact solution does not decompose neatly into isolated contributions from classical solutions.…”