Twisting it up the Quantum Way : On Matrix Models, q-deformations and Supersymmetric Gauge Theories
Abstract: The mathematical framework which quantum field theory constitutes has been very successful in describing nature. As an extension of such a framework, the idea of supersymmetry was introduced. This greatly simplified the mathematical description of the theories, making them more tractable. Recently, the method of supersymmetric localisation, in which one can compute infinite dimensional integrals exactly, enabled computations of partition functions for different supersymmetric gauge theories in various dimensions. Such partition functions sometimes resulted in the form of matrix models or even q-deformed matrix models, where the latter are not very well-studied. Classical, or un-deformed, matrix models on the other hand are studied in much greater detail. One particular tool that is used in the study of classical matrix models is the Ward identities called Virasoro constraints. Motivated by firstly the desire to understand q-deformed matrix models better and secondly the gauge theory applications of the results, we studied the derivation of and solution to such q-deformed Virasoro constraints. We also explored the implications of partition functions taking the form of q-deformed matrix models in the case of three and four dimensional supersymmetric gauge theories. Furthermore, we studied various generalisations of the classical matrix model, such as having different limits of integration and different potentials, in order to see how the Virasoro constraints and its solution changed. Finally, we made a connection with the area of integrability and investigated how classical matrix model satisfying the Virasoro constraints could be related to certain tau-functions satisfying the Hirota equations.
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