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# Brief Introduction to Quantum Field Theories

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Prof. Sengupta's research work has centered around the following topics: (1) mathematical problems connected with geometric quantum field theories, (2) problems involving infinite-dimensional stochastic analysis, and (3) study of the Segal-Bargmann transform. The mathematical ideas involved include probability theory and stochastics, differential geometry and topology, and symplectic structures. Some of the motivation for the problems comes from quantum physics. Sengupta's work has involved the construction of a probability measure on an infinite dimensional non-linear space whose points represent geometric objects called connections. The study of this measure is motivated by quantum gauge theory in two dimensions. The construction involves an infinite-dimensional Gaussian measure conditioned to satisfy certain constraints imposed by topology and geometry. As a limiting case of the measure described above, one obtains a measure on the moduli space of flat connections over a surface. This space is of interest from a variety of points of view, including topology and algebraic geometry. Sengupta has been studying primarily the symplectic structure on this moduli space (when the underlying surface is oriented). The Chern-Simons integral is an object which arises again from a geometric quantum field theory and it was shown by Witten that this integral provides a way to determine topological invariants associated with knots in three dimensional manifolds. In work done jointly with Professor S. Albeverio (Bonn), Sengupta has also constructed a rigorously meaningful version of the Chern-Simons functional integral as an infinite-dimensional distribution. Further work needs to be done to evaluate this distribution on functions of interest, thereby establishing the connection with topology in a mathematically rigorous way. Sengupta is also interested in infinite-dimensional distribution theory (white noise analysis) and, along with Professors Cochran and Kuo (both of LSU), has investigated general classes of such distributions and the effect of a standard transform called the S-transform on such distributions. Sengupta has also been investigating, mainly in collaboration with Professor Brian Hall(Notre Dame), the Segal-Bargmann transform, a unitary isomorphism connecting spaces of square-integrable functions with spaces of holomorphic square-integrable functions, in a Gaussian background.