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      Lectures on Batalin-Vilkovisky formalism and its applications in topological quantum field theory

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          Abstract

          Lecture notes for the course "Batalin-Vilkovisky formalism and applications in topological quantum field theory" given at the University of Notre Dame in the Fall 2016 for a mathematical audience. In these lectures we give a slow introduction to the perturbative path integral for gauge theories in Batalin-Vilkovisky formalism and the associated mathematical concepts.

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          Quantum field theory and the Jones polynomial

          Edward Witten (1989)
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            Feynman diagrams for the Yang-Mills field

            L.D. Faddeev, V.N. Popov (1967)
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              Deformation quantization of Poisson manifolds, I

              Maxim Kontsevich (1997)
              I prove that every finite-dimensional Poisson manifold X admits a canonical deformation quantization. Informally, it means that the set of equivalence classes of associative algebras close to the algebra of functions on X is in one-to-one correspondence with the set of equivalence classes of Poisson structures on X modulo diffeomorphisms. In fact, a more general statement is proven ("Formality conjecture"), relating the Lie superalgebra of polyvector fields on X and the Hochschild complex of the algebra of functions on X. Coefficients in explicit formulas for the deformed product can be interpreted as correlators in a topological open string theory, although I do not use explicitly the language of functional integrals. One of corollaries is a justification of the orbit method in the representation theory.
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                Publication date Created: 25 July 2017
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                ArXiV ID: 1707.08096
                SO-VID: 20632d3a-be7c-4b64-8048-a35a497947a8
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                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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                Categories math-ph hep-th math.MP

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