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제목 Higher algebra in SUSY QFT III: Line operators and geometry in 3d N=4 gauge theory
발표자 Dimofte, Tudor ( UC Davis ) 날짜 2018-08-03
Host KIAS Place 8101
Abstract A mathematical treatment of TQFT, based on category theory, was initiated in the early 90's. In more modern times the mathematics of TQFT has come to use advanced techniques from derived geometry and higher algebra (? la Lurie). This series of talks is loosely aimed at explaining how some of these advanced techniques apply in familiar examples of supersymmetric field theory and its topological twists. For physics, this will lead to some surprising new structure, as well as some useful organizing principles. The first and second lectures are based on work with C. Beem, D. Ben-Zvi, M. Bullimore, and A. Neitzke. The first lecture will re-examine operator algebras in topological twists of supersymmetric field theories. In d dimension, the algebras naturally come equipped with a Lie bracket of degree 1-d, which can be realized very concretely via topological descent. We'll look at examples of this bracket, in 2d, 3d, and 4d. We'll also use topological descent to give a new interpretation of the statement that turning on on Omega background is a form of quantization. The second lecture focuses on another sort of homological/descent operation, this time in the context of SUSY quantum mechanics with G symmetry. We'll find that Hilbert spaces in (de Rham) SQM come equipped with a natural homological action of G, i.e. an action of the exterior algebra H_*(G). This action controls the process of gauging the G symmetry. A nice physical way to understand the H_*(G) action and gauging/ungauging comes from considering SUSY boundary conditions for 2d G gauge theory; this will lead us to a definition of the "Fukaya category of point/G." The mathematical structures involved come from work of Goresky, Kottwitz, and MacPherson. Given time, we'll discuss some higher-dimensional examples, and the physics of Koszul duality. The third lecture applies some of the ideas from the first two in the specific context of 3d N=4 gauge theories. Our goal will be to define the category of line operators in the A and B twists of these theories, and to understand -- in a concrete and computational way -- the algebras of local operators bound to a line. These algebras get quantized in an Omega background; and they act on modules defined by boundary conditions. In the special case of an A twist and the trivial (identity) line operator, we will recover the Braverman-Finkelberg-Nakajima construction of the Coulomb-branch chiral ring.

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