Browsing by Subject "Topological field theories"
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Item Geometry of integrable hierarchies and their dispersionless limits(2014-05) Safronov, Pavel; Ben-Zvi, David, 1974-This thesis describes a geometric approach to integrable systems. In the first part we describe the geometry of Drinfeld--Sokolov integrable hierarchies including the corresponding tau-functions. Motivated by a relation between Drinfeld--Sokolov hierarchies and certain physical partition functions, we define a dispersionless limit of Drinfeld--Sokolov systems. We introduce a class of solutions which we call string solutions and prove that the tau-functions of string solutions satisfy Virasoro constraints generalizing those familiar from two-dimensional quantum gravity. In the second part we explain how procedures of Hamiltonian and quasi-Hamiltonian reductions in symplectic geometry arise naturally in the context of shifted symplectic structures. All constructions that appear in quasi-Hamiltonian reduction have a natural interpretation in terms of the classical Chern-Simons theory that we explain. As an application, we construct a prequantization of character stacks purely locally.Item Taking topological field theory at phase value(2021-05-05) Debray, Arun Albert; Freed, Daniel S.; Ben-Zvi, David; Distler, Jacques; Neitzke, Andrew; Raskin, SamIn this thesis, we use methods of topological field theory to model and study topological phases of matter. This includes computing TFTs that capture low-energy information for the GDS model and for the Majorana chain with time-reversal symmetry. We then investigate phases of matter with spatial symmetries that mix with the internal symmetry type; we provide a mathematical model for these phases and prove a “fermionic crystalline equivalence principle” theorem as predicted in the physics literature. Some of our computations lead to a bonus theorem on the classification of some unorientable 4-manifolds up to stable diffeomorphism.