Extending Landau-Ginzburg Models to the Point
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Communications in
Mathematical Physics
Extending Landau-Ginzburg Models to the Point Nils Carqueville , Flavio Montiel Montoya Fakultät für Physik, Universität Wien, Wien, Austria. E-mail: [email protected]; [email protected] Received: 12 December 2018 / Accepted: 4 August 2020 Published online: 7 October 2020 – © The Author(s) 2020
Abstract: We classify framed and oriented 2-1-0-extended TQFTs with values in the bicategories of Landau-Ginzburg models, whose objects and 1-morphisms are isolated singularities and (either Z2 - or (Z2 × Q)-graded) matrix factorisations, respectively. For this we present the relevant symmetric monoidal structures and find that every object W ∈ k[x1 , . . . , xn ] determines a framed extended TQFT. We then compute the Serre automorphisms SW to show that W determines an oriented extended TQFT if the associated category of matrix factorisations is (n − 2)-Calabi-Yau. The extended TQFTs we construct from W assign the non-separable Jacobi algebra of W to a circle. This illustrates how non-separable algebras can appear in 2-1-0-extended TQFTs, and more generally that the question of extendability depends on the choice of target category. As another application, we show how the construction of the extended TQFT based on W = x N +1 given by Khovanov and Rozansky can be derived directly from the cobordism hypothesis. Contents 1. 2.
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Introduction . . . . . . . . . . . . . . . . . . Bicategories of Landau-Ginzburg Models . . . 2.1 Definition of LG . . . . . . . . . . . . . 2.2 Monoidal structure for LG . . . . . . . . 2.3 Symmetric monoidal structure for LG . . 2.4 Duality in LG . . . . . . . . . . . . . . . 2.4.1 Adjoints for 1-morphisms . . . . . . 2.4.2 Duals for objects . . . . . . . . . . . 2.5 Graded matrix factorisations . . . . . . . Extended TQFTs with Values in LG and LG gr 3.1 Framed case . . . . . . . . . . . . . . . . 3.2 Oriented case . . . . . . . . . . . . . . .
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N. Carqueville, F. Montiel Montoya
1. Introduction Fully extended topological quantum field theory is simultaneously an attempt to capture the quantum field theoretic notion of locality in a simplified rigorous setting, and a source of functorial topological invariants. In dimension n, such TQFTs have been formalised as symmetric monoidal (∞, n)-functors from certain categories of bordisms with extra geometric structure to some symmetric monoidal (∞, n)-category C. The fact that such functors must respect structure and relations among bordisms of all dimensions from 0 to n is highly restrictive. Specifically, the cobordism hypothesis of [BD] as formalised in [Lu,AF] states t
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