Adjunctions and defects in Landau-Ginzburg models

Abstract

We study the bicategory of Landau-Ginzburg models, which has potentials as objects and matrix factorisations as 1-morphisms. Our main result is the existence of adjoints in this bicategory and a description of evaluation and coevaluation maps in terms of Atiyah classes and homological perturbation. The bicategorical perspective offers a unified approach to Landau-Ginzburg models: we show how to compute arbitrary correlators and recover the full structure of open/closed TFT, including the Kapustin-Li disk correlator and a simple proof of the Cardy condition, in terms of defect operators which in turn are directly computable from the adjunctions.

58 pages; v2: Fixed typos and references, removed comments about graded matrix factorisations; v3: exposition improved and shortened, main result now holds over an arbitrary ring k; v4: many improvements to exposition

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• Towards R-matrix construction of Khovanov-Rozansky polynomials. I. Primary $T$-deformation of HOMFLY
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• Lecture notes on 2-dimensional defect TQFT
• Introductory lectures on topological quantum field theory
• Topological defects
• Realizing IR theories by projections in the UV
• Fully extended $\mathbf{r}$-spin TQFTs
• Extending Landau-Ginzburg models to the point
• Hybrid models for homological projective duals and noncommutative resolutions
• Differential models for B-type open-closed topological Landau-Ginzburg theories
• Matrix factorisations for rational boundary conditions by defect fusion
• Calabi-Yau completions and orbifold equivalences
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• Truncated affine Rozansky--Witten models as extended TQFTs
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• Orbifold autoequivalent exceptional unimodal singularities
• Planar diagrammatics of self-adjoint functors and recognizable tree series
• Gerstenhaber brackets on Hochschild cohomology of quantum symmetric algebras and their group extensions
• A toolkit for defect computations in Landau-Ginzburg models
• Fusion of interfaces in Landau-Ginzburg models: a functorial approach
• B-type D-branes in hybrid models
• Necessary conditions for the existence of Morita Contexts in the bicategory of Landau-Ginzburg Models
• Chern character for matrix factorizations via Chern-Weil
• Foams, iterated wreath products, field extensions and Sylvester sums
• Truncated affine Rozansky--Witten models as extended defect TQFTs
• Some applications of the multiplicative tensor product of matrix factorizations
• Constructing $A_{\infty }$-categories of matrix factorisations