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Cohomological field theories, Chern characters and matrix factorizations (introductory lecture)

Banff International Research Station for Mathematical Innovation and Discovery

Many properties of an algebraic variety X can be expressed in terms of the derived category of coherent sheaves on X (or its differential-graded enhancement). Kontsevich proposed to view arbitrary smooth and proper dg-categories as non-commutative analogs of smooth projective varieties. I will show how holomorphic functions with isolated singularities fit into this picture. In the first part of the lecture we will talk about Chern characters and the Hirzebruch-Riemann-Roch theorem for dg-categories and will see how classical invariants of singularities appear via dg-categories of matrix factorizations. Then we will turn to quantum invariants and cohomological field theories - an algebraic structure underpinning formal properties of the Gromov-Witten invariants. For a quasi-homogeneous singularity W and a finite group G of its symmetries we will describe a CohFT whose state space is the equivariant local algebra (Milnor ring) of W and whose correlators can be viewed as analogs of Gromov-Witten invariants for the non-commutative space associated with the pair (W,G). The role of the virtual fundamental class from the Gromov-Witten theory is played here by a "fundamental matrix factorization" over a certain moduli space.

Mathematics; Algebraic geometry; Quantum theory; Mathematical physics

video/mp4

Author affiliation: University of Oregon

2019-03-25

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/69172

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/