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3-dimensional mirror symmetry and elliptic cohomology

Course Type: 
PhD Course
Academic Year: 
20 h

This is an advanced course that will explore connections between recent developments in geom- etry, mathematical physics and homotopy theory. Its format will be that of a working seminar. Mirror symmetry has been at the center stage of geometry for the last thirty years. It is a sophisticated dictionary relating the symplectic geometry of a variety X and the algebraic ge- ometry of its mirror Y , and vice versa. The current picture is that at bottom this dictionary is an equivalence of categories between the Fukaya category of X and the derived category of coherent sheaves of Y . Taking Hochschild cohomology on both sides, yields an isomorphism between the quantum cohomology of X and the algebra of polyvector fields on Y . This recovers enumerative identities relating curve counts on Y and periods of X. The theory of 3d mirror symmetry is much more recent. Let me stress 3d does not refer to the dimension of X or Y , but rather to the fact that this theory is a three-dimensional field theory rather than a two-dimensional field theory (such as classical mirror symmetry). At bottom 3d mirror symmetry should be an equivalence of 2-categories. One expectation is that, on one side of the correspondence, the 2-category will be the Rozansky-Witten 2-category. Then one can compute invariants of these 2-categories, and obtain isomorphisms which are the analogue of the enumerative isomorphisms in ordinary mirror symmetry. Elliptic cohomology is expected to arise precisely as an invariant of these 2-categories. In this course we will try to understand various threads coming together in 3d mirror sym- metry and elliptic cohomology. The specific topics we will tackle will depend on the interests of the participants, but here are some pointers to relevant literature:

  •  Costello, K., 2010. A geometric construction of the Witten genus, I. In Proceedings of the International Congress of Mathematicians 2010 (ICM 2010) (In 4 Volumes) Vol. I: Plenary Lectures and Ceremonies Vols. II–IV: Invited Lectures (pp. 942-959).
  • Gammage, B., Hilburn, J. and Mazel-Gee, A., 2022. Perverse schobers and 3d mirror symmetry. arXiv preprint arXiv:2202.06833.
  • Teleman, C., 2014. Gauge theory and mirror symmetry. arXiv preprint arXiv:1404.6305.
  • • Rimányi, R., Smirnov, A., Varchenko, A., Zhou, Z., 2019. 3d mirror symmetry and elliptic stable envelopes. arXiv:1902.03677.
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