Seminar on noncommutative Hodge theory, Spring 2021

Overview

This seminar is focused on applications of the theory of non-commutative Hodge structures, as formalized by Kontsevich-Katzarkov-Pantev. One perspective on this theory begins from the observation that all of the information contained in the Hodge structure of a variety X can actually be read off of the dg category of coherent sheaves Coh(X). Accordingly, the KKP theory proposes to read off a "noncommutative Hodge structure" directly from a dg category, which may be the category of coherent sheaves on X, or it may be a Fukaya-type category. This categorical approach to Hodge theory is able to produce quite a lot of information, including genus 0 Gromov-Witten invariants, and it has become even more popular recently thanks to its role in the claimed proof of irrationality for a general cubic fourfold.

Although the phrase "noncommutative Hodge structure" is relatively new, actual examples of such structures existed for decades before, including the "tt*-geometry" of Cecotti-Vafa, "weight systems" of K. Saito, "twistor structures" of Simpson, and the "variations of semi-infinite Hodge structure" of Barannikov-Kontsevich. One basic way to understand the theory of NC Hodge structures is as a generalization of the fact that the Hodge structure associated to a variety X may be recovered through consideration of Hochschild invariants of the category Coh(X) of coherent sheaves on X. The same techniques may be applied to more general dg categories to yield new types of invariants. For instance, from the Fukaya category Fuk(X), it is possible to recover Gromov-Witten invariants in this way.

The seminar will begin with introductory talks on non-commutative Hodge structures, Frobenius manifolds, and cohomological field theories. Afterward, some topics that may be covered include: E-mail me (Ben G.) if you want on the mailing list.

The seminar meets 2pm Wednesdays.

Schedule

Date Speaker Topic
Jan. 27 Benjamin Gammage Introduction and organization
Feb. 3 Benjamin Gammage Non-commutative Hodge structures [1]
Feb. 10 Tim Large Frobenius manifolds [3,4]
Feb. 17 Dori Bejleri Cohomological field theories [4,14]
Feb. 24 Kyoung-Seog Lee Saito primitive form theory [6,7,10]
Mar. 3 Oleg Lazarev Curve counts from categories [15,15a,18]
Mar. 10 Alex Takeda Categorical primitive forms [9]
Mar. 17 Semon Rezchikov Givental reconstruction [4,12,13,14]
Mar. 24 Surya Raghavendran Classical BCOV theory [16,11]
Mar. 31 Surya Raghavendran Quantum BCOV theory [16, 11]
Apr. 7 Dori Bejleri Higher-genus reconstruction: a worked example [14]
Apr. 14 Benjamin Gammage Gamma(ge?) conjectures [19]



References

[1] Kontsevich-Katzarkov-Pantev, Hodge theoretic aspects of mirror symmetry.
[2] Simpson, Mixed twistor structures.
[3] Dubrovin, Geometry of 2d topological field theories.
[4] Teleman, The structure of 2d semi-simple field theories.
[5] Barannikov-Kontsevich, Frobenius manifolds and formality of Lie algebras of polyvector fields.
[6] K. Saito, Period mapping associated to a primitive form.
[7] K. Saito, Duality for regular systems of weights: a précis.
[8] Steenbrink, Mixed Hodge structure on the vanishing cohomology.
[9] Caldararu, Tu, et al: papers on categorical Saito theory and categorical enumerative invariants.
[9a] Tu-Amorim, Categorical primitive forms of Calabi-Yau A∞-categories with semi-simple cohomology
[10] C. Li-S. Li-K. Saito, Primitive forms via polyvector fields.
[11] Costello, S. Li, et al.: papers on BCOV theory.
[11a] S. Li, Renormalization method and mirror symmetry.
[12] Givental, Semisimple Frobenius structures at higher genus.
[13] Givental, Gromov-Witten invariants and quantization of quadratic Hamiltonians.
[14] Pandharipande, Cohomological field theory calculations.
[15] Ganatra-Perutz-Sheridan, Mirror symmetry: from categories to curve counts.
[15a] Sheridan, Formulae in noncommutative Hodge theory
[16] Costello, TCFTs and Calabi-Yau categories.
[17] Costello, The partition function of a topological field theory.
[18] Shklyarov, Matrix factorizations and higher residue pairings
[19] Galkin-Golyshev-Iritani, Gamma classes and quantum cohomology of Fano manifolds: Gamma conjectures


Organized by Ben G. (that's me!)