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:

A detailed study of simple examples. One excellent class of examples to study is are the B-side singularity categories associated to simple hypersurface singularities. This involves Saito's primitive form theory and has connections to more recent developments in Hodge-theoretic invariants of singularities.
The research program of Caldararu-Tu on categorical Gromov-Witten invariants.
Si Li's work on higher genus B-model invariants via BCOV theory. (And its relations to the above.)
Ganatra-Perutz-Sheridan's derivation of "classical mirror symmetry" from homological mirror symmetry.
Quantum D-modules of Fano varieties (especially with an eye to the Katzarkov-Kontsevich-Pantev program for irrationality proofs).

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!)