Derived Categories and Mirror Symmetry

Title: Generalized Riemann-Hilbert correspondence

Abstract: Riemann-Hilbert correspondence for regular singular holonomic D-modules on a complex manifold is an equivalence between the corresponding derived category of such D-modules and the derived category of constructible sheaves. The natural abelian heart on the D-module side corresponds to the abelian category of perverse sheaves. In this form it was proved by Kashiwara in 80's. There is a recent upgrade of this result to irregular holonomic D-modules (D'Agnolo-Kashiwara).

I am going to discuss current status of the generalized Riemann-Hilbert correspondence proposed by Kontsevich and myself 10 years ago as a part of our program ``Holomorphic Floer Theory". It is a claim (conjectural in general) about derived equivalence of the appropriate category of holonomic DQ-modules on a general complex symplectic manifold (DQ here stands for ``deformation quantization") and the Fukaya category of this manifold. The natural abelian heart on the DQ-module side corresponds to the full subcategory of the Fukaya category in which objects are supported on complex Lagrangian subvarieties. In this form generalized Riemann-Hilbert correspondence can be applied not only to differential equations (i.e. D-modules) but also to q-difference equations, elliptic difference equations, etc. unifying different known versions of the Riemann-Hilbert correspondence under the same roof of Holomorphic Floer Theory.

Title: Gopakumar-Vafa invariants via Wilson loops

Abstract: A compact Calabi-Yau threefold X can be considered to approach a local Calabi-Yau threefold Y in certain limits of Kahler moduli. The enumerative geometry of Y is simpler and physical tools of 5-dimensional gauge theory apply, particularly the use of Wilson loops. A general proposal is made for the structure of the refined BPS numbers of X in terms of refined Wilson loops. Using low degree geometric computation and methods of physics including a refined holomorphic anomaly equation for the Wilson loop partition function, refined BPS numbers of elliptically fibered X are computed for many examples, with h^{1,1}(X) as large as 5. In particular, the unrefined limit produces the Gopakumar-Vafa invariants of X. This talk is based on joint work with Minxin Huang, Albrecht Klemm, and Xin Wang.

Title: Finding the walls in GIT

Abstract: Geometric Invariant Theory (GIT) can be used to construct familiar varieties, including many toric varieties, as well as flag varieties or more generally moduli of quiver representations. The version of GIT I will use in this talk depends on a complex reductive group G, a linear representation of G, and a homomorphism from G to the group of complex units. While complex reductive groups include general linear (type A) groups as well as orthogonal and symplectic (types B, C, D) groups, the examples of GIT mentioned in the first sentence all arise from representations type A groups. Even for representations of the general linear group, the GIT procedure sometimes “goes wrong” and produces an Artin stack instead of a variety or Deligne-Mumford stack.

Why don’t we have examples of GIT using symplectic or orthogonal groups? When the group is GLn (or more generally type A), can we tell when GIT will produce a variety or Deligne-Mumford instead of an Artin stack? The answers to these questions both amount to “finding the walls” in variation of GIT. I will present partial answers, some of which are joint with Hans Franzen, and others with Riku Kurama, Ruoxi Li, and Henry Talbott.

Title: On the Torelli problem for Calabi–Yau 3-folds

Abstract: A global Torelli theorem does not hold for Calabi–Yau 3-folds: the best counterexample, due independently to Ottem and Rennemo and to Borisov, Caldararu, and Perry, gives pairs of CY3s in the same deformation family that have equivalent derived categories of coherent sheaves, hence isomorphic polarized Hodge structures on H^3(X,Z), but are not birational. One might then ask whether a derived Torelli theorem holds. I will discuss an example of Aspinwall, Morrison, and Szendrői, which was expected to provide a counterexample; it turns out not to, but fits into the whole picture in a surprising way. This is joint work with Ben Tighe, arXiv:2407.11176.

Title: Motivic Classes of Character Stacks via TQFT

Abstract: In this talk, I will present a Topological Quantum Field Theory (TQFT) framework for computing motivic classes of character stacks. Focusing on the case of the special linear group $SL_2$, I will explain how to derive these motivic classes using a finite field variant of the TQFT together with a motivic decomposition theorem that decomposes motivic classes of $G$-varieties (with $G$ being finite) in terms of rational representations of $G$. This work is joint with Ángel González-Prieto and Jesse Vogel. If time permits, I will also discuss a conjectural extension to linear reductive groups, based on ongoing work with Lucas de Amorin, Ángel González-Prieto, and Martín Mereb.

Title: Counting pseudo-holomorphic sections of Lefschetz fibrations

Abstract: Joint work with Riccardo Pedrotti (U. Mass. Amherst). A sequence of simple closed curves on a surface determines a word in the mapping class group – the positive Dehn twists along the curves - and a 4-dimensional Lefschetz fibration over the disc, with the composite of the twists as its outer monodromy. In the case that the word is trivial, this closes up - uniquely, when the fiber-genus is at least 2 - to a Lefschetz fibration over the 2-sphere. This 4-manifold has Seiberg-Witten invariants, which have a known symplectic interpretation as Gromov-Witten-type invariants counting pseudo-holomorphic multisections of the Lefschetz fibration (Donaldson-Smith invariants). Is there a “formula” for the Seiberg-Witten invariants in terms of the configuration of curves on the surface? We show that there is a recipe which, at least for the portion of the story that counts sections, gives such a formula. The story involves a quilted Fukaya category of cyclic sequences of Lagrangian correspondences, and analogs of Hochschild homology (an A-infinity version of Barr-Beck or cotriple homology).

Title: Braided Koszul duality and Lie algebras

Abstract: Braided operads (introduced by Fiedorowicz) are gadgets that control algebraic structures on objects in braided monoidal categories. There are natural braided analogues of the familiar associative and commutative operads in the symmetric monoidal setting. While one can define a braided Lie operad by pattern-matching with the symmetric Lie operad, it doesn’t correctly describe the operations on the primitives in a braided Hopf algebra, unlike its symmetric counterpart. In this talk, I’ll explain how one can come to a correct braided analogue of the Lie operad using Koszul duality for braided operads. The result has nice interpretations in terms of a Cohen-Macaulay braided partition poset and the dualizing module for the braid groups.

Title: TQFT-like operations for Fano varieties

Abstract: The Hochschild chain complex of a Calabi-Yau variety (or more generally category) famously admits the structure of a framed E_2 algebra (the genus zero operations of the corresponding 2d TQFT). I will describe a new algebraic point of view on these operations which admits generalizations to the setting of relative Calabi-Yau structures, which do not seem to fit into the framework of TQFTs. In particular, we obtain analogous operations on Hochschild homology of Fano varieties. This is joint work with Chris Brav.

Regsitration will take place in room Van Vleck 901 between 2:00 and 3:30 on Wednesday 5/28.

Title: Open FJRW theory

Abstract: FJRW theory is an enumerative theory built from characteristic classes corresponding to the moduli space of W-spin curves, a natural generalisation of higher spin curves. They can be interpreted as the enumerative geometry of gauged Landau-Ginzburg models. In this talk, I will describe an enumerative theory for an open version of FJRW invariants for maximally graded Landau-Ginzburg models with Fermat superpotentials and discuss their properties.

Title: Categorical and geometric enumerative invariants

Abstract: Costello constructed categorical enumerative invariants associated to a smooth and proper Calabi--Yau category equipped with a splitting of the non-commutative Hodge--de Rham spectral sequence. I will discuss the question of extending this to a chain-level CohFT structure on the cyclic homology of the category. I will describe the implication of this construction for comparing the categorical CohFT of the Fukaya category with the Gromov--Witten CohFT, when a suitable chain-level open-closed Gromov--Witten CohFT can be constructed. Partly based on joint work with A. Hirschi and K. Hugtenberg.

Title: Open-Closed Categorical Enumerative Invariants

Abstract: I will discuss ongoing work, with Junwu Tu, aiming at defining an open-closed version of Categorical Enumerative Invariants. For Fukaya categories these should recover open Gromov-Witten invariants (when both are defined). I will describe the first step in this project: an open-closed version of the Sen-Zwiebach DGLA and its resolution by a L-infinity algebra. I will then explain how to extract enumerative invariants from this setup in genus zero and then discuss the higher genus situation.

Title: Galkin's Conjecture for Cubic Fourfolds

Abstract: A smooth cubic fourfold gives rise to two kinds of hyperkähler fourfolds: one is classical --the Fano variety of lines on the cubic; and the other is "non-commutative" --arising from the symmetric square of the Kuznetsov component. Galkin conjectured that these two objects should be derived equivalent. In this talk, I’ll explain a proof of this conjecture using ideas from matrix factorizations and window categories. This is joint work with Ed Segal.

Title: King's Conjecture is Dead, Long Live the King's Conjecture

Abstract: King's conjecture proposed that sheaves on projective toric varieties can be understood using noncommutative algebra. Or more formally, that the derived category of a toric variety is equivalent to modules over a noncommutative ring. While this conjecture has been proven in many cases, it turns out that it is false in general with counterexamples by Hille-Perling, Efimov, and Michalek. Inspired by homological mirror symmetry, I will discuss a modification of the conjecture where it turns out to be true as well as some consequences. The talk is based on joint works with Matthew Ballard, Michael Brown, Lauren Cranton Heller, Daniel Erman, Sheel Ganatra, Andrew Hanlon, Jesse Huang, and Mykola Sapronov.

Title: Homological mirror symmetry for Batyrev mirror pairs

Abstract: I will overview joint work with Hanlon-Hicks-Pomerleano-Sheridan proving Kontsevich’s homological mirror symmetry conjecture for a large class of mirror pairs of (compact) Calabi–Yau hypersurfaces in toric varieties. These mirror pairs were first constructed by Batyrev from dual reflexive polytopes, and our result holds in characteristic zero and in all but finitely many positive characteristics.

Title: Categorical enumerative invariants of the ground field

Abstract: Categorical enumerative invariants, first introduced by K. Costello around 2004, are certain invariants associated with smooth and proper Calabi-Yau categories. These invariants conjecturally should recover the Gromov-Witten invariants when applied to Fukaya categories. In this talk, we prove this comparison result in the simplest case when the category under consideration is the ground field. We shall also discuss how this result might be useful in proving the comparison with Gromov-Witten invariants in all semi-simple cases.

Title: On the differentials of the Hochschild-Kostant-Rosenberg spectral sequence

Abstract: Hochschild homology is an invariant of derived categories. When applied to the derived category of modules over a commutative ring, the Hochschild-Kostant-Rosenberg theorem gives a formula for Hochschild homology in terms of differential forms. This formula extends to the Hochschild-Kostant-Rosenberg decomposition for complex algebraic varieties. In this talk, we quantitatively explain the failure of this decomposition in positive characteristic.