北京雁栖湖应用数学研究院

The quantum period and toric Landau-Ginzburg models can be defined for many Fano varieties. They are powerful invariants for Fano varieties up to Q-Gorenstein deformation. I will explain some recent results on how to compute these invariants under extremal contractions. This work is joint with A. Sheshmani.

In the talk we will discuss complete intersections in weighted projective spaces and more general ambient spaces from the point of view of classification of Fano varieties, Mirror Symmetry, and arithmetics.

In 1983, Feingold and Frenkel posed a question about possible relations between affine Lie algebras, hyperbolic Kac-Moody algebras and Siegel modular forms. We give an automorphic answer to this question and its generalization. We classify Borcherds-Kac-Moody algebras whose denominators define reflective Borcherds products of singular weight. We prove that there are exactly 81 affine Lie algebras which have nice extensions to BKM algebras. We find that 69 of them appear in Schellekens' list of holomorphic CFT of central charge 24, 8 of them correspond to the N=1 structures of holomorphic SCFT of central charge 12, and the last 4 cases are related to exceptional modular invariants from nontrivial automorphisms of fusion algebras. We further discuss the generalization to hyperbolization of affine Lie superalgebras. This is based on joint works with Haowu Wang and Brandon Williams.

In this talk I will consider a moduli space of projective varieties enhanced with a certain frame of their cohomology bundles. In many examples such as elliptic curves, abelian varieties, Calabi-Yau varieties, and conjecturally in general, this moduli space is a quasi-affine scheme over Z[1/N] for some natural number N. The ring of global regular functions on this quasi-affine variety is called the ring of CY modular forms and it gives us natural generalizations of modular and quasi-modular forms. In the case of Calabi-Yau threefolds, it includes the genus g topological string partition function encoding genus g Gromov-Witten invariants. We can also rewrite the BCOV anomaly equation using certain vector fields on this moduli space which are algebraic incarnation of differential equations of automorphic forms. After taking modulo p of this moduli space for p coprime with N, I will discuss an outline of a project how to prove the p-integrality of Calabi-Yau modular forms beyond the established cases of hypergeometric Calabi-Yau varieties. The talk is based on my book "Modular and Automorphic Forms & Beyond, Monographs in Number Theory, World Scientific (2021)" in which the Tupi name ibiporanga (pretty land) for such a moduli space is suggested.

Using ideas from gauged linear sigma model (GLSM) we write noncommutative resolutions of branched double covers, focusing on the Calabi-Yau (CY) case. We use the GLSM data to write central charges of B-branes, i.e. certain functions from a category of matrix factorizations associated to the noncommutative resolution to the complex numbers, termed A-periods. In addition, I will show that such A-periods can also be realized as A-periods of a certain smooth CY family X, and a finite quotient of this family recovers the branched double cover CY we have started with (hence X is termed the pre-quotient).

We prove that the supersymmetric deformed CP1 sigma model admits an equivalent description as a generalized Gross-Neveu model. Remarkably we find new Nahm-type conditions, which guarantee renormalizability and supersymmetric invariance. Moreover it provides formalism, which is useful for the study of renormalization properties and particularly for calculation of such observables as β and correlation functions. We study the RG flow of the new class and find special UV conformal points from both sides of the new Chiral/Sigma model correspondence. We further explore novel relations of our construction through mirror symmetry and dimensional reductions. We demonstrate its emergence from the four-dimensional TQFT with defects, investigate surface observables and provide the associated spin chain system.

This talk will discuss generating functions for Hecke systems of Poincaré series, their relationship to the higher automorphic Green's function on the product of two modular curves, and a conjectured relation to single-valued polylogarithms based on the action of Maass operators.

We consider a particular example of a discrete Painlevé equation arising from a construction of quantum minimal surfaces by Arnlind, Hoppe and Kontsevich. Observing that this equation corresponds to a very special choice of parameters (root variables) in the Space of Initial Conditions for the differential Painlevé V equation, we show that some explicit special function solutions, written in terms of modified Bessel functions, for d-PV yield the unique positive solution for some initial value problem for the discrete Painlevé equation needed for quantum minimal surfaces. This is a joint work with Peter Clarkson, Andy Hone, and Ben Mitchell.

We report on recent series of joint works with Jacob Kryczka and Shing-Tung Yau on construction of derived moduli spaces of solutions to nonlinear PDE. We construct a parameterizing space of ideal sheaves of involutive and formally integrable non-linear partial differential equations in the algebraic-geometric setting. We elaborate on the construction of a D-geometric analog of Grothendieck's Quot (resp. Hilbert) functor and prove that its is represented by a D-scheme which is suitably of finite type. A natural derived enhancement of the so-called D-Quot (resp. D-Hilbert) moduli functor is constructed and its representability by a differentially graded D-manifold with corresponding finiteness properties is studied. If time permits, we further elaborate on construction of universal variational tri-complexes over these derived D-hilbert schemes and their induced BV structure.