Inaugural conference for the Laboratory of Mirror Symmetry and Automorphic forms


Serguei Barannikov

Title: Operads and matrix integrals

Abstract: Matrix integral naturally associated with the simplest cyclic associative algebra, consisting of one element, was shown in the work of the speaker to be the Airy-Kontsevich matrix integral of the 2-dimensional topological gravity. This in particular defines an equivariant super extension of the latter matrix integral. The asymptotic expansion of the extended integral gives a representation of products of psi-classes as cocycles on stable ribbon graph complex. The matrix integrals generalizing this matrix integral were introduced in the work of the speaker for any associative/A-infinity algebras. Their asymptotic expansions define interesting cocycles on the compactified moduli space of curves. These cocycles describe conjecturally the all genus Gromov-Witten invariants of compact Calabi-Yau manifolds. Different aspects of this will be discussed, if the time permits, in particular, the BV operator on symmetric/exterior powers of cyclic cochains and the construction of solutions to the noncommutative quantum master equation from the degeneration of the Hodge to de Rham spectral sequence.

Alexey Bondal

Title: Categorifying the noncommutative deformation theory

Abstract: We consider a categorification of the infinitesimal noncommutative deformation theory for a system of objects in an abelian category. By categorification we mean here that the underlying noncommutative deformation space for the system of objects is understood as a suitable category as an avatar for the spectrum of a noncommutative algebra. We define the deformation functor in this set-up and prove that it is ind-representable. We give an explicit description of the ind-representing category and describe some applications in geometry.

Chris Brav

Title: Relative Calabi-Yau structures on dg functors and Lagrangian structures on maps of moduli

Abstract: Given a Calabi-Yau structure on a smooth dg category S, I’ll explain how to construct a shifted symplectic structure on its derived moduli space of objects M(S), and given a “relative Calabi-Yau structure” on a dg functor S → T, how to construct a shifted Lagrangian structure on the corresponding map of moduli spaces M(T) → M(S). After describing a natural relative Calabi-Yau structure related to quivers of type A, I’ll then focus on the rich structure induced on the moduli spaces of all type A quiver representations. This is joint work with Tobias Dyckerhoff.

Alexander Efimov

Title: On the geometric t-structures on derived categories of finite-dimensional algebras

Abstract: We will adress a question of Bondal and Polishchuk on existence of geometric t-structures on certain triangulated categories with full exceptional collections. In the classical case of Beilinson’s exceptional collection on the derived category of a projective space one recovers the t-structure whose core consists of coherent sheaves.

In the general case the existence of such a t-structure turns out to be a very difficult question. I will show that it is equivalent to the vanishing of higher self-extensions of a certain infinite-simensional module. Also, in the case of the tensor product of two Kronecker quivers, the question will be interpreted in terms of the limit points (in the lower limit topology) of the union of sets of Harder-Narasimhan slopes of certain sequences of Kronecker representations.

Sergei Gorchinskiy

Title: Categorical measures for varieties with finite group actions.

Abstract: The talk is based on a common work with D. Bergh, M. Larsen, and V. Lunts. Given a variety with a finite group action, we compare categorical measures of the corresponding quotient stack and the extended quotient. Using weak factorization for orbifolds, we show that for a wide range of cases, these two measures coincide, which implies, in particular, a conjecture of Galkin and Shinder on categorical and motivic zeta-functions of varieties. We provide examples showing that in general, these two measures are not equal.

Sergey Gukov

Title: Modularity and 3-manifolds

Abstract: In this talk I will describe a new bridge between number theory and low-dimensional topology, motivated by physics (more precisely, by the recent work arXiv:1701.06567). In topology, one way to characterize spaces is by their fundamental group, which unfortunately is too complicated in general and, therefore, one might prefer to consider its representations into various Lie groups. In the case of 2-manifolds, it turns out that such representation spaces for a group G and its Langlands dual provide examples of mirror manifolds. In this talk we explore the question: What is the analogue for 3-manifolds?

Albrecht Klemm

Title: Topological String theory and the ring of Jacobi forms

Abstract: We show that the all genus topological string partition function on compact elliptically fibered Calabi-Yau manifolds can be written in terms of meromorphic Jacobi forms whose weight grows linearly and whose index grows quadratically with the base degrees. For suitable contractible curve configurations in the base, the BPS spectrum of the corresponding superconformal 6d-theory is completly determined by this topological string calculation. In particular this applies to the E-string, E-String chains and chains involving 6d superconformal theories with non-abelian gauge symmetries.

Alexander Kuznetsov

Title: Derived categories of families of sextic del Pezzo surfaces

Abstract: I will give a description of the derived category of coherent sheaves on a flat family of normal sextic del Pezzo surfaces with du Val singularities.

Alexey Litvinov

Title: On the dual description of the deformed O(N) sigma model

Abstract: TBA

Valery Lunts

Title: Hodge numbers of Landau-Ginzburg models

Abstract: Abstract: Homological mirror symmetry predicts in particular an equivalence of categories Db(coh X)=Fuk(Y, f), where X is a smooth projective Fano variety and (Y, f) is the “mirror symmetric” Landau-Ginzburg model. The Hodge numbers of X can be reconstructed from the category Db(coh X). I will discuss a conjectural description of these Hodge numbers (suggested by Katzarkov-Kontsevich-Pantev) in terms of the geometry of (Y, f). This is work in progress with V. Przyjalkowski, A. Harder, and L. Katzarkov.

Nikita Nekrasov

Title: Magnificent Four

Abstract: Supersymmetric gauge theories in various spacetime dimensions lead to enumerative problems: counting of holomorphic curves in Calabi-Yau manifolds, intersection theory on moduli spaces of instantons, limit shapes for random Young diagrams in two and three dimensions. The latter problem is related to dimer models and the models of crystal melting. We shall report on the recent developments in four dimensions, the ultimate dimension accessible by supersymmetric gauge theory. The string theory motivation for the problem is the counting of bound states of D0 branes in the presence of the D8-antiD8 branes and a B-field. The random configurations are the so-called solid partitions, which can also be viewed as tesselations of the 3-space by four types of squashed cubes. The similar problem of D0-D6 brane counting led to the partition function which was conjectured in 2004 to be given by the (square of the) Witten index of 11d supergravity. The conjecture was proven in 2015 by A.Okounkov. I will present the conjecture on the partition function of the new model. The M-theory interpretation of the problem remains a mystery. Mathematically we are proposing an explicit formula for the sheaf counting on toric Calabi-Yau fourfolds. We conclude with some speculations on representation theory of quantum toroidal algebras.

Vyacheslav Nikulin

Title: Lorentzian Kac–Moody algebras with Weyl groups of 2-reflections.

Abstract: This is a continuation of our papers with V. A. Gritsenko in 1997–2002 where we mainly considered the case of hyperbolic lattices of the rank 3. Here, for all ranks, we classify 2-reflective hyperbolic lattices S of elliptic type with a lattice Weyl vector. They define the corresponding hyperbolic Kac–Moody algebras of arithmetic type which are graded by S. For most of them, we construct Lorentzian Kac–Moody algebras which give their automorphic corrections: they are graded by the S, have the same simple real roots, but their denominator identity is given by automorphic forms with 2-reflective divisors. We give exact construction of these automorphic forms.

All these considerations are related to interesting classes of algebraic K3 surfaces and their mirror symmetry.

See our recent preprints with V. A. Gritsenko arXiv:1602.08359 and arXiv:1702.07551 for some details.

Alena Pirutka

Title: Irrationality problems

Abstract: Let X be a projective algebraic variety, the set of solutions of a system of homogeneous polynomial equations. Several classical notions describe how “unconstrained” the solutions are, i.e., how close X is to projective space: there are notions of rational, unirational and stably rational varieties. Over the field of complex numbers, these notions coincide in dimensions one and two, but diverge in higher dimensions. In the last years, many new classes of non stably rational varieties were found, using a specialization technique, introduced by C. Voisin. This method also allowed to prove that the rationality is not a deformation invariant in smooth and projective families of complex varieties, this is a joint work with B. Hassett and Y. Tschinkel. In the first part of my talk I will describe some classical examples, as well as the recent examples obtained by the specialization method. I will give more details on this method in the second part of my talk.

Alexander Polishchuk

Title: Mirror symmetry for punctured surfaces and Auslander orders

Abstract: This is a joint work with Yanki Lekili. We consider partially wrapped Fukaya categories of punctured surfaces with stops at their boundary. We prove equivalences between such categories and derived categories of modules over the Auslander order on certain nodal stacky curves. As an application, we prove equivalences between perfect derived categories of such stacky curves and compact Fukaya categories of the punctured surfaces (that could be of arbitrary genus).

Samson Shatashvili

Title: G2 mirror symmetry revisited

Abstract: I will discuss the world-sheet supersymmetric CFTs corresponding to the sigma models with target space being G2 (or Spini7) holonomy manifolds, and corresponding mirror symmetry conjecture.

Theodore Spaide

Title: Calabi-Yau Structures, Spherical Functors, and Gluing

Abstract: I will discuss Calabi-Yau structures on categories and their relation to homological mirror symmetry. These Calabi-Yau structures have relative versions which can be related to the notion of spherical functors. I will describe ways these functors can be glued and the potential relation to Fukaya-Seidel constructions. This is joint work with Ludmil Katzarkov and Pranav Pandit.

Vyacheslav Spiridonov

Title: Elliptic Bailey, Fourier and Yang-Baxter

Abstract: An elliptic Fourier transform was introduced in 2003 (as an integral generalization of Bailey’s approach to proving q-series identities) and used for derivation of symmetries of elliptic hypergeometric functions. In the talk I will sketch its applications to the Yang-Baxter equation and quantum field theory.

Atsushi Takahashi

Title: Orbifold Jacobian algebras for invertible polynomials

Abstract: Motivated by K. Saito’s theory of primitive form and some algebraic structures on the pair of Hochshild cohomology and homology groups, we propose an axiom for the “orbifold Jacobian algebra”, the Jacobian algebra for a pair of isolated hypersurface singularity and a finite group preserving the defining polynomial of the singularity, in physics terminology, the B-model chiral algebra for Landau-Ginzburg orbifolds. We show the existence and the uniqueness of the orbifold Jacobian algebra for an invertible polynomial and its symmetry group. The relation to the category of equivariant matrix factorizations will also be given if it is possible. This is a joint work with Alexey Basalaev and Elisabeth Werner.

Yukinobu Toda

Title: Donaldson-Thomas invariants on abelian 3-folds and Fourier-Mukai transforms

Abstract: I will show that Donaldson-Thomas invariants on abelian 3-folds have symmetric properties with respect to their autoequivalence groups, using wall-crossing formula in the space of Bridgeland stability conditions. This result is compatible with the conjecture of Bryan-Oberdieck-Pandharipande-Yin on curve counting DT invariants on abelian 3-folds. This is a work in progress with G. Oberdieck and D. Piyaratne.

Yuri Tschinkel

Title: Rationality problems

Abstract: I will report on recent advances in the study of rationality properties of algebraic varieties that allowed to essentially settle the problem of stable rationality of very general threefolds and led to the discovery of new effects in dimension 4.

Kazushi Ueda

Title: An update on tropical coamoebas

Abstract: Tropical coamoebas are introduced as higher-dimensional generalizations of dimer models, which conjecturally give combinatorial descriptions of the complex side and the symplectic side of homological mirror symmetry simultaneously. Higher-dimensional generalizations of dimer models are also studied recently by string theorists under the name “brand bricks”. In the talk, we will discuss tropical coamoebas from the point of view of Weinstein surgery and microlocal sheaf theory.

Pierre Vanhove

Title: Local mirror symmetry and the sunset Feynman integral

Abstract: We study the Feynman integral for the sunset graph defined as the scalar two-point self-energy at two-loop order. The Feynman integral is evaluated for all inequal internal masses in two space-time dimensions. Two calculations are given for the Feynman integral; one based on an interpretation of the integral as an inhomogeneous solution of a classical Picard-Fuchs differential equation, and the other using arithmetic algebraic geometry, motivic cohomology, and Eisenstein series. Both methods use the rather special fact that the Feynman integral is a family of regulator periods associated to a family of elliptic curves. Using an Hodge theoretic (B-model) approach, we show that the integral is given by a sum of elliptic dilogarithms evaluated at the divisors determined by the punctures. Secondly we associate to the sunset elliptic curve a local non-compact Calabi-Yau 3-fold, obtained as a limit of elliptically fibered compact Calabi-Yau 3-folds. By considering the limiting mixed Hodge structure of the Batyrev dual A-model, we arrive at an expression for the sunset Feynman integral in terms of the local Gromov-Witten prepotential of the del Pezzo surface of degree 6. This expression is obtained by proving a strong form of local mirror symmetry which identifies this prepotential with the second regulator period of the motivic cohomology class.

Ilia Zharkov

Title: Thimbles and spheres in phase tropical hypersurfaces

Abstract: To a smooth affine hypersurface we associate a homeomorphic object — the phase tropical hypersurface. It possesses many naturally “immerced” spheres and thimbles in it which have a very simple combinatorial description. They conjecturally represent Lagrangian objects in the original complex hypersurface.