# GEOQUANT 2017

Gaëtan Borot (MPIM, Bonn)
Title: Quantizing quadratic Lagrangians via topological recursion

Abstract: Kontsevich and Soibelman introduced the notion of classical Airy structure (Lagrangians in a symplectic vector space defines by certain quadratic equations), and its quantum counterpart (Lie algebra of differential operators quantizing the above Lagrangian), and showed how topological recursion computes a preferred wave function for this quantization. We will review this construction, and exhibit some examples of quantum Airy structures - based on a joint work with Jørgen Ellegaard Andersen, Leonid Chekhov and Nicolas Orantin.

Bohui Chen (Sichuan University)
Title: On equivariant Donaldson theory

Abstract: In this talk, I will explain a smooth compactification on Donaldson moduli space. Moreover, by considering moduli spaces with marked points, one may realize $\mu$ maps used in Donaldson theory by using deRham forms. The construction may be extended to equivariant case as well. This is a joint work with Bai-Ling Wang.

Mario Garcia Fernandez (ICMAT)
Title: Ricci flow, Killing spinors, and T-duality in generalized geometry

Abstract: In this talk we will overview recent work in arXiv:1611.08926, where we introduce a notion of Ricci flow in generalized geometry, extending a previous definition by Gualtieri on exact Courant algebroids. Special stationary points of the flow are given by solutions to first-order differential equations, the Killing spinor equations, which encompass special holonomy metrics with solutions of the Strominger system. We then consider T-duality between possibly topologically distinct torus bundles endowed with Courant structures, and demonstrate that solutions of the equations are exchanged under this symmetry. As applications, we give a mathematical explanation of the "dilaton shift" and prove that the Strominger system is preserved by heterotic T-duality, as defined by Baraglia and Hekmati.

Sergei Gukov (Caltech)
Title: New q-series invariants of 3-manifolds with integer coefficients

Abstract: Quantum invariants of 3-manifolds (the so-called WRT invariants) are defined at roots of unity. In this talk, I will introduce a new class of 3-manifold invariants, motivated by physics, with the following three properties: 1) for a given root system and a 3-manifold, the invariant is a q-series convergent inside a unit disk, 2) the limiting values at roots of unity agree with WRT invariants, 3) the coefficients of q-expansion exhibit integrality. The last property holds the key to "categorification" of quantum group invariants of 3-manifolds. This talk is based on work done during last year in collaboration with Marcos Marino, Du Pei, Pavel Putrov, and Cumrun Vafa.

Brian Hall (University of Notre Dame, USA)
Title: Old and new unitarity results in "quantization commutes with reduction"

Abstract: A famous result of Guillemin and Sternberg says that in the context of geometric quantization of compact Kahler manifolds, there is a natural one-to-one and onto map between the “first quantize and then reduce” space and the “first reduce and then quantize” space. Guillemin and Sternberg, however, do not address the important question of the unitarity of this map. In earlier work with Will Kirwin, I showed that the Guillemin-Sternberg map is not unitary in general, indeed, not even asymptotically unitary as Planck’s constant tends to zero. On the other hand, Kirwin and I showed that if one includes half-forms in the quantization process, one still has a bijective Guillemin-Sternberg-type map, and this map is asymptotically unitary for small values of Planck’s constant. There remains a question of whether there are interesting cases in which the half-form corrected Guillemin-Sternberg map is actually exactly unitary. I will review my results with Kirwin and discuss new results with Benjamin Lewis giving examples where exact unitarity is achieved.

Jianxun Hu (Sun Yat-sen Univ.)
Title:  Gromov-Witten invariant and birational geometry

Abstract: In this talk，I will first introduce moduli method, Gromov-Witten invariant and its degeneration formula. Then I will explain how to generalize Mori's theory of the birational classification of projective varieties to symplectic geometry. Finally, I will talk about some recent progress.

Rinat M. Kashaev (University of Geneva, Switzerland)
Title: Teichmüller TQFT: old and new formulations

Abstract: Based on the combinatorics of shaped ordered Delta triangulations, Teichmüller TQFT suggests mathematically precise de finitions and calculation recipes for quantum Chern-Simons theory with non-compact gauge groups $PSL(2;R)$ and $PSL(2;C)$. There exists two different formulations of the theory which we call old and new, and the relation between them is a topic of ongoing research. In this talk I will explain the equivalence of the two formulations in the case of integer homology spheres. The talk is based on joint works with Jørgen Ellegaard Andersen.

Ryoichi Kobayashi (Nagoya Universit, Japan)
Title: A Quantization of Osserman's theory of algebraic minimal surfaces

Abstract: I propose a quantization scheme of Osserman's theory on algebraic minimal surfaces. Starting with an observation that the basic quantity in Osserman's theory is interpreted as a sort of partition function in which the limit h to 0 comes fi rst before summation, I will answer the question what happens if we perform summation first and then take the limit h to 0. We arrive at a geometry completely different from Osserman's.

Leonid Polterovich (Tel Aviv University)
Title: Quantum speed limit vs. classical displacement energy

Abstract: We discuss a link (in the context of the Berezin-Toeplitz quantization) between displacement energy, a fundamental notion of symplectic topology, and the quantum speed limit, a universal constraint on quantum-mechanical processes. Joint work with Laurent Charles.

Laura Schaposnik (University of Illinois, Chicago)
Title: On some singular fibres of the Hitchin fibration

Abstract: The Hitchin fibration is a natural tool through which one can study the moduli space of Higgs bundles and its interesting subspaces (branes). We shall dedicate this talk to the study of certain singular fibres of the Hitchin fibrations, obtain correspondences between fibres, and provide a geometric description of branes which lie entirely over the singular loci (based partially on work in collaboration with David Baraglia, Steve Bradlow and Sebastian Heller).

Ekaterina Shemyakova (University of Toledo, Ohio)

Title: Differential Operators and Darboux Transformations in Supergeometry

Abstract: In this talk we consider Darboux transformations (DT), which are certain non-group symmetries of linear partial differential equations, in the setting of supergeometry, an area on the borderline between quantum physics and differential geometry. All our results in this talk will be for the superline (by which we mean a supermanifold of dimension $1|1$, i.e. the case of one even and one odd variable). We show that linear partial differential operators on the superline are close by their properties to ordinary differential operators; in particular, we show that every DT on the superline corresponds to an invariant subspace of the source operator and upon a choice of a basis is given by a super-Wronskian formula. On the way we establish important properties of Berezinians (or superdeterminants) which are analogs of textbook formulas for ordinary determinants but seem to be not known in the super case.

Tatsuya Tate (Tohoku Univ.)
Title: Quantum walks defined by periodic unitary transition operators

Abstract: The term “quantum walks" is a generic word used to mean certain probability distributions on a parameter space of an orthonormal basis on a Hilbert space defined by the powers of unitary operators, called unitary transition operators. It is introduced in quantum physics in 1993 and reformulated in computer science around early 2000s. They sometimes regarded as a quantum counterpart of classical random walks but its behavior is drastically different from them. For example, it is known that the variance has a linear order in the time parameter and they usually have ballistic behavior. They also has “localization” phenomenon even in a very simple one-dimensional model, which is an example of “lazy Grover walks". In the first half of the talk, some of interesting and famous results in this area will be reviewed. Recently the notion of periodic unitary transition operators are introduced, and it is proved that they have no singular continuous spectrum. This result gives an explanation for a well-known results on localization phenomenon of one-dimensional “lazy Grover walk". Furthermore, some localization results in higher dimension are obtained in a recent work with T. Komatsu. For example, the existence of the limit of “lazy Grover walks” in any dimension has been shown. In the second half of the talk, these recent results will be introduced.

Nikolay A. Tyurin (BLTPh JINR (Dubna) & NRU HSE (Moscow))
Title: Moduli space of special Bohr - Sommerfeld lagrangian cycles for algebraic varieties

Abstract: Combinig ideas of N. Hitchin (Special lagrangian geometry for Calabi - Yau manifolds) and A. Tyurin (moduli space of Bohr - Sommerfled cycles for 1- connected compact symplectic manifolds with integer symplectic form) we construct certain finite dimensional moduli spaces for 1- connected compact algebraic varieties in terms of their lagrangian geometry if one fixes any ample line bundle and take the corresponding Kahler form of the Hodge type as the symplectic form. The construction is based on the observation which was called "lagrangian shadow of ample algebraic divisor". In the talk we present the definition of the moduli spaces, the simplest properties of these spaces and the first examples.

Theodore Voronov (University of Manchester, UK)
Title: Microformal geometry, Poisson manifolds and homotopy algebra

Abstract: Given two homotopy Poisson manifolds, it is necessary for some problems to construct L- in finity morphisms between the algebras of functions. Recall that L-infi nity morphisms are described geometrically as generally non-linear maps between Q-manifolds. Therefore, the problem, in particular, is to have some general construction giving non-linear mappings between algebras of functions viewed as in finite-dimensional supermanifolds. In the talk, I will describe the solution of this problem, which is as follows. There is a natural generalization of smooth maps of (super)manifolds, which are not maps in the ordinary sense but rather certain type canonical relations between the cotangent bundles. I call them "microformal morphisms" or "thick morphisms". They induce pullbacks of smooth functions, which are, in general, non-linear mappings. (More precisely, formal nonlinear differential operators.) Such nonlinear pullbacks have interesting properties, including the fact that their derivatives are algebra homomorphisms. In homotopy Poisson setting, they yield desired L-infi nity morphisms. Another application is a construction of adjoints for nonlinear operators on vector bundles. This can be applied to L-infi nity algebroids. Thick morphisms form a formal category which is a formal neigh- borhood of the ordinary category of supermanifolds. (To be more precise, there are two parallel constructions, for even and odd functions or "bosonic" and "fermionic" fields.) Finally, there is a quantum version of all that, where "quantum thick morphisms" are represented by oscillatory integral operators.

Siye Wu (National Tsing Hua University)
Title: Deformation of the prequantum action

Abstract: We construct a family of non‐formal star products of functions on polarised sections of the prequantum line bundle over a symplectic vector space as a deformation of the standard prequantum action in geometric quantisation. We show that the star products are compatible with the flat connection defined by intertwining operators on functions and the projectively flat connection on the bundle of Hilbert spaces.

Alexander Zheglov (Lomosov State University)
Title: Cohen-Macaulay modules over the algebra of planar quasi-invariants and Calogero-Moser systems

Abstract: My talk (based on a joint work with Igor Burban) is devoted to the algebraic analysis of planar rational Calogero-Moser systems. This class of quantum integrable systems is known to be superintegrable. This means that the underlying Schrödinger operator with Calogero-Moser potential can be included into a large family of pairwise commuting partial differential operators such that the space of joint power series eigenfunctions is generically one-dimensional. More algebraically, any such system is essentially determined by a certain algebro-geometric datum: the projective spectral surface (defined by the algebra of planar quasi-invariants with natural filtration) and the spectral sheaf (defined by a module known to be Cohen-Macaulay of rank one). This geometric datum has very special algebro-geometric properties, the most important of which is a very special form of the Hilbert polynomial of the module (sheaf). Moreover, the spectral variety appears to be rational but very singular (only Cohen-Macaulay, even not normal). It turns out that all rank one Cohen-Macaulay modules over the algebra of planar quasi-invariants can be explicitly described in terms of very natural moduli parameters, and this description looks in some sence very similar to to the description of the generalised Jacobian for singular rational curves. The spectral module of a planar Calogero-Moser system is actually projective, and its underlying moduli parameters are explicitely determined. Unlike the case of curves, not every Cohen-Macaulay module is spectral. The moduli space of spectral sheaves appears to be much more subtle, but its structure indicates the existence of integrable deformations of Calogero-Moser systems. I am going to explain how the classification of CM modules, combined with tools of the algebraic inverse scattering method, leads to certain new integrable deformations of Calogero-Moser systems in the algebra of differential-difference operators.