# QUANTMOD2 — Quantization and Moduli Spaces

June 4–8, 2018, University of Luxembourg

Supported by:

## Program

• Pierre Bieliavsky (University of Louvain) , "Quantum differential surfaces of higher genera "
Abstract: We construct a real family of $SL(2,{\bf R})$-invariant symbol composition products $\{\sharp_\theta \}_{\theta\in{\bf R}}$ on the analogue of the Schwartz space ${\mathcal{S}}({\bf D})$ on the hyperbolic plane ${\bf D}\;:=\;SL(2,{\bf R})/SO(2)$. The value $\theta=0$ consists of the pointwise commutative product of functions on ${\bf D}$ and admits an asymptotic expansion that deforms the pointwise product in the direction of the canonical $SL(2,{\bf R})$-invariant Kahler two-form on ${\bf D}$. \noindent We then extend this construction to any (non-homogeneous) compact surface by considering the left action of an arithmetic Fuschian group $\Gamma\subset SL(2,{\bf R})$ on ${\bf D}$ with the associated Riemann surface $\Sigma_\Gamma\;:=\;\Gamma\backslash{\bf D}$. More precisely, we prove the following:
1. the product $\sharp_\theta$ extends from ${\mathcal{S}}({\bf D})$ to ${\it B}^0_\infty({\bf D})$, a smooth $SL(2,{\bf R})$- sub-module of $C^\infty({\bf D})$ that contains the $\Gamma$-invariants $C^\infty({\bf D})^\Gamma\simeq C^\infty(\Sigma_\Gamma)$ in $C^\infty({\bf D})$. In particular, $\sharp_\theta$ defines a Fr\'echet algebra structure on $C^\infty(\Sigma_\Gamma)$.
2. We give a \emph{tracial} (or closed" in the terminology of Connes-Flato-Sternheimer) version $\sharp^2_\theta$ of $\sharp_\theta$ i.e. such that $\int_{\Sigma_\Gamma}u\sharp^2v\;=\;\int_{\Sigma_\Gamma}u.v$ ($u,v\in C^\infty(\Sigma_\Gamma)$). The algebra $(C^\infty(\Sigma_\Gamma),\sharp^2_\theta)$ is moreover equipped with the involution that consists in the complex conjugation of the smooth functions on $\Sigma_\Gamma$.
3. As a byproduct of item 2, the integral on $\Sigma_\Gamma$ defines a continuous trace on $(C^\infty(\Sigma_\Gamma),\sharp^2_\theta)$.
4. We also represent our surfaces as pre-$C^\star$-algebras. Namely, the algebra $(C^\infty(\Sigma_\Gamma),\sharp^2_\theta)$ continuously acts on $L^2({\bf D})$ by regular multiplication. The resulting operator norm on $(C^\infty(\Sigma_\Gamma),\sharp^2_\theta)$ is then a $C^\star$-norm.
5. The (pre-) $C^\star$-algebra mentioned in item 4 naturally acts on a discrete series projective representation ${\it D}_{{\left(\frac{\theta+2}{\theta}\right)}}$ of $SL(2,{\bf R})$ by $\Gamma$-commuting operators.
• Martin Bordemann (Université de Haute Alsace, Mulhouse, France) , "A generalization of T.Voronov's higher bracket construction and applications "
Abstract: In his well-known 1997 work, Maxim Kontsevich formulated the deformation quantization problem as an L-infinity quasi-isomorphism between the Hochschild complex of an associative algebra and its cohomology and solved it for the commutative smooth function algebra of an arbitrary Poisson manifold. I shall report on a generalization (2015) of a construction found in 2005 by T.Voronov: there are problems in deformation and quantization which can no longer be described by differential graded Lie algebras, but require L-infinity structures with higher order brackets, for instance the problem of deforming two associative algebras and a given morphism (Fr\'{e}gier/Zambon 2015) due to the cubic identities arising in the problem. T.Voronov supposed that the graded vector space $V$ on which the L-infinity structure ought to be build is an abelian subalgebra in a graded Lie algebra $\mathfrak{g}$ complementing a subalgebra $\mathfrak{h}$. In a second step the structure can be extended to the ambient Lie algebra $\mathfrak{g}$. We generalize his construction to an L-infinity structure on the quotient $\mathfrak{g}/\mathfrak{h}$ (and the extension) without assuming that there is an abelian subalgebra complement to $\mathfrak{h}$ in $\mathfrak{g}$. The construction simplifies a bit to some `graded dressing transformation' if there is a (non)abelian subalgebra complement (R.Bandiera 2015). The main idea is the observation that the quotient $U(\mathfrak{g})/(U(\mathfrak{g})\mathfrak{h})$ (which is also a graded Verma module) of the universal envelopping algebra $U(\mathfrak{g})$ of $\mathfrak{g}$ is a graded cofree connected coalgebra on which $\mathfrak{g}$ acts from the left by coderivations. This quotient had recently been studied in the trivially graded case by Calaque, Caldararu and Tu: using their result we can show that the generalized Voronov L-infinity structure is isomorphic just to a differential (no higher brackets) iff the (graded) Atiyah (or Nguyen-van Hai) class of the Lie algebra pair $(\mathfrak{g},\mathfrak{h})$ vanishes. We sketch a possible application to the deformation quantization problem of co-isotropic submanifolds.
• Andrea Brini (CNRS & Imperial College, London) , "Chern--Simons theory and mirror symmetry "
Abstract: I will present a conjectural general correspondence relating quantum invariants of knots in three-space to a version of higher genus mirror symmetry, the computation of the former being solved by the latter. The correspondence rests on the identification of a natural group of piecewise linear transformations on both sides of the correspondence. The talk will be largely introductory and mainly focused on the enumerative geometry/B-model side of story.
• Ugo Bruzzo (SISSA) , "Crepant resolutions of 3-dimensional orbifold singularities and superconformal Chern-Simons gauge theory "
Abstract: We shall describe crepant resolutions of singularities of the type $\mathbb{C}^3/\Gamma$, where Gamma is a finite abelian subgroup of $SL(3,\mathbb{C})$, comparing the toric resolution with a Kronheimer-like construction, and we shall relate this geometry with the field content and interaction structure of a superconformal Chern-Simons Gauge Theory.
• Laurent Charles (Université Paris VI) , "Berezin-Toeplitz operators and entanglement entropy "
Abstract: We consider Berezin-Toeplitz operators on compact Kahler manifolds whose symbols are characteristic functions. When the support of the characteristic function has a smooth boundary, we prove a two-term Weyl law, the second term being proportional to the Riemannian volume of the boundary. As a consequence, we deduce the area law for the entanglement entropy of integer quantum Hall states.
• Grégory Ginot (Université Paris XIII) , "Large N-limit for Chern-Simons and String Topology "
Abstract: This is joint work in progress with Gwilliam and Zeinalian. String topology arised as a higher dimensional generalisation of Goldman-Turaev Lie bialgebra structure on free loops on a surface which is closely related to Poisson algebra of character varieties. Our aim is to consider a higher version of this relation relating string topology and quantization of Chern-Simons field theory.
• Simone Gutt (ULB) , "$L_\infty$- formality problem for the universal Lie algebra of semi simple Lie algebras. "
Abstract: We show that the Hochschild complex of the universal enveloping algebra of a nonabelian reductive Lie algebra is not formal. In the case of $\mathfrak{so}(3)$, we show that adding one higher bracket of order $3$ restores a $L-\infty$-quasi-isomorphism. This is a joint work with Martin Bordemann, Olivier Elchinger and Abdenacer Makhlouf.
• Benjamin Hennion (Université d'Orsay - Paris Sud) , "Gelfand--Fuks cohomology for algebraic varieties "
Abstract: Given a smooth affine algebraic variety over $\mathbf C$, we prove that the Chevalley-Eilenberg cohomology of its Lie algebra of global vector fields is a topological invariant of the underlying complex manifold and is finite dimensional in every degree. The proof uses methods from factorization homology. In this talk, we will first explain the case of smooth real manifolds as studied in the 70's (Gelfand, Fuks, Bott--Segal, Haefliger, Guillemin, ...). We will show how to transpose those methods to complex algebraic varieties.
• Christian Kassel (CNRS & Université de Strasbourg) , "The Hilbert scheme of $n$ points on a torus and modular forms "
Abstract: Looking for a way to produce $q$-analogues by passing from groups to their group algebras over the finite field~$\mathbb{F}_q$, Christophe Reutenauer (UQAM) and I found a family of polynomials with nice properties: they are palindromic, their coefficients are non-negative integers and their values at $1$ and at roots of unity of order $2, 3, 4$ and $6$ can be expressed in terms of well-known modular forms. As a consequence we explicitly determined the zeta function of the Hilbert scheme of $n$~points on a two-dimensional torus. (See arXiv:1505.07229v4 and 1610.07793v2.)
• Semyon Klevtsov (University of Cologne) , "TBA "
Abstract: TBA.
• St\'ephane Korvers %(University of Luxembourg) , "Deformation quantization method for bounded symmetric domains "
Abstract: I will revisit a method for explicitly realizing any invariant deformation quantization on an arbitrary rank $r$ bounded symmetric domain of $\mathbb{C}^n$.
• Laurent La Fuente-Gravy (University of Luxembourg) , "Moment map and closed Fedosov star products "
Abstract: I will describe a moment map on the space of symplecic connections on a given closed symplectic manifold. The value of this moment map at a symplectic connection is contained in the trace density of the Fedosov star product attached to this connection. Moreover, this Fedosov star product can only be closed when the symplectic connection lies in the vanishing set of the moment map. Considering closed Kähler manifolds, I will show that the problem of finding zeroes of the moment map is an elliptic partial differential equation. On complex tori and complex projective spaces, I will show that part of the zero set of the moment map has the structure of a finite dimensional manifold. I will also discuss obstructions to the existence of zeroes of the moment map, which means obstructions to the closedness of the Fedosov star product attached to the considered Kähler data.
• Giovanni Landi (University of Trieste) , "The quantum Yang--Baxter equation and noncommutative Euclidean spaces "
Abstract: We present natural families of coordinate algebras of noncommutative Euclidean spaces and noncommutative products of Euclidean spaces. These coordinate algebras are quadratic ones associated with an R-matrix which is involutive and satisfies the quantum Yang--Baxter equation. As a consequence they enjoy a list of nice properties, being regular of finite global dimension. Notably, we have spherical manifolds, and noncommutative quaternionic planes as well as noncommutative quaternionic tori. On these there is an action of the classical quaternionic torus $SU(2) \times SU(2)$ in parallel with the action of the torus $U(1) \times U(1)$ on a complex noncommutative torus.
• Semyon Klevtsov (University of Cologne) , "Quantum Hall Effect and Quillen metric "
Abstract: In this talk we explore the role of the Quillen metric in the theory of Quantum Hall Effect.
• Alessandro Malusà (QGM--Aarhus University) , "TBA "
Abstract: TBA
• Martin Möller (Goethe-Universität Frankfurt) , "Compactification of the moduli space of flat surfaces "
Abstract: Flat surfaces are Riemann surfaces together with a holomorphic one-form. The moduli space of flat surfaces carries an ${\rm SL}_2({\mathbb R})$-action with interesting dynamical properties. To compute its invariants, it is desirable to compactifiy the moduli space of flat surfaces. \par We highlight some obstacles towards compactification and propose a compactification with properties as nice as the Deligne-Mumford compactification of the moduli space of curves.
• Hugo Parlier (University of Luxembourg) , "Hyperbolic surfaces, simple closed geodesics and moduli spaces "
Abstract: This talk will be about various properties of simple closed geodesics on hyperbolic surfaces and how this relates to underlying moduli spaces. A particular focus will be on quantifications of a result of Birman and Series which states that simple closed geodesics are nowhere dense on a given closed hyperbolic surface.
• François Petit (University of Luxembourg) , "The codimension three conjecture "
Abstract: The codimension three conjecture for microdifferential modules was formulated by M. Kashiwara at the beginning of the eighties. It is concerned with the extension of microdifferential modules through certain complex analytic subsets and was proved by M. Kashiwara and K. Vilonen in 2012. We prove an analogue result for Deformation Quantization modules and show how to use it to deduce the codimension three for microdifferential modules with the help of a result we recently obtained, stating an equivalence between the category of DQ-modules endowed with an holomorphic Frobenius action and the category microdifferential modules.
• Marco Robalo (Université Paris VI) , "TBA "
Abstract: TBA
• Paolo Rossi (Université de Bourgogne) , "Quantum integrable systems from the moduli space of curves "
Abstract: The quantum double ramification hierarchy is a construction that uses the intersection theory in the moduli space of stable curves to produce a vast and diverse family of quantum integrable systems in 1+1 dimensions. This construction uses various types of cohomology classes on the moduli spaces, namely the double ramification cycles, other natural tautological classes like the Hodge and psi classes, and cohomological field theories. I will explain how this works and give explicit examples. This is a joint work with A. Buryak.
• Boris Shoikhet (Universiteit Antwerpen) , "Twisted tensor product of dg categories "
Abstract: We introduce a new non-symmetric monoidal product on the category of small dg categories over a field, called the {\it twisted tensor product} and denoted by $C\overset{\sim}{\otimes}D$. The construction mimics the Gray product of 2-categories for the dg enrichment. The functor $-\overset{\sim}{\otimes}D$ is left adjoint to $Coh(D,-)$, where $Coh(A,B)$ stands for the dg category whose objects are dg functors $A\to B$, and whose morphisms are {\it coherent natural transformations}. This adjunction holds in the category of small dg categories but not in their homotopy category. Thus, it partially upgrades the Toen adjunction from the homotopy category to the category of small dg categories itself. Our main result determines the homotopy type of $C\overset{\sim}{\otimes}D$, for the case when $C,D$ are cofibrant. It says that in this case $C\overset{\sim}{\otimes}D$ is isomorphic to the ordinary tensor product $C\otimes D$, as an object of the homotopy category. A direct proof of this statement is hardly possible; we suggest a conceptual proof, in which we prove that the above mentioned adjunction descends to an adjunction on the level of the homotopy category, when $C,D$ are cofibrant.
• Katrin Wendland (Albert-Ludwigs-Universität Freiburg) , "Invariants of superconformal field theories "
Abstract: For certain classes of superconformal quantum field theories, like K3 theories, the global form of the moduli space is known. It then is natural to search for invariants, or structures that all theories share which are accounted for in such a moduli space. Traditionally, the elliptic genus yields such an invariant. We introduce refinements of the elliptic genus which encode the generic space of states of K3 theories and beyond.
• Yannick Voglaire (University of Luxembourg) , "TBA "
Abstract: TBA
• Hao Xu (Zhejiang University) , "Asymptotic Expansion of Bergman Kernel and Deformation Quantization "
Abstract: The coefficients of asymptotic expansion of Bergman kernel on Kahler manifolds give important geometric information. We show that they could be expressed in a compact form as a summation over strongly connected graphs. Similar graph formulas exits for Karabegov-Schlichenmaier asymptotic expansion of Berezin star product. The relationship to heat kernel and applications will be discussed.