# QUANTMOD — Quantization and Moduli Spaces

January 9–13, 2017, University of Luxembourg

Supported by:

## Programme

• Jorgen Andersen (Aarhus University), "The Verlinde formula for Higgs bundles "
Abstract: In this talk we will present a Verlinde formula for the quantization of the Higgs bundle moduli spaces and stacks for any simple and simply-connected group. We further present a Verlinde formula for the quantization of parabolic Higgs bundle moduli spaces and stacks. We will explain how all these dimensions fit into a 2D TQFT, encoded in a Frobenius algebra, which we will construct.
• Philip Boalch (Université Paris-Sud), "Wild character varieties, meromorphic Hitchin systems and Dynkin diagrams"
Abstract: Recall that Atiyah-Bott/Goldman constructed symplectic structures on spaces of connections on Riemann surfaces, and in the case of complex groups this was upgraded to a hyperkahler structure by Hitchin. Combined with Riemann-Hilbert this shows moduli spaces of algebraic connections have at least two other natural algebraic structures: as Higgs bundles or as spaces of fundamental group representations. In this talk I will discuss various questions and describe some simple examples relating to the generalisation of this rich picture which occurs when meromorphic connections are considered. If time permits I will explain how this leads to links to Catalan numbers and triangulations, and in particular how simple examples of gluing wild boundary conditions for Stokes data leads to duplicial algebras in the sense of Loday. The new results to be discussed are joint work with R. Paluba and/or D. Yamakawa.
• Alberto Cattaneo (Institut für Mathematik Universität Zürich), "Perturbative BV-BFV theories on manifolds with boundary "
Abstract: According to Segal and Atiyah, a quantum field theory on manifolds with boundary should be thought of as, roughly speaking, the assignment of a vector space (space of states) to the boundary and an element thereof (the state or the evolution operator) to the bulk, in a way that is compatible with gluing. In this talk (based on joint work with P. Mnev and N. Reshetikhin) I will describe how this has to be reformulated when working in perturbation theory. In particular, I will discuss the perturbative quantization of gauge theories on manifolds with boundary. It turns out that, under suitable assumptions, the bulk symmetries, treated in the BV formalism, naturally give rise to a cohomological description of the reduced phase space (BFV formalism) in a correlated way that can be quantized.
• Ozgur Ceyhan (University of Luxembourg), "Backpropagation, its geometry and tropicalisation"
Abstract: The algorithms that make current artificial neural networks successes possible are decades old. They became applicable only recently as these algorithms demand huge computational power. Any technique which reduces the needs for computation have a potential to make great impact. In this talk, I am going to discuss the basics of backpropagation techniques and tropicalisation of the problem that promises to reduce the time complexity and accelerate computations.
• Vladimir Fock (Université de Strasbourg), "Higher complex structures on Riemann surfaces (Joint work with A.Thomas)."
Abstract: The ordinary Teichmüller space can be defined either as the space of complex structures or as a space of hyperbolic structures or as a component of the space of discrete representations of the fundamental group in PSL(2,R). The second and the third definitions were generalized to groups of higher rank providing a definition of a Higher Teichmüller space. In this talk we suggest a construction of a structure generalizing the first definition by constructing a local structure on the surface which can be considered as a complex structure for higher rank.
• Jochen Heinloth (Universität Duisburg-Essen), "On supports of the cohomology of Hitchin's fibrations "
Abstract: Hitchin's fibrations have a very rich geometry and the singularites of the fibers contain arithmetically interesting information. For genearlizations of the original fibration Ngo's theorem showed that miracously one can often obtain all of this information from the smooth part of the fibration alone.
However, these arguments break down for in the original symplectic moduli space of Higgs bundles, which is particularly interesting. It turns out that in this case there are other subspaces that do support summands of the cohomology of the fibers.
After introducing this circle of questions, I will report on some answers, obtained in joint work in progress with M. de Cataldo and L. Migliorini.
• Lotte Hollands (Heriot-Watt University) , " Spectral networks and higher rank Fenchel-Nielsen coordinates"
Abstract: A spectral network is a collection of trajectories on a (punctured) Riemann surface. Given a spectral network we can define a notion of "abelianization", which relates flat SL(K) connections on the Riemann surface to flat $\mathbb{C}^*$ connections on a covering. For any spectral network abelianization gives a construction of a local Darboux coordinate system on the moduli space of flat SL(K) connections. In this seminar we focus on a particularly rich example for rank K=3, with interesting applications to WKB analysis and to quantum physics. This is based on work (in progress) with Andy Neitzke.
• Oleksandr Iena (University of Luxembourg), "On vector bundles on curves and 1-dimensional sheaves"
Abstract: Some examples of varieties of degerenations of vector bundles on planar curves will be discussed.
• Kenji Iohara (University Lyon 1), "A characterization of holomorphic unstable principal bundles over an elliptic curve"
Abstract: After recalling some basic facts about principal bundles over an ellpitic curve, a characterization of a unstable point of the space of $\overline{\partial}$-connections will be given. This is a joint work with H. Yamada.
• Etienne Mann (University of Angers, Angers), "Catgeorification of Gromov-Witten invaraints"
Abstract: In this talk, we give a new structure that is satisfied by the Gromov-Witten invariants in completely geometrical terms that generalizes the notion of cohomological field theory. Namely that any projective variety is a algebra over the moduli space of stable curves. We use derived algrebraic geometry to state our result. This is a joint work with Marco Robalo.
• Johan Martens (University of Edinburgh), " Toric degenerations, Hamiltonian contractions and real polarisations"
Abstract: In 1950 Gel’fand and Cetlin constructed the first example of what we now call canonical bases for the representations of a semi-simple Lie algebra. Much later Guillemin and Sternberg showed how these could be understood as arising out of the geometric quantization of flag varieties or co-adjoint orbits in various polarisations, relying on a (real) integrable system that is closely related to toric degenerations of these flag varieties. We shall outline this story, and then show how this symplectic story generalises using contractions of Hamiltonian spaces. This is based on joint work with Joachim Hilgert and Chis Manon.
• Motohico Mulase (University of California, Davis), "Holomorphic quantization of Higgs bundles"
Abstract: The mathematical process of quantization is never unique. However, many examples of quantum curves exhibit the unique choice of quantization of spectral curves dictated by the requirement from the mirror A-side of the story. This is because a quantum curve is expected to be the result of quantization of holomorphic geometry of the B-model, that represents the mirror of the totality of the A-model invariants for all genera. A typical example is the quantization of A-polynomials in knot theory. How do we find the unique quantization from the B-model geometry alone?
For the case of quantizing Hitchin spectral curves, there is indeed a canonical mechanism due to the method of scaling limit of Gaiotto. In this talk, the algebro-geometric mechanism of this quantization is explained. The talk is based on the joint work with Olivia Dumitrescu.
• Christian Pauly (Université de Nice Sophia Antipolis), "Hitchin's connection and non-abelian Prym varieties."
Abstract: First I will introduce Hitchin's connection on the sheaf of generalized theta functions of level l on the moduli space of rank-r vector bundles for a family of smooth projective curves and show that its monodromy representation is infinite, except in some special cases. Then I will discuss these special cases. Some of them (l=1, any r) and (l=4, r=2) are related to abelian theta functions (having finite monodromy). The latter case will involve abelian and non-abelian Prym varieties. This is work in progress with Thomas Baier, Michele Bolognesi and Johan Martens.
• Du Pei (QGM and Caltech), "Wild Hitchin Characters"
Abstract: In this talk, I will discuss the quantization of wild Hitchin moduli spaces. Although these non-compact moduli spaces lead to infinite-dimensional Hilbert spaces after the quantization, in many cases a $\mathbb{C}^*$ "Hitchin action" exists, enabling us to talk about the "Hitchin character" of the Hilbert space. This quantity can be computed using string theory and quantum field theory, and encodes much geometric and topological information about the moduli space. This talk is based on joint work with Laura Fredrickson, Wenbin Yan and Ke Ye.
• Francois Petit (University of Luxembourg), " Tempered algebraization theorem"
Abstract: The problem of comparing algebraic and complex analytic geometry is a very classical question. In the case of proper algebraic varieties, the question has been settled by Serre's famous GAGA theorem. In the non-proper case, this theorem does not hold but some comparison results between analytic and algebraic objects have been obtained when the properness assumption is replaced by a growth condition on the analytic functions considered. In this talk, we will present an approach based on the theory of tempered holomorphic functions which allows to obtain a partial GAGA type theorem for non-proper smooth algebraic varieties.
• Pavel Safronov (Max Planck Institute for Mathematics, Bonn), "Poisson-Lie theory and shifted Poisson structures"
Abstract: Shifted Poisson structures were developed by Calaque-Pantev-Toen-Vaquie-Vezzosi as a homotopical generalization of Poisson structures to derived stacks. In this talk I will explain how they provide a natural framework for many structures in Poisson-Lie theory such as dynamical r-matrices, reflection equation algebras and so on.
• Armen Sergeev (Steklov Mathematical Institute, Moscow), "Non-smooth strings and noncommutatitve geometry"
Abstract: The phase space of the $d$-dimensional theory of smooth closed strings may be identified with the space of smooth loops with values in the $d$-dimensional Minkowsky space $R_d$. However, the symplectic form of this theory extends to the Sobolev space $V_d=H_0^{1/2}(S^1,R_d)$ of half-differentiable loops in $R_d$. The group $\text{QS}(S^1)$ of reparameterizations of such loops consists of quasisymmetric homeomorphisms of the circle. The action of this group on the Sobolev space Vd preserves the symplectic form. The phase space of the theory of non-smooth strings is now identified with the pair $(V_d; \text{QS}(S^1))$. To quantize this system, we use methods borrowed from the noncommutative geometry.
• Oleg K. Sheinman (Steklov Mathematical Institute, Moscow), "Moduli of matrix divisors on Riemann surfaces."
Abstract: Matrix divisors are introduced by A.Weil (1938) and are considered as a chronologically first approach to the theory of holomorphic vector bundles on Riemann surfaces. Their interrelation with holomorphic vector bundles is similar to the interrelation between conventional divisors and linear bundles. The matrix divisor approach by A.N.Tyurin to classification of holomorphic vector bundles provides invariants not only of stable bundles but also of families of smaller dimension. Moreover, it provides explicit coordinates (given the name of Tyurin parameters) which are helpful in integration of soliton equations.
In my talk I will gain attention to another interrelation between matrix divisors and integrable systems. I will set a certain classification problem of matrix divisors with given discrete invariants and support, associate a Lax operator algebra with the same data, and give two descriptions of the tangent space to the moduli space, one of them as a quotient of the space of M-operators by the Lax operator algebra.
• Joerg Teschner (University of Hamburg), "A variant of the geometric Langlands correspondence"
Abstract: The goal of my talk will be to propose a variant of the geometric Langlands correspondence suggested by some recent developments in the study of supersymmetric field theories. This variant extends the scope of the usual geometric Langlands program in the following way: It aims to construct and classify families of certain sections of the D-modules usually considered in the geometric Langlands correspondence rather than the D-modules themselves.
A one-parameter deformation related to conformal field theory plays an important role.
• Richard Wentworth (University of Maryland), " On level-rank and strange duality maps for odd orthogonal bundles"
Abstract: The talk will present results on spaces of generalized theta functions for odd orthogonal bundles and corresponding conformal blocks. One of these is a proof of a Verlinde type formula and a dimension equality that was conjectured by Oxbury-Wilson. I will outline a notion of strange duality for odd orthogonal bundles, and indicate how the naive conjecture fails in general. A consequence of this is the reducibility of the projective representations of spin mapping class groups arising from the Hitchin connection for these moduli spaces. I will also address a question of Nakanishi-Tsuchiya about level-rank duality for conformal blocks on the pointed projective line with spin weights. This is joint work with Swarnava Mukhopadhyay