Invited Speakers - European Location

**Damien Calaque**, Montpellier University

*Vertex models and En-algebras *

I will explain and state a conjecture of Kontsevich, that relates vertex models from statistical mechanics to En-algebras. I will also give the main ingredients of the proof of Kontsevich’s conjecture, which is a joint work in progress with Damien Lejay.

**Tobias Dyckerhoﬀ**, Bonn University

*A categoriﬁed Dold–Kan correspondence*

Various recent developments, in particular in the context of topological Fukaya categories, seem to be glimpses of an emerging theory of categoriﬁed homotopical and homological algebra. The increasing number of meaningful examples and constructions make it desirable to develop such a theory systematically. In this talk, we discuss a step towards this goal: a categoriﬁcation of the classical Dold–Kan correspondence.

**Lotte Hollands**, Heriot-Watt University

*Spectral problems for the E6 Minahan–Nemeschansky theory*

According to Nekrasov and Shatashvili the Coulomb vacua of four-dimensional N=2 theories of “class S”, subjected to the Omega background in two of the four dimensions, correspond to the eigenstates of a quantisation of the Hitchin integrable system. The vacua may be found as the intersection between two Lagrangian branes in the Hitchin moduli space, one of which is the space of opers (or quantum Hamiltonians) and one is deﬁned in terms of a system of Darboux coordinates on the corresponding moduli space of ﬂat connections. I will introduce such a system of Darboux coordinates on the moduli space of SL(3) ﬂat connections on the three-punctured sphere through a procedure called abelianization and describe the spectral problem characterising the corresponding quantum Hitchin system. This talk is based on work to appear with Andrew Neitzke.

**Sylvie Paycha**, Potsdam University

*An algebraic locality principle to renormalise higher zeta functions *

According to the principle of locality in physics, events taking place at diﬀerent locations should behave independently of each other, a feature expected to be reﬂected in the measurements. We propose an algebraic locality framework to keep track of the independence, where sets are equipped with a binary symmetric relation we call a locality relation on the set, this giving rise to a locality set category. In this algebraic locality setup, we implement a multivariate regularisation, which gives rise to multivariate meromorphic functions. In this case, independence of events is reﬂected in the fact that the multivariate meromorphic functions involve independent sets of variables. A minimal subtraction scheme deﬁned in terms of a projection map onto the holomorphic part then yields renormalised values. This multivariate approach can be implemented to renormalise at poles, various higher multizeta functions such as conical zeta functions (discrete sums on convex cones) and branched zeta functions (discrete sums associated with rooted trees). This renormalisation scheme strongly relies on the fact that the maps we are renormalizing can be viewed as locality algebra morphisms. This talk is based on joint work with Pierre Clavier, Li Guo and Bin Zhang.

**Jörg**** Teschner**, Hamburg University

*Geometric Langlands: Comparing the views from CFT and TQFT *

The goal of my talk will be to discuss the relation between two approaches to the geometric Langlands program. The ﬁrst has been proposed by Beilinson and Drinfeld, using ideas and methods from conformal ﬁeld theory (CFT). The second was initiated by Kapustin and Witten based on a topological version of four-dimensional maximally supersymmetric Yang–Mills theory and its reduction to a two-dimensional topological sigma model. After discussing some issues complicating a direct comparison we will formulate a proposal for a precise relation between two main ingredients in the two approaches.

**Bertrand ****Toën**, Toulouse University

*Moduli of connexions on open varieties*

This is a join work with T. Pantev. In this talk, we will discuss moduli of ﬂat bundles on smooth algebraic varieties, with possibly irregular singularities at inﬁnity. For this, we use the notion of “formal boundary”, previously studied by Ben Bassat-Temkin, Eﬁmov and Hennion– Porta–Vezzosi, as well as the moduli of ﬂat bundles at inﬁnity. We prove that the ﬁbers of the restriction map to inﬁnity are representable. We also prove that this restriction map has a canonical Lagrangian structure in the sense of shifted symplectic geometry.

**Katrin Wendland**, Freiburg University

*A natural reﬁnement of the Euler characteristic *

The Euler characteristic of a compact complex manifold M is a classical cohomological invariant. Depending on the viewpoint, it is most natural to interpret it as an index of an elliptic diﬀerential operator on M, or as a supersymmetric index in superconformal ﬁeld theories “on M”. Reﬁning the Euler characteristic but keeping with both index theoretic interpretations, one arrives at the notion of complex elliptic genera. We argue that superconformal ﬁeld theory motivates further reﬁnements of these elliptic genera which result in a choice of several new invariants, all of which have lost their interpretation in terms of index theory. However, at least if M is a K3 surface, then superconformal ﬁeld theory and higher algebra select the same new invariant as a natural reﬁnement of the complex elliptic genus.

Invited Speakers - North American Location

**Davide Gaiotto**, Perimeter Institute

*N=1 supersymmetric vertex algebras of small index *

I will describe examples of holomorphic N=1 super-symmetric vertex algebras with small (non-zero) values of the elliptic genus. I will speculate on a relation to certain patterns in the theory of topological modular forms.

**Lisa Jeﬀrey**, University of Toronto

*The Duistermaat–Heckman distribution for the based loop group *

The based loop group is an inﬁnite-dimensional manifold equipped with a Hamiltonian action of a ﬁnite dimensional torus. This was studied by Atiyah and Pressley. We investigate the Duistermaat–Heckman distribution using the theory of hyperfunctions. In applications involving Hamiltonian actions on inﬁnite-dimensional manifolds, this theory is necessary to accommodate the existence of the inﬁnite order diﬀerential operators which aries from the isotropy representation on the tangent spaces to ﬁxed points. (Joint work with James Mracek)

**Matilde Marcolli**, Perimeter Institute & University of Toronto

*Homotopy types and geometries below Spec(Z) *

This talk is based on joint work with Yuri Manin. The idea of a “geometry over the ﬁeld with one element F1” arises in connection with the study of properties of zeta functions of varieties deﬁned over Z. Several diﬀerent versions of F1 geometry (geometry below Spec(Z)) have been proposed over the years (by Tits, Manin, Deninger, Kapranov–Smirnov, etc.) including the use of homotopy theoretic methods and “brave new algebra” of ring spectra (To¨en–Vaqui´e). We present a version of F1 geometry that connects the homotopy theoretic viewpoint, using Zakharevich’s approach to the construction of spectra via assembler categories, and a point of view based on the Bost–Connes quantum statistical mechanical system, and we discuss its relevance in the context of counting problems, zeta-functions and generalised scissors congruences.

**David Nadler**, University of California, Berkeley

*Cutting and gluing branes *

I’ll discuss some results and expectations about the behavior of branes in Betti geometric Langlands under cutting and gluing Riemann surfaces.

**Andrew Neitzke**, University of Texas

*Higher operations in supersymmetric ﬁeld theory *

I will review the construction of “higher operations” on local and extended operators in topological ﬁeld theory, and some applications of this construction in supersymmetric ﬁeld theory. In particular, the higher operation on supersymmetric local operators in a 3d N=4 theory turns out to be induced by the holomorphic Poisson structure on the moduli space of the theory. This leads to a new way of establishing the non-renormalization properties of this Poisson structure, and also to a simple topological reason for the appearance of its deformation quantization when the theory is placed in Omega-background. This is an account of joint work with Christopher Beem, DavidB en-Zvi, Mathew Bullimore, and Tudor Dimofte.

**Stephan Stolz**, University of Notre Dame

*Invertible topological ﬁeld theories are SKK manifold invariants*

Topological ﬁeld theories in the sense of Atiyah–Segal are symmetric monoidal functors from a bordism category to the category of complex (super) vector spaces. A ﬁeld theory E of dimension d associates vector spaces to closed (d-1)-manifolds and linear maps to manifolds of dimension d. It turns out that if E is invertible, i.e., if the vector spaces associated to (d-1)-manifolds have dimension one, then the complex number E(M) that E associates to a closed d-manifold M, is an SKK manifold invariant. Here these letters stand for schneiden=cut, kleben=glue and kontrolliert=controlled, meaning that E(M) does not change when modifying the manifold by cutting and gluing along hypersurfaces in a controlled way. The main result of this joint work with Matthias Kreck and Peter Teichner is that the map described above gives a bijection between topological ﬁeld theories and SKK manifold invariants.

**Valerio Toledano Laredo**, Northeastern University

*Elliptic quantum groups and their ﬁnite-dimensional representations *

I will describe joint work with Sachin Gautam where we give a deﬁnition of the category of ﬁnite-dimensional representations of an elliptic quantum group which is intrinsic, uniform for all Lie types, and valid for numerical values of the deformation and elliptic parameters. We also classify simple objects in this category in terms of elliptic Drinfeld polynomials. This classiﬁcation is new even for sl(2), as is our deﬁnition outside of type A.

Participants - North American Location

**Arun Debray**, University of Texas

*The low-energy TQFT of the generalized double semion model*

The generalized double semion model, introduced by Freedman and Hastings, is a lattice field theory similar to the toric code, with a gapped Hamiltonian whose space of ground states depends on the topology of the ambient manifold. In this talk, I’ll explain how to calculate its low-energy limit, which forms part of a topological field theory, in terms of characteristic classes of the ambient manifold.

**Justin Hilburn**, University of Pennsylvania

*Symplectic duality and geometric Langlands*

In this talk I would like to briefly sketch how one can use the tools of derived symplectic geometry and holomorphically twisted gauge theories to derive a relationship between symplectic duality and local Langlands. Our starting point will be an observation due to Gaiotto-Witten that a 3d N=4 theory with a G-flavor symmetry is a boundary condition for 4d N=4 SYM with gauge group G.

By examining the relationship between boundary observables and bulk lines we will be able to derive constructions originally due to Braverman, Finkelberg, Nakajima. By examine the relationship between boundary lines and bulk surface operators one can derive new connections to local geometric Langlands.

This is based on joint work with Philsang Yoo, Tudor Dimofte, and Davide Gaiotto

**Omar Kidwai**, University of Toronto

*Higher length-twist coordinates and applications - effective superpotentials from the geometry of opers*

We describe joint work with L. Hollands on the geometry of the moduli space of flat connections over a Riemann surface. On the one hand, we generalize and compute certain "complexified Fenchel-Nielsen" coordinates for SL(2)-connections to higher rank using the spectral network "abelianization" approach of Gaiotto-Moore-Neitzke. We then use these coordinates to compute superpotentials, following a conjecture of Nekrasov-Rosly-Shatashvili which roughly states the following: a certain low energy effective twisted superpotential arising from compactifying a theory of class S is equal to the generating function (in the sense of symplectic geometry), in some special coordinates, of the Lagrangian submanifold of opers in the associated moduli space of flat connections.

**Eugene Rabinovich**, University of California, Berkeley

*Perturbative Anomalies of the Massless Free Fermion and Formal Moduli Problems*

It is conventional wisdom among physicists that anomalies of fermionic theories measure an obstruction to the existence of a well-defined (gauge-invariant) partition function. The aim of this talk is to use the formalism of Costello and Gwilliam to show how this wisdom is instantiated for perturbative anomalies of the massless free fermion. We will show how an action of a dg Lie algebra L on the massless free fermion theory gives rise to a line bundle over the formal moduli problem corresponding to L; the anomaly is precisely the failure of this line bundle to be trivial. Our running example will be the axial symmetry of the massless free fermion.

**Alex Takeda**, University of California, Berkeley

*Stability conditions on Fukaya categories of surfaces: Some new techniques and results*

In this talk I will present some upcoming work on Bridgeland stability conditions on partially wrapped Fukaya categories of topological surfaces. The main result is a proof that the stability conditions defined by Haiden, Katzarkov and Kontsevich using quadratic differentials cover the entire stability space. This proof uses a definition of the new concept of relative stability conditions, which is a relative version of Bridgeland's definition, with functorial behavior analogous to compactly supported cohomology. This definition is exclusive to the setting of these categories, and I will discuss problems and possibilities regarding generalization to other types of categories.

**Laura Wells**, University of Notre Dame

*G-equivariant factorization algebras*

There are various ways to define factorization algebras: one can define a factorization algebra that lives over the open subsets of some fixed manifold; or, alternatively, one can define a factorization algebra on the site of all manifolds of a given dimension (possibly with a specified geometric structure). In this talk, I will outline a comparison between G-equivariant factorization algebras on a fixed model space M to factorization algebras on the site of all manifolds equipped with a (M, G)-structure, given by an atlas with charts in M and transition maps given by elements of G. I will introduce the definitions of these two concepts and then sketch the proof that there is a quasi-equivalence between these dg-categories. This is work in progress