(pure) gauge theories on $mathbb{C}^2$. In the present talk I will describe K-theoretic/Cohomological Hall algebras associated with stacks of torsion sheaves on A-type ALE spaces and their representations; the study of these algebras will provide eventually an approach to prove the AGT conjecture for (pure) gauge theories on A-type ALE spaces. (Work in progress with Olivier Schiffmann.)

]]>of N=4 boundary conditions in four-dimensional supersymmetric Yang Mills theory.

The action of S-duality on such boundary conditions can be understood

in terms of symplectic duality. ]]>

In this talk I will describe a project, joint with Satyan Devadoss, Stefan Forcey, and Katrin Wehrheim, which attempts to relate the Fukaya categories of different symplectic manifolds via a notion of functoriality. After mentioning some analytic results about the singular quilts necessary for this construction, I will describe the combinatorial component: with Devadoss and Forcey we are constructing a family of polytopes that specialize to the associahedra in two different ways, and can be thought of as the 2-categorical version of associahedra.

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]]>We study a moduli problem on a nodal curve of arithmetic genus 1, whose solution is an open subscheme in the zastava space for projective line. This moduli space is equipped with a natural Poisson structure, and we compute it in a natural coordinate system. We compare this Poisson structure with the trigonometric Poisson structure on the transversal slices in an affine flag variety.

We conjecture that certain generalized minors give rise to a cluster structure on the trigonometric zastava.

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It is interesting to note that, unlike Chern–Simons theories whose gauge group can be disconnected, the gauge group of a chiral WZW model is necessarily connected. This raises the question of what is the chiral WZW model associated to a disconnected gauge group? Whatever the answer turns out to be, mathematicians will probably need to enlarge the class of objects that they agree to call ‘chiral conformal field theories’ in order to accommodate these yet to be defined models.

I will present some speculations on the matter.

]]>They appear in a variety of contexts in geometry, representation theory, and physics. We reinterpret the definition as: from a quiver (and some additional choices) one builds an exact symplectic 4-manifold from which the cluster variety is recovered as a component in its moduli space of Lagrangian branes. In particular, structures from cluster algebra govern the classification of exact Lagrangian surfaces in Weinstein 4-manifolds.

This is joint work with Vivek Shende and David Treumann.

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The algebras produced in this way for the annulus and punctured torus are the so-called ``reflection equation algebra" and "quantum differential equation algebra", respectively. When we close up punctures, a variation on the our formalism naturally reproduces the framework of quantum Hamiltonian reduction, and leads to simultaneous deformations of categories D(g/G) of character sheaves on g, on the one hand, and categories QC(Ch_G(T^2)), of quasi-coherent sheaves on the character variety of T^2, on the other. We call these deformations "quantum character varieties", and they form the two-dimensional part of a four dimensional TFT related to Kapustin-Witten's geometric Langlands TFT, or "topologically twisted N=4 SYM".

]]>We will show how to construct two examples of such structure on the integrable category $matcal{O}$ representations of a symmetrisable Kac–Moody algebra $mathfrak{g}$, the first one arising from the quantum group $U_hbar(mathfrak{g})$, and the second one encoding the monodromy of the KZ and Casimir connections of $mathfrak{g}$.

The rigidity of this structure, proved in the framework of $mathsf{PROP}$ categories, implies in particular that the monodromy of the Casimir connection is given by the quantum Weyl group operators of $U_hbar(mathfrak{g})$.

This is a joint work with Valerio Toledano Laredo. ]]>

Abstract: The moduli spaces of local systems on marked surfaces enjoy many nice properties. In particular, it was shown by Fock and Goncharov that they form examples of cluster varieties, which means that they are Poisson varieties with a positive atlas of toric charts, and thus admit canonical quantizations. I will describe joint work with A. Shapiro in which we embed the quantized enveloping algebra U_q(sl_n) into the quantum character variety associated to a punctured disk with two marked points on its boundary. The construction is closely related to the (quantized) multiplicative Grothendieck-Springer resolution for SL_n. I will also explain how the R-matrix of U_q(sl_n) arises naturally in this topological setup as a (half) Dehn twist. Time permitting, I will describe some potential applications to the study of positive representations of the split real quantum group U_q(sl_n,R) ]]>

In this talk we will give a quick introduction to the theory of periods and motives, relate said theory to special values of zeta functions, and discuss a graphical definition of the associated category of motives.

Any original work discussed in this talk is joint with Susama Agarwala. ]]>

known as topological recursion. Starting with a ``spectral curve" and some ``initial data"

(roughly, meromorphic differentials of order one and two) the topological recursion produces by induction

a collection of symmetric meromorphic differentials on the spectral curves parametrized by

pairs of non-negative integers (g,n) (g should be thought of as a genus and n as the number of punctures).

Despite of many applications of the topological recursion (matrix integrals, WKB expansions, TFTs, etc.etc.)

the nature of the recursive relations was not understood.

Recently, in a joint work with Maxim Kontsevich we found a simple underlying structure of the recursive relations of Eynard and Orantin. We call it ``Airy structure". In this talk I am going to define this notion and explain how the recursive relations of Eynard and Orantin follow from the quantization of

a quadratic Lagrangian subvariety in a symplectic vector space. ]]>

procedure for the construction of effective field theories can be

extended to manifolds with boundary. ]]>

The second half of the talk will concern applications to affine Macdonald theory. I will present refinements of the denominator and evaluation conjectures for affine Macdonald polynomials proposed by Etingof-Kirillov Jr. I will then explain how to prove the first non-trivial cases of these conjectures by combining the methods of the first half and well-chosen applications of the elliptic beta integral. The second half of this talk is joint work with E. Rains and A. Varchenko.

]]>can be used to construct many other new finite-dimensional quasitriangular Hopf algebras and their duals that could be fed into applications in quantum gravity and quantum computing. ]]>

Based on joint works with P.Koroteev, A.Smirnov and A.Zeitlin ]]>

After the historical introduction, I will focus the example of the Euclidean Schwarzschild (ES) manifold from Quantum gravity. The ES manifold was first introduced by Hawking in the late 70's. It is an asymptotically locally flat solution of the Euclidean Einstein equation, but not hyper-Kahler, which makes instanton theory harder. Throughout the past four decades, related results have been sporadic: there have been only finitely many (at most 2) known solutions for each energy value.

In our work, making use of an old idea of Witten, and subsequently Taubes and Garcia-Prada, we produce infinitely many unit energy solutions, and also solutions of continuously varying energy, a feature (seemingly) unique to asymptotically locally flat spaces. These instantons describe vortex-like pseudoparticles; at the end of the talk I will present an intuitive picture in terms of planar (Ginzburg-Landau) vortices.

This is joint work with Gonçalo Oliveira (IMPA).

]]>Natural targets for extended topological field theories are higher Morita categories: generalizations of the bicategory of algebras, bimodules, and homomorphisms.

After giving an introduction to topological field theories, I will explain how one can use geometric arguments to obtain results on dualizablity in a ``factorization version’’ of the Morita category and using this, examples of low-dimensional field theories “relative” to their observables. An example will be given by polynomial differential operators, i.e. the Weyl algebra, in positive characteristic and its center. This is joint work with Owen Gwilliam. ]]>

interpolate between bundles with connections on a curve and

Higgs bundes (precise formulations requires some additional data like parabolic structure and stability structure).

I will explain the framework for a generalization of the non-abelian Hodge theory

which unifies Simpson's story ("rational case") with those for q-difference

equations ("trigonometric case") and elliptic difference equations

("elliptic case").

This unification leads to a class of examples of the new notion of ``twistor families of categories".

In the rational,trigonometric and elliptic cases twistor families of categories involve partially wrapped Fukaya categories of certain complex symplectic surfaces, categories of

holonomic modules over quantizations of these surfaces and categories of coherent sheaves on the surfaces

with certain restrictions on the support.

In the trigonometric and elliptic cases doubly and triply periodic monopoles give an alternative description of harmonic objects, hence playing the same role

as harmonic bundles play in the case of Simpson theory. ]]>

In particular, the sheafified elliptic quantum group is an algebra object in a certain monoidal category of sheaves on the colored Hilbert scheme of an elliptic curve. This monoidal structure is related to Mirkovic’s refinement of the factorization structure on semi-infinite affine Grassmannian over an elliptic curve. If time permits, I will also talk about a work in progress, joint with Mirkovic and Yang, towards a construction of a double loop Grassmannian and vertex representations of the toroidal algebra, which in turn is related to representations of the elliptic quantum groups

]]>cation. We show that the closure in overline{G} of the centralizer of a regular nilpotent is isomorphic to the Peterson variety. We generalize this result to show that for any regular , the closure of the centralizer in is isomorphic to the closure of a general -orbit in the flag variety. We consider the family of all such centralizer closures, which is a partial compactication of the universal centralizer. We show that it has a natural log-symplectic Poisson structure that extends the usual symplectic structure on the universal centralizer. ]]>

This is a repeat of a talk I gave in McGill University in February, 2017. A video recording, a handout, and some further links are at McGill-1702 ]]>

Given the difficulty of the Connes embedding problem, it makes sense to look at an easier problem: how fast does the dimension of approximate representations grow (as a function of how close the defining relations are to the identity) when we require a given set of elements to be bounded away from the identity. These growth rates are called the hyperlinear profile of the group.

In this talk, I will explain our best lower bounds on hyperlinear profile, as well as the connection to entanglement requirements for non-local games (joint work with Thomas Vidick). Time permitting, I will also mention some other approaches to looking for non-hyperlinear groups, including the recent work of De Chiffre, Glebsky, Lubotzky, and Thom on a group which is not approximable in the unnormalized Hilbert-Schmidt norm.

]]>n-dimensional fully local topological field theories, valued in some

higher category C, and the fully dualizable object of C equipped with

the structure of SO(n)-fixed point. In this talk I'll explain recent

works of Haugseng, Johnson-Freyd and Scheimbauer which construct a

Morita 4-category of braided tensor categories, and I'll report on joint

work with Brochier and Snyder which identifies two natural subcategories

therein which are 3- and 4-dualizable. These are the rigid braided

tensor categories with enough compact projectives, and the braided

fusion categories, respectively. I'll also explain work in progress by

us to construct SO(3)- and SO(4)-fixed point structures in each case,

starting from ribbon and pre-modular categories, respectively.

Applying the cobordism hypothesis, we obtain 3- and 4-dimensional fully

local TFT's, which extend the 2-dimensional TFT's we constructed with

Ben-Zvi and Brochier, and which conjecturally relate to a number of

constructions in the literature, including: skein modules, quantum

A-polynomials, Crane-Kauffmann-Yetter invariants; hence our construction

puts these on firm foundational grounds as fully local TFT's. A key

feature of our construction in dimension 3 is that we require the input

braided tensor category neither to be finite, nor semi-simple, so this

opens up new examples -- such as non-modularized quantum groups at roots

of unity -- which were not obtainable by earlier methods.

]]> - VOA[M4] from transgression of EFTs;

- SW and Donaldson invariants as chiral algebra correlators;

- Massey triple products from trisections.

These topics are based, respectively, on work with A.Gadde, P.Putrov (and work to appear with B.Feigin); recent paper with M.Dedushenko and P.Putrov; and a solo paper of the speaker. ]]>

equipped with a meromorphic ``canonical form" whose residues reflect

the boundary structure of the space. Familiar examples include

polytopes and the positive parts of toric varieties. A central, but

conjectural, example is the amplituhedron of Arkani-Hamed and Trnka.

In this case, the canonical form should essentially be the tree

amplitude of N=4 super Yang-Mills.

I will talk about the definition and examples of positive geometries,

and discuss what is known about the geometry and combinatorics of the

amplituhedron. The talk will be based on various joint works with

Arkani-Hamed, Bai, Galashin, and Karp. ]]>

Roughly speaking, one finds vortex lines in the A-twist and Wilson lines in the B-twist. I will propose a geometric identification of the corresponding categories, and explain how geometric calculations can be used to find the vector spaces of local operators at junctions of lines.

Combined with 3d mirror symmetry, this will lead to new dualities of braided tensor categories. [Joint work with N. Garner, M. Geracie, and J. Hilburn.]

]]>Namely, we explicate a precise relationship between K-theoretic Donaldson-Thomas theory and the refined topological vertex of Iqbal, Kosçaz and Vafa. Applying such results to specific toric threefolds, we deduce dualities satisfied by certain generating series that control integrals over the Hilbert scheme of points on a surface. We then explain how to use these dualities to evaluate certain Euler characteristics of tautological bundles on the Hilbert scheme of points on a general surface.

]]>The elliptic quantum (toroidal) group U_{q,p}(g) is an elliptic and dynamical analogue of the Drinfeld realization

of the affine quantum (toroidal) group U_q(g). I will discuss an interesting connection of its representations with

a geometry such as an identification of the elliptic weight functions derived by using the vertex operators with

the elliptic stable envelopes in [Aganagic- Okounkov ’16] and correspondence between the Gelfand-Tsetlin bases

of a finite dimensional representation of U_{q,p} with the fixed point classes in the equivariant elliptic cohomology.

I will describe some general mathematical structures expected to arise from field theories with boundary conditions in terms of factorization algebras, and outline some results and future directions in the study of boundary chiral algebras for 3d N=4 theories following the work of Costello and Gaiotto.

Given by Dylan Butson.

]]>In this talk I shall consider a variation of this construction for the case of a hyper-Kähler vector space V, in which case the symplectic form and the pre-quantum line bundle are allowed to change together with the complex structure. In a joint work with Andersen and Rembado, we construct a bundle of Hilbert spaces on the sphere of Kähler structures, on which a flat connection is naturally defined. The group of isometries of V which preserve this sphere globally also acts on the bundle, and the construction determines a quantisation of this group. Time permitting, I shall discuss our main application of this construction—the quantisation of the circle action on the moduli space of reductive Higgs bundles over an elliptic curve. ]]>

Despite the importance of these phases, many of the computational techniques for working with fusion categories have not percolated into condensed matter physics. Many of these techniques are "folk theorems"

and have not appeared in the literature in a digestible form.

Jacob Bridgeman and myself have spent the last two years extracting these computational techniques from Corey Jones, and have been using them to understand defects in Levin-Wen phases. Our papers 1806.01279,

1810.09469, 1901.08069, 1907.06692 document the development of our understanding, and demonstrate how to do physically relevant fusion category computations. ]]>

Then, I will present work in progress on a general TFT-type procedure for calculating the factorization algebras describing 2d CFTs which arise as compactifications of such configurations. I will show that this method correctly computes the class S chiral algebras, matching the construction of Arakawa, and discuss potential applications to computing the vertex algebras associated to toric divisors in toric CY3s, following Gaiotto-Rapcak.

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]]>the conformal blocks/chiral cohomology of 2d chiral algebras.

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]]>In 2014 we started a joint project with Maxim Kontsevich which we named "Holomorphic Floer Theory" (HFT for short) in order to study all these (and other) phenomena as a part of a bigger picture.

Aim of my talk is to discuss aspects of HFT related to deformation quantization of complex symplectic manifolds, including the conjectural Riemann-Hilbert correspondence. Although some parts of this story have been already reported elsewhere, the topic has many ramifications which have not been discussed earlier.

]]>gl(N) skein algebra of M to the gl(1) skein algebra of a covering three-manifold M'. In the special case of M=R^3, this map computes well-known link invariants in a new way. As a physical application, the q-nonabelianization map computes protected spin character counting BPS ground states with spin for line defects in 4d N=2 theories of class-S. I will also mention possible extension to more general three-manifolds, as well as further physical applications to class-S theories. This talk is based on joint work with Andrew Neitzke, and ongoing work with Gregory Moore and Andrew Neitzke. ]]>

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