# Representation Theory and Mathematical Physics seminar

## UC Berkeley, Fall 2017

All talks in Evans Hall 939, 4 to 5 pm.

Sep 06 - Monica Vazirani (UC Davis)

An elliptic Schur-Weyl construction of the rectangular representation of the DAHA.

Building on the work of Calaque-Enriquez-Etingof, Lyubashenko-Majid, and Arakawa-Suzuki, Jordan constructed a functor from quantum D-modules on general linear groups to representations of the double affine Hecke algebra (DAHA) in type A. When we input quantum functions on GL(N) the output is L($k^N$), the irreducible DAHA representation indexed by an N by k rectangle. For the specified parameters, L($k^N$) is Y-semisimple, i.e. one can diagonalize the Dunkl operators. We give an explicit combinatorial description of this module via its Y-weight basis. This is joint work with David Jordan.

Sep 13 - No seminar.

"Integrability Across Mathematics and Physics" Workshop.

Sep 20 - Julia Pevtsova (University of Washington)

Supports and tensor ideals in stable module categories.

Classifying modules up to direct sums in modular representation theory is usually a hopeless task – there are too many indecomposable modules (over a field of characteristic $p$) even for such a seemingly small group as $Z/3 \times Z/3$ or a three dimensional Heisenberg Lie algebra. Inspired by ideas from stable homotopy theory and algebraic geometry we suggest a different way of organizing our understanding of modular representations. Namely, we seek to classify modules up to homological operations: not only direct sums, but also extensions, syzygies and tensor products with simple modules. I will describe both the problem and the answer which involves cohomology and support varieties. Based on joint work with D. Benson, S. Iyengar, H. Krause.

Sep 27 - Alexander Alldridge (University of Cologne)

A cohomological approach to the Berezin fibre integral.

The fibrewise Berezin integral is an important tool in mathematical physics, e.g., to derive supersymmetric field equations from variational principles. For a supermanifold, the de Rham complex is underbounded, and differential forms cannot be integrated; the Berezinian sheaf was introduced by Berezin to address this problem. However, the definition is ad hoc, the resulting integral is defined coordinate-independently only for compact supports, and "boundary corrections" appear when changing coordinates for non-compactly supported integrands. So far, there was no systematic and conceptual understanding of these terms.

On the other hand, by the work of Penkov and of Verbovetsky, it has been known for some time that the Berezin sheaf can be obtained as the cohomology of a natural complex of D-modules. However, it was not known how to use this to define the Berezin integral, except in the non-fibrewise case (i.e. over a trivial base) by the work of Rothstein. Unfortunately, his approach uses the trivial base (and the boundedness of the de Rham complex) in an essential way and does not generalise to the case where the base is a supermanifold.

We address these issues and give a definition of the Berezin integral in terms of the cohomology of a complex of D-modules. This is based on the observation that although arbitrary differential forms cannot be integrated on a supermanifold, closed forms can be. Using higher order differential forms, we derive a higher order Stokes's theorem for relative supermanifolds with corners and show how this gives a systematic derivation of the boundary terms. Joint work with Joachim Hilgert and Tilmann Wurzbacher.

Oct 4 - No Seminar

Oct 11 - Shamil Shakirov (Harvard University)

Mapping class groups and difference operators.

We review the representation of $SL(2,Z)$ - the mapping class group of the torus - by automorphisms of a simple algebra of difference operators. The algebra, known as spherical double affine Hecke algebra (DAHA) plays an important role in many developments in modern representation theory and mathematical physics. We will define a new algebra which is a direct analogue of spherical DAHA for a genus two surface, and sketch the proof of the corresponding mapping class group action. Time permitting, we will explain the connection to the Reshetikhin-Turaev construction, and possible generalizations to higher genus.

Oct 18 - NO SEMINAR

Oct 25 - Andrey Smirnov (UC Berkeley)

Bethe ansatz from geometry I.

This talk is the summary of new geometric approch to the quantum integrable spin chains. As a warm up, I will illustrate these ideas on the example of $sl(2)$ XXZ spin chain: we will obtain conventianal $sl(2)$ Bethe ansatz for this model from geometry of cotangent bundles over grassmannians. In the second part we use same ideas to derive Bethe ansatz for moduli spaces of instantons. As a byproduct we get a new look at the theory of symmetric polynomials.

Nov 1 - Andrey Smirnov (UC Berkeley)

Bethe ansatz from geometry II.

This talk is the summary of new geometric approch to the quantum integrable spin chains. As a warm up, I will illustrate these ideas on the example of $sl(2)$ XXZ spin chain: we will obtain conventianal $sl(2)$ Bethe ansatz for this model from geometry of cotangent bundles over grassmannians. In the second part we use same ideas to derive Bethe ansatz for moduli spaces of instantons. As a byproduct we get a new look at the theory of symmetric polynomials.

Nov 8 - Jeffrey Kuan (Columbia University)

Algebraic constructions of Markov duality functions.

Markov duality in spin chains and exclusion processes has found a wide variety of applications throughout probability theory. We review the duality of the asymmetric simple exclusion process (ASEP) and its underlying algebraic symmetry: in particular, there is an underlying quantum group and affine Lie algebra symmetry. We then explain how the algebraic structure leads to a wide generalization of models with duality, such as higher spin exclusion processes, zero range processes, stochastic vertex models, and their multi-species analogues.

Nov 9 - Konstantin Wernli (University of Zurich) - Extra Talk on THURSDAY - 2 PM in 732 EVANS

Perturbative Chern-Simons invariants from quantum BV-BFV formalism.

I will report on recent developments in the computation of Perturbative Chern-Simons invariants via cutting and gluing in the quantum BV-BFV formalism. In particular, I will present results on theta-invariants of lens spaces that agree with earlier works of Kuperberg, Thurston and Lescop.
This is ongoing joint work with A. Cattaneo and P. Mnev.

Nov 15 - Theo Johnson-Freyd (Perimeter Institute)

Higher categories, generalized cohomology, and condensed matter.

I will report on joint work in progress with Davide Gaiotto on the classification of gapped phases of matter. I will explain what symmetry protected phases are and why they are classified by reduced generalized group cohomology. I will also introduce the notion of "condensable n-algebra," and the higher category thereof, as an axiomatization of the algebraic structure enjoyed by gapped phases that can be condensed from the vacuum. Finally, I will interpret the Cobordism Hypothesis as the equivalence between (condensable) topological field theories and (condensable) gapped phases.

Nov 16 - Drazen Petrovic (Purdue University) - Extra Talk on THURSDAY - 2 PM in 732 EVANS

Pfaffian Sign Theorem for the Dimer Model on a Triangular Lattice.

We prove the Pfaffian Sign Theorem for the dimer model on a triangular lattice embedded in the torus. More specifically, we prove that the Pfaffian of the Kasteleyn periodic-periodic matrix is negative, while the Pfaffians of the Kasteleyn periodic-antiperiodic, antiperiodic-periodic, and antiperiodic-antiperiodic matrices are all positive. The proof is based on the Kasteleyn identities and on small weight expansions. As an application, we obtain an asymptotics of the dimer model partition function with an exponentially small error term. This is a joint work with Pavel Bleher and Brad Elwood.

Nov 22 - No Seminar

Nov 29 - Ivan Contreras (UI Urbana Champaign)

Integration of poly-Poisson structures.

Poly-Poisson geometry can be traced back to de-Donder and Weyl in 1930's. This approach leads to a poly-symplectic formulation of Lagrangian field theories, with several applications to mechanics. In this talk we address the problem of integration of poly-Poisson manifolds via Lagrangian field theories with boundary, which is a natural extension of the Poisson sigma model. Joint work with N. Martinez Alba (arXiv: 1706.0614).

Dec 6 - David Hernandez (Paris 7)

Spectra of quantum integrable systems, Langlands duality and category O
(based on joint works with M. Jimbo, E. Frenkel and B. Leclerc).

The spectrum of a quantum integrable system is crucial to understand its properties. R-matrices give power tools to study such spectra. A better understanding of transfer-matrices obtained from R-matrices led us to the proof of several results for the corresponding quantum integrable systems. In particular, their spectra can be described in terms of "Baxter polynomials" as conjectured by Frenkel-Reshetikhin. They appear naturally in the study of a category O of representations of a Borel subalgebra of a quantum affine algebra (in the $sl_2$-case, this is due to Bazhanov-Lukyanov-Zamolochikov). The properties of geometric objects attached to the Langlands dual Lie algebra (the affine opers) led us to establish new relations in the Grothendieck ring of this category O, from which one can derive the generic Bethe Ansatz equations between the roots of the Baxter polynomials. They are also related to a cluster algebra structure on the Grothendieck ring. ​

Supported by the European Research Council under the European Union's Framework Programme H2020 with ERC Grant Agreement number 647353 Qaffine.

Dec 13

TBA