Tag - Vertex operator algebras

Slaven Kožić: Representations of the quantum affine vertex algebra associated with the trigonometric R-matrix of A

14th October, 2024

One important problem in the vertex algebra theory is to associate certain vertex algebra-like objects, the quantum vertex algebras, to various classes of quantum groups, such as quantum affine algebras or double Yangians. In this talk, I will discuss this problem in the context of Etingof-Kazhdan's quantum affine vertex algebra Vc(𝔤𝔩N) associated with the trigonometric R-matrix of type A. The main focus will be on the explicit description of the centre of Vc(𝔤𝔩N) at the critical level c = -N and, furthermore, on the connection between certain classes of Vc(𝔤𝔩N)-modules and representation theories of the quantum affine algebra of type A and the orthogonal twisted h-Yangian.

Naihuan Jing: Characters of GLn(q) and vertex operators

3rd March, 2024

Irreducible characters of the finite group GLn(q) were determined by Green in a remarkable paper that has influenced representation theory greatly. In this talk, I will discuss a vertex algebraic approach to construct and compute all complex irreducible characters of GLn(q). Green's theory is recovered and enhanced under the realization of the Grothendieck ring of representations R(G)=⨁​n≥0​​R(GLn(q)) as two isomorphic Fock spaces. Under this picture, the irreducible characters are realized by the Bernstein vertex operators for Schur functions, the characteristic functions of the conjugacy classes are realized by the vertex operators for the Hall-Littlewood functions, and the character table is completely given by matrix coefficients of vertex operators of these two types. This offers a simplification to identify the Fock space R(G) as the Hall algebra of symmetric functions. We will also discuss how to compute the characters in general.

Andrew Linshaw: Universal vertex algebras beyond the W∞-algebras

31st August, 2023

The principal W-algebras Wk(𝔰𝔩n), as well as the cosets Com(Vk(𝔤𝔩n), Vk(𝔰𝔩n+1)) for n ≥ 2, are all quotients of a universal 2-parameter vertex operator algebra (VOA) which is freely generated of type W(2,3,4, ...). This VOA admits many other interesting 1-parameter quotients which can be identified (up to an extra Heisenberg field) with the Gaiotto-Rapcak Y-algebras. We consider a similar construction in type C, namely, the cosets Com(Vk(𝔰𝔭2n), Vk(𝔰𝔭2n+2)), for n ≥ 2. This gives rise to a 2-parameter VOA which is freely generated of type W(13, 2, 33, 4,...), which we expect to be the universal VOA of this type. The universal algebra admits 8 infinite families of 1-parameter quotients, which are analogues of the Gaiotto-Rapcak Y-algebras. Assuming that the universal algebra has exactly two parameters, which is ongoing work to prove, we present some applications including new rationality results for W-(super)algebras.

Anne Moreau: Functorial constructions of double Poisson vertex algebras

30th August, 2023

To any double Poisson algebra we produce a double Poisson vertex algebra using the jet algebra construction. We show that this construction is compatible with the representation functor which associates to any double Poisson (vertex) algebra and any positive integer a Poisson (vertex) algebra. We also consider related constructions, such as Poisson reductions and Hamiltonian reductions. This allows us to provide various interesting examples of double Poisson vertex algebras, in particular from double quivers.

Tomoyuki Arakawa: Hilbert Schemes of the points in the plane and quasi-lisse vertex superalgebras

29th August, 2023

For each complex reflection group Γ one can attach a canonical symplectic singularity ℳΓ. Motivated by the 4D/2D duality discovered by Beem et al., Bonetti, Menegheli and Rastelli conjectured the existence of a supersymmetric vertex operator algebra WΓ whose associated variety is isomorphic to ℳΓ. We prove this conjecture when the complex reflection group Γ is the symmetric group SN, by constructing a sheaf of ℏ-adic vertex algebras on the Hilbert schemes of N points in the plane. In physical terms, the vertex operator algebra WSN corresponds, by the 4D/2D duality, to the 4-dimensional N=4 super Yang-Mills theory with gauge group SLN.

Anne Moreau: Associated varieties and finite extension of vertex algebras

20th March, 2023

Motivated by recent works joint with Tomoyuki Arakawa and Jethro Van Ekeren on collapsing levels, we conjectured that if W is a finite extension of a vertex subalgebra V, then the natural morphism between the corresponding associated varieties is dominant. In the case where W is a simple W-algebra and V is its simple affine vertex algebra, the conjecture is deeply related with the singularities of nilpotent Słodowy slices. In this talk, I will explain some results toward the conjecture and interesting examples.

Thomas Creutzig: Vertex Operator Algebras

1st August, 2022

This is a 20-lecture course, with each lecture being about 45 minutes or so, given by Thomas Creutzig. It gives an introduction to vertex operator algebras from the point of view of quantum mechanics.

Vertex operator algebras (VOAs) first appeared in the 1980s as the rigorous notion of chiral algebras (the symmetry algebras) of two-dimensional conformal quantum field theories. Since then, they have been employed as key ingredients in many modern problems of mathematical physics and pure mathematics, ranging from monstrous moonshine to knot theory and geometry. The older problems have been mostly concerned with the simplest type of VOAs, so‐called rational theories.

In the last few years, it has been realized that VOAs and their representation theories yield rich invariants of three and four‐dimensional supersymmetric quantum field theories. This provides new insights into low‐dimensional topology and the quantum geometric Langlands programme. Involved VOAs are however not rational (often called logarithmic) and so their representation theory is rich and exciting.

These lectures will be a very modern introduction to the theory of VOAs. We will use techniques from representation theory (especially Lie theory), geometry and topology; no knowledge of VOAs is needed. The lectures will be a mix of general theory and illustrating it with the most important examples, that is free field theories, affine and W‐algebras; and the school will end with an exposition of the very recent use and appearance of VOAs in physics, geometry, and low‐dimensional topology.

Katrin Wendland: How do quarter BPS states cease being BPS?

8th June, 2022

The so-called BPS states in a conformal field theory with extended supersymmetry are key when assigning a geometric interpretation to the theory. Standard invariants for such theories arise from a net count of BPS, half or quarter BPS states, according to the ℤ2 grading into ‘bosons' and ‘fermions'. This allows for boson-fermion pairs of states to cease being BPS under deformation of the theory. The talk will give a review of this phenomenon, arguing that it is ubiquitous in theories with geometric interpretation by a K3 surface. For a particular type of deformations, we propose that the process is channelled by the action of SU(2) on an appropriate subspace of the space of states.

Damien Calaque: Vertex models and En-algebras

7th June, 2022

I will explain and state a conjecture of Kontsevich, that relates vertex models from statistical mechanics to En-algebras (i.e., algebras for the n-dimensional little disks operad). I will also give the main ingredients of the proof of Kontsevich's conjecture, that involve discretized versions of the little disks operad. This is a work in progress with Damien Lejay.