Liouville quantum gravity in probability theory

In probability theory, Liouville quantum gravity is a model for a random surface. There are two main directions of study: random conformal geometry and Liouville conformal field theory. I will give an introduction to these two directions and present some recent developments.

Duality-invariant (super)conformal higher-spin models

Three-point functions of conserved currents in superconformal field theory

The Yang-Baxter Paradox

Yang-Baxter techniques are well-known to be fundamental in the study of integrable quantum systems, particularly for the derivation of exact solutions via Bethe Ansatz methods. In this talk I will argue that there exist Yang-Baxter integrable systems that do not admit exact solutions. The conclusion is drawn by setting up a formal framework and the examination of a particular bosonic model.

Analysis of a quantum integrable four-site Bose-Hubbard model

Representation Theory of the One-Boundary Temperley-Lieb Algebra

Cluster algebras and its applications to rational maps

The cluster algebra was introduced around 2000 by Fomin and Zelevinsky. The heart of the algebra is algebraic operations called `mutations’ acting on pairs of a quiver and a set of variables. Surprisingly, the mutations directly or indirectly appear in various area of mathematics and mathematical physics. One idea is to find an interesting sequence of mutations which preserves the initial quiver, from which we get a rational map for the variables. In this talk, I explain the basic notion of cluster algebras (CA), and review some applications of CA to rational maps related to integrable systems and representation theory of quantum groups.

Relating W-algebras using inverse quantum hamiltonian reduction

W-algebras are interesting and useful examples of vertex operator algebras constructed from affine vertex algebras by quantum hamiltonian reduction. Inverses to this construction, originating in string-theoretic work by Semikhatov for sl2 and later by Adamovic for certain W-algebras, have proved very useful in the investigation of logarithmic conformal field theories. In this talk, I will describe the ‘spirit’ of inverse quantum hamiltonian reduction, detail some recent results and make some claims about the future.

Transition Probabilities in the Multi-species Asymmetric Exclusion Process

AdS (super)projectors in three dimensions

Gapped interpolations between even-spin qAKLT states

The Calabi-Yau landscape and supersymmetric gauge theory

The first part of the talk gives a brief overview of Calabi-Yau spaces in the context of string theory and reviews the basics of a standard construction in terms of toric geometry. In the second part it will be shown how Calabi-Yaus can be analysed via the physics of certain supersymmetic gauge theories. This approach provides a pathway to construct new types of Calabi-Yaus beyond the standard methods. A class of such new Calabi-Yaus has been constructed in recent work in collaboration with E.Scheidegger and T.Schimannek and has applications in topological string theory, F-theory and pure mathematics. This will be outlined in the third part of the talk.

An effective action approach for superconformal higher spin gauge theories

Shifted quantum groups and branes dynamics

Quantum groups were introduced in the 80s to describe the mathematical structure responsible for the integrable properties of quantum systems. More recently, the refined notion of “shifted quantum groups” has played an important role in algebraic geometry. This subtle modification of the original definition brings more flexibility in the representation theory. In this talk, I will present several representations for simple shifted quantum groups, namely the shifted quantum affine sl(2) and quantum toroidal gl(1) algebras. Then, I will introduce an application to “Algebraic Engineering”, a newly developed technique for studying the low-energy dynamics of brane systems in string theory.

On the construction of invariants for quantum algebras

Banach-Tarski Paradox and related Infinity Analysis

Algebraic Integrability

A random matrix integral and the moduli space of super Riemann surfaces.

Perhaps the simplest random matrix integral is given by the Selberg integral. It produces a random matrix integral over Hermitian matrices with eigenvalues contained in an interval, known as the Legendre ensemble. We will describe how this random matrix integral and the related Legendre polynomials produce intersection numbers over the compactification of the moduli space of Riemann surfaces. This uses Gromov-Witten invariants of the sphere coupled to a class related to the moduli space of super Riemann surfaces.

On the monodromy manifold of q-difference Painlevé VI

The monodromy manifolds of classical Painlevé equations are well-known to be identifiable with affine cubic surfaces. Points on any such surface are in one-to-one correspondence with solutions of the corresponding Painlevé equation. It is expected that analogous surfaces exist for discrete Painlevé equations but their constructions have so far largely remained elusive. In this talk, I will present the construction of such a surface for the q-difference Painlevé VI equation and detail some of its properties.

Higher rank quadratic algebra of the $N$-dimensional Smorodinsky-Winternitz model

A superintegrable system is a system that allows more integrals of motion than degrees of freedom, i.e., in a classical mechanics, an n-dimensional Hamiltonian system with Hamiltonian
\begin{eqnarray*}
H=\frac{1}{2}g_{ik}p_i p_j+V(\vec{x},\vec{p}),\quad X_a=f_a(\vec{x},\vec{p}), a=1,\dots, n-1,
\end{eqnarray*}
is called completely integrable (Liouville integrable) if it allows $n$ integrals of motion (including $H$) that are well-defined functions on phase space, are in involution
\begin{eqnarray*}
\{H,X_a\}_{p}=0,\quad \{X_a, X_b\}_{p}=0, a, b=1,\dots, n-1
\end{eqnarray*}
and are functionally independent. The system is superintegrable if it is integrable and allows additional integrals of motion $Y_b(\vec{x},\vec{p})$, $\{H,Y_b\}_{p}=0, b=1,2,\dots, k$, $1\leq k\leq n-1$, that are also well-defined functions on phase space and the integrals $\{H, X_1,\dots, X_{n-1}, Y_n,\dots, Y_{n+k}\}$ are functionally independent.
In this talk, I present higher rank quadratic algebra of the $N$-dimensional quantum Smorodinsky-Winternitz system, which is a maximally superintegrable and exactly solvable model. It is shown that the model is multiseparable and the wave function can be expressed in terms of Laguerre and Jacobi polynomials. We present a complete symmetry algebra ${\cal SW}(N)$ of the system, which it is a higher-rank quadratic one containing Racah algebra ${\cal R}(N)$ as subalgebra. The substructures of distinct quadratic Q(3) algebras and their related Casimirs are also studied. The energy spectrum of the $N$-dimensional Smorodinsky-Winternitz system is obtained algebraically via the different set of subalgebras based on the Racah algebra ${\cal R}(N)$.

On a three-state R-matrix without difference property

We consider a three-state 19-vertex solution of the Yang-Baxter equation with a spectral curve of genus five proposed by Martins. After a special gauge transformation we show that it can be reduced to a genus one solution without a difference property. We parameterize this solution in terms of theta-functions and investigate its trigonometric limit.

Towards a unified framework for the mathematics of conformal field theory

Two-dimensional chiral conformal field theories have applications in several distinct contexts in mathematical physics, and they have been linked to a broad range of (seemingly!) unconnected mathematical applications. As with other flavors of quantum field theories, there are significant challenges involved in rigorously axiomatizing chiral CFTs, although there are several approaches which have been heavily studied and which provide partial axiomatizations, with the best known being vertex operator algebras and conformal nets. In this talk I will describe a program to develop a unified mathematical framework which bridges these approaches. This framework is closely linked to Segal’s geometric definition of conformal field theory. I will also touch on recent and ongoing work which uses these ideas to resolve open problems in the mathematical study of conformal field theories (e.g. showing that WZW models produce finite-index subfactors in the sense of V. Jones, and showing that every conformal net has an associated vertex operator algebra).

New multiplets of off-shell conformal supergravity

Conformal supergravity plays an important role in the construction of general off-shell supergravity-matter systems and locally supersymmetric invariants. A starting point of superconformal techniques applied to supergravity is a supersymmetric multiplet containing the metric and transforming off-shell under the local/gauged superconformal algebra. Focusing on the case of eight real supercharges, we present new multiplets of off-shell conformal supergravity. We will then comment on new constructions of electric and magnetic Fayet-Iliopoulos terms and applications to supergravity models possessing spontaneously (partially) broken supersymmetry.

Superintegrable systems with coalgebra symmetry and Racah algebras

Weakly self-avoiding walk on the complete graph

We study a weakly self-avoiding walk model on the complete graph, in which return visits to the same vertex are penalised by a parameter $\lambda \in [0,1]$. The cases $\lambda =0$ and 1 correspond to the self-avoiding walk and the simple random walk. We will discuss universal limit theorems for the walk length when $\lambda<1$, in various regimes of fugacity. We will describe connections to certain sums of Stirling numbers, which may be of independent interest.

Exact solvability and superintegrability: algebraic constructions

It was discovered how polynomial algebras appear naturally as symmetry algebra of superintegrable quantum systems. They provide insight into their degenerate spectrum, in particular for models involving Painleve transcendents. I will discuss an alternative perspective on those algebraic structure based on Lie algebras and their related enveloping algebras. I will discuss how the symmetry algebra of the generic superintegrable systems on the 2-sphere can be generated from an underlying Lie algebra. I will also discuss how exact solvability also can be connected with such approach and how it differs from using differential operator realizations.