Attendees of the meeting are listed below. For attendees giving a talk, click on their talk title to see the abstract.
|Name||Talk title (if applicable)|
|Nina Holden||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.
|Rei Inoue||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.
|Johanna Knapp||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.
|Zhengwei Liu||Quantum Fourier Analysis
Quantum Fourier Analysis is a subject that combines an algebraic Fourier transform (pictorial in the case of subfactor theory and TQFT) with analytic estimates. This provides interesting tools to investigate phenomena such as quantum symmetry. In this talk, we will introduce its background, development and outlook, based on representative examples, results and applications.
|James Tener||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).
|Lachlan Bennett||Analysis of a quantum integrable four-site Bose-Hubbard model
For this talk, we will be discussing a particular quantum integrable four-site Bose-Hubbard model — a model which describes the physics of interacting bosons on a four-site lattice. Using the Bethe Ansatz technique, we will find exact expressions for leading order terms of the eigenvalues and eigenstates of the Hamiltonian. This enables us to find formulae describing the quantum dynamics of the system. With what we derive, we can show that given specific initial conditions, there exists a time when we have a quantum entangled state that approximates a NOON state. NOON states are used for parameter estimation that can outperform any classical interferometer. We will quantify exactly how close we come to creating a NOON state and compare the results of our analytic formulae to what we expect from numerical calculations.
|Jean-Emile Bourgine||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.
|Jan de Gier|
|Zachary Fehily||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.
|Md Fazlul Hoque|
|Daniel Hutchings||AdS (super)projectors in three dimensions
Given a tensor field which satisfies the massive Klein-Gordon equation on four-dimensional Minkowski space, it decomposes into a sum of constrained fields describing irreducible representations of the Poincaré group with varying spin. Over sixty years ago, Behrends and Fronsdal derived spin projection operators which extract the component of this decomposition corresponding to the representation with the highest spin. Since then, numerous applications have been found for these projectors within the landscape of theoretical physics. For example, they were vital in the original formulation of conformal higher-spin gauge actions. In this talk, I will detail how these operators can be constructed on three-dimensional anti-de Sitter space and study several novel applications. I will then discuss the supersymmetric generalisations of these results.
|Phillip Isaac||On the construction of invariants for quantum algebras
We discuss recent results on the construction of quantum (super)group invariants and explain some details in the context of historical results. The methods involved employ the characteristic identities. One of the aims of the talk is to share insights into these techniques.
|James La Fontaine||An effective action approach for superconformal higher spin gauge theories
Conformal higher-spin gauge (CHS) theories and their supersymmetric extensions have recently attracted considerable interest. CHS theories are higher-spin extensions (s > 2) of Maxwell’s electrodynamics (s = 1) and conformal gravity (s = 2). Formally, CHS theories can be identified as the logarithmically UV divergent part of the effective action (called the induced action) for a complex scalar field coupled to a higher-spin background. However, this induced action has never been computed in closed form. In 2017, linearised superconformal higher-spin gauge (SCHS) theories were constructed and a question arose of whether these models and their non-linear completions can be obtained as an induced action. This talk will give a pedagogical review of our method for computing SCHS actions as induced ones.
|Danilo Latini||Superintegrable systems with coalgebra symmetry and Racah algebras
The rank-1 Racah algebra R(3) plays a pivotal role in the theory of superintegrable systems. It appears as the symmetry algebra of the generic 3-parameter model on the 2-sphere, from which all second-order conformally flat superintegrable systems in 2D can be obtained as limiting cases. A higher rank generalization of R(3) has been considered and showed to be the symmetry algebra of the generic superintegrable model on the (n−1)-sphere. It has recently been shown that it also characterizes the symmetry algebra of generic superintegrable systems defined on pseudo-spheres.
In this talk, I will discuss how such a higher rank generalization of the Racah algebra naturally emerges for a class of nD quasi-maximally superintegrable systems endowed with sl(2,R) coalgebra symmetry.
Reference: D. Latini, I. Marquette and Y.-Z. Zhang. Annals of Physics 426, 168397 (2021)
|Jon Links||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.
|Vladimir Mangazeev||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.
|Ian Marquette||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.
|William Mead||Transition Probabilities in the Multi-species Asymmetric Exclusion Process
The asymmetric simple exclusion process (ASEP) is a stochastic model of indistinguishable particles with KPZ-like limiting behaviour. There are very few mathematically rigorous results for the limiting behaviour of multi-species models. In this talk we describe a method for evaluating transition probabilities for a multi-species version of the ASEP. This is performed via a reduction from a family of integrable stochastic vertex models. From this, we derive expressions for bulk crossing probabilities and discuss their implications. Based on joint work with Jan de Gier and Michael Wheeler.
|Paul Norbury||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.
|Madeline Nurcombe||Representation Theory of the One-Boundary Temperley-Lieb Algebra
The Temperley-Lieb algebra arises in many lattice models in statistical mechanics, from quantum spin chains to systems of polymers. It admits a useful diagrammatic presentation and has a rich and well-studied representation theory. In this talk, I will discuss a one-boundary generalisation of the algebra that depends on three complex parameters. Using a diagrammatic presentation, I will introduce a family of its representations and identify which of them are irreducible. I will also determine for what parameter values the algebra is semisimple, and discuss the structure of the representations when the algebra is non-semisimple.
|Xavier Poncini||Loop models on causal triangulations
Causal dynamical triangulations (CDT) is a non-perturbative approach to quantum gravity where the formal path integral is regularised by a triangulation. Of particular interest is the coupling of matter fields to the geometry of pure triangulations and the question whether the coupling influences the macroscopic features of the ensemble, such as the Hausdorff dimension. In this talk, I will present three models on 2D causal triangulations: a pure CDT model, a dense loop model, and a dilute loop model. The two loop models introduce a matter coupling to the pure CDT model, reminiscent respectively of the fully-packed and dilute loop models on regular lattices. After introducing correspondences with labelled planar trees and developing a transfer-matrix formalism, the critical behaviour of each model is analysed. The pure CDT and dense loop models are shown to exhibit similar behaviour — indicating the absence of a significant interaction between matter and geometry. The critical behaviour of the dilute loop model is found to be different, with evidence of a shift in Hausdorff dimension from 2 to 1.
|Thomas Quella||Gapped interpolations between even-spin qAKLT states
We study $U_q(sl_2)$-invariant interpolations between even-spin $q$AKLT states and fully dimerized states and show that the associated transfer matrix has a spectral gap. This confirms an earlier suggestion by one of the authors that even-spin $q$AKLT states are topologically trivial, even if quantum group invariance is preserved. All computations are carried out analytically for general even-integer physical spin and arbitrary $q>0$. An analytical verification of earlier numerical results for even-spin AKLT states by Pollmann et al is included as a special case for $q=1$.
|Emmanouil Raptakis||Duality-invariant conformal higher-spin models
The study of symmetries of a given field theory yields nontrivial information regarding its structure and often provides important insights into potential extensions or generalisations. In particular, a novel class of models for non-linear electrodynamics was recently discovered by Bandos et al. by requiring that they preserve the conformal and U(1) duality invariance obeyed by the free theory. Since such symmetries are also shared by the free theories of conformal higher-spin fields, it is an interesting question to determine what nonlinear deformations consistent with conformal and duality invariance the latter admit. In this presentation, I will present new families of nonlinear models for conformal fields of spins greater than 1.
|Pieter Roffelsen||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.
|Benjamin Stone||Three-point functions of conserved currents in superconformal field theory
Two-point and three-point correlation functions of primary operators are the main observables in conformal field theories. Of particular interest are correlation functions of conserved currents such as the vector current, the energy momentum tensor, and more generally, higher-spin conserved currents. An interesting feature of conformal field theories in three-dimensions is that correlation functions of conserved currents contain parity-violating structures which do not appear in four-dimensional theories. In superconformal field theories the vector current and energy-momentum tensor are contained within the flavour current and supercurrent multiplets respectively. In this talk I will discuss the properties of the three-point functions of the supercurrent and flavour current multiplets in three-dimensional superconformal field theories; in particular I will show that they do not-contain parity violating structures. Hence there is an apparent tension between supersymmetry and the existence of parity violating structures in conformal field theories.
|Gabriele Tartaglino-Mazzucchelli||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.
|Zongzheng Zhou||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.