Participants

I will discuss a coupled Temperley-Lieb algebra featuring N ‘coupled’ copies of the standard Temperley-Lieb algebra. The generalised cubic relations and the diagrammatic interpretation will be introduced as well as the relation to the chiral Potts and staggered XX quantum spin chains.

In this talk we will review the current state of modelling of folliculogenesis from the perspective of statistical mechanics.

Women are born with between 0.5-2 million eggs encased in follicles, which decrease exponentially throughout life, leading to menopause when the number reaches around 1000 [1]. While this decrease is exponential, it is quite striking that the ovulation number is a constant, one egg a month. It is even more striking when one considers that there are two ovaries.

Only a total of around 500 eggs are ovulated in a lifetime; the vast majority of eggs atrophy and die in a process called atresia. The initial pool of follicles is in a primordial, dormant state. Once they proceed to a growing state, they grow from primary, to secondary, and then antral follicle states, a process called folliculogenesis. Many eggs die at each stage, leading to fewer at each step. Nevertheless, this large pool of eggs contributes to a group dynamic before they die, marked by the appearance of order: the ovulation of one and only one egg each month. This group dynamic affects growing eggs in both ovaries simultaneously, since serum levels of hormones are shared between them.

Mid-menstrual cycle, follicle stimulating hormone (FSH) is released by the pituitary, which protects follicles, helping them to keep growing. As they grow, the follicles develop receptors to luteinizing hormone (LH), which helps them express the hormone estradiol. As the follicles continue to grow, the levels of estradiol rise, which has the effect of decreasing the FSH levels, which in turn hinders further growth. In other words, follicle growth is dependent on a feedback system, several in fact.

To first order, one could consider the growth of ovarian follicles as a many-body problem with spatially independent interactions. There have been several mathematical models developed based on hormone levels which show the emergence of order from straightforward assumptions (e.g. [2]). Recently multi-scale models have been developed that consider the growth of individual cells and their receptivity to hormones or their expression of hormones (see e.g. [3]). One motivation for studying these models is to understand disorders that possibly arise from a dysfunctional feedback system, for example polycystic ovarian syndrome (PCOS). Nevertheless, current models leave out much new research on the roles of various proteins in folliculogenesis, some having their own feedback systems. The roles of these proteins could be elucidated through exploration within more accurate models.

It is the goal of this talk to summarize and review the current state of the understanding of folliculogenesis and its modelling from the perspective of statistical mechanics in the hope of involving more of the mathematical community, including students, into this fascinating endeavour.

1. Wallace, W. Hamish B., and Thomas W. Kelsey. “Human ovarian reserve from conception to the menopause.” PloS one 5.1 (2010): e8772.

2. Michael Lacker, H., and Allon Percus. “How do ovarian follicles interact? A many-body problem with unusual symmetry and symmetry-breaking properties.” Journal of statistical physics 63.5 (1991): 1133-1161.

3. Clément, Frédérique, and Danielle Monniaux. “Mathematical modeling of ovarian follicle development: A population dynamics viewpoint.” Current Opinion in Endocrine and Metabolic Research 18 (2021): 54-61.

The Baxter-Fendley model of Z(N) parafermions is a relatively simple N-state generalization of the quantum Ising chain. Subject to open boundary conditions, the energy eigenspectrum of this non-Hermitian model is composed of free parafermions, which are a natural generalization of free fermions to the complex plane. In this talk, based on work with Alex Henry in arXiv:2301.11031, I will discuss the appearance of exceptional points in the eigenspectrum.

We consider a solvable model of self-avoiding walks (SAWs) which cross an $L \times L$ box, namely partially directed walks (PDWs).

For SAWs, the number is conjectured to be asymptotic to $\Lambda^{L^2+bL+c}\cdot L^g$, for constants $\Lambda, b, c, g$. Moreover, when a Boltzmann weight $t$ is associated with the length of the walks, a phase transition occurs at $t=\mu^{-1}$, where $\mu$ is the connective constant for the lattice. For $t<\mu^{-1}$ the average walk is of length $O(L)$, while for $t>\mu^{-1}$ it is of length $O(L^2)$.

Here we solve the PDW version of this model and compute the asymptotic behaviour for all $t$. The phase transition occurs at $t=1$, and we use quite different methodology for $t$ below, above and at the critical point.

For this presentation, I’ll introduce a family of quantum integrable models. These models, characterised by a Hamiltonian, describe boson tunnelling in multi-well systems. After discussing the properties of these models, I will demonstrate how a Bethe Ansatz technique can be applied to find exact solutions. These solutions allow us to analyse the quantum dynamics and measurement outcomes at specified times. If physically realised, this family of Hamiltonians can be beneficial in studying entanglement.

The algebraic engineering consists in constructing observables of supersymmetric gauge theories within the representation theory of a quantum group. It is based on the branes system realization in string theory, this system being mapped to a network of modules on which act intertwining operators. The algebraic construction brings new perspectives on many important properties of gauge theories (e.g. AGT-correspondence, dualities, integrability,…). In this talk, I will briefly review the recent advances on this topic, and then use the underlying algebra to revisit the relation between topological strings and the KP integrable hierarchy. This talk will be based on the preprint arXiv:2101.09925.

In this talk I will show how T-duality for circle bundles over tori with background H-flux can be reformulated in terms group cohomology. This will then be generalised to T-duality for circle bundles over toroidal orbifolds with background flux. The geometric counterpart of this T-duality will be discussed in a talk by my PhD student Jaklyn Crilly.

The Hermitian two-band SSH model proposed by Su, Schrieffer and Heeger is the simplest topological insulator model. It describes the single spinless non-interacting fermion Hamiltonian on a one-dimensional finite lattice with staggered hopping amplitudes. The topological properties of this system have been thoroughly researched, including bulk-boundary correspondence, the existence of edge states (also zero-energy states), and topological invariants calculated using Berry curvature.

There have been numerous Hermitian and non-Hermitian extensions of the SSH model. The Hermitian extensions include long-range hoppings, extended unit cells and the inclusion of on-site potentials and spin-orbit interaction. The non-Hermitian SSH models can be roughly divided into two main classes: the ones with asymmetrical hopping (either long-range or short-range) and the ones with complex onsite potentials. There are also other higher dimensional SSH models proposed in the area of superconductivity.

In this talk, I explore the possibility of using topological data analysis to detect the topological phases in Hermitian and non-Hermitian SSH models. I will review the current literature and introduce my in-house program. I will also sketch out some directions for future analysis with more complicated models using this method.

In this talk, I will focus on global aspects of T-duality applied to geometric backgrounds and explore how T-duality affects a group action on such background. This will naturally lead to exploring how T-duality applies to orbifolds, and backgrounds in the presence of a discrete torsion factor.

The topological vertex is a beautiful theory that was inspired by topological strings and allows one to explicitly compute Gromov-Witten invariants of toric Calabi-Yau threefolds. In this talk, we briefly describe some of the rich algebraic, combinatorial, and geometric structures underlying the theory. Finally, we state a recent result obtained with Brett Parker, which presents symmetries for the topological vertex that are captured by a quantum torus Lie algebra.

The accepted paradigm for understanding the nature of dark matter is based on the existence of an intrinsically weakly interacting, “as yet unknown”, particle beyond the standard model. Calculations from gravitational quantum theory, however, show quite conclusively that ordinary protons and electrons can similarly exhibit reduced interaction cross sections in the weak gravity regions of large gravitational wells like galaxy halos, by virtue of their gravitational eigenspectral composition [1,2]. A galaxy halo consisting entirely of baryonic gas would appear largely invisible, its fraction of “dark matter” depending on the proximity to equilibrium, the halo particles’ position and uncertainty in phase space, and the size and depth of the gravitational well they are in. This environmentally induced darkness is in some ways analogous to an electronic wavefunction in mixtures of dark atomic eigenstates.
A halo’s erroneously inferred dark matter fraction depends on the “quantum temperature”. Over a narrow range of halo temperatures, a halo will be composed predominantly of atomic hydrogen, the correct gas mass being obtained from the internal 21 cm line, and no dark matter required to make up a mass shortfall. Such galaxies have already been observed [3]. Dark matter-rich, hot ionized halos have reduced scattering cross sections, causing their gas mass fraction to be underestimated, hence requiring a need for dark matter when it may not be necessary. The low quantum temperature halos of the super-dark matter dominated dwarf spheroidal galaxies have predominantly molecular hydrogen whose internal molecular transitions may soon be detectable with the James Webb Space Telescope. This presentation will elaborate further on weak-gravity quantum theory and its implications for dark matter.
[1] A. D. Ernest, J. Phys. A: Math. Theor. 42, 115207 (2009).
[2] A. D. Ernest, J. Phys. A: Math. Theor. 42, 115208 (2009).
[3] P. E. M. Pina et al., arXiv:2112.00017v2 (2021)

W-algebras are an important class of vertex operator algebras that appear frequently in both mathematics and physics. Understanding their structure and representation theory is therefore a fruitful endeavour. In this talk, I will discuss how free field realisations and screening operators can help.

I will talk about some recent developments related to connections between integrable lattice models and shuffle algebras associated to quantum algebras.

To probe detailed phenomena predicted by the AdS/CFT correspondence, quantum corrections (i.e., higher-derivative corrections) to gauged supergravity must be constructed, the classification of which remains an open problem. For instance, only two curvature squared invariants are currently known in the presence of a cosmological constant (gauged supergravity) in five dimensions which is dual to four-dimensional quantum field theories. In 2014, a third invariant was constructed in superspace, but its component field structure has only now been constructed. Importantly, this third invariant is key to obtain the extension of the Gauss-Bonnet term which is expected to describe the first quantum correction of compactified string theory in five dimensions. In this talk, I review aspects of our new analysis of the 5D N=1 Gauss-Bonnet term and its applications. For example, recent studies of supersymmetric black-hole entropy by quantum corrected 5D gauged supergravity were performed using only two curvature-squared invariants. The off-shell supersymmetric extension of the Gauss-Bonnet term in gauged supergravity allows one to extend these results which has relevance in studying the entropy of asymptotically AdS5 black holes.

Modular tensor categories arise as representation categories of rational conformal field theories, and in recent years have also attracted interest for their role in topological quantum computation. Given a modular tensor category, there is associated a pair of matrices $S$ and $T$ called the modular data. The $S$ and $T$ matrices generate a projective unitary representation of $SL(2,\mathbb{Z})$, and the fusion rules of the category can be recovered from the $S$ matrix via the Verlinde formula. In this talk we will discuss recent discoveries of large classes of modular data defined in terms of pairs of involutive metric groups. This is joint work with Masaki Izumi, generalizing work of Evans and Gannon.

It has been demonstrated in the literature that combining techniques from number theory and mathematical physics can produce useful and interesting, or even unexpected results for both areas. During our study of Gepner’s coupled free fermions from the coset construction and coupled free fermions constructed from scaled root lattices, we observe some q-identities, including some of Rogers-Ramanujan type, from their charaters and universal chiral partition functions, which are then rigorously proved using various techniques of q-series. These identities further motivate us to investigate more general q-identities and help reveal a connection between these two constructions.

(Super)spin projection operators have found numerous applications within the landscape of high energy physics. In particular, recent studies of these projectors in anti-de Sitter (AdS) space have revealed an innate connection to partially massless fields. This observation yields a novel method to derive the characteristics of these exotic fields. In this talk, we will explore this relationship in the context of four-dimensional N=2 AdS superspace.

In (super)conformal field theory, two- and three-point correlation functions of conserved (super)currents are important physical observables. This talk will review the recent results of arXiv:2106.14498 and 2208.07057. I will first describe a formalism aimed at deriving all constraints imposed by N=1 superconformal symmetry and conservation laws on the three-point function of higher-spin supercurrents. This formalism is then applied to constrain several new mixed three-point functions involving higher-spin supercurrents and the flavour current multiplet.

We study the geometry of confocal families on hyperboloids in pseudo-Euclidean spaces of dimension four in all signatures. The aim is to completely classify and describe them, and to prove Chasles’ theorem in this ambient. The methodology we use includes concepts of pseudo-Euclidean and Euclidean geometry, and linear algebra as well. We also give a clue about the natural characteristics confocal families possess to be applied in a billiard theory. This research is done as a part of a Ph.D. project at the University of Sydney.

We study the class of indecomposable representations of superalgebras in which a given finite dimensional representation is repeated with arbitrary multiplicity. Such “replicant” or “matryoschka” modules have dimension rD, with r composition factors equivalent to the fixed D-dimensional module. For the case of the classical Lie superalgebras sl(m/n) and osp(2/2n) and r=2, we prove by a cohomology argument that for each Kac module there is a 1-parameter family of indecomposable doubles. For general r, we provide an explicit construction of the replicant Kac modules. In conclusion, we give some illustrative examples from physics, including a possible application to family generation structure in the standard model.

Lie superalgebras attract substantial research for applications in modelling systems of fermions. For integrable systems, previous research has shown a novel method for constructing R-matrices with non-additive spectral parameters associated with the general linear Lie superalgebras gl(m|n). We explore properties of the Gaudin (super)algebra derived from a classical R-matrix related to the Lie superalgebra gl(2|1). We look in detail at the case corresponding to the representation with highest weight (0,0|alpha), where alpha is a free parameter. We also discuss the associated commuting transfer matrices.

The elliptic Painlevé is the top level equation that arises from Sakai’s classification. Recently, Noumi has given the construction of hypergeometric tau-functions for the elliptic Painlevé equation by solving the appropriate forms of the Hirota bilinear equations. Such tau-functions are defined on an infinite sequence of hyperplanes parallel to the highest root in $E_8$ and their restriction to each individual hyperplane is invariant under the action of the Weyl group of type $E_7$. In this talk, I will present an extended version of such Hirota bilinear equations, and their solutions, which involve an additional 8 independent discrete parameters taking values from the $E_8$ root lattice. Based on A.P. Kels, M. Yamazaki, Int. Math. Res. Not. 1, 110-151 2021

Topological invariance is an extremely important concept in physics. On the one hand, it leads to a classification of systems that will share similar behaviour in some regimes. On the other hand, topological properties are especially robust against perturbations. In a system, where the Hamiltonian is chiral and shows a spectral gap about the origin, one particular quantity is the winding number of the determinant of the off-diagonal block matrix. The spectrum of this off-diagonal block is complex ad shows interesting and universal behaviour. We investigated the local statistic of this winding number with the help of a Gaussian random matrix field on a one dimensional Brillouin zone. I will report on these developments in my presentation.

The most natural and efficient setting for analysing the properties of superconformal field theories is conformally compactified Minkowski superspace. Amongst the most powerful formulations of the latter are those utilising supertwistor techniques. These supertwistor methods have recently been extended to three and four dimensional AdS superspaces. In this talk I will discuss a supertwistor realisation of four dimensional N-extended AdS superspace, as well as the procedure to develop field theory on such a space using a variant of Cartan’s coset construction.

The Stäckel transform of natural Hamiltonian systems gives rise to a conformally invariant notion of second order superintegrability. Non-degenerate second order superintegrable systems have been classified on three-dimensional conformally flat spaces, but extending methods used were not easily extended to higher dimensions. In this talk, simple algebraic conditions describing these systems in arbitrary dimensions will be given. It is hoped that this formulation will lead to a similar classification of non-degenerate second order superintegrable systems in all dimensions.

The study of integrable quantum systems has recently made inroads into the field of quantum technology.

Examples include general results towards a deeper understanding of quantum circuits, and specific investigations such as simulation of the ground-state for the 1-d Heisenberg model on a quantum computer.

NOON states are “all and nothing” examples of Schroedinger-cat states. They have been well-studied over the last 20 years, for both fundamental tests of quantum theory and potential applications in quantum metrology.

In this presentation I will describe a simple protocol, designed around Hamiltonian time evolution of an integrable system and local measurement, to produce high-fidelity NOON states.

We study the 2D Ising model in a complex magnetic field in the vicinity of the Lee-Yang edge singularity. Using Baxter’s CTM method combined with analytic techniques, we obtain the scaling function together with an accurate estimate of the location of the Lee-Yang singularity. Our results are in excellent agreement with the Ising field theory calculations by Fonseca, Zamolodchikov (2001) and Zamolodchikov, Xu (2022).

It was discovered how polynomial algebras appear naturally as symmetry algebra of quantum superintegrable quantum systems. They provide insight into their degenerate spectrum, in particular for models involving Painlevé transcendents for which usual approaches of solving ODEs and PDEs cannot be applied. Those algebraic structures extend the scope of usual symmetries in context of quantum systems, but they also been connected to different areas of mathematics such as orthogonal polynomials. Among them, the well-known Racah algebra which also admit various generalisations.

I will take a different perspective on those algebraic structures which is based on Lie algebras, their related enveloping algebras, partial Casimir and commutant. I will discuss how such approach differs from using differential operator realizations and why this framework offers advantages for their classification. I will point out as well how different methods from the study of Casimir invariant of non semisimple algebras which involves solving systems of PDEs can be applied in this context and greatly facilitate making calculations.

The talk will present various explicit examples, and in particular the symmetry algebra of the generic superintegrable systems on the 2-sphere which can be understood in a purely algebraic manner using an underlying Lie algebra.

The talk is based on following works:

R Campoamor-Stursberg, I Marquette, Hidden symmetry algebra and construction of quadratic algebras of superintegrable systems, Annals of Physics 424 168378 (2021), arXiv 2020.168378

F Correa, MA del Olmo, I Marquette, J Negro, Polynomial algebras from su(3) and a quadratically superintegrable model on the two sphere, J.PhysA. Math. and Theor 54 015205 (2021) arXiv:2007.11163

R Campoamor-Stursberg, Ian Marquette, Quadratic algebras as commutants of algebraic Hamiltonians in the enveloping algebra of Schrodinger algebras, Annals of Physics. 437 168694 (2022) arXiv 2021.168694

D Latini, I Marquette, YZ Zhang, Construction of polynomial algebras from intermediate Casimir invariants of Lie algebras, J.PhysA Math. and Theor. (2022) arXiv:2204.06840

Rutwig Campoamor-Stursberg, Danilo Latini, Ian Marquette, Yao-Zhong Zhang, Algebraic (super-)integrability from commutants of subalgebras in universal enveloping algebras, arXiv:2211.04664

Work of Penrose and Rindler in the 1908s developed a formalism for spinors in relativity theory. In their work they gave geometric interpretations of 2-component spinors in terms of Minkowski space. We present some extensions of this work, involving 3-dimensional hyperbolic geometry.

We demonstrate a method for obtaining expectation values of duality observables in the asymmetric simple exclusion process. This approach is then mimicked using an integrable vertex model which allows for some generalisations of the duality observables for higher-rank exclusion processes.

Mirzakhani produced recursion relations between polynomials that give Weil-Petersson volumes of moduli spaces of hyperbolic surfaces. Stanford and Witten described an analogous construction for moduli spaces of super hyperbolic surfaces producing Mirzakhani-like recursion relations between polynomials that give super volumes. This was achieved in the so-called Neveu-Schwarz case. Both of these stories have an algebro-geometric description, and in particular this led Mirzakhani to a new proof of Witten’s conjecture on intersection numbers over the moduli space of stable curves. In this lecture, via the algebro-geometric description, I will describe what occurs in the Ramond case of the super construction. It produces deformations of the Neveu-Schwarz polynomials again satisfying Mirzakhani-like recursion relations.

Superintegrable systems are physical systems with the maximal amount of symmetry. A large portion of the literature deals with non-degenerate systems, which are the ‘nicest’ superintegrable systems. In this talk we discuss previous and current research efforts into semi-degenerate superintegrable systems, which can be considered as the ‘second-nicest’ class of superintegrable systems.

Polygonal billiard channels are examples of pseudo-chaotic dynamics, a combination of integrable evolution and sudden jumps due to conical singular points that arise from the corners of the polygon. Such pseudo-chaotic behaviour, often characterised by an algebraic separation of nearby trajectories, is believed to be linked to the wild dependence that particle transport has on the billiard geometry. We focus on a two-parameter family of right-angled parallel billiard channels having either finite or infinite horizon and consider a scattering problem defined on the periodic boundaries of an elementary cell. Through studying singular points of the billiard flow, we partition the phase space into eighteen subsets for the finite horizon and sixteen for the infinite horizon, where each subset is associated with a unique itinerary. The corresponding eighteen and sixteen branch scattering maps are presented in explicit form with natural extensions enabling the study of transport from the scattering dynamics.

Self-avoiding walks (SAW) confined in a square admit a different window on the behaviour of SAW than the usual length scaling considered. Previously the endpoints of the walks have been fixed to the corners of the square or perhaps the sides of the square. A proof that the endpoints drive only subdominant behaviour can be made using an expanded set of moves of the endpoint that arises in a Monte Carlo algorithm for Hamiltonian walks. A separate Monte Carlo method and advanced series analysis confirm and expand on this result.

We consider polynomial Lie algebras with two generators obtained from the set of symmetries of an exactly soluble Hamiltonian by the ladder operator approach. We develop the method for explicit computations of the algebra, obtain its Casimir and its spectrum. We show their realisations as differential operators and deformed oscillator algebras.

Conformal higher-spin (CHS) gravity is a rare example of a local Lagrangian theory involving bosonic fields of all spins interacting with one another. It arises as the logarithmically divergent part of the effective action associated with scalar matter coupled to background CHS fields, or, in other words, as an induced action. In this talk, I will give a brief overview of this theory, and discuss its N=1 supersymmetric generalisation which was proposed recently.

B-twisted (2,2) Hybrid models are a class of superconformal field theories with string theoretic relevance. They can be understood as Landau-Ginzburg models fibred over non-linear sigma models with on a compact Kahler manifold. The bulk theory of these hybrid models is relatively well understood. In particular, the spectra and correlators for several examples of topologically B-twisted theories are known. On the other hand, the boundary theory and the associated D-branes are still relatively unexplored. In this talk, I will introduce these hybrid models, as well as some of their key features. I will then discuss D-branes in B-twisted (2,2) Landau-Ginzburg and non-linear sigma models, as well as some results about D-branes in B-twisted (2,2) Hybrid models. In particular, I will focus on how these D-branes arise physically, as well as their categorical interpretation and relevance.

The recent and continuing success of deep learning in many real-world problems has motivated an intense effort to theoretically understand the dynamical principles of deep learning in the training and generalization of complex tasks. Although empirical data has suggested that the weights of deep neural networks acquire heavy-tailed statistics after training, most theoretical studies have based their analysis on random networks with coupling weights obeying Gaussian statistics. In this work, we investigate the phenomenon of heavy-tailed coupling weights in deep neural networks (DNNs) across fully- connected, convolutional and residual architectures. After verifying the emergence of heavy-tailed coupling across many common pretrained neural networks, we analyse the propagation of signals through DNNs with random, heavy-tailed weights using mean-field and random matrix theory. Importantly, we introduce a criterion for criticality using the entire set of Jacobian eigenvalues of random, heavy-tailed DNNs which extends the classical, edge-of-chaos notion in a consistent fashion while accounting for empirical observations on signal propagation. In this manner, we establish an extended heavy-tailed regime of criticality across those architectures which classically have a fixed critical point under Gaussian coupling. We show that this extended critical regime allows networks already initialised with random, heavy-tailed coupling to learn real-world tasks faster without fine-tuning the weight statistics. Surprisingly, our empirical simulations reveal that despite the fully-connected mean-field analysis, the predictions of our theory are largely shared by residual networks which continue to benefit from heavy-tailed initialisation from a classically chaotic regime.

We study the fate of certain discrete symmetries of spin chains under quantum group deformation of their continuous symmetry and report on the implications on the classification of symmetry-protected topological phases.

In this talk we show how to construct large classes of Lotka-Volterra ODEs in Rn with n-1 first integrals. The building blocks we use will be linear Darboux Polynomials of the ODE. In the talk these concepts will be defined, and the procedure explained.

Local superconformal symmetry has played a major role in string theory and supergravity in two dimensions (2D). In particular, the N=1 and N=2 spinning strings may be formulated as a 2D linear sigma model coupled to conformal supergravity. In this talk I will report on some recent results in constructing superspace formulations for 2D conformal (p,q) supergravity as the gauge theory of the superconformal group OSp_0(p|2;R) x OSp_0(q|2;R) and some applications of this formalism.

Galilean algebras are infinite-dimensional symmetry algebras for conformal field theories in two dimensions. One way of obtaining these algebras is through a parametric contraction of conformal symmetry algebras such as the Virasoro algebra or affine Lie algebras. However, it does not generally apply to algebras such as W-algebras, which are of great interest in the literature. We provide an introduction to the problems that arise and discuss how to give a uniform construction of Galilean W-algebras via quantum hamiltonian reduction.

A fundamental result due to M. Jimbo (1982), relates Painlevé VI to a family of affine cubic surfaces via the Riemann-Hilbert correspondence. In recent work with Nalini Joshi, a q-analog of this result was obtained, relating q-Painlevé VI to a family of affine Segre surfaces. I will explain this result and some of its consequences.

Recently, we reviewed [1] some properties of the affine Weyl group in the context of their applications to discrete integrable systems such as the discrete Painleve equations [2]. In particular, a dual representation is used to discuss translational elements of the Weyl groups. They are found to give rise to the dynamics of various discrete integrable equations.

Quantum field theory (QFT) is one of the most successful frameworks to describe a wide array of physical phenomena from particle physics to condensed matter systems. It is also the core description of models of (quantum) gravity. Despite its success, the understanding of strongly coupled, interacting QFTs remains an outstanding mathematical problem. One route to make progress is to study exactly-solvable models and deformations thereof, together with symmetries, to move within the set of QFTs. The TT deformation is an exciting tool which aids in this exploration. Defined as the determinant of the stress-energy tensor for a two-dimensional QFT, it has proven to preserve integrability, (super-)symmetries, and it has shed new light on various areas of research including: non-local QFT, string theory, and holographic (AdS/CFT) dualities. TTbar-like deformations have been proposed also in D>2 dimensions finding surprising relations with interesting effective actions, such as the Born-Infeld theory of non-linear Electrodynamics, that describe universal sectors of string theory at low-energy. A sqrt{TT} type of deformation have recently also been proven to lead to the ModMAX theory of non-linear Electrodynamics in D=4 that has attracted substantial attention in the last couple of years. This talk will summarise our results in finding theories in higher dimensions which obey a TT “like” flow equation, as well as pioneering work on understanding the new aforementioned sqrt{TT} deformation, which, in the D=2 case, has been shown to preserve classical integrability for a large class of theories. Supersymmetric extensions will also be presented.

We study a frequency downshifting in wavetrains propagating in an ocean covered by ice floes. Using the empirical model which suggests that small-amplitude surface waves in such an environment decay exponentially with the decay rate depending on frequency as ki ~ 3 [1, 2], we derive the frequency downshifting of a wavetrain within the framework of the linear theory. We show that the apparent downshifting appears due to the faster decay of high-frequency components compared to the low-frequency components with no energy flux along the spectrum. The alternative model is also considered within the framework of the nonlinear Schrödinger (NLS) equation [3] augment by the empirical dissipative terms. This model describes the propagation and decay of weakly nonlinear wavetrains and accounts for an energy flux along the spectrum down to lower frequencies. Assuming that the dissipation is relatively small compared to the nonlinear and dispersive terms in the NLS equation and using the asymptotic approach [4], we derive the frequency downshift for the decaying envelope soliton. The comparison of the downshifting obtained within the framework of linear and nonlinear models shows that in the latter case the frequency downshifting is much greater. In conclusion, estimates for the real oceanic conditions will be provided.

References

[1] Meylan M.H., Bennetts L.G., Mosig J.E.M., Rogers W.E., Doble M.J., Peter M.A. Dispersion relations, power laws, and energy loss for 845 waves in the marginal ice zone. Journal of Geophysical Research: Oceans, 2018, v. 123, 3322–3335. DOI: 10.1002/2018JC013776
[2] Squire V.A. Ocean wave interactions with sea ice: a reappraisal. Annu. Rev. Fluid Mech., 2020, v. 52, 37–60. DOI: 10.1146/annurev-fluid-830 010719-060301
[3] Slunyaev A.V., Stepanyants Y.A. Modulation property of flexural-gravity waves on a water surface covered by a compressed ice sheet. Phys. Fluids, 2022, v. 34, 077121. DOI: 10.1063/5.0100179
[4] Fabrikant A.L., Stepanyants Yu.A. Propagation of waves in shear flows. World Scientific, Singapore, 1998, 287 p. DOI: 10.1142/2557

A simplistic overview of the moduli space of the heterotic string. In a physically unrealistic subset of heterotic vacuum solutions, we have special geometry, which is essentially the space of Calabi Yau manifolds. In a more general case, we have universal geometry, a fibration, which makes understanding the general case somewhat tractable.

It is a well known fact that the general structure of two- and three-point correlation functions of primary operators is fixed up to finitely many parameters by conformal symmetry. In particular, correlation functions of conserved currents, such as the energy-momentum tensor, vector current and more generally, higher-spin currents, are of fundamental importance as they possess properties associated with spacetime and internal symmetries. Deriving the explicit form of three-point functions of conserved currents for arbitrary spin remains an open problem. In this talk we will discuss a general formalism for constructing three-point functions of conserved currents for arbitrary (super)spin in 3D (S)CFT.

Anti-de Sitter (AdS) superspaces originate as maximally symmetric solutions of supergravity theories. In three dimensions, the isometry supergroup of N-extended AdS superspace is labeled by two positive integers (p,q) with N=p+q. When working with AdS supergroups, supertwistor techniques serve as an effective tool, and have recently been used to construct three-dimensional (p,q) AdS superspace. In this talk, I will discuss this supertwistor formulation, and also present the supercoset construction of three-dimensional (p,q) AdS superspace. It is shown that this formalism is ideal for extracting geometric information which is crucial for developing field theories on these superspaces.

Central spin models are closely related to Richardson-Gaudin models and have many present and potential physical applications. Recently the XX central spin model was shown to be integrable for arbitrary spin and the eigenstates and eigenvalues found using a Bethe-Ansatz method of solution. We have shown that integrability carries over to XY models albeit restricting to spin-1/2. The conserved charges were found and a quadratic relation in these charges have been used to obtain the Bethe-Ansatz equations.

Superintegrable systems in 2D Darboux spaces were classified and it was found that there exist 12 distinct classes of superintegrable systems with quadratic integrals of motion (and quadratic symmetry algebras generated by the integrals) in the Darboux spaces. In this talk, I will explore obtaining exact solutions via purely algebraic means for the energies of all the 12 existing classes of superintegrable systems in four different 2D Darboux spaces. This is achieved by constructing the deformed oscillator realization and finite-dimensional irreducible representation of the underlying quadratic symmetry algebra generated by quadratic integrals respectively for each of the 12 superintegrable systems.

The Ising model is one of the most fundamental models in statistical physics and condensed matter. It is well-known that, from the renormalization-group theory, the upper critical dimension of the Ising model is 4, above which the critical behaviour follows mean-field theory. However, under the geometric Fortuin-Kasteleyn random-cluster representation, we argue that the Ising model simultaneously exhibits two upper critical dimensions: 4 and 6. In this talk, we will show strong numerical evidence to support this argument.