Responsable : Alessandra Occelli

Le séminaire a lieu usuellement le mardi à 15h30.**Séminaires à venir**

Four-dimensional Wess-Zumino-Witten (4dWZW) models are analogous to the two dimensional WZW models and possesses aspects of conformal field theory and twistor theory [Losev-Moore-Nekrasov-Shatashvili, Inami-Kanno-Ueno-Xiong,...]. Equation of motion of the 4dWZW model is the Yang equation which is equivalent to the anti-self-dual Yang-Mills (ASDYM) equation. It is well known as the Ward conjecture that the ASDYM equations can be reduced to many classical integrable systems, such as the KdV eq. and Toda eq. [Ward, Mason-Woodhouse,...]. On the other hand, 4d Chern-Simons (CS) theory has connections to many solvable models such as spin chains and principal chiral models [Costello-Witten-Yamazaki, ...]. These two theories (4dCS and 4dWZW) have been derived from a 6dCS theory like a double fibration [Costello, Bittleston-Skinner]. This suggests a nontrivial duality correspondence between the 4dWZW model and the 4dCS theory. We note that the Ward conjecture holds mostly in the split signature (+,+,-,-) and then the 4dWZW model describes the open N=2 string theory in the four-dimensional space-time. Hence a unified theory of integrable systems (6dCS --> 4dCS/4dWZW) can be proposed in the split signature. In this talk, I would like to discuss the soliton/instanton solutions of the 4dWZW model. We calculate the 4dWZW action density of the soliton solutions and found that the solutions behaves as the KP-type solitons, that is, the one-soliton solution has localized action (energy) density on a 3d hyperplane in 4-dimensions (soliton wall) and the N-soliton solution describes N intersecting soliton walls with phase shifts. Our solutions would describe a new-type of physical objects in the N=2 string theory. If time permits, I would mention reduction to lower-dimensions and extension to noncommutative spaces. This talk is based on our works: [arXiv:2212.11800, 2106.01353, 2004.09248, 2004.01718] and forthcoming papers.

**Séminaires passés**

I am going to describe a class of integrable systems on Poisson cluster varieties, which is based on combinatorial construction due to Goncharov and Kenyon and has alternative formulation on Poisson submanifolds in Lie groups. Deautonomization of these cluster integrable systems leads to non-autonomous systems of q-Painleve type, whose solutions are constructed using 5d supersymmetric gauge theories.

We provide sufficient conditions such that the time evolution of a mesoscopic tight-binding open system with a local Hartree-Fock non-linearity converges to a self-consistent non-equilibrium steady state, which is independent of the initial condition from the "small sample". We also show that the steady charge current intensities are given by Landauer-Büttiker-like formulas, and make the connection with the case of weakly self-interacting many-body systems. This is a joint work with Horia D. Cornean.

Recently Diaconis and Ethier[1] discussed the following game with three players. The players 1, 2 and 3 start to play this game with some respectively positive capitals A,B and C dollars. At each round a pair of players is chosen uniformly at random and then a fair coin is tossed to decide who win in this round. After that one dollar is transferred from one of players in this pair to another. Players are eliminated when their capital becomes zero. The game ends when only one player is left. The question of interest is the probability of winning this game for player 1 when A and B are fixed and total capital N=A+B+C increases to infinity. We use the Brownian approximation for random walks in cones to find exact asymptotics for this probability. The talk is based on a joint work with V. Wachtel [2]. [1] Diaconis and Ethier (2022). Gambler’s ruin and ICM. Statistical Science, 37(3), 289-305, https://doi.org/10.1214/21-STS826. [2] Denisov and Wachtel (2024). Harmonic measure in a multidimensional gambler’s problem. Accepted by Ann. Appl. Probab. https://arxiv.org/abs/2212.11526.

Random matrix models appear naturally in several areas. Big data techniques, logarithm gas and quantum chaos are some examples of systems that can be modelled by eigenvalues in random matrix ensembles. Consequently, the asymptotic behaviour of such problems can be achieved by studying the eigenvalues in random matrix models. Aiming to study the asymptotic behaviour of such eigenvalues, the present seminar is divided into two parts. Firstly, we present some applications and standard techniques in RMT. Next, we present two asymptotic results: the first one is related to the Painlevé XXXIV equation and the other one aims at some asymptotics for logarithm-type integrals. Acknowledgements: I would like to thank Sao Paulo Research Foundation (FAPESP) for the financial support to the project, grant #2021/10819-3.

First I report on recent numerical experiments with time-dependent tree tensor network algorithms for the approximation of quantum spin systems. I will then describe the basics in the design of time integration methods that are robust to the usual presence of small singular values, that have good structure-preserving properties (norm, energy conservation or dissipation), that allow for rank (= bond dimension) adaptivity and also have parallelism among the nodes. This discussion of basic concepts will be done for the smallest possible type of tensor network differential equations, namely low-rank matrix differential equations. Once this simplest case is understood, there is a systematic path to the extension of the integrators and their favourable properties to general tree tensor networks.