Mathematical Physics Group

Program: The one-workshop includes lectures and seminars as detailed below.

• 10.25 - 10.30 Opening

• 10.30 - 11.15 Curved translation principle in generalised conformal calculus (Slovak)

• 11.15 - 11.30 Break

• 11.30 - 12.15 Atiyah sequences of braided Lie algebras and their splittings (Aschieri)

• 12.20 - 12.50 Not all non-metricities are born equal: when symmetric metric-affine gravity meets Finsler (Voicu)

• 12.50 - 14.30 Lunch Break

• 14.30 - 15.15 Dual Lagrangian Theories (Ferraris)

• 15.20 - 16.05 Cauchy problems in the Holst model: a canonical analysis in covariant Lagrangian field theories (Fatibene)

• 16.05 - 16.20 Break

• 16.20 - 16.40 Geometric aspects of Lepage equivalents and related conservation laws (Zanello)

• 16.45 - 17.10 Separation of variables and superintegrability on Riemannian coverings: classical and quantum (Rastelli)

Program: The workshop revolves around far reaching generalizations of the classical concept of Lie algebgras, i.e. quantum groups, Hopf algebras etc., and their applications to diverse areas of mathematical physics, especially String Theory and Quantum Integrability. The workshop includes introductory lectures and seminars on the latest developments.

Lectures:

Hall algebras and Hopf Algebras (Francesco Sala)

Abstract: The goal of this 2-hour talk is to provide a brief introduction to the theory of Cohomological Hall Algebras (CoHAs) from the perspective of Hopf algebras. I will focus in particular on the example of the quantum toroidal algebra of gl(1), presenting its definition via generators and relations, and explaining its realization as a K-theoretic Hall algebra. Along the way, I will describe a geometric representation of this algebra in terms of the equivariant K-theory of Hilbert schemes of points on the affine plane, and outline its connection to quantum W-algebras.

In the final part of the talk, I will place this example in a broader context, discussing the theory of two-dimensional Cohomological Hall algebras of quivers and the associated Maulik–Okounkov Yangians, as well as the two-dimensional K-theoretic Hall algebras of quivers and the Okounkov–Smirnov quantum loop algebras.

No prior knowledge of Hall algebras or their refined versions (cohomological, K-theoretic, etc.) will be assumed. However, some familiarity with the basic notions of Hopf algebras may be helpful.

Quantum Toroidal Symmetries in Gauge and String Theory (Jean-Emile Bourgine)

Abstract: The following topics will be covered.

1. Quantum toroidal symmetries: What is a quantum toroidal algebra? Applications in mathematics and physics.

2. Quantum toroidal gl(1): Definition, representations (vector, Fock/horizontal & vertical, MacMahon), vertex operators. 

3. (p,q)-brane web and algebraic engineering: 5d N=1 gauge theories, (p,q)-brane realization, correspondence with webs of representations. 

4. 5d AGT correspondence: From free field representations to q-deformed W-algebras, role of Miki's automorphism. 

5. Orbifolds and Bethe/gauge correspondence: Surface defect partition functions, quantum toroidal gl(N) with a cyclic shift.

Seminars:

Infinitesimal braidings and quasitriangular bialgebras (Lucrezia Bottegoni)

Abstract: In this talk we present an infinitesimal version of a braided monoidal category, that we call braided pre-Cartier, whose corresponding infinitesimal braiding is a first-order deformation of the given braiding. This extends the (symmetric) Cartier categories, where an additional commutativity of the infinitesimal braiding with the involutive braiding is required. It is known that the monoidal category of modules over a bialgebra is braided precisely if the bialgebra is quasitriangular, i.e. it is equipped with a so-called universal R-matrix which is a solution of the quantum Yang-Baxter equation. We consider the pre-Cartier structure for the categories of modules over a quasitriangular bialgebra and we characterize the resulting infinitesimal R-matrices on the bialgebra. Among their main properties, infinitesimal R-matrices are 2-cocycles in Hochschild cohomology and they satisfy an infinitesimal quantum Yang-Baxter equation. We provide explicit examples of infinitesimal R-matrices on some known Hopf algebras, such as on the Hopf algebras E(n) which are a generalization of the Sweedler’s Hopf algebra E(1).

Free field representations of quantum groups and q-deformed W-algebras through cluster algebras (Jean-Emile Bourgine)

Abstract: Following the development of the AGT correspondence, new relations between free field representations of quantum groups and W-algebras were obtained. The simplest one is the homomorphism between the level (N,0) horizontal representation of the quantum toroidal gl(1) algebra and (dressed) q-deformed W(N) algebras. In this talk, I will explain how to extend this type of relations to the Wakimoto representations of quantum affine sl(N) algebras using the 'surface defect' deformation of the quantum toroidal sl(N) algebra.

COHAs of coherent sheaves on smooth surfaces and affine Yangians (Francesco Sala)

Abstract: The aim of this talk is to present a (conjecturally) new half of the affine Yangian of type ADE, which arises naturally from the theory of Cohomological Hall Algebras (COHAs) associated to coherent sheaves on a minimal resolution X of a Kleinian singularity. The construction uses the derived McKay correspondence and the variation of t-structures to establish a connection between the COHA of the affine ADE quiver and the COHA of the surface X.

Spiralling branes, R-matrices and elliptic integrable systems (Yegor Zenkevich)

Abstract: I will explain how R-matrices of quantum toroidal algebras arise from a certain brane setup in Type IIB string theory. String theory implies relations fo the R-matrices and eventually determines their explicit form. I will then show that such R-matrices lead to a new compact formulation of quantum trigonometric and elliptic Ruijsenaars-Schneider type integrable systems. They will also provide a link to some new elliptic integrable systems recently proposed by Koroteev and Shakirov.

Schedule:

Day 1

• 10.00 - 11.00 Lecture (Sala)

• 11.00 - 11.30 Coffee Break

• 11.30 - 12.30 Lecture (Sala)

• 12.30 - 14.30 Lunch Break

• 14.30 - 15.15 Seminar (Bottegoni)

• 15.15 - 15.45 Coffee Break

• 15.45 - 16.30 Seminar (Sala)

Day 2

• 10.00 - 11.00 Lecture (Bourgine)

• 11.00 - 11.30 Coffee Break

• 11.30 - 12.30 Lecture (Bourgine)

• 12.30 - 14.30 Lunch Break

• 14.30 - 15.15 Seminar (Zenkevich)

• 15.15 - 15.45 Coffee Break

• 15.45 - 16.30 Seminar (Bourgine)

Program: TBA

Program: I am going to review recent progress in the mathematical understanding of the low-energy properties of dilute quantum systems. In particular, I am going to present a rigorous version of Bogoliubov theory and I am going to show how it can be used to obtain precise estimates for the ground state energy and the low-energy excitation spectrum and also to approximate the time evolution of Bose gases in the so-called Gross-Pitaevskii regime.

Program: These are two introductory lectures on the broad notion of generalized symmetries and their applications to Quantum Field Theory. The covered topics include:

1. Non-invertible topological lines in (1+1)d Ising CFT and Kramers-Wannier duality;

2. Selection rules and physical consequences;

3. High-form global symmetries;

4. Non-invertible symmetries in (3+1)d QFT, including their relation to the Adler-Bell-Jackiw anomaly.

Ref.s:

- Sub-topics 1 and 2: [arXiv:1802.04445], [arXiv:2308.00747], [arXiv:2307.02534], [arXiv:2401.12281]

- Sub-topics 3 and 4: [arXiv:1412.5148], [arXiv:2204.09025], [arXiv:2205.05086], [arXiv:2205.01104]