Tag - Conformal field theories

Fabienne Chouraqui: Connections between the Yang-Baxter equation and Thompson’s group F

22nd June, 2023

The quantum Yang-Baxter equation is an equation in mathematical physics and it lies in the foundation of the theory of quantum groups. One of the fundamental problems is to find all the solutions of this equation. Drinfeld suggested the study of a particular class of solutions, derived from the so-called set-theoretic solutions. A set-theoretic solution of the Yang-Baxter equation is a pair (X,r), where X is a set and

r : X ⨯ X → X ⨯ X     r(x,y)=(σx(y),γy(x))

is a bijective map satisfying r12r23r12 = r23r12r23, where r12 = r ⨯ IdX and r23 = IdX ⨯ r. We define non-degenerate involutive partial solutions as a generalization of non-degenerate involutive set-theoretical solutions of the quantum Yang-Baxter equation (QYBE). The induced operator is not a classical solution of the QYBE, but a braiding operator as in conformal field theory. We define the structure inverse monoid of a non-degenerate involutive partial solution and prove that if the partial solution is square-free, then it embeds into the restricted product of a commutative inverse monoid and an inverse symmetric monoid. Furthermore, we show that there is a connection between partial solutions and the Thompson's group F. This raises the question of whether there are further connections between partial solutions and Thompson's groups in general.

Dario Benedetti: Old and new conformal field theories at large N

15th June, 2023

The 1/N expansion is a well established approach to studying interacting fixed points of the renormalization group, and the associated conformal field theories. In this talk, I will review old and new results on the conformal limit of the O(N) (vector) and O(N)3 (tensor) models at large N.

Sabine Harribey: Extraordinary Interfaces and Boundaries in (4 − ε)-dimensional O(N) models

14th June, 2023

The critical O(N) models are one of the most thoroughly studied classes of conformal field theories (CFTs) in three dimensions. Indeed, there exists a variety of approximation methods applicable to them, such as epsilon expansion, large N expansion or conformal bootstrap. It is then of interest to study the 3-dimensional O(N) models with interfaces (co-dimension 1 defects), as well as on spaces with boundaries. In particular, we are interested in "extraordinary" critical interfaces and boundaries which break the O(N) symmetry to O(N −1). Such models were studied recently with a D-dimensional bulk and surface defects quadratic in the fields. In this talk, we will adopt a different approach and study defects that are always of co-dimension 1. More precisely, I will present the renormalization group analysis for a quartic O(N) model in 4 − ε dimensions with cubic interactions on an interface. For sufficiently large N, we find stable IR fixed points with purely imaginary cubic couplings. I will also review the special case N = 1 corresponding to a boundary Yang-Lee model.

Scott Sheffield: An Introduction to Random Surfaces

28th November, 2022

The theory of 'random surfaces' has emerged in recent decades as a significant field of mathematics, lying somehow at the interface between geometry, probability, and mathematical physics. I will give a friendly (I hope) colloquium-level overview of the subject with lots of pictures. Topics will include random planar maps (interpreted as discrete random surfaces), Liouville quantum gravity surfaces, conformal field theory. and the random fractal curves produced from the Schramm-Loewner evolution.  Many of these topics are motivated by physics (statistical physics, string theory, quantum field theory, etc.) but they also have simple mathematical definitions that can be understood without a lot of physics background.

Katrin Wendland: How do quarter BPS states cease being BPS?

8th June, 2022

The so-called BPS states in a conformal field theory with extended supersymmetry are key when assigning a geometric interpretation to the theory. Standard invariants for such theories arise from a net count of BPS, half or quarter BPS states, according to the ℤ2 grading into ‘bosons' and ‘fermions'. This allows for boson-fermion pairs of states to cease being BPS under deformation of the theory. The talk will give a review of this phenomenon, arguing that it is ubiquitous in theories with geometric interpretation by a K3 surface. For a particular type of deformations, we propose that the process is channelled by the action of SU(2) on an appropriate subspace of the space of states.

Rajesh Gopkumar: Deriving Gauge-String Duality

1st February, 2022

Gauge (or Yang-Mills) theories are the building blocks of our current physical understanding of the universe. In parallel, string theory is a framework for a consistent quantum description of gravity. Gauge-String duality a.k.a. the AdS/CFT correspondence is a remarkable connection between these two very different classes of theories. This has, in fact, been one of the main engines driving progress in theoretical physics over the last two decades. I will begin by discussing why it is important to arrive at a first principles understanding of the underlying mechanism of this duality relating quantum field theories and string theories (or other theories of gravity). I will then proceed to discuss a very general approach which aims to relate large N QFTs and string theories, starting from free field theories. This corresponds to a tensionless limit of the dual string theory on AdS spacetime. Finally, I will discuss specific cases of this limit for AdS3/CFT2 and AdS5/CFT4, where one has begun to carry this programme through to fruition, going from the string theory to the field theory and vice versa.

Yuly Billig: Towards Kac-van de Leur Conjecture

2nd July, 2020

Superconformal algebras are graded Lie superalgebras of growth 1, containing a Virasoro subalgebra. They play an important role in Conformal Field Theory. In 1988 Kac and van de Leur made a conjectural list of simple superconformal algebras, which since has been amended with an exceptional superalgebra CK(6). It has been proposed to use conformal superalgebras to attack this conjecture, and Fattori and Kac established a classification of finite simple conformal superalgebras. It still needs to be proved that one can associate a finite conformal superalgebra to each simple superconformal algebra. In this talk we will show how to use the results of Billig-Futorny to prove that every simple superconformal algebra is polynomial, which implies that one can attach to it an affine conformal superalgebra. We will discuss the difference between finite and affine conformal algebras. We also introduce quasi-Poisson algebras and show how to use them to construct known simple superconformal algebras. Quasi-Poisson algebras may be viewed as a refinement of the notion of Novikov algebras. Quasi-Poisson algebras may be used for computations of automorphisms and twisted forms of superconformal algebras.

Yujiro Kawamata: Birational geometry and derived categories

30th April, 2017

I will talk about the recent progress on the DK conjecture connecting birational geometry and the
derived categories, and related conjectures such as DL conjecture, etc. I will also discuss two kinds of
factorizations of birational maps; those into flips, flops and divisorial contractions according to the
minimal model programme, and more traditional factorizations into blow-ups and blow-downs with
smooth centres.

Si Li: Vertex algebras, quantum master equation and mirror symmetry

30th April, 2017

We develop the effective Batalin-Vilkovisky quantization theory for chiral deformation of 2-dimensional conformal field theories. We establish an exact correspondence between renormalized quantum master equations for effective functionals and Maurer-Cartan equations for chiral vertex operators. As an application, we explain a universal approach to KdV type integrable hierarchies via B-twisted topological string field theory. This leads to an exact solution of quantum B-model (BCOV theory) in complex one dimension that solves the higher genus mirror symmetry conjecture on elliptic curves.