Tag - Yang-Mills theory

Sourav Chatterjee: Mass Generation by the Higgs Mechanism at All Couplings

20th November, 2023

The Higgs mechanism is a part of the Standard Model of quantum mechanics that allows certain kinds of particles to have non-zero mass. In spite of its great importance, there is no rigorous proof that the Higgs mechanism can indeed generate mass in situations that are relevant for the Standard Model. In technical terms, this corresponds to the "coupling parameter" of the model being small, and the "gauge group" being non-abelian (the most important cases are SU(2) or SU(3)). I will present the first rigorous proof in this direction, showing that SU(2) lattice Yang-Mills theory coupled to a Higgs field transforming in the fundamental representation of SU(2) has a mass gap at any value of the coupling parameter, provided that the interaction of the Higgs field with the gauge field is strong enough. No background is needed.

Martin Hairer: Stochastic Quantisation of Yang-Mills

4th October, 2023

We report on recent progress on the problem of building a stochastic process that admits the hypothetical Yang-Mills measure as its invariant measure. One interesting feature of our construction is that it preserves gauge-covariance in the limit even though it is broken by our UV regularisation.

Tomoyuki Arakawa: Hilbert Schemes of the points in the plane and quasi-lisse vertex superalgebras

29th August, 2023

For each complex reflection group Γ one can attach a canonical symplectic singularity ℳΓ. Motivated by the 4D/2D duality discovered by Beem et al., Bonetti, Menegheli and Rastelli conjectured the existence of a supersymmetric vertex operator algebra WΓ whose associated variety is isomorphic to ℳΓ. We prove this conjecture when the complex reflection group Γ is the symmetric group SN, by constructing a sheaf of ℏ-adic vertex algebras on the Hilbert schemes of N points in the plane. In physical terms, the vertex operator algebra WSN corresponds, by the 4D/2D duality, to the 4-dimensional N=4 super Yang-Mills theory with gauge group SLN.

Ilya Chevyrev: Invariant measure and universality of the 2D Yang-Mills Langevin dynamic, II

13th June, 2023

In this talk, I will present a recent work on the invariance of the 2D Yang-Mills measure for its Langevin dynamic. The Langevin dynamic both in 2D and 3D had previously been constructed in joint work with Chandra-Hairer-Shen, but it was an open problem to show the existence of an invariant measure even in 2D. In establishing this invariance, we follow Bourgain’s invariant measure argument by taking lattice approximations, but with several twists. An important one, which I will focus on, is that the approximating invariant measures require gauge-fixing, which we achieve by developing a rough version of Uhlenbeck compactness combined with rough path estimates of random walks. I will also present several corollaries of our main result, including a representation of the YM measure as a perturbation of the Gaussian free field, and a new universality result for its discrete approximations.

Hao Shen: Invariant measure and universality of the 2D Yang-Mills Langevin dynamic, I

13th June, 2023

In an earlier work with Chandra, Chevyrev and Hairer, we constructed the local solution to the stochastic Yang-Mills equation on 2D torus, which was shown to have gauge covariance property and thus induces a Markov process on a singular space of gauge equivalent classes. In this talk, we discuss a more recent work with Chevyrev, where we consider the Langevin dynamics of a large class of lattice gauge theories on 2D torus, and prove that these discrete dynamics all converge to the same limiting dynamic. A novel step in the argument is a geometric way to identify the limit using Wilson loops. This universality of the dynamics is crucial for obtaining a sequence of important results for 2D Yang-Mills, including for instance the invariance of the 2D Yang-Mills measure for its Langevin dynamic, which will be discussed by Ilya Chevyrev.

Martin Hairer: The role of symmetry in renormalisation

13th June, 2023

There are several interesting situations where the solutions to singular SPDEs exhibit a symmetry at a formal level that could in principle be broken by the renormalization procedure required to define them. We’ll discuss a relatively simple argument showing that, in many cases, the renormalization can be chosen in such a way that the symmetry does indeed hold and we’ll apply it to the stochastic quantization of the 3D Yang-Mills theory.

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.