Supersymmetry Meets Random Matrices

Computing observables in quantum field theory in the regime when the particles interact strongly represents a fundamental challenge in theoretical physics. Supersymmetric quantum field theories, which are characterised by interactions that preserve a symmetry relating bosons (integer spin) and fermions (half-integer spin), offer a powerful training ground for developing the computational tools that are required to address this problem.

A team consisting of Gregory Korchemsky (Institut de Physique Théorique, CNRS) and two Hungarian collaborators, Zoltan Bajnok and Bercel Bordis (*HUN-REN Wigner Research Center for Physics*), has recently developed a new, general method to systematically compute a special class of important observables in strongly coupled four-dimensional supersymmetric gauge theories. Their findings has recently been published in the prestigious journal *Physics Review Letters* as Editors' Suggestion [1].

A distinguished feature of these observables is that, in the limit of a large gauge group rank (planar limit) and for any 't Hooft coupling, they can be expressed as determinants of certain semi-infinite matrices. Crucially, the same determinants have previously appeared in the context of random matrix theory, where they were computed exactly in terms of a well-known probability distribution known as the *"Tracy-Widom distribution"* (or a more general version of it).

This distribution is a powerful tool for analyzing a wide range of complex systems in physics (quantum chaos, directed polymers, surface growth, turbulence, etc.), revealing the underlying connections between seemingly disparate physical phenomena.

The Tracy-Widom distribution proves surprisingly versatile in supersymmetric gauge theories. It describes observables like the free energy on a sphere and correlations of specific operators and even more applications are anticipated. The team's findings provide a systematic treatment and new insights into the strong coupling regime of four-dimensional supersymmetric gauge theories, inviting further investigations in related theoretical frameworks. The technique developed by the authors is quite general and can be applied to determine the asymptotic behavior of a general class of determinants involving the so-called *"Bessel operators"*, this opening a new avenue for many physical and mathematical applications.

[1] Zoltan Bajnok, Bercel Boldis, and Gregory P. Korchemsky. *Tracy-Widom distribution in four- dimensional supersymmetric Yang-Mills theories. *Phys. Rev. Lett. **133**, 031601. https://doi.org/10.1103/PhysRevLett.133.031601

R. Guida, 2024-07-04 15:21:00