68th North British Mathematical Physics Seminar

Abstract: In this talk, I will demonstrate how discrete cluster integrable systems can provide a comprehensive understanding of the stable BPS particle spectrum in five-dimensional theories on a circle. These theories emerge from the geometric engineering of M-theory on toric Calabi-Yau threefolds. While their BPS spectrum is typically "wild", with an infinite number of states of arbitrarily high spin, specific algebraic solutions of the integrable systems indicate the existence of "tame" chambers. In these chambers, the BPS spectrum can be expressed in closed form and comprises vector multiplets, infinite towers of hypermultiplets, and Kaluza-Klein states. By utilizing wall-crossing formulas, it becomes possible to obtain the spectrum in any other chamber. To illustrate this methodology, I will examine the concrete example of local del Pezzo threefolds, geometrically engineering SU(2) Super Yang-Mills with matter, and their corresponding q-Painleve' cluster integrable system. Additionally, time permitting, I will briefly discuss the interpretation of the aforementioned results through the lens of WKB approximation for q-difference equations.

Abstract: In the functional approach to constructing (classical and quantum) field theories, the space of observables is modelled using smooth functions on an infinite-dimensional space of configurations (also called fields). By using functionals, this method seeks to go beyond perturbative treatments, in which usually only polynomials in the fields or local functionals are used.
The algebra of quantum observables is constructed using a *-product of functionals. Because of the distributional nature of our functionals however, this is only defined if one restricts their singular structure. The usual approach is to use microcausal functionals.
We have been investigating the time-slice axiom in the functional approach. It is straightforward to prove that it is satisfied with arbitrary smooth functionals, but the proof does not work with microcausal functionals, because this class is not closed under integration.
We discuss how this problem can be remedied by sharpening the requirements on the singular structure of the microcausal functionals. We will define in this talk the class of equicausal functionals. With these functionals, we can prove the time-slice axiom and show that the *-product closes on this subspace, giving it the structure of a differential graded algebra resolving the algebra of on-shell functionals.

Abstract: The aim of this talk is to demonstrate how finite group gauge theories lead to non-invertible symmetries in dimensions greater than two, based on two recent works of the same name by T. Bartsch, M. Bullimore, A. E. V. Ferrari and myself. I will begin by introducing the fact that symmetries of quantum field theories correspond to topological defects, and that the most general mathematical structure for describing symmetry is that of a (higher) fusion category. Starting with the well-understood case of two dimensions, I will recap how topological lines of a theory with finite gauge group are Wilson lines and so the symmetry of the theory is characterised by a category of representations. In the second half of this talk I will discuss our recent generalisation of this picture to three dimensions, introducing higher dimensional analogues of Wilson lines supported on surfaces as an intuitive stepping stone to higher representation theory.

Abstract: Recently, with my collaborator Stefano Negro and my PhD student Fabio Sailis, we have released a series of three preprints, where we propose a form factor program for integrable quantum field theories perturbed by irrelevant operators. We see the development of this program as a natural further step in the development of the bootstrap program for these theories, following from the computation of an exact S-matrix and its study by thermodynamic Bethe ansatz. In our work we have shown that, similar as for the S-matrix, exact matrix elements of operators in generalised $T\bar{T}$-perturbed theories can be computed systematically and that they factorise into those of the unperturbed theory and a new factor depending on the perturbation couplings. In this talk I will give a summary of our main findings and an indication of the problems that this approach may allow us to tackle.

Abstract: Several classical relativity theorems, including the famous singularity theorems, have in their assumptions pointwise energy conditions. Those conditions bound the energy density (or similar quantities) on every spacetime point and are easily violated by quantum fields. One way to examine the applicability of those theorems in semiclassical gravity is to replace them with an averaged version, where the energy density is bounded on a segment of a causal geodesic. The index form method, used instead of the Raychaudhuri equation, provides a direct way of using those weakened conditions. In this talk I will apply this method to the Penrose singularity theorem and the Hawking area theorem and present progress and challenges for both.