Fifty-Seventh Meeting of the North British Mathematical Physics Seminar

Aspects of RG flows of four-dimensional N=2 theories will be discussed. Particular emphasis will be given on the effect on the spectrum of BPS states, and on predicting Maruyoshi-Song types of RG flows.

The study of chiral CFTs in 2 dimensions through axiomatic approaches such as Vertex Operator Algebras (VOAs) and Conformal Nets has lead to the understanding of a great wealth of qualitative features, for example, the prevalence of the Virasoro algebra. My work (joint with Kasia Rejzner and Benoit Vicedo) has been focussed on trying to obtain such qualitative features within the framework of Perturbative Algebraic Quantum Field Theory (pAQFT), a mathematically rigorous formulation of perturbation theory in QFT. In this talk, I will explain how it is possible to obtain generators of the Virasoro algebra from the free scalar field on a cylindrical spacetime, without recourse to Wick rotation, complexification of spacetime coordinates or compactification of null rays.

Over half a century ago, Bacry and Lévy-Leblond asked “What are the possible kinematics”? In their pioneering work they introduced the notion of a kinematical Lie algebra in four-dimensional spacetime and gave a classification. Some of their conditions were relaxed by Bacry and Nuyts twenty years later arriving at a classification of kinematical Lie algebras. They observed that each such Lie algebra acts transitively on some four-dimensional homogeneous spacetime and arrived at 11 classes of kinematical spacetimes. In this talk I will summarise recent work with Stefan Prohazka and Ross Grassie, where we refine and extend this pioneering work to arrive at a classification of simply-connected homogeneous kinematical spacetimes in arbitrary dimension, determine their (often infinite-dimensional) Lie algebras of symmetries and, in the case of four-dimensional spacetimes, we classify their N=1 superspaces.

The elliptic genus of K3 surfaces encrypts an intriguing connection between the sporadic group Mathieu 24 and non-linear sigma models on K3, dubbed `Mathieu Moonshine'. By restricting to Kummer K3 surfaces, which may be constructed as Z2 orbifolds of complex 2-tori with blown up singularities, and in work done in collaboration with Katrin Wendland, it has been possible to devise a framework in which our concept of symmetry surfing can be explored and tested in a concrete way.

This talk focusses on what has been learned so far that supports the symmetry surfing idea when lifting the Kummer construction to the level of conformal field theory, with particular emphasis on quarter BPS states. Some of these states enter the elliptic genus with opposite signs thus cancelling each other when counted by t his index, yet they carry interesting information that should help understand Mathieu Moonshine.

A four-dimensional abelian gauge theory can be coupled to a 3d CFT with a U(1) symmetry living on a boundary. This coupling gives rise to a continuous family of boundary conformal field theories (BCFTs) parametrized by the gauge coupling \tau and by the choice of the CFT in the decoupling limit. Upon performing an Electric-Magnetic duality in the bulk and going to the decoupling limit in the new frame, one finds a different 3d CFT on the boundary, related to the original one by Witten's SL(2, Z) action. In particular the cusps on the real \tau axis correspond to the 3d gauging of the original CFT. We study general properties of this family of BCFTs. We show how to express bulk one and two-point functions, and the hemisphere free-energy, in terms of the two-point functions of the boundary electric and magnetic currents. Finally, upon assuming particle-vortex duality (and its fermionic version), we show how to turn this machinery into a powerful computational tool to study 3d gauge theories.

The quantized Coulomb branch of 3d N=4 U(N) theories can be studied by computing the correlators of monopole operators that are inserted in an Omega background. In this talk, I will describe the computation of such correlators by using a brane realisation from type IIB string theory. In particular, I will consider an example of a correlator involving the product of monopoles of minimal positive and negative charges. This is based on 1910.01650 with Benjamin Assel and Stefano Cremonesi.