Talks

Principal bundles and their moduli have been important in various aspects of physics and geometry for many decades. It is perhaps not so well-known that a substantial portion of the original motivation for studying them came from number theory, namely the study of Diophantine equations. I will describe a bit of this history and some recent developments.

I discuss a geometric interpretation of the twisted indexes of 3d (softly broken) $\cN=4$ gauge theories on $S^1 \times \Sigma$ where $\Sigma$ is a closed genus $g$ Riemann surface, mainly focussing on quivers with unitary gauge groups. The path integral localises to a moduli space of solutions to generalised vortex equations on $\Sigma$, which can be understood algebraically as quasi-maps to the Higgs branch. I demonstrate that the twisted indexes computed in previous work reproduce the virtual Euler characteristic of the moduli spaces of twisted quasi-maps. I investigate 3d $\cN=4$ mirror symmetry in this context, which implies an equality of enumerative invariants associated to mirror pairs of Higgs branches under the exchange of equivariant and degree counting parameters. I will conclude with some remarks about how holomorphic Morse theory can be used to access the spaces of supersymmetric ground states in limits where $\cN=4$ supersymmetry is fully restored. These spaces of ground states may be related to the spaces of conformal blocks for the VOAs introduced by Costello and Gaiotto.

The discovery of a Higgs like boson at LHC in 2012 was a development of fundamental importance in particle physics. With this newly discovered state included in theoretical predictions of scattering events,  the Standard Model can be extrapolated  to much higher experimental energies. This implies that a degree of scale separation will be present between the discovered particles of the Standard Model, and any new particles required to address the outstanding mysteries that the Standard Model leaves unanswered. As a result, interest in treating the Standard Model as a consistent Effective Field Theory (the SMEFT) has dramatically increased. This powerful theory is the state of the art in studying experimental results model independently, up to energies that LHC probes in a robust fashion experimentally. Rather remarkably, as interest in this theory has increased in recent years, a number of subtleties has also been discovered in the physics of the SMEFT.  In this talk, I will review this rapidly advancing area.