I will describe some recent progress on the correspondence between four-dimensional ${\cal N=2}$ superconformal field theories (SCFTs) and two-dimensional vertex operator algebras (VOAs). In particular I will introduce the notion of the “Higgs scheme”, an extension by nilpotent elements of the standard Higgs variety of an ${\cal N=2}$ SCFT, which plays a natural role in the associated VOA. Unlike the Higgs variety, theHiggs scheme appears to be a perfect invariant, i.e. it conjecturally fully characterizes the SCFT.

In the past decades, String Theory has emerged as the prime candidate for quantum unification of electromagnetic, nuclear, and weak forces with gravitational ones It has shed light on important fundamental questions of theoretical physics, such as the microscopic structure of black holes and the geometric origin of particle physics. We review these developments, such as the introduction of extended objects - Dirichlet branes - and highlight the important geometric role these objects play in deriving the Standard Model of particle physics and the microscopic structure of black holes. We also highlight progress made in deriving particle physics from F-theory, a geometric domain of string theory at finite string coupling, and recent systematic exploration of the landscape of the quadrillion Standard Models with three families of quarks and leptons.

We consider matter density effects in theories with a false ground state. Large and dense systems, such as stars, can destabilize a metastable minimum and allow for the formation of bubbles of the true minimum. Interestingly these bubbles are not necessarily confined to the dense region, but can escape to infinity. This leads to a phase transition in the universe after the formation of stars, and therefore has significant impact on e.g. solutions to the electroweak hierarchy problem based on dynamical selection of the electroweak vacuum. We work out some of the phenomenological consequences of such density triggered late phase transitions and put new constraints on the parameter space of some benchmark relaxion models.

Gravitational waves will be an important probe of physics beyond the Standard Model, as they would be produced at possible first order phase transitions in the early universe. I will discuss the characteristic spectrum of gravitational radiation from phase transitions, how it is connected to underlying physics, and prospects for probing physics beyond the Standard Model at the future space-based gravitational wave detector LISA.
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