Dualities and Categorical Structures from 2D Up

Abstract

String Theory is the most promising candidate unifying theory of fundamental interactions so far; however, the Standard Model (SM) still features many open questions. The present work aims at providing a step further towards reconciling the two, analysing part of the richness that underlying mathematical structures and dualities are able to provide in, both, gravitating systems and Quantum Field Theories (QFTs) alike. In doing so, our approach will be of the top-down kind. In particular, we will be relying upon the key tools of holographic duality and categorical algebraic geometry. The use of the former is justified by the lack of a non-perturbative formulation of String Theory, whereas the latter is dictated by the great advancement there has been in the past decades in studying algebraic varieties associated to moduli spaces, specifically Higgs and Coulomb branches. A fundamental step towards studying string theory vacua, and, ultimately their stability, is that of understanding the underlying mathematical structure of the QFT resulting from its dimensional reduction on Calabi-Yau (CY) manifolds, the latter being complex manifolds admitting a category theory description. In particular, the work of Kapustin, Rozansky and Saulina (KRS) has shown how this can be achieved in terms of a 3D TFT equipped with a 2-categorical structure. Our analysis develops in two main directions, namely on the gravitational, and supersym metric quiver gauge theory side. In both cases, our treatment focuses on lower-dimensional structures necessitating extensions and generalisations of well-established dualities and correspondences, specifically, holographic duality, homological mirror symmetry, and 3D mirror symmetry. As we shall see, the common ground in between the two paths taken in this treatment is the role played by amplitutdes in studying fundamental interactions and the properties of the vacuum structure, as well as the role played by dualities in understanding analytic results.