Gauge theories and Courant algebroids

• Grant number: 24/06658-2
• Support Opportunities: Scholarships in Brazil - Doctorate
• Start date: September 01, 2024
• End date: August 31, 2028
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Henrique Nogueira de Sá Earp
• Grantee: Agnaldo Alessandro da Silva Junior
• Host Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil
• Associated research grant: 21/04065-6 - BRIDGES: Brazil-France interplays in Gauge Theory, extremal structures and stability, AP.TEM

Abstract

This doctoral program delves into the intricate realm of coupled instanton equations, exploring their manifestations in metric, spinor, three-form, and connection fields over spin manifolds. Notably, special solutions emerge in dimensions $6$ and $7$, originating from the Hull--Strominger and heterotic $\mathrm{G}_2$ systems, respectively. Motivated by recent advancements in theoretical physics, these equations undergo a transformative analysis through the lens of generalized geometry. Building upon the candidate's prior research during their master's degree, which investigated the relationship between coupled instantons, generalized Ricci-flat metrics, and Killing spinors on Courant algebroids, this study embarks on a natural continuation, addressing two open questions concerning the intricate interplay of these geometric conditions.Anticipating affirmative responses buoyed by recent breakthroughs in both physics and mathematics literature, particularly in the domain of Calabi--Yau manifolds, the candidate has already made significant strides by providing a complete solution to these problems for $\mathrm{G}_2$-structures with torsion coupled to $\mathrm{G}_2$-instantons. Furthermore, the candidate sets their sights on delving deeper into these concepts, particularly in the unexplored terrain of $Sp(k)Sp(1)$-structures, an area still nascent in the existing literature.