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Symmetries in exceptional holonomy problems

Grant number: 21/07249-0
Support type:Scholarships in Brazil - Post-Doctorate
Effective date (Start): November 01, 2021
Effective date (End): October 31, 2023
Field of knowledge:Physical Sciences and Mathematics - Mathematics - Geometry and Topology
Principal researcher:Henrique Nogueira de Sá Earp
Grantee:Udhav Fowdar
Home Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil
Associated research grant:18/21391-1 - Gauge theory and algebraic geometry, AP.TEM

Abstract

G2 and Spin (7) manifolds are special classes of Einstein manifolds occurring in dimensions 7 and 8 respectively. Aside from being mathematically interesting, they are also of interest to physicists as they occur in M- and F-theories: generalisations of supersymmetric string theories. This project is broadly concerned with studying problems pertaining to these so-called exceptional holonomy manifolds in the presence of continuous group actions. Since exceptional holonomy manifolds are Ricci-flat, the hypothesis that they admit Killing vector fields implies that we must consider non-compact examples. A key advantage that the non-compact setting has over the compact one is that one can often expect to find explicit solutions. More specifically the main problems that we want to address about these manifolds are as follows: 1. Given certain torus actions on a G2 or Spin (7) manifold, what are the geometric properties of the quotient space? Can one characterise/classify all such examples? Can one construct new examples of G2 and Spin (7) metrics starting from suitable data on the quotient space?; 2. Calibrated submanifolds are special classes of minimal submanifolds introduced by Harvey-Lawson. In G2 manifolds these are known as associatives and co-associatives, and in Spin (7) manifolds they are called Cayley submanifolds. We want to construct examples of such objects which are invariant under certain G action. In particular, one of our main goal is to construct explicit examples of Cayley fibrations; 3. G2 and Spin (7) instantons are higher dimensional generalisations of ASD instantons. These are conjectured to play an important role in defining topological invariants just like ASD instantons on 4-manifolds. We want to construct explicit examples of such instantons which are invariant under certain torus actions and study their properties; 4. The G2 Laplacian flow is a geometric flow introduced by Robert Bryant as a way of potentially deforming a closed G2-structure to a torsion free one. It can be viewed as an analogue of the Kähler Ricci flow for G2 manifolds. We want to investigate a cohomogeneity one version of the flow on the spinor bundle of the 3-sphere (which is known to admit a 1-parameter family of G2 metrics). (AU)

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