Bhoja, Niren
(2024)
Aspects of Spinorial G-Structures.
PhD thesis, University of Nottingham.
Abstract
There is a remarkable isomorphism between a pair (metric, spinor), and a certain collection of differential forms on a manifold $M$. This isomorphism holds for numerous examples from four dimensions to eight dimensions.
In this thesis, we aim to understand reductions of the spin-frame bundle to various orbits of spinors on a spin manifold, called spinorial $G$-structures, and then develop a schematic for exploring the most ``natural'' second-order partial differential equation on the collection of differential forms arising from a pair (metric, spinor).
The first part of the thesis deals with real and complexified stabilisers of Weyl spinors. Understanding real stabilisers is done through the study of pure spinors, the real index, and spinor bilinears. These three tools generate a large class of spinorial $G$-structures, for which we construct a new class called mixed structures. We also apply this machinery to investigate previously unexplored stabilisers in ten and twelve dimensions. Regarding complexified stabilisers of Weyl spinors, we develop an elementary approach using $k$-simplices and combinatorics. The novelty of this simpler scheme is two-fold. First, the study of the stabiliser of the spinor is shifted to studying the group that leaves the collection of differential forms, arising from the metric and spinor, invariant. Second, this method allows one to examine complexified stabilisers in arbitrarily high dimensions, which classical methods do not allow.
The second part of the thesis explores the most natural second-order partial differential equations on a pair (metric, spinor). We are inspired by Plebanski's theory of gravity in 4 dimensions. He constructs an action functional, which when extremised, results in Einstein conditions on the curvature. In this thesis, we construct all families of diffeomorphism invariant action functionals in the examples of $\text{SU}(2)$- and $\text{SU}(3)$-structures. From our methods, we recover Plebanski's most natural second-order partial differential equations (PDEs) on the collection of differential forms coming from the metric and spinor in the case of $\text{SU}(2)$-structures. Furthermore, we conjecture a schematic for natural second-order PDEs on the collection of differential forms coming from the metric and spinor in the case of $\text{SU}(3)$-structures. In both examples, we show that the linearised action functionals are completely determined by representation theory, and that there is a family of Einstein-Hilbert actions of gravity in a vacuum.
| Item Type: |
Thesis (University of Nottingham only)
(PhD)
|
| Supervisors: |
Krasnov, Kirill |
| Keywords: |
spinor, stabiliser, orbit, g-structure, differential form, gravity, einstein-hilbert, functional, plebanski, octonions, mathematical physics |
| Subjects: |
Q Science > QA Mathematics > QA440 Geometry
Q Science > QA Mathematics > QA611 Topology |
| Faculties/Schools: |
UK Campuses > Faculty of Science > School of Mathematical Sciences |
| Item ID: |
80011 |
| Depositing User: |
Bhoja, Niren
|
| Date Deposited: |
13 Dec 2024 04:40 |
| Last Modified: |
13 Dec 2024 04:40 |
| URI: |
https://eprints.nottingham.ac.uk/id/eprint/80011 |
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