Spin Geometry and Some Applications

TL;DR

This paper reviews spin geometry, its mathematical foundations, and diverse applications in physics such as supersymmetry, supergravity, and condensed matter, highlighting new methods for analyzing spinor equations and topological phases.

Contribution

It provides a comprehensive overview of spin geometry concepts and introduces novel techniques for constructing symmetry operators and analyzing topological insulators.

Findings

Construction of symmetry operators from spinor bilinears

Method for deriving harmonic spinors from twistor spinors

Summary of the topological insulators classification via Clifford algebra

Abstract

In this review, basic definitions of spin geometry are given and some of its applications to supersymmetry, supergravity and condensed matter physics are summarized. Clifford algebras and spinors are defined and the first-order differential operators on spinors which lead to the definitions of twistor and Killing spinors are discussed. Holonomy classification for manifolds admitting parallel and Killing spinors are given. Killing-Yano and conformal Killing-Yano forms resulting from the spinor bilinears of Killing and twistor spinors are introduced and the symmetry operators of special spinor equations are constructed in terms of them. Spinor bilinears and symmetry operators are used for constructing the extended superalgebras from twistor and Killing spinors. A method to obtain harmonic spinors from twistor spinors and potential forms is given and its implications on finding solutions of the Seiberg-Witten equations are discussed. Supergravity Killing spinors defined in bosonic supergravity theories are considered and possible Lie algebra structures satisfied by their spinor bilinears are examined. Spin raising and lowering operators for massless field equations with different spins are constructed and the case for Rarita-Schwinger fields is investigated. The derivation of the periodic table of topological insulators and superconductors in terms of Clifford chessboard and index of Dirac operators is summarized.