spin geometry

string geometry

bundles

# Contents

## Definition

A spinor bundle on a smooth manifold with spin structure is a $\rho$-associated bundle associated to the spin group-principal bundle lifting the tangent bundle, for $\rho : \mathbf{B} Spin \to$ Vect a representation of the spin group.

A section of a spinor bundle is called a spinor.

A Dirac operator acts on sections of a spinor bundle.

In physics, sections of spinor bundles model matter particles: fermion. See spinors in Yang-Mills theory.

standard model of particle physics and cosmology

| | gravity | electroweak and strong nuclear force | fermionic matter | scalar field | | field content: | vielbein field $e$ | principal connection $\nabla$ | spinor $\psi$ | scalar field $H$ | | Lagrangian: | scalar curvature density | field strength squared | Dirac operator component density | field strength squared + potential density | | $L =$ | $R(e) vol(e) +$ | $\langle F_\nabla \wedge \star_e F_\nabla\rangle +$ | $(\psi , D_{(e,\nabla)} \psi) vol(e) +$ | $\nabla \bar H \wedge \star_e \nabla H + \left(\lambda {\vert H\vert}^4 - \mu^2 {\vert H\vert}^2 \right) vol(e)$ |

## References

Revised on October 15, 2013 19:48:26 by Urs Schreiber (80.237.234.132)