Graded Manifolds with Spin-Conformal Structure

Details Graded Manifolds with Spin-Conformal Structure Graded Manifolds with Spin-Conformal Structure

Jan 1994

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1994phdt........34k&link_type=abstract

Thesis (PH.D.)--UNIVERSITY OF MARYLAND COLLEGE PARK, 1994.Source: Dissertation Abstracts International, Volume: 55-10, Section:

Physics

Manifolds

Scientific paper

A (Kostant) graded manifold is roughly a smooth manifold M along with a (Z/2Z-graded commutative) sheaf of algebras {cal A} which replaces the usual sheaf of algebras {cal C}infty of locally defined smooth functions on M. This new sheaf of algebras admits a homomorphism ~ onto {cal C}infty which is locally split. Derivations of { cal A}>=neralize the notion of vector field on M; {cal D}er({cal A}) itself is Z/2Z-graded; and { cal D}er({cal A}) itself admits a ~ map. Motivated by physics, we desire that the odd derivations have some spinor-like properties. For M an orientable Lorentzian manifold of dimension 2n, we accomplish this by introducing the notion of an almost spin-conformal structure on the graded manifold. This is an odd (complex) distribution {cal S} (i.e. graded sub-bundle of { cal D}er({cal A}) otimes C) of rank 2^{n -1} with certain conditions placed on the Lie bracket of elements therein: specifically the graded Lie bracket of elements in {cal S} with elements in ={cal S}, taken modulo {cal S } otimes ={cal S}, is a map from {cal S} otimes ={cal S} to { cal D}er({cal A}) otimes C, and we require that when the ~ functor is applied this map, fiber-wise, mimics the half-spinor representation pairing Delta^+ otimes overline{Delta ^+} to V for the representations Delta^+ and V of Spin(2n) and SO(2n) respectively. We explore certain differential -geometric consequences for M of such a structure, in particular the conformal structure it naturally imposes. While the obstruction to integrability is a natural object to use in constructing various tensors on (M, {cal A}), we show that many four-dimensional non -conformally flat M admit what we call spin-conformal structures (i.e. an integrable almost spin-conformal structures).

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