Conformal Geometry of the Supercotangent and Spinor Bundles

Abstract

We study the actions of local conformal vector elds X ∈ conf(M, g) on the spinor bundle of (M, g) and on its classical counterpart: the supercotangent bundle M of (M, g). We rst deal with the classical framework and determine the Hamiltonian lift of conf(M, g) to M. We then perform the geometric quantization of the supercotangent bundle of (M, g), which constructs the spinor bundle as the quantum representation space. The Kosmann Lie derivative of spinors is obtained by quantization of the comoment map. The quantum and classical actions of conf(M, g) turn, respectively, the space of di erential operators acting on spinor densities and the space of their symbols into conf(M, g)-modules. They are ltered and admit a common associated graded module. In the conformally at case, the latter helps us determine the conformal invariants of both conf(M, g)-modules, in particular the conformally odd powers of the Dirac operator.Keywords: onform l geometryD symple ti superm nifoldD spin geometryD geometri qu nE tiz tionD onform lly inv ri nt di'erenti l oper torsF

Conformally Equivariant Quantization for Spinning Particles

“…In addition, the obtained symplectic structure on M corresponds to the one coming from the Lagrangian of a free pseudo-classical spinning particle on (M, ě) [39,27], so that the graded Poisson algebra O(M) is a classical counterpart to D(M, S). This is confirmed by the geometric quantization scheme, which associates the Hilbert space of square integrable spinors to the supercotangent bundle M [34].…”

“…The g-module of Hamiltonian symbols S ν . In contradistinction with the cotangent bundle case, the natural Lift (2.11) of Vect(M ) by Lie derivatives does not lead to Hamiltonian vector fields on M. Besides, the preservation of the potential 1-form α, see (2.2), is not a strong enough condition to determine a unique lift of X ∈ Vect(M ) to M. In [34], we ask in addition for preservation of the direction of the 1-form β = ě ij ξ i dx j . Both conditions can be satisfied only for vector fields X ∈ g, and fix a unique lift…”

“…In order to keep a self-contained exposition, this section presents a recollection of our previous work [34]. Namely, we introduce a family of o(p + 1, q + 1)-module structures on the space of spinor differential operators, denoted by (D λ,µ ) λ,µ∈R , and on its two spaces of symbols, denoted by (S ν ) ν∈R and (T ν ) ν∈R .…”

“…This supermanifold is precisely the phase space introduced by Berezin and Marinov [5] to deal with pseudo-classical spinning particles on M = R n . Besides, the author proved in [34] the existence of a Hamiltonian filtration on D(M, S), assigning order 2 to spinor covariant derivatives ∇ X , with X ∈ Vect(M ), and order 1 to the Clifford elements γ(ξ), with ξ ∈ Ω 1 (M ). This filtration is compatible with the commutator of D(M, S) and induces a super Poisson bracket and a new gradation on O(M).…”

An Approach to Classical Quantum Field Theory Based on the Geometry of Locally Conformally Flat Space-Time

“…There is large literature on the action of conformal vector fields on spinor bundles (e.g., [20,21]) and conformally invariant differential operators on Minkowski space [22]. Our work is different to this work; we do not construct differential operators with respect to given vector fields on the manifold (Lie derivatives).…”