Symplectic connections and Fedosov's quantization on supermanifolds

Abstract

A (biased and incomplete) review of the status of the theory of symplectic connections on supermanifolds is presented. Also, some comments regarding Fedosov's technique of quantization are made. system, it can be formally expressed, by using the inverse Fourier transform, aŝWeyl then defines the operator corresponding to f as the W f obtained by substituting q, p forq,p in the formula (1), soand this acts on a function u ∈ L 2 (R n ) through its kernel, giving 1 the result:Let us remark that W f acts on functions defined on R n (the configuration space) and not on the full phase-space R 2n . The mathematical details justifying these formal manipulations can be found in [28,31]. What we want to remark now, is the fact that the correspondence f → W f is C−linear, but the space of classical observables is not just a vector space, it also has the structure of a commutative algebra, with the point-wise product of functions. The space of self-adjoint operators such as W f , when endowed with the composition of operators, also has an algebra structure, but this time a non-commutative one. To describe Quantum Mechanics in phase space following the initial motivation, one should be able to establish also a morphism between these two algebras, but this can not be done without modifying the commutative product of classical observables. Thus, the problem was to find a non-commutative product ⋆, on the algebra of functions on R 2n , such that the operator corresponding to f ⋆ g is precisely W f • W g . Clearly, this can be done by definingThe expression for the inverse operator W −1 was found by Wigner short after the ideas of Weyl were published, in 1932 (see [74]). However, the formulae found by Wigner (in terms of a trace operator) did not allow for a direct physical interpretation, so J. E. Moyal undertook the task of finding that interpretation for f ⋆ g. He (and, independently, H. Groenewold) discovered what nowadays we call the Moyal product (see [33,54]). In modern terminology, Moyal realized that the star product can be written in terms of the Poisson bi-vector determined by the classical bracket on C ∞ (R 2n ),After some algebraic manipulations, the use in the Baker-Campbell-Hausdorff formula of the commutator p d dq , iηq· = iηpI, and the fact that e i∂q is the generator of translations in q.

Symplectic scalar curvature on supermanifolds

“…In this setting, a graded connection is a mapping ∇ ∇ ∶ Der ) is a supermanifold, the triple (( , Γ ⋀ ), , ∇ ∇) is a Fedosov supermanifold when ∇ ∇ = 0. In dealing with symplectic supercurvatures, as in the non-graded case, particular attention will be paid to compatible connections, that is, to Fedosov supermanifolds [1,13,16]. The Riemann supercurvature of a graded connection ∇ ∇ can be defined as usual, with the aid of the graded canonical commutator of endomorphisms of superalgebras, [[⋅, ⋅]]:…”

“…The (super) differential 1−forms on ( , Ω( )) are defined as the duals Der * Ω( ), and −forms are defined by taking exterior products as usual, and noting that they are bigraded objects; if, for instance, ∈ Ω 2 ( , Ω( )) (that is the way of denoting the space of 2−superforms, sometimes we will use a notation such as Ω 2 ( )), its action on two supervector fields , ′ ∈ Der Ω( ) will be denoted ⟨ , ′ ; ⟩, a notation well adapted to the fact that Der Ω( ) is considered here as a left Ω( )−module and Ω 2 ( , Ω( )) as a right one. Other objects such as the graded exterior differential can be defined as in the classical setting, but taking into account the ℤ 2 −degree (see [16] for details). Thus, if ∈ Ω 0 ( , Ω( )), its graded differential is given by ⟨ ; ⟩ = ( ), and if ∈ Ω 1 ( , Ω( )), we have a 2−form ∈ Ω 2 ( , Ω( )) whose action is given by…”

Supermanifolds, Symplectic Geometry and Curvature

“…The (super) differential 1−forms on (M, Ω(M )) are defined as the duals Der * Ω(M ), and k−forms are defined by taking exterior products as usual, and noting that they are bigraded objects; if, for instance, ω ∈ Ω 2 (M, Ω(M )) (that is the way of denoting the space of 2−superforms), its action on two supervector fields D, D ′ ∈ Der Ω(M ) will be denoted D, D ′ ; ω , a notation well adapted to the fact that Der Ω(M ) is considered here as a left Ω(M )−module and Ω 2 (M, Ω(M )) as a right one. Other objects such as the graded exterior differential can be defined as in the classical setting, but taking into account the Z 2 −degree (for details in the spirit of this paper, see [16]). Thus, if α ∈ Ω 0 (M, Ω(M )), its graded differential d is given by D; dα = D(α), and if β ∈ Ω 1 (M, Ω(M )), we have a 2−form dβ ∈ Ω 2 (M, Ω(M )) whose action is given by…”