The structure of Fedosov supermanifolds

Monterde, J.; Muñoz-Masqué, J.; Vallejo, José Antonio

doi:10.1016/j.geomphys.2009.02.001

Cited by 4 publications

(8 citation statements)

References 15 publications

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“…Thus, for example, starting only from a usual symplectic formω ∈ Ω 2 (M ), one could construct an odd symplectic form ω on the geometric supermanifold (M, Ω(M ) (taking K 1 : T M → T * M as the inverse Poisson map associated tõ ω). The results in [50] provide a way to construct also a symplectic connection ∇ ∇ fromω, and the resulting odd symplectic scalar curvature is solely determined in terms of geometric objects defined on M . These ideas may be useful to study the geometric meaning of the odd symplectic scalar curvature and its relationship to the physical interpretation presented in [5].…”

Section: The Geometry Of Fedosov Supermanifoldsmentioning

confidence: 99%

“…The possibility of constructing a formal deformation quantization on a supermanifold through the Fedosov formalism, has led in a natural way to the study of the geometric properties of Fedosov supermanifolds. References on this topic are [30,44,1,50]. Given a Fedosov supermanifold ((M, E), ∇ ∇, ω), these properties are encoded in the symplectic curvature tensor…”

Section: The Geometry Of Fedosov Supermanifoldsmentioning

confidence: 99%

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Symplectic connections and Fedosov's quantization on supermanifolds

Vallejo

2012

J. Phys.: Conf. Ser.

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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.

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Section: The Geometry Of Fedosov Supermanifoldsmentioning

confidence: 99%

Section: The Geometry Of Fedosov Supermanifoldsmentioning

confidence: 99%

Symplectic connections and Fedosov's quantization on supermanifolds

Vallejo

2012

J. Phys.: Conf. Ser.

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“…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, [[⋅, ⋅]]:…”

Section: Whenmentioning

confidence: 99%

Symplectic scalar curvature on supermanifolds

Hernández-Amador,

Vallejo,

Vorobiev

2020

Preprint

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We study the notion of symplectic scalar curvature on the supermanifold over an ordinary Fedosov manifold whose structural sheaf is that of differential forms. In this purely geometric context, we introduce two families of odd super-Fedosov structures, the first one is very general and uses a graded symmetric connection, leading to a vanishing odd symplectic scalar curvature, while the second one is based on a graded non-symmetric connection and has a non-trivial odd symplectic scalar curvature. As a simple example of the second case, we determine that curvature when the base Fedosov manifold is the torus.

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“…In particular, we will need the analog of the Levi-Cività theorem concerning the existence of superconnections such that ∇ω = 0 for a supersymplectic form ω, and also their corresponding structure theorem. We follow here the approach in [14], although with some differences, the main one being that we do not assume that ∇ ∇ is adapted to the splitting H (also, see The definition of torsion and curvature also mimics the non-graded case:…”

Section: Fedosov Supermanifoldsmentioning

confidence: 99%

“…[14] Let ∇ be a linear connection on M . A superconnection ∇ ∇ on (M, Ω(M )) is symmetric if and only if…”

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confidence: 99%