Minkowski superspaces and superstrings as almost real-complex supermanifolds

We introduce a wide category of superspaces, called locally finitely generated, which properly includes supermanifolds but enjoys much stronger permanence properties, as are prompted by applications. Namely, it is closed under taking finite fibre products (i.e. is finitely complete) and thickenings by spectra of Weil superalgebras. Nevertheless, in this category, morphisms with values in a supermanifold are still given in terms of coordinates. This framework gives a natural notion of relative supermanifolds over a locally finitely generated base. Moreover, the existence of inner homs, whose source is the spectrum of a Weil superalgebra, is established; they are generalisations of the Weil functors defined for smooth manifolds.Comment: v3: final version prior to publication, examples added, corrections, numbering adapted, 47 pages, Math. Z. (2014

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“…Then the complex structure on the odd part of T X defines a non-involutive distribution, as observed in Ref. [3].…”

Section: Relative Supermanifoldsmentioning

confidence: 83%

Singular superspaces

2014

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“…It is the maximal transitive prolongation of the consistently Z-graded Lie superalgebra m = m −2 ⊕ m −1 , where m −2 = (C 5 ) * and m −1 = Π(Λ 2 (C 5 )), with bracket given by [α, β] = ı α∧β vol, for any α, β ∈ m −1 . The subalgebra g 0 is gl (5) acting in the obvious way on m; in particular, h 0 = 0.…”

Section: Preliminary Resultsmentioning

confidence: 99%

“…Our motivation to study super-Poincaré structures relies on the interesting fact that supergravity theories admit, besides the traditional "component formalism" formulations (see [14,29]), more geometric presentations in terms of super-Poincaré structures (M, D) (see [5,16,26,30,31,32,34,35]).…”

Section: Introductionmentioning

confidence: 99%

Classification of maximal transitive prolongations of super-Poincaré algebras

Altomani

Santi

2014

Advances in Mathematics

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Let V be a complex vector space with a non-degenerate symmetric bilinear form and S an irreducible module over the Clifford algebra Cℓ(V ) determined by this form. A supertranslation algebra is a Z-graded Lie superalgebra m = m−2 ⊕m−1, where m−2 = V and m−1 = S⊕· · ·⊕S is the direct sum of an arbitrary number N ≥ 1 of copies of S, whose bracket. We consider the maximal transitive prolongations in the sense of Tanaka of supertranslation algebras. We prove that they are finitedimensional for dim V ≥ 3 and classify them in terms of super-Poincaré algebras and appropriate Z-gradings of simple Lie superalgebras.the Clifford algebra determined by this form, with its natural Z 2 -grading. We adopt the conventions used in [24], in particular, the product in Cℓ(V ) satisfies vu + uv = −2(v, u)1 for any v, u ∈ V and the natural inclusion of the orthogonal Lie algebra in the Clifford algebra is given by the mapwhere v ∧ u is the anti-symmetric endomorphism (v, ·)u − (u, ·)v. Note that some authors prefer to use different conventions, cf. Deligne's lectures [12].

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“…The Lie superalgebra is of dimension n|n, therefore it admits an odd involution. Then, all Lie supergroups of dimension n|n can be equipped with an odd involution, i.e., the module of vector fields is Π-symmetric in the language of Manin [29] and others [5].…”

Section: On the Existence Of Odd Connectionsmentioning

confidence: 99%

Odd connections on supermanifolds: existence and relation with affine connections

Bruce¹,

Grabowski²

2020

J. Phys. A: Math. Theor.

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The notion of an odd quasi-connection on a supermanifold, which is loosely an affine connection that carries non-zero Grassmann parity, is examined. Their torsion and curvature are defined, however, in general, they are not tensors. A special class of such generalised connections, referred to as odd connections in this paper, have torsion and curvature tensors. Part of the structure is an odd involution of the tangent bundle of the supermanifold and this puts drastic restrictions on the supermanifolds that admit odd connections. In particular, they must have equal number of even and odd dimensions. Amongst other results, we show that an odd connection is defined, up to an odd tensor field of type (1, 2), by an affine connection and an odd endomorphism of the tangent bundle. Thus, the theory of odd connections and affine connections are not completely separate theories. As an example relevant to physics, it is shown that N = 1 super-Minkowski spacetime admits a natural odd connection.

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Minkowski superspaces and superstrings as almost real-complex supermanifolds

Cited by 8 publications

References 17 publications

Singular superspaces

Singular superspaces

Classification of maximal transitive prolongations of super-Poincaré algebras

Odd connections on supermanifolds: existence and relation with affine connections

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