Supersymmetry for Mathematicians: An Introduction

Varadarajan, V. S.

doi:10.1090/cln/011

Cited by 297 publications

(433 citation statements)

References 3 publications

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“…It is shown that this scheme can be applied to the N = 2 supersymmetry theory [47] provided one must take care of anticommutative properties of fermions. In this work we reexamine the proof of N = 2 supersymmetric KdV equation using Grozman prescription.…”

Section: Motivation Results and Planmentioning

confidence: 99%

Virasoro Action on Pseudo-Differential Symbols and (Noncommutative) Supersymmetric Peakon Type Integrable Systems

Guha

2008

Acta Appl Math

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Using Grozman's formalism of invariant differential operators we demonstrate the derivation of N = 2 Camassa-Holm equation from the action of Vect(S 1|2 ) on the space of pseudo-differential symbols. We also use generalized logarithmic 2-cocycles to derive N = 2 super KdV equations. We show this method is equally effective to derive CamassaHolm family of equations and these system of equations can also be interpreted as geodesic flows on the Bott-Virasoro group with respect to right invariant H 1 -metric. In the second half of the paper we focus on the derivations of the fermionic extension of a new peakon type systems. This new one-parameter family of N = 1 super peakon type equations, known as N = 1 super b-field equations, are derived from the action of Vect(S 1|1 ) on tensor densities of arbitrary weights. Finally, using the formal Moyal deformed action of Vect(S 1|1 ) on the space of Pseudo-differential symbols to derive the noncommutative analogues of N = 1 super b-field equations.

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Section: Motivation Results and Planmentioning

confidence: 99%

Virasoro Action on Pseudo-Differential Symbols and (Noncommutative) Supersymmetric Peakon Type Integrable Systems

Guha

2008

Acta Appl Math

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“…It is known then -see for instance [22], §7.2, with the few, obvious changes needed to take into account the relations of type z 2 − z 2 = 0 (that are superfluous in the setting therein) -that one has splitting(s) of k-supermodules (actually, even of k-supercoalgebras)…”

Section: The Representation G P −−→ Gl(v )mentioning

confidence: 99%

Global splittings and super Harish-Chandra pairs for affine supergroups

Gavarini

2015

Trans. Amer. Math. Soc.

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This paper dwells upon two aspects of affine supergroup theory, investigating the links among them. First, I discuss the "splitting" properties of affine supergroups, i.e. special kinds of factorizations they may admit -either globally, or point-wise. Almost everything should be more or less known, but seems to be not as clear in literature (to the author's knowledge) as it ought to.Second, I present a new contribution to the study of affine supergroups by means of super HarishChandra pairs (a method already introduced by Koszul, and later extended by other authors). Namely, Iprovide a new functorial construction Ψ which, with each super Harish-Chandra pair, associates an affine supergroup that is always globally strongly split (in short, gs-split) -thus setting a link with the first part of the paper. One knows that there exists a natural functor Φ from affine supergroups to super Harish-Chandra pairs: then I show that the new functor Ψ -which goes the other way round -is indeed a quasi-inverse to Φ , provided we restrict our attention to the subcategory of affine supergroups that are gs-split. Therefore, (the restrictions of) Φ and Ψ are equivalences between the categories of gs-split affine supergroups and of super Harish-Chandra pairs. Such a result was known in other contexts, such as the smooth differential or the complex analytic one, via different approaches (see [16], [19], [7]): nevertheless, the novelty in the present paper lies in that I construct a different functor Ψ and thus extend the result to a much larger setup, with a totally different, more geometrical method. In fact, this method (very concrete, indeed) is universal and characteristic-free: I present it here for the algebro-geometric setting, but actually it can be easily adapted to the frameworks of differential or complex-analytic supergeometry.The case of linear supergroups is treated also as an intermediate, inspiring step. Some examples, applications and further generalizations are presented at the end of the paper. 1

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“…Of course, there is an extensive literature on Supergeometry and related topics and it is impossible to give complete references; let us at least mention [BBHR91], [Ber79], [Ber87], [DeW84], [Kos75], [Rog07], [Tuy04]. The sources that had the highest impact on the present text are: [Lei80], [Rog07], [Tuy04], [Lei11], [Var04], [Man02], [DM99], [CCF11], [DSB03], [Vor12], [BP12], [GKP09], [GKP10], [Sch97].…”

Section: Introductionmentioning

confidence: 99%

The category of Z2n-supermanifolds

Covolo

Grabowski

Poncin

2016

Journal of Mathematical Physics

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In Physics and in Mathematics Z n 2 -gradings, n ≥ 2, appear in various fields. The corresponding sign rule is determined by the 'scalar product' of the involved Z n 2 -degrees. The Z n 2 -Supergeometry exhibits challenging differences with the classical one: nonzero degree even coordinates are not nilpotent, and even (resp., odd) coordinates do not necessarily commute (resp., anticommute) pairwise. In this article we develop the foundations of the theory: we define Z n 2 -supermanifolds and provide examples in the ringed space and coordinate settings. We thus show that formal series are the appropriate substitute for nilpotency. Moreover, the class of Z • 2 -supermanifolds is closed with respect to the tangent and cotangent functors. We explain that any n-fold vector bundle has a canonical 'superization' to a Z n 2 -supermanifold and prove that the fundamental theorem describing supermorphisms in terms of coordinates can be extended to the Z n 2 -context.

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Supersymmetry for Mathematicians: An Introduction

Cited by 297 publications

References 3 publications

Virasoro Action on Pseudo-Differential Symbols and (Noncommutative) Supersymmetric Peakon Type Integrable Systems

Virasoro Action on Pseudo-Differential Symbols and (Noncommutative) Supersymmetric Peakon Type Integrable Systems

Global splittings and super Harish-Chandra pairs for affine supergroups

The category of Z2n-supermanifolds

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