This is the first of two papers in which we construct the Hodge dual for supermanifolds by means of the Grassmannian Fourier transform of superforms. In this paper we introduce the fundamental ...concepts and a method for computing Hodge duals in simple cases. We refer to a subsequent publication 12 for a more general approach and the required mathematical details. In the case of supermanifolds it is known that superforms are not sufficient to construct a consistent integration theory and that integral forms are needed. They are distribution-like forms which can be integrated on supermanifolds as a top form can be integrated on a conventional manifold. In our construction of the Hodge dual of superforms they arise naturally. The compatibility between Hodge duality and supersymmetry is exploited and applied to several examples. We define the irreducible representations of supersymmetry in terms of integral and super forms in a new way which can be easily generalized to several models in different dimensions. The construction of supersymmetric actions based on the Hodge duality is presented and new supersymmetric actions with higher derivative terms are found. These terms are required by the invertibility of the Hodge operator.
The integral form of supergravity Castellani, L.; Catenacci, R.; Grassi, P. A.
The journal of high energy physics,
10/2016, Letnik:
2016, Številka:
10
Journal Article
Recenzirano
Odprti dostop
A
bstract
By using integral forms we derive the superspace action of
D
= 3,
N
= 1 supergravity as an integral on a supermanifold. The construction is based on target space picture changing operators, ...here playing the rôle of Poincaré duals to the lower-dimensional spacetime surfaces embedded into the supermanifold. We show how the group geometrical action based on the group manifold approach interpolates between the superspace and the component supergravity actions, thus providing another proof of their equivalence.
Supergravity actions with integral forms Castellani, L.; Catenacci, R.; Grassi, P.A.
Nuclear physics. B,
December 2014, 2014-12-00, 2014-12-01, Letnik:
889, Številka:
C
Journal Article
Recenzirano
Odprti dostop
Integral forms provide a natural and powerful tool for the construction of supergravity actions. They are generalizations of usual differential forms and are needed for a consistent theory of ...integration on supermanifolds. The group geometrical approach to supergravity and its variational principle are reformulated and clarified in this language. Central in our analysis is the Poincaré dual of a bosonic manifold embedded into a supermanifold. Finally, using integral forms we provide a proof of Gates' so-called “Ectoplasmic Integration Theorem”, relating superfield actions to component actions.
A
bstract
We reconstruct the action of
N
= 1
, D
= 4 Wess-Zumino and
N
= 1
,
2
, D
= 4 super-Yang-Mills theories, using integral top forms on the supermanifold
ℳ
4
4
. Choosing different Picture ...Changing Operators, we show the equivalence of their rheonomic and superspace actions. The corresponding supergeometry and integration theory are discussed in detail. This formalism is an efficient tool for building supersymmetric models in a geometrical framework.
Hodge dualities on supermanifolds Castellani, L.; Catenacci, R.; Grassi, P.A.
Nuclear physics. B,
October 2015, 2015-10-00, 2015-10-01, Letnik:
899, Številka:
C
Journal Article
Recenzirano
Odprti dostop
We discuss the cohomology of superforms and integral forms from a new perspective based on a recently proposed Hodge dual operator. We show how the superspace constraints (a.k.a. rheonomic ...parametrization) are translated from the space of superforms Ω(p|0) to the space of integral forms Ω(p|m) where 0≤p≤n, n is the bosonic dimension of the supermanifold and m its fermionic dimension. We dwell on the relation between supermanifolds with non-trivial curvature and Ramond–Ramond fields, for which the Laplace–Beltrami differential, constructed with our Hodge dual, is an essential ingredient. We discuss the definition of Picture Lowering and Picture Raising Operators (acting on the space of superforms and on the space of integral forms) and their relation with the cohomology. We construct non-abelian curvatures for gauge connections in the space Ω(1|m) and finally discuss Hodge dual fields within the present framework.
A
bstract
We construct N=1 d=3 AdS supergravity within the group manifold approach and compare it with Achucarro-Townsend Chern-Simons formulation of the same theory. We clarify the relation between ...the off-shell super gauge transformations of the Chern- Simons theory and the off-shell worldvolume supersymmetry transformations of the group manifold action. We formulate the Achucarro-Townsend model in a double supersymmetric action where the Chern-Simons theory with a supergroup gauge symmetry is constructed on a supergroup manifold. This framework is useful to establish a correspondence of degrees of freedom and auxiliary fields between the two descriptions of d=3 supergravity.
We thank P. Fré and C. Maccaferri for fruitful discussions. This research is original and has a financial support of the Università del Piemonte Orientale (Fondi Ricerca Locale).
Primary total knee arthroplasty (TKA) is a reliable procedure with reproducible long-term results. Nevertheless, there are conditions related to the type of patient or local conditions of the knee ...that can make it a difficult procedure. The most common scenarios that make it difficult are discussed in this review. These include patients with many previous operations and incisions, and those with severe coronal deformities, genu recurvatum, a stiff knee, extra-articular deformities and those who have previously undergone osteotomy around the knee and those with chronic dislocation of the patella. Each condition is analysed according to the characteristics of the patient, the pre-operative planning and the reported outcomes. When approaching the difficult primary TKA surgeons should use a systematic approach, which begins with the review of the existing literature for each specific clinical situation.
We reformulate super-quantum mechanics in the context of integral forms. This framework allows to interpolate between different actions for the same theory, connected by different choices of picture ...changing operators (PCO). In this way we retrieve component and superspace actions and prove their equivalence. The PCO are closed integral forms and can be interpreted as super-Poincaré duals of bosonic submanifolds embedded into a supermanifold. We use them to construct Lagrangians that are top integral forms, and therefore can be integrated on the whole supermanifold. The
D
=
1
,
N
=
1
and the
D
=
1
,
N
=
2
cases are studied, in a flat and in a curved supermanifold. In this formalism, we also consider coupling with gauge fields, Hilbert space of quantum states, and observables.
We present a noncommutative version of D = 5 Chern‐Simons supergravity, where noncommutativity is encoded in a *‐product associated to an abelian Drinfeld twist. The theory is invariant under ...diffeomorphisms, and under the *‐gauge supergroup SU(2,2|4), including Lorentz and N = 4 local supersymmetries.
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