We present few types of integral transforms and integral representations that are very useful for extending to supergeometry many familiar concepts of differential geometry. Among them we discuss the construction of the super Hodge dual, the integral representation of picture changing operators of string theories and the construction of the super-Liouville form of a symplectic supermanifold.
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 appro
ach to supergravity and its variational principle are reformulated and clarified in this language. Central in our analysis is the Poincare 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.
We provide the classification of real forms of complex D=4 Euclidean algebra $mathcal{epsilon}(4; mathbb{C}) = mathfrak{o}(4;mathbb{C})) ltimes mathbf{T}_{mathbb{C}}^4$ as well as (pseudo)real forms of complex D=4 Euclidean superalgebras $mathcal{eps
ilon}(4|N; mathbb{C})$ for N=1,2. Further we present our results: N=1 and N=2 supersymmetric D=4 Poincare and Euclidean r-matrices obtained by using D= 4 Poincare r-matrices provided by Zakrzewski [1]. For N=2 we shall consider the general superalgebras with two central charges.
We construct the Hodge dual for supermanifolds by means of the Grassmannian Fourier transform of superforms. In the case of supermanifolds it is known that the superforms are not sufficient to construct a consistent integration theory and that the in
tegral 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 superforms in a new way which can be easily generalised 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.
We present a new construction for the Hodge operator for differential manifolds based on a Fourier (Berezin)-integral representation. We find a simple formula for the Hodge dual of the wedge product of differential forms, using the (Berezin)-convolut
ion. The present analysis is easily extended to supergeometry and to non-commutative geometry.
The AGT motivated relation between the tensor product of the N = 1 super-Liouville field theory with the imaginary free fermion (SL) and a certain projected tensor product of the real and the imaginary Liouville field theories (LL) is analyzed. Using
conformal field theory techniques we give a complete proof of the equivalence in the NS sector. It is shown that the SL-LL correspondence is based on the equivalence of chiral objects including suitably chosen chiral structure constants of all the three Liouville theories involved.
L. Castellani
,R. Catenacci
,P.A. Grassi
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(2016)
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"Integral representations in supermanifolds: super Hodge duals, PCOs and Liouville forms"
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Pietro Antonio Grassi
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