Narrow Results

We give a self-contained exposition of the differential geometry of Grassmann algebras. We also study elementary properties of these algebras from the point of view of Hochschild and cyclic cohomologies.

We study the functor of points and different local functors of points for smooth and holomorphic supermanifolds, providing characterization theorems and fully discussing the representability issues. In the end we examine applications to differential calculus including the transitivity theorems.

Superspace is considered as space of parameters of the supercoherent states defining the basis for oscillator-like unitary irreducible representations of the generalized superconformal group $SU(2m,2n|2N)$ in the field of quaternions $ℍ$. The specific construction contains naturally the supertwistor one of the previous work by Litov and Pervushin [1] and it is shown that in the case of extended supersymmetry such an approach leads to the separation of a class of superspaces and its groups of motion. We briefly discuss this particular extension to the domain of quaternionic superspaces as nonlinear realization of some kind of the affine and the superconformal groups with the final end to include also the gravitational field [6] (this last possibility to include gravitation, can be realized on the basis of Ref. 12 where the coset $\frac{Sp(8)}{SL(4R)}\sim$$\frac{SU(2,2)}{SL(2C)}$ was used in the non supersymmetric case). It is shown that this quaternionic construction avoid some unconsistencies appearing at the level of the generators of the superalgebras (for specific values of $p$ and $q$; $p+q=N$) in the twistor one.

In a attempt to treat a supergravity as a tensor representation, the four-dimensional $N$-extended quaternionic superspaces are constructed from the (diffeomorphyc) graded extension of the ordinary Penrose-twistor formulation, performed in a previous work of the authors [D. J. Cirilo-Lombardo and V. N. Pervushin, Int. J. Geom. Methods Mod. Phys., doi: http://dx.doi.org/10.1142/S0219887816501139.], with $N=p+k.$ These quaternionic superspaces have $4+k(N-k)$ even-quaternionic coordinates and $4N$ odd-quaternionic coordinates, where each coordinate is a quaternion composed by four $ℂ$-fields (bosons and fermions respectively). The fields content as the dimensionality (even and odd sectors) of these superspaces are given and exemplified by selected physical cases. In this case, the number of fields of the supergravity is determined by the number of components of the tensor representation of the four-dimensional $N$-extended quaternionic superspaces. The role of tensorial central charges for any $N$even$USp(N)=Sp(N,{ℍ}_{ℂ})\cap U(N,{ℍ}_{ℂ})$ is elucidated from this theoretical context.

In this paper, geometries and supergeometries are analyzed from the point of view of the coset coherent states. To this end, before detailing the different factorizations of the simplest cosets, the dynamics and structure of Supercoherent states of the Klauder–Perelomov type (that are defined taking into account the geometry of the coset based on the simplest supergroup $SU(2|1)$ in the context of extensions of the standard model that were given previously in [A. B. Arbuzov and D. J. Cirilo-Lombardo, Phys. Scr. 94 (2019) 125302] as structural basis of the electroweak sector of the standard model (SM)) are used. The supergeometrical descriptions with such coherent superstates which uses group representation from [Y. Ne’eman, S. Sternberg and D. Fairlie, Phys. Rept. 406 (2005) 303–377] for a beyond SM model, are also defined for the noncompact case $SU(1,1|1).$

This is an introductory review of topological field theories (TFTs) called AKSZ sigma models. The AKSZ construction is a mathematical formulation for the construction and analysis of a large class of TFTs, inspired by the Batalin-Vilkovisky formalism of gauge theories. We begin by considering a simple two-dimensional topological field theory and explain the ideas of the AKSZ sigma models. This construction is then generalized and leads to a mathematical formulation of a general topological sigma model. We review the mathematical objects, such as algebroids and supergeometry, that are used in the analysis of general gauge structures. The quantization of the Poisson sigma model is presented as an example of a quantization of an AKSZ sigma model.