Narrow Results

Using the notions of gauge symmetry and gauge connection on ordinary superspaces, we derive a class of generalized supertranslation algebras in the case of N = 1, D = 2 Euclidean superspace with a U(1) gauge group. This generalizes the ordinary algebra by inclusion of some additional bosonic and fermionic operators which are interpreted as the generators of the U(1) gauge symmetry on superspace. The generalized superalgebra closes only for very particular configurations of the gauge connection superfield. This provides a unified framework for a variety of generalizations of the ordinary superalgebra such as its central extension and its noncommutative deformation found in an earlier work.

We consider the generic non-anticommutative model of chiral–antichiral superfields on superspace. The model is formulated in terms of an arbitrary Kählerian potential, chiral and antichiral superpotentials and can include the non-anticommutative supersymmetric sigma-model as a partial case. We study a component structure of the model and derive the component Lagrangian in an explicit form with all auxiliary fields contributions. We show that the infinite series in the classical action for generic non-anticommutative model of chiral–antichiral superfields in D = 4 dimensions can be resumed in a compact expression which can be written as a deformation of standard Zumino's Lagrangian and chiral superpotential. Problem of eliminating the auxiliary fields in the generic model is discussed and the first perturbative correction to the effective scalar potential is obtained.

We propose to construct the off-shell supersymmetric structure present in quantized D=4 BF theories in terms of a BRST–VSUSY superspace formalism. The method is based on the introduction of constrained scalar superfields. We also discuss how to find the D=4 BF action from D=N=4 supersymmetric action by an appropriate twisting process.

We propose an N = 2 twisted superspace formalism with a central charge in four dimensions by introducing a Dirac–Kähler twist. Using this formalism, we construct an off-shell twisted hypermultiplet action and find an explicit form of fermionic scalar, vector and tensor transformations. We construct an off-shell Donaldson–Witten theory coupled to the twisted hypermultiplet. We show that this action possesses N = 4 twisted supersymmetry at the on-shell level. It turns out that the four-dimensional Dirac–Kähler twist is equivalent to Marcus' twist.

A short introduction to N = 1 supergravity in four dimensions in the superspace approach is given emphasizing on all steps to obtain the final Lagrangian. In particular, starting from geometrical principles and the introduction of superfields in curved superspace, the action coupling matter and gauge fields to supergravity is derived. This paper is based on the book [M. Rausch de Traubenberg and M. Valenzuela, A Supergravity Primer: From Geometrical Principles to the Final Lagrangian (World Scientific, 2020)] and on several lectures given at the doctoral school of Strasbourg.

Based on an argument for the noncommutativity of momenta in noncommutative directions, we arrive at a generalization of the super E² algebra associated to the deformation of translations in a noncommutative Euclidean plane. The algebra is obtained using appropriate representations of its generators on the space of superfields in a D = 2, "noncommutative superspace." We find that the (anti)commutators of several (super)translation generators are no longer vanishing, but involve a new set of generators which together with the (super)translation and rotation generators form a consistent closed algebra. We then analyze the spectrum of this algebra in order to obtain its fundamental and adjoint representations.

We propose a twisted D=N=2 superspace formalism. The relation between the twisted super charges including the BRST charge, vector and pseudoscalar super charges and the N=2 spinor super charges is established. We claim that this relation is essentially related with the Dirac–Kähler fermion mechanism. We show that a fermionic bilinear form of twisted N=2 chiral and anti-chiral superfields is equivalent to the quantized version of BF theory with the Landau type gauge fixing while a bosonic bilinear form leads to the N=2 Wess–Zumino action. We then construct a Yang–Mills action described by the twisted N=2 chiral and vector superfields, and show that the action is equivalent to the twisted version of the D=N=2 super Yang–Mills action, previously obtained from the quantized generalized topological Yang–Mills action with instanton gauge fixing.

In this talk we present a field theoretical model constructed in Minkowski superspace with a deformed supercoordinate algebra. Our study is motivated in part by recent results from super-string theory, which show that in a particular scenario in Euclidean superspace the spinor coordinates θ do not anticommute. Field theoretical consequences of this deformation were studied in a number of articles. We present a way to extend the discussion to Minkowski space, by assuming non-vanishing anticommutators for both θ, and . We give a consistent supercoordinate algebra, and a star product that is real and preserves the (anti)chirality of a product of (anti)chiral superfields. We also give the Wess-Zumino Lagrangian that gains only Lorentz-invariant corrections due to non(anti)commutativity within our model. The Lagrangian in Minkowski superspace is also always manifestly Hermitian.

The recently formulated Bagger-Lambert-Gustavsson (BLG) theory in three dimensions is described in terms of a constrained chiral superfield in light-cone superspace. We discuss the use of Superconformal symmetry to determine the form of its interactions, in complete analogy with N = 4 SuperYang-Mills in four dimensions.

We review the new theory of modified supergravity, dubbed the supergravity, and some of its recent applications to inflation and reheating in the early universe cosmology. The supergravity is the N = 1 locally supersymmetric extension of the f(R) gravity in four space–time dimensions. A manifestly supersymmetric formulation of the supergravity exist in terms of N = 1 superfields, by using the (old) minimal Poincaré supergravity in curved superspace. We find the conditions for stability, the absence of ghosts and tachyons. Three models of the supergravity are studied. The first example is devoted to a recovery of the standard (pure) N = 1 supergravity with a negative cosmological constant from the supergravity. As the second example, a generic supergravity is investigated, and the existence of the AdS bound on the scalar curvature is found. As the third (and most important) example, a simple viable realization of chaotic inflation in supergravity is found. Our approach is minimalistic since it does not introduce new exotic fields or new interactions, beyond those already present in (super)gravity. The universal reheating mechanism is automatic. We establish the consistency of our approach and also apply it to preheating and reheating after inflation. The Higgs inflation and its correspondence to the Starobinsky inflation are established in the context of supergravity. We briefly review other relevant issues such as non-Gaussianity, CP-violation, origin of baryonic asymmetry, lepto- and baryo-genesis. The supergravity has promise for possible solutions to those outstanding problems too.

We construct the extended BRST and anti-BRST transformation by considering a shift symmetry both in Abelian and non-Abelian Chern–Simons theories coupled to scalar fields using the approach of Batalin–Vilkovisky quantization. The extended BRST invariant Lagrangian density is able to be addressed in the framework of superfield formalism with one fermionic coordinate. In the case of extended BRST–anti-BRST symmetry, it is shown that the Batalin–Vilkovisky action of the theory can be expressed in a covariant form with two fermionic coordinates in the superspace.

A framework is developed enabling the global analysis of the stability of cosmological models using the local geometric characteristics of the infinite-dimensional superspace, i.e. using the generalized Jacobi equation reformulated for pseudo-Riemannian manifolds. We give a direct formalism for dynamical analysis in the superspace, the requisite equation pertinent for stability analysis of the universe by means of generalized covariant and Fermi derivative is derived. Then, the relevant definitions and formulae are retrieved for cosmological models with a scalar field.

In this paper, we obtain a plane wave decomposition for the delta distribution in superspace, provided that the superdimension is not odd and negative. This decomposition allows for explicit inversion formulas for the super Radon transform in these cases. Moreover, we prove a more general Radon inversion formula valid for all possible integer values of the superdimension. The proof of this result comes along with the study of fractional powers of the super Laplacian, their fundamental solutions, and the plane wave decompositions of super Riesz kernels.

The arbitrary dimensional BF theory is presented in a superconnection formalism. This makes the off-shell BRST-VSUSY superalgebra more obvious. We also discuss how to build the BRST–VSUSY exact quantum action.

A discussion is presented on aspects of supersymmetry in the context of quantum mechanical models. Rather than dealing with the details of specific models, the presentation aims to draw out general features of such systems.

The recently formulated Bagger-Lambert-Gustavsson (BLG) theory in three dimensions is described in terms of a constrained chiral superfield in light-cone superspace. We discuss the use of Superconformal symmetry to determine the form of its interactions, in complete analogy with N = 4 Super Yang-Mills in four dimensions.

We review the features of the background field method for three-dimensional gauge theories in N = 2 superspace. As an application, we compute one-loop low-energy effective actions in N = 2, 4, 8, d = 3 SYM theories.