The supersymmetric standard model (SSM) appears to be firmly grounded in superspace. For example, it would be natural to assume that all the physically important composite operators can be made by combining superfields and superspace derivatives. But even for the simplest possible, free, massless and unbroken SUSY theory in 3+1 dimensions, this is not true. This paper shows that there is a large set of physically important composite operators in the SSM that require explicit factors of the Grassmann odd `$θ$' parameters of superspace. These explicitly break superspace invariance. These composite operators will be called `Outfields', because they are intrinsically `outside' of superspace. It is not possible to write the Outfields using only superfields and superspace derivatives. The Outfields can be found in the BRST cohomology space of the theory. The calculation of the BRST cohomology space for these theories is performed in this paper using a spectral sequence analysis, starting with the free massless theory, and then adding interactions, and then masses.
The aim of this paper is to present a possible framework for incorporating a superspace formulation of supersymmetry into the formalism of noncommutative geometry à la Alain Connes. In analogy with the almost-commutative (AC) manifold construction of field theory, a base space is taken to be the superspace $\mathbb{R}^{3 | 2}$, with associated 2-point (Grassmann valued) finite space. The data of the spectral triple is explored, including decorations, i.e. algebra, Hilbert space, grading, real structure, and Dirac operator. The gauge fields arising from the inner fluctuations of the Dirac opera…
We investigate quantum corrections in two-dimensional CP^{N-1} supersymmetric nonlinear sigma model on noncommutative superspace. We show that this model is renormalizable, the N=2 SUSY sector is not affected by the C-deformation and that the non(anti)commutativity parameter C receives infinite renormalization at one-loop order. And it is the renormalizability of the model at one-loop order.
We investigate SUSY of Wess-Zumino models in non(anti-)commutative Euclidean superspaces. Non(anti-)commutative deformations break 1/2 SUSY, then non(anti-)commutative Wess-Zumino models do not have full SUSY in general. However, we can recover full SUSY at specific coupling constants satisfying some relations. We give a general way to construct full SUSY non(anti-)commutative Wess-Zumino models. For a some example, we investigate quantum corrections and $β$-functions behavior.
In our recent paper arXiv:0704.1185 [hep-th], we presented the differential geometry of five-dimensional N=1 anti-de Sitter superspace AdS^{5|8}=SU(2,2|1)/SO(4,1) x U(1) and developed the harmonic and the projective superspace settings to formulate off-shell supersymmetric theories in AdS^{5|8}. Here we give a brief review of the geometry and the harmonic superspace construction, and also elaborate on vector supermultiplets.
These lectures from the TASI 2012 summer school outline the basics of supersymmetry (SUSY) in 3+1 dimensions. Starting from a ground-up development of superspace, we develop all of the tools necessary to construct SUSY lagrangians. While aimed at an introductory level, these lectures incorporate a number of "super-tricks" for SUSY aficionados, including SUSY-covariant derivatives, equations of motion in superspace, background field methods, and non-linear realizations of goldstinos.
We propose N=4 twisted superspace formalism in four dimensions by introducing Dirac-Kahler twist. In addition to the BRST charge as a scalar counter part of twisted supercharge we find vector and tensor twisted supercharges. By introducing twisted chiral superfield we explicitly construct off-shell twisted N=4 SUSY invariant action. We can propose variety of supergauge invariant actions by introducing twisted vector superfield. We may, however, need to find further constraints to identify twisted N=4 super Yang-Mills action. We propose a superconnection formalism of twisted superspace where co…
We focus on a nonlinear supersymmetry (NL SUSY) in curved spacetime introduced in the superon-graviton model (SGM) towards a SUSY composite unified model based on SO(10) super-Poincaré symmetry, and we consider for $N = 1$ SUSY a systematic procedure to linearize the NL SUSY. By introducing modified superspace translations of superspace coordinates and their specific coordinate transformations both depending on a Nambu-Goldstone fermion, we show a homogeneous transformation's law of superfields which is important in the relation between linear (L) and NL SUSY. Furthermore, as a preliminary to …
We present a systematic analysis of the N=6 superspace constraints in three space-time dimensions. The general coupling between vector and scalar supermultiplets is encoded in an SU(4) tensor $W^i_j$ which is a function of the matter fields and subject to a set of algebraic and super-differential relations. We give a genuine N=6 classification for superconformal models with polynomial interactions and find the known ABJM and ABJ models. We further study the issue of supersymmetry enhancement to N=8 and the role of monopole operators in this scenario. To this end we assume the existence of a co…
We construct the Lagrangian of the ${\cal N}=1$ four-dimensional generalized supersymmetric Nambu$-$Jona-Lasinio (SNJL) model, which has ${\cal N}=1/2$ supersymmetry (SUSY) on non(anti)commutative superspace. A special attention is paid to the examination on the nonperturbative quantum dynamics: The phenomenon of dynamical-symmetry-breaking/mass-generation on the deformed superspace is investigated. The model Lagrangian and the method of SUSY auxiliary fields of composites are examined in terms of component fields. We derive the effective action, examine it, and solve the gap equation for self…
As the necessary background to construct from the aspect of Grothendieck's Algebraic Geometry dynamical fermionic D3-branes along the line of Ramond-Neveu-Schwarz superstrings in string theory, three pieces of the building blocks are given in the current notes: (1) basic $C^\infty$-algebrogeometric foundations of $d=4$, $N=1$ supersymmetry and $d=4$, $N=1$ superspace in physics, with emphasis on the partial $C^\infty$-ring structure on the function ring of the superspace, (2) the notion of SUSY-rep compatible hybrid connections on bundles over the superspace to address connections on the Chan-…
A few years ago, we found the supersymmetric(SUSY) counterpart of the spectral triple which specified noncommutative geometry(NCG). Based on "the triple", we considered the SUSY version of the spectral action principle and had derived the action of super Yang--Mills theory, minimal supersymmetric standard model, and supergravity. In these theories, we used vector notation in order to express a chiral or an anti-chiral matter superfield. We also represented the NCG algebra and the Dirac operator by matrices which operated on the space of matter field. In this paper, we represent the triple in t…
R-matrix method is used to construct supersymmetric extensions of theta - Euclidean group preserving N = 1/2 supersymmetry and its three- parameter generalization. These quantum symmetry supergroups can be considered as global counterparts of appropriately twisted Euclidean superalgebras. The corresponding generalized global symmetry transformations act on deformed superspaces as the usual ones do on undeformed spaces. However, they depend on non(anti)commuting parameters satisfying (anti)commutation relations defined by relevant R matrix.
Harmonic superspace can be used to construct higher derivative terms in N=2 supersymmetric effective actions despite the infinite redundancy in their description due to the infinite number of auxiliary fields. We are able to write down all of the 3- and 4-derivative terms on the Higgs, Coulomb, and mixed branches, modulo the possible existence of superspace Chern-Simons-like terms, which we discuss. Many of the terms we find are holomorphic, and at least one is shown to not receive quantum corrections.
We discuss a deformation of the Hopf algebra of supersymmetry (SUSY) transformations based on a special choice of twist. As usual, algebra itself remains unchanged, but the comultiplication changes. This leads to the deformed Leibniz rule for SUSY transformations. Superfields are elements of the algebra of functions of the usual supercoordinates. Elements of this algebra are multiplied by using a $\star$-product which is noncommutative, hermitian and finite when expanded in power series of the deformation parameter. Chiral fields are no longer a subalgebra of the algebra of superfields. One po…
We discuss the bi-local collective theory for the $\mathcal{N}=1,2$ supersymmetric Sachdev-Ye-Kitaev (SUSY SYK) models. We construct a bi-local superspace, and formulate the bi-local collective superfield theory of the one-dimensional SUSY vector model. The bi-local collective theory provides systematic analysis of the SUSY SYK models. We find that this bi-local collective theory naturally leads to supermatrix formulation in the bi-local superspace. This supermatrix formulation drastically simplifies the analysis of the SUSY SYK models. We also study $\mathcal{N}=1$ bi-local superconformal gen…
In this work, we adopt the simplest model that spontaneously breaks supersymmetry, namely, the minimal O'Raifeartaigh model. The effective potential is computed in the framework of the linear delta expansion (LDE) approach up to the order $δ^2$, conjugated with superspace and supergraph techniques. The latter can be duly mastered even if supersymmetry is no longer exact and the efficacy of the superfield approach in connection with the LDE procedure is confirmed according to our investigation. That opens up a way for a semi-nonperturbative superspace computation which allows us to deal with sp…
Perimeter Institute for Theoretical Physics（Perimeter理论物理研究所）；Università degli Studi di Milano–Bicocca（米兰比可卡大学）；INFN, Sezione di Milano–Bicocca（意大利国家核物理研究所米兰比可卡分部）；Università del Piemonte Orientale（皮埃蒙特大西洋大学）；INFN, Sez. Torino（意大利国家核物理研究所都灵分部）
We shortly review our new superfield formalism in the framework of Fujita, Kugo, Ohashi 5D conformal supergravity, in particular with an $S^1/Z_2$ orbifold. The radion of the fifth dimension is embedded in two related superfields, a chiral and a general multiplet and is linked to the radion superfield of rigid SUSY. The superspace action of the gauge sector is of the Chern-Simons type. We also present the superspace action for hypermultiplets and discuss the role of compensators. The presented formalism should be very useful for applications. We demonstrate this for obtaining the RS solution.
Super Feynman rules for any superspin are given for massive $ \mathcal{N}=1 $ supersymmetric theories, including momentum superspace on-shell legs. This is done by extending, from space to superspace, Weinberg's perturbative approach to quantum field theory. Superfields work just as a device that allow one to write super Poincare-covariant superamplitudes for interacting theories, relying neither in path integral nor canonical formulations. Explicit transformation laws for particle states under finite supersymmetric transformations are offered. $ \mathit{C}, \mathit{P}, \mathit{T}, $ and $ \ma…
We present an explicit formulation of supersymmetric Yang-Mills theories from $\D=$ 5 to 10 dimensions in the familiar $\N=1,\D=4$ superspace. This provides the rules for globally supersymmetric model building with extra dimensions and in particular allows us to simply write down $\N=1$ SUSY preserving interactions between bulk fields and fields localized on branes. We present a few applications of the formalism by way of illustration, including supersymmetric ``shining'' of bulk fields, orbifolds and localization of chiral fermions, anomaly inflow and super-Chern-Simons theories.
We review shortly present status of quantum deformations of Poincaré and conformal supersymmetries. After recalling the $κ$-deformation of $\hbox{D=4}$ Poincaré supersymmetries we describe the corresponding star product multiplication for chiral superfields. In order to describe the deformation of chiral vertices in momentum space the integration formula over $κ$-deformed chiral superspace is proposed.
The existence of a linearized SUSY invariant for N=8 supergravity whose gravitational components are usually called R^4 was established long ago by on-shell superspace arguments. Superspace and string theory methods have also established analogous higher dimensional D^{2k} R^4 invariants. However, very little is known about the SUSY completions of these operators which involve other fields of the theory. In this paper we find the detailed component expansion of the linearized R^4 invariant starting from the corresponding superamplitude which generates all component matrix elements of the opera…
We study possible relations between the full Green's functions of softly broken supersymmetric theories and the full Green's functions of rigid supersymmetric theories on the example of the supersymmetric quantum mechanics and find that algebraic relations can exist and can be written in a simple form. These algebraic relations between the Green's functions have been derived by transforming the path integral of the rigid theory. In this approach soft terms appear as the result of general changes of coordinates in the superspace.
BPS wall solutions are obtained for N=2 supersymmetric nonlinear sigma model with Eguchi-Hanson target manifold in a manifestly supersymmetric manner. The model is constructed by a massive hyper-Kahler quotient method both in the N=1 superfield and in the N=2 superfield (harmonic superspace). We describe the model in simple terms and give relations between various parameterizations which are useful to describe the model and the solution. Some more details can be found in our previous paper [hep-th/0211103]. This article is dedicated to Professor Hiroshi Ezawa on the occasion of his seventieth …
Models like the Supersymmetric Standard Model (SSM) possess simple, but well-hidden, `Outfields'. These Outfields are composite operators that violate superspace invariance, but in a special way. A new mechanism for SUSY breaking arises from the Outfields, for a special non-minimal version of the SSM, which will be called the CSSM. The CSSM has right neutrinos and a Higgs singlet, which we call J, in addition to the usual SSM. This breaking of SUSY cannot be avoided, because it arises from the local BRST cohomology of the theory, which is also the origin of the Outfields. It can also be seen t…
We make a first step to extend to the supersymmetric arena the effective action method, which is used to covariantly deduce the low energy dynamics of topological defects directly from their parent field theory. By focussing on two-dimensional supersymmetric theories we are able to derive the appropriate change of variables that singles out the low energy degrees of freedom. These correspond to super-worldline embeddings in superspace which are subject to a geometrical constraint. We obtain a supersymmetric and $κ$--invariant low energy expansion, with the standard superparticle action as the …
We present a new formalism for finding the low-energy effective Lagrangian of Goldstinos and other fields. This Lagrangian is written using standard superspace and the superfields are constrained to include only the light degrees of freedom. The Goldstino resides in a (constrained) chiral superfield X which is naturally identified at short distances. This allows us to exactly compute the IR behavior of some correlation functions even in strongly coupled theories with SUSY breaking. The Goldstino couplings above the scale of the matter superpartners are determined by identifying X with the stan…
We present an alternative method to explore the off-shell component structure of theories that describe half-integer super-helicities Y=s+1/2 (where sis any positive integer). We use it to derive the component action, component SUSY transformation laws, and count the component-level degrees of freedom involved. This counting will give us clues about N=2 representations. The foundation of the process relies on the superfield equations of motion, generated by variation of a superspace action expressed in terms of prepotentials. Using this approach we reproduce the half-integer super-helicity sup…
We present an alternative method of exploring the component structure of an integer super-helicity Y=s (for any integers) irreducible representation of the Super-Poincare group. We use it to derive the component action and the SUSY transformation laws. The effectiveness of this approach is based on the equations of motion and their properties, like Bianchi identities. These equations are generated by the superspace action when it is expressed in terms of prepotentials. For that reason we reproduce the superspace action for integer superspin, using unconstrained superfields. The appropriate, to…
Using the methods developed in earlier papers we analyze a new type of deformation of the superspace. The twist we use to deform the N=1 SUSY Hopf algebra is non-hermitian and is given in terms of the covariant derivatives $D_α$. A SUSY invariant deformation of the Wess-Zumino action is constructed and compared with results already known in the literature. Finally, by calculating divergences of the two-point Green functions a preliminary analysis of renormalizability properties of the constructed model is done. As expected, there is no renormalization of mass and no tadpole diagrams appear.
The local SUSY symmetry of the loop dynamics of QCD is found. The remarkable thing is, there is no einbein-gravitino on this theory, which makes it a 1D topological supergravity, or locally SUSY quantum mechanics. Using this symmetry, we derive the large $N_c$ loop equation in momentum superloop space. Introducing as before the position operator $\Xμ$ we argue that the superloop equation is equivalent to invariance of correlation functions of products of these operators with respect to certain quadrilinear transformation. The applications to meson and glueball sectors as well as the chiral sym…
In this paper, using a Hopf-algebraic method, we construct deformed Poincaré SUSY algebra in terms of twisted (Hopf) algebra. By adapting this twist deformed super-Poincaré algrebra as our fundamental symmetry, we can see the consistency between the algebra and non(anti)commutative relation among (super)coordinates and interpret that symmetry of non(anti)commutative QFT is in fact twisted one. The key point is validity of our new twist element that guarantees non(anti)commutativity of space. It is checked in this paper for N=1 case. We also comment on the possibility of noncommutative central …
In these introductory lectures, we review the theoretical tools used in constructing supersymmetric field theories and their application to physical models. We first introduce the technology of two-component spinors, which is convenient for describing spin-$1/2$ fermions. After motivating why a theory of nature may be supersymmetric at the TeV energy scale, we show how supersymmetry (SUSY) arises as an extension of the Poincaré algebra of spacetime symmetries. We then obtain the representations of the SUSY algebra and discuss its simplest realization in the Wess-Zumino model. In order to have …
