We construct the unintegrated vertex operator at the first mass level of the open superstring from the OPE of massless vertices. Using BRST cohomology manipulations, the tree amplitude of two massless and one massive state is rewritten in terms of the pure spinor superspace kinematic expression of the massless four-point amplitude at the ${α^\prime}^2$ level. A generalization relating the partial $n$-point tree amplitudes with one massive state and linear combinations of the ${α^\prime}^2$ corrections to n+1 massless amplitudes is found and shown to be consistent with unitarity.
We discuss the higher superspin superfield formalism of Kuzenko et al from the basis of a unified treatment of an unconstrained spinorial superfield prepotential, its action and a restricted set of gauge transformations. We recover previous results as distinct limits of this unified treatment and give the first derivations of the complete set of Bianchi identities associated with these equations.
Although the chiral dotted spinor superfield should describe a Massive Superspin One Half multiplet, it has not been obvious how to derive this from an action. In this paper this is done by including a chiral undotted spinor superfield, finding the BRST transformations that govern both of these, and then finding the action as an invariant of the transformations. It turns out that both kinds of spinor superfields are needed. Moreover, the BRST transformations for the two kinds of chiral spinor superfields are generated from each other by a special involution that exchanges Grassmann odd (even) …
Four dimensional N=1 supersymmetric Yang-Mills theory action is written in terms of the spinor superfields in transverse gauge. This action is seemingly first order in space-time derivatives. Thus, it suggests that the generalized fields approach of obtaining Batalin-Vilkovisky quantization can be applicable. In fact, generalized fields which collect spinor superfields possessing different ghost numbers are introduced to obtain the minimal solution of its Batalin-Vilkovisky master equation in a compact form.
Universidade Federal do Amapá（阿马帕联邦大学）；Universidade Federal da Paraíba（帕拉伊巴联邦大学）；Instituto de Física Corpuscular (IFIC), CSIC‐Universitat de València（瓦伦西亚大学-西班牙国家研究委员会粒子物理研究所）；Universidade Federal do Ceará（塞阿拉联邦大学）
We present a method for introducing and analysing higher-derivative deformations of maximally supersymmetric field theories. Such terms are built in the pure spinor superfield framework, using a set of operators representing physical fields. The action for abelian Born-Infeld theory becomes polynomial in this language, and contains only a four-point interaction in addition to the free action. Simplifications also occur in the non-abelian case.
Chiral superfields with multiple dotted Lorentz spinor indices (`dotspinors') are important in the analysis of supersymmetry breaking through the mechanisms of Cybersusy. This paper describes the actions for massive dotspinors coupled to supersymmetric gauge theory and to chiral matter. It analyzes the free equations of motion and mass spectra for the case of unbroken supersymmetry. The general form of the Cybersusy algebra for dotsupers with multiple indices is also discussed briefly.
We study the duality of the supersymmetric self-dual and Maxwell-Chern-Simons theories coupled to a fermionic matter superfield, using a master action. This approach evades the difficulties inherent to the quartic couplings that appear when matter is represented by a scalar superfield. The price is that the spinorial matter superfield represents a unusual supersymmetric multiplet, whose main physical properties we also discuss.
This is the first installment of a series of three papers in which we describe a method to determine higher-point correlation functions in one-loop open-superstring amplitudes from first principles. In this first part, we exploit the synergy between the cohomological features of pure-spinor superspace and the pure-spinor zero-mode integration rules of the one-loop amplitude prescription. This leads to the study of a rich variety of multiparticle superfields which are local, have covariant BRST variations, and are compatible with the particularities of the pure-spinor amplitude prescription. Se…
Maximally supersymmetric theories do not allow off-shell superspace formulations with traditional superfields containing a finite set of auxiliary fields. It has become clear that off-shell supersymmetric action formulations of such models can be achieved by the introduction of pure spinors. In this talk, an overview of this formalism is given, with emphasis on D=10 super-Yang-Mills theory and D=11 supergravity. This a somewhat expanded version of a talk presented at the workshop "Breaking of supersymmetry and ultraviolet divergences in extended supergravity" (BUDS), Laboratori Nazionali di Fr…
I review and discuss the construction of supersymmetry multiplets and manifestly supersymmetric Batalin-Vilkovisky actions using pure spinors, with emphasis on models with maximal supersymmetry. The special cases of D=3, N=8 (Bagger-Lambert-Gustavsson) and N=6 (Aharony-Bergman-Jafferis-Maldacena) conformal models are treated in detail. Most of the material is covered by the papers arXiv:0808.3242 and arXiv:0809.0318. This is the written version of a talk given at 4th Baltic-Nordic workshop "Algebra, Geometry and Mathematical Physics", Tartu, Estonia, October 9-11, 2008, to appear in the Procee…
We describe the ``universal'' action for massless superfields of all superspins in N = 1, D = 4 anti-de Sitter superspace as a gauge theory of unconstrained superfields taking their values in the commutative algebra of analytic functions over a one-sheeted hyperboloid in $R^{3,1}$. The action is invariant under N = 2 supersymmetry transformations which form a closed algebra off the mass-shell.
A modification of the harmonic superfield formalism in $D=4, N=2$ supergravity using a subsidiary condition of covariance under the background supersymmetry with a central charge ($B$-covariance) is considered. Conservation of analyticity together with the $B$-covariance leads to the appearance of linear gravitational superfields. Analytic prepotentials arise in a decomposition of the background linear superfields in terms of spinor coordinates and transform in a nonstandard way under the background supersymmetry. The linear gravitational superfields can be written via spinor derivatives of no…
We study the relation between a supermatrix model and the free 4D, N=1 supersymmetric field theory of a massless supermultiplet with spins (3, 5/2). In order to do this, we construct a superfield formulation of the theory. We show that solutions of the equations of motion for the supermultiplet (3, 5/2) satisfy the equations of motion of a supermatrix model.
We construct unitary representations of (1,0) and (2,0) superconformal algebras in six dimensions by using superfields defined on harmonic superspaces with coset manifolds USp(2n)/[U(1)]^n, n=1,2. In the spirit of the AdS_7/CFT_6 correspondence massless conformal fields correspond to "supersingletons" in AdS_7. By tensoring them we produce all short representations corresponding to 1/2 and 1/4 BPS anti-de Sitter bulk states of which "massless bulk" representations are particular cases.
Massive arbitrary spin supermultiplets and massless (scalar and spin one-half) supermultiplets of the N=2 Poincare superalgebra in three-dimensional flat space are considered. Both the integer spin and half-integer spin supermultiplets are studied. For such massive and massless supermultiplets, a formulation in terms of light-cone gauge unconstrained superfields defined in a momentum superspace is developed. For the supermultiplets under consideration a superspace first derivative representation for all cubic interaction vertices is obtained. A superspace representation for dynamical generator…
We study theories of 4D, N=1 supersymmetric massless, arbitrary integer superspins. A new state-of-the art is being established by the discovery of a new series of such theories for arbitrary superspin Y (Y=s for any integer s) The lowest member of the series is surprisingly found to be a previously established formulation of the spin (3/2, 1) supermultiplet.
We study theories of 4D, N=1 supersymmetric massless, arbitrary higher half odd-integer superspins. A new series of such theories is found to exist for arbitrary superspin Y (Y=s+1/2 for any integer s). The lowest member (s=1) of the series is the original off-shell formulation of 4D, N=1 supergravity first presented by Breitenlohner in 1977.
This is an overview of the method of pure spinor superfields, written for "Handbook of Quantum Gravity", eds. C. Bambi, L. Modesto and I. Shapiro. The main focus is on the use of the formalism in maximal supergravity on a flat background. The basics of pure spinor superfields, and their relation to standard superspace, is reviewed. The pure spinor superstring model of Berkovits is briefly discussed. Consequences for divergence properties of loop diagrams in maximal supergravity are restated. Some final remarks are made concerning desirable development of the theoretical framework.
A Batalin-Vilkovisky action for D=6, N=1 super-Yang--Mills theory, including coupling to hypermultiplets, is given. The formalism involves pure spinor superfields. The geometric properties of the D=6, N=1 pure spinors (which differ from Cartan pure spinors) are examined. Unlike the situation for maximally supersymmetric models, the fields and antifields (including ghosts) of the vector multiplet reside in separate superfields. The formalism provides an off-shell superspace formulation for matter hypermultiplets, which in a traditional treatment are on-shell.
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…
We consider the interacting system of D=4 N=1 supergravity and the Brink-Schwarz massless superparticle as described by the sum of their superfield actions, and derive the complete set of superfield equations of motion for the coupled dynamical system. These include source terms given by derivatives of a vector superfield current density with support on the worldline. This current density is constructed from the spin 3/2 and spin 2 current density `prepotentials'. We analyze the gauge symmetry of the coupled action and show that it is possible to fix the gauge in such a way that the equations …
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 examine the BRS invariance of the open pure spinor superstring in the presence of background superfields on a Dp-brane. It is shown that the BRS invariance leads not only to boundary conditions on the spacetime spinors, but also to supersymmetric DBI equations of motion for the background superfields on the Dp-brane. These DBI equations are consistent with the supersymmetric DBI equations for a D9-brane.
The manifestly supersymmetric pure spinor formulations of the Bagger-Lambert-Gustavsson models with N=8 supersymmetry and the Aharony-Bergman-Jafferis-Maldacena models with N=6 supersymmetry are given. The structures of the pure spinors are investigated in both cases, and non-degenerate measures are formed using non-minimal sets of variables, allowing for the formulation of an action principle.
An earlier paper introduced an action for a new kind of irreducible massive superspin one half multiplet, using BRST cohomological techniques including `BRST Recycling'. A mass term was introduced in the earlier paper. A second mass term is discussed in this paper. This new mass invariant is an `Extraordinary Invariant'--it has Zinn sources in it. The natural treatment for this situation is to `Complete the Action' so that the new action yields zero for the BRST Poisson Bracket. In the present case, this Completion meets a BRST Obstruction. Setting the coefficient of this `Completion Obstructi…
D=4, N=4 super-Yang-Mills theory has an off-shell superspace formulation in terms of pure spinor superfields, which is directly inherited from the D=10 theory. That superspace, in particular the choice of pure spinor variables, is less suitable for dealing with fields that are inherently 4-dimensional, such as the superfields based on the scalars, which are gauge-covariant, and traces of powers of scalars, which are gauge-invariant. We give a reformulation of D=4, N=4 super-Yang-Mills theory in N=4 superspace, using inherently 4-dimensional pure spinors. All local degrees of freedom reside in …
In this talk, we review how the superspace formulation of maximally supersymmetric field theories (including supergravity) naturally leads to introduction of pure spinors and pure spinor superfields, and why the formalism provides off-shell formulations. This approach to pure spinor superfields thus stresses field-theoretic aspects rather than the first-quantised ones normally used e.g. in superstring theory. We discuss how the BRST operator arises and the principles behind constructions of actions, as well as the general Batalin-Vilkovisky framework. D=11 supergravity and its recently constru…
A variant for the superspin one-half massive superparticle in $ 4D $, $ \mathcal{N}=1 $, based on Dirac superfields, is offered. As opposed to the current known models that use spinor chiral superfields, the propagating fields of the supermultiplet are those of the lowest mass dimensions possible: scalar, Dirac and vector fields. Besides the supersymmetric chiral condition, the Dirac superfields are not further constrained, allowing a very straightforward implementation of the path-integral method. The corresponding superpropagators are presented. In addition, an interaction super Yukawa poten…
Using non-minimal pure spinor superspace, Cederwall has constructed BRST-invariant actions for $D=10$ super-Born-Infeld and $D=11$ supergravity which are quartic in the superfields. But since the superfields have explicit dependence on the non-minimal pure spinor variables, it is non-trivial to show these actions correctly describe super-Born-Infeld and supergravity. In this paper, we expand solutions to the equations of motion from Cederwall's actions to leading order around the linearized solutions and show that they correctly describe the interactions of $D=10$ super-Born-Infeld and $D=11$ …
We reformulate the Bagger-Lambert-Gustavsson model using an N=8 superspace, thus making the full supersymmetry manifest. The formulation is based on appropriate "pure spinor wave functions" for the Chern-Simons and matter multiplets. The Lagrangian has an extremely simple structure, essentially containing a Chern-Simons-like term for the gauge field wave function and a minimally coupled matter wave function. No higher order interactions than cubic are present. The consistency of the setup relies on an interplay between the algebraic structures of the 3-algebra and of the pure spinors.
We explicitly calculate the one-loop effective potential for the supersymmetric topologically massive gauge theory in four dimensions, where the chiral scalar superfield is directly coupled to the field strength for the gauge spinor superfield.
In this paper a multiparticle generalization of linearized ten-dimensional super Yang--Mills superfields is proposed. Their equations of motions are shown to take the same form as in the single-particle case, supplemented by contact terms. A recursive construction of these superfields is inspired by the iterated OPEs among massless vertex operators in the pure spinor formalism. An enlarged set of BRST-covariant pure spinor blocks is then defined in a streamlined fashion and combined to multiparticle vertex operators. The latter can be used to universally describe tree-level subdiagrams in the …
N-extended massless arbitrary integer and half-integer spin supermultiplets in four dimensional flat space are studied in the framework of light-cone gauge formalism. For such multiplets, by using light-cone momentum superspace, we build unconstrained light-cone gauge superfield formulation. The superfield formulation is used to develop a superspace representation for all cubic interactions vertices of the N-extended massless supermultiplets. Our suitable treatment of the light-cone gauge superfields allows us to obtain attractively simple superspace representation for the cubic interaction ve…
It is shown that canonical quantization of the $4d$ Siegel superparticle yields massless Wess-Zumino model as an effective field theory. Quantum states of the superparticle are realized in terms of the real scalar superfields which prove to be decomposed into the sum of on-shell chiral and antichiral superfields.
We provide the covariant superspace equations that are sufficient to determine the complete $θ$ expansion of the vertex operator of the open string massive states with $(mass)^2=1/α'$ in pure spinor formalism of superstring theory. These equations get rid of the redundant degrees of freedom in superfields and are consistent with the BRST conditions derived in [1]. Further, we give the explicit $θ$ expansion of the superfields appearing in the unintegrated vertex to $O(θ^3)$. Finally, we compute the contribution to a 3-point tree amplitude with the resulting vertex operator upto $O(θ^3)$ and fi…
We describe a pure spinor BRST cohomology framework to compactly represent ten-dimensional one-loop amplitudes involving any number of massless open- and closed-string states. The method of previous work to construct scalar and vectorial BRST invariants in pure spinor superspace signals the appearance of the hexagon gauge anomaly when applied to tensors. We study the systematics of the underlying BRST anomaly by defining the notion of pseudo-cohomology. This leads to a rich network of pseudo-invariant superfields of arbitrary tensor rank whose behavior under traces and contractions with extern…
In the $\mathcal{N}=1$, $d=3$ superspace, we propose a massive superfield theory formulated in terms of a spinor gauge superfield, whose component content includes a two-form field, and a real scalar matter superfield. For this model, we explicitly calculate the one-loop correction to the superfield effective potential. In particular, we show that the one-loop effective potential is independent the gauge-fixing parameters.
We propose a modification of the Sp(2) covariant superfield quantization to realize a superalgebra of generating operators isomorphic to the massless limit of the corresponding superalgebra of the osp(1,2) covariant formalism. The modified scheme ensures the compatibility of the superalgebra of generating operators with extended BRST symmetry without imposing restrictions eliminating superfield components from the quantum action. The formalism coincides with the Sp(2) covariant superfield scheme and with the massless limit of the osp(1,2) covariant quantization in particular cases of gauge-fix…
The pure spinor formalism for the superstring can be formulated as a twisted N=2 worldsheet theory with fermionic generators $j_{BRST}$ and composite $b$ ghost. After untwisting the formalism to an N=1 worldsheet theory with fermionic stress tensor $j_{BRST}+b$, the worldsheet variables combine into N=1 worldsheet superfields $X^m$ and $Θ^α$ together with a superfield constraint relating $DX^m$ and $DΘ^α$. The constraint implies that the worldsheet superpartner of $θ^α$ is a bosonic twistor variable, and different solutions of the constraint give rise to the pure spinor or extended RNS formali…
In this note, we study, formalize, and generalize the pure spinor superfield formalism from a rather nontraditional perspective. To set the stage, we review the notion of a multiplet for a general super Lie algebra, working in the context of the BV and BRST formalisms. Building on this, we explain how the pure spinor superfield formalism can be viewed as constructing a supermultiplet out of the input datum of an equivariant graded module over the ring of functions on the nilpotence variety. We use the homotopy transfer theorem and other computational techniques from homological algebra to rela…
In the framework of light-cone gauge formulation, massless arbitrary spin N=1 supermultiplets in four-dimensional flat space are considered. We study both the integer (super)spin and half-integer (super)spin supermultiplets. For such supermultiplets, formulation in terms of unconstrained light-cone gauge superfields defined on chiral momentum superspace is used. Superfield representation for all cubic interaction vertices of the supermultiplets is obtained. Representation of the cubic vertices in terms of component fields is derived. Realization of relativistic symmetries of N=1 Poincare super…
The genuine Kaluza-Klein-like theories (with no fields in addition to gravity) have difficulties with the existence of massless spinors after the ompactification of some of dimensions of space\cite{witten}. We assume a $M^{(1+3)} \times$ a flat finite disk in $(1+5)$-dimensional space, with the boundary allowing spinors of only one handedness. Massless spinors then chirally couple to the corresponding background gauge gravitational field, which solves equations of motion for a free field, linear in the Riemann curvature.
In this letter we suggest a natural spinor-helicity formalism for massless fields in AdS${}_4$. It is based on the standard realization of the AdS${}_4$ isometry algebra $so(3,2)$ in terms of differential operators acting on $sl(2,\mathbb{C})$ spinor variables. We start by deriving the AdS counterpart of plane waves in flat space and then use them to evaluate simple scattering amplitudes. Finally, based on symmetry arguments we classify all three-point amplitudes involving massless spinning fields. As in flat space, we find that the spinor-helicity formalism allows to construct additional cons…
The technique for representing spinors and the definition of the discrete symmetries is used to illustrate on a toy model properties of massless and massive spinors states, in the first and the second quantized picture. Since in this toy model the number of the starting massless representations is well defined as well as the origin of masses and charges in $d=(3+1)$ space, this contribution might help to clarify the problem about Dirac, Weyl and Majorana kinds of representations in physically more interesting cases.
We describe a Lagrangian quantization of the free massless gauge superfield theories of higher superspins both in the anti-de Sitter and flat global superspaces.
A FORM program which is used to efficiently expand in components pure spinor superfield expressions of kinematic factors is presented and comments on how it works are made. It is highly customizable using the standard features of FORM and can be used to help obtaining superstring effective actions from the scattering amplitudes computed with the pure spinor formalism.
We compute the massless five-point amplitude of open superstrings using the non-minimal pure spinor formalism and obtain a simple kinematic factor in pure spinor superspace, which can be viewed as the natural extension of the kinematic factor of the massless four-point amplitude. It encodes bosonic and fermionic external states in supersymmetric form and reduces to existing bosonic amplitudes when expanded in components, therefore proving their equivalence. We also show how to compute the kinematic structures involving fermionic states.
We show that a certain superfield formalism can be used to find an off-shell supersymmetric description for some supersymmetric field theories where conventional superfield formalism does not work. This "new" formalism contains even auxiliary variables in addition to conventional odd super-coordinates. The idea of this construction is similar to the pure spinor formalism developed by N.Berkovits. It is demonstrated that using this formalism it is possible to prove that the certain Chern-Simons-like (Witten's OSFT-like) theory can be considered as an off-shell version for some on-shell supersym…
The genuine Kaluza-Klein-like theories--with no fields in addition to gravity--have difficulties with the existence of massless spinors after the compactification of some space dimensions \cite{witten}. We proposed in previous paper a boundary condition for spinors in d=(1+5) compactified on a flat disk that ensures masslessness of spinors (with all positive half integer charges) in d=(1+3) as well as their chiral coupling to the corresponding background gauge gravitational field. In this paper we study the same toy model, proposing a boundary condition allowing a massless spinor of one handed…
Using the pure spinor formalism we prove identities which relate the tree-level, one-loop and two-loop kinematic factors for massless four-point amplitudes. From these identities it follows that the complete supersymmetric one- and two-loop amplitudes are immediately known once the tree-level kinematic factor is evaluated. In particular, the two-loop equivalence with the RNS formalism (up to an overall coefficient) is obtained as a corollary.
The $b$ ghost, or $b$ operator, used for fixing Siegel gauge in the pure spinor superfield formalism, is a composite operator of negative ghost number, satisfying $\{q,b\}=\square$, where $q$ is the pure spinor differential (BRST operator). It is traditionally constructed using non-minimal variables. However, since all cohomology has minimal representatives, it seems likely that there should be versions of physically meaningful operators, also with negative ghost number, using only minimal variables. The purpose of this letter is to demonstrate that this statement holds by providing a concrete…
For the gauge massless higher spin 4D, N = 1 off-shell supermultiplets previously developed, we provide evidence of a twistor-like oscillator realization that is intrinsically related to the superfield structure of the dynamical variables and gauge transformations. Gauge invariant field strengths and linearized Bianchi identities for these multiplets are worked out. It is further argued, inspired by earlier non- supersymmetric constructions due to Klishevich and Zinoviev, that a massive superspin-$s$ multiplet can be described as a gauge-invariant dynamical system involving massless multiplets…
The quantum field theory is expanding in de Sitter space-time. The ambient space formalism let us to develop the quantum field theory in a careful mathematical framework. Using the gauge-covariant derivative in the de Sitter ambient space, the gauge invariant Lagrangian density has been found. In this paper, the equations of the conjugate spinor for massless spin-$\frac{3}{2}$ gravitational field is obtained by Euler-Lagrange equation. Finally, these equations can be written in terms of the operator of the Casimir group de Sitter.
The genuine Kaluza-Klein-like theories (with no fields in addition to gravity with torsion) have difficulties with the existence of massless spinors after the compactification of some of dimensions of space\cite{witten}. We demonstrate in this letter on an example of a flat torus - as a compactified part of an (1+5)-dimensional space - that for constant fields and appropriate boundary conditions there exists in the 1+3 dimensional space a massless solution, which is mass protected and chirally coupled with a Kaluza-Klein charge to the corresponding gauge field.
Free massless higher-superspin superfields on the N=1, D=4 anti-de Sitter superspace are introduced. The linearized gauge transformations are postulated. Two families of dually equivalent gauge-invariant action functionals are constructed for massless half-integer-superspin s+1/2 (s >= 2) and integer-superspin s (s >= 1) superfields. For s=1, one of the formulations for half-integer superspin multiplets reduces to linearized minimal N=1 supergravity with a cosmological term, while the other is the lifting to the anti-de Sitter superspace of linearized non-minimal n=-1 supergravity.
We study contact four-point amplitudes in the spinor-helicity formalism in anti-de Sitter space. We find that these amplitudes can be brought to an especially simple form, which we call canonical. Next, we classify consistent contact amplitudes by requiring correct transformation properties with respect to the AdS isometry algebra. Finally, we establish a connection between the canonical form of AdS amplitudes and scalar multi-trace conformal primaries in flat space.
