Search

The algebra of spacetime supersymmetry generators in the RNS formalism for the superstring closes only up to a picture-changing operation. After adding non-minimal variables and working in the "large" Hilbert space, the algebra closes without picture-changing and spacetime supersymmetry can be made manifest. The resulting non-minimal version of the RNS formalism is related by a field redefinition to the pure spinor formalism., Comment: 10 pages

Previous attempts to determine the worldsheet origin of the pure spinor formalism were not completely successful, but introduced important concepts that seem to be connected to its fundamental structure, e.g., emergent supersymmetry and the role of reparametrization symmetry. In this work, a new proposal towards the underlying gauge theory of the pure spinor superstring is presented, based on an extension of Berkovits' twistor-like constraint. The gauge algebra is analyzed in detail and worldsheet reparametrization is shown to be a redundant symmetry. The master action is built with a careful account of the intrinsic gauge symmetries associated with the pure spinor constraint and a consistent gauge fixing is performed. After a field redefinition, spacetime supersymmetry emerges and the resulting action describes the pure spinor superstring., v1: 12 pages, short version presenting the results; v2: 36 pages, extended version containing a thorough analysis of the gauge algebra, the explicit construction of the pure spinor master action and some side results; v3: 37 pages, title changed, minor improvements and references added, published version

The $D=11$ pure spinor superparticle has been shown to describe linearized $D=11$ supergravity in a manifestly covariant way. A number of authors have proposed that its correlation functions be used to compute amplitudes. The use of the scalar structure of the eleven-dimensional pure spinor top cohomology introduces a natural measure for computing such correlation functions. This prescription requires the construction of ghost number one and zero vertex operators. In these notes, we construct explicitly a ghost number one vertex operator but show the incompatibiliy of a ghost number zero vertex operator satisfying a standard descent equation for $D=11$ supergravity., Comment: 13 pages

At small AdS radius, the superstring on $AdS_5\times S^5$ was conjectured by Maldacena to be equivalent to ${\cal N}=4$ super-Yang-Mills at small `t Hooft coupling where thickened Feynman diagrams can be used to compute scattering amplitudes. It was previously shown that the pure spinor worldsheet action of the $AdS_5\times S^5$ superstring can be expressed as the sum of a BRST-trivial term and a `B-term' which is antisymmetric in worldsheet derivatives. Using the explicit form of the pure spinor vertex operators, it will be argued here that the free super-Yang-Mills Feynman diagrams are described by the BRST-trivial term where the thickened propagators are the regions of the string worldsheet near the AdS boundary and the holes are the regions near the AdS horizon. Evidence will then be presented that the antisymmetric B-term generates the super-Yang-Mills vertex so that, at small radius and arbitrary genus, the superstring amplitudes correctly reproduce the super-Yang-Mills Feynman diagram expansion., Comment: 17 pages with 2 figures. Added references

The general fluctuations, in the form of vertex operators, for the type II superstring in the pure spinor formalism are considered. We review the construction of these vertex operators in flat space-time. We then review the type II superstrings in curved background in the pure spinor formalism to finally construct the vertex operators on a generic type II supergravity background., Comment: harvmac-big, 13+1 pages. Added a section

Pure spinor formalism implies that supergravity equations in space-time are equivalent to the requirement that the worldsheet sigma-model satisfies certain properties. Here we point out that one of these properties has a particularly transparent geometrical interpretation. Namely, there exists an odd nilpotent vector field on some singular supermanifold, naturally associated to space-time. All supergravity fields are encoded in this vector field, as coefficients in its normal form. The nilpotence implies, modulo some zero modes, that they satisfy the SUGRA equations of motion., LaTeX, 22pp; v2: corrected mistakes, added explanations, new Section 7; v3: we show that the nilpotence of Q implies all SUGRA equations of motion, and discuss div Q in the Introduction

The OPE between the composite $b$ ghost and the unintegrated vertex operator for massless states of the pure spinor superstring is computed and shown to reproduce the structure of the bosonic string result. The vanishing of the double pole singularity is Lorentz gauge and the single pole is shown to be equal to the corresponding integrated vertex operator., Latex, 24 pages, 1 figure. Reference added. Minor corrections

We provide a linearised superfield description of the exotic non-metric $N=(4,0)$ supergravity in $D=6$, by using a pure spinor superfield formalism. The basic field $\Psi$ is a ghost number 2 scalar, transforming in the same R-symmetry module as the tensor fields. Partial results for the $N=(3,1)$ model are presented., Comment: v2. 14 pp. Expanded version, added results on N=(3,1)

For several classes of BPS vacua, we find a procedure to modify the PDEs that imply preserved supersymmetry and the equations of motion so that they still imply the latter but not the former. In each case we trace back this supersymmetry-breaking deformation to a distinct modification of the pure spinor equations that provide a geometrical interpretation of supersymmetry. We give some concrete examples: first we generalize the Imamura class of Mink6 solutions by removing a symmetry requirement, and then derive some local and global solutions both before and after breaking supersymmetry., 30 pages, references added

The construction of the ghost number zero and one vertex operators for the 11D pure spinor superparticle will be revisited. In this sense, an alternative way of defining the ghost number one vertex operator will be given after introducing a ghost number -2 operator made out of physical operators defined on the 11D non-minimal pure spinor superspace. This procedure will make explicit and transparent the relation between the ghost number three and one vertex operators. In addition, using a non-Lorentz covariant b-ghost, ghost number zero and two vertex operators satisfying standard descent equations will be presented in full form., 16 pages. Added reference. Small modification to the published version

We present a novel twistor formulation of the ten-dimensional massless superparticle. This formulation is based on the introduction of pure spinor variables through a field redefinition of another model for the superparticle, and in the new description we find that the super-Pauli-Lubanski three-form naturally arises as a constraint. Quantization is studied in detail for both models and they are shown to correctly describe the $D = 10$ super-Yang-Mills states., 23 pages. Fixed typos, appendix added. Published version

We construct the integrated vertex operator for the first massive states of open superstrings with $(mass)^2=1/\alpha'$ in the pure spinor formalism of the superstring theory. This vertex operator is expressed in terms of the ten dimensional $\mathcal{N}=1$ superfields describing the massive supermultiplet which appear in the unintegrated vertex operator of the same states., Comment: 29 pages, some clarifications added, matches with the published version

The pure spinor formulation of superstring theory includes an interacting sector of central charge $c_{\lambda}=22$, which can be realized as a curved $\beta\gamma$ system on the cone over the orthogonal Grassmannian $\text{OG}^{+}(5,10)$. We find that the spectrum of the $\beta\gamma$ system organizes into representations of the $\mathfrak{g}=\mathfrak{e}_6$ affine algebra at level $-3$, whose $\mathfrak{so}(10)_{-3}\oplus {\mathfrak u}(1)_{-4}$ subalgebra encodes the rotational and ghost symmetries of the system. As a consequence, the pure spinor partition function decomposes as a sum of affine $\mathfrak{e}_6$ characters. We interpret this as an instance of a more general pattern of enhancements in curved $\beta\gamma$ systems, which also includes the cases $\mathfrak{g}=\mathfrak{so}(8)$ and $\mathfrak{e}_7$, corresponding to target spaces that are cones over the complex Grassmannian $\text{Gr}(2,4)$ and the complex Cayley plane $\mathbb{OP}^2$. We identify these curved $\beta\gamma$ systems with the chiral algebras of certain $2d$ $(0,2)$ CFTs arising from twisted compactification of 4d $\mathcal{N}=2$ SCFTs on $S^2$., Comment: 8 pages, 1 figure; v2: added references, minor updates

It is well known in NSR string theory, that vertex operators can be constructed in various ``pictures''. Recently this was discussed in the context of pure spinor formalism. NSR picture changing operators have an elegant super-geometrical interpretation. In this paper we provide a generalization of this super-geometrical construction, which is also applicable to the pure spinor formalism., Comment: 14 pages, typos corrected, comments added

We evaluate closed string disk amplitudes in the pure spinor formalism. We focus on low point functions (two- and one-point functions) but our analysis is also relevant for higher n-point functions. Amongst others we discuss issues arising due to the gauge fixing of the conformal Killing group of the disk and due to the zero mode prescription in the pure spinor formalism. As expected, in the end we find agreement with the known results from the RNS formalism., Comment: 28 pages, v2: Small changes; in particular added footnote 2 and reference [11]; version published by JHEP

After constructing a simplified four-dimensional version of the b ghost, topological multiloop amplitudes in type II superstring theory compactified on a six-dimensional orbifold are computed using the non-minimal pure spinor formalism. These pure spinor amplitude computations preserve manifest $N = 2$ $D = 4$ supersymmetry and, unlike the analogous topological multiloop amplitude computations using the hybrid formalism, can be extended to non-topological amplitudes., 32 pages

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. Several objects related to these superfields, such as their non-local counterparts and the so-called BRST pseudo-invariants, are thoroughly reviewed and put into new light. Their properties will turn out to be mysteriously connected to products of one-loop worldsheet functions in packages dubbed "generalized elliptic integrands", whose prominence will be seen in the later parts of this series of papers., Comment: 72 pp, v2 published version

An attempt is made to apply the pure spinor formalism to the non-relativistic IIA D2-brane. The fermionic constraints corresponding to the rescaled fermionic coordinates are given. Two commuting spinor fields are introduced, each one corresponding to a fermionic constraint. A BRST charge is constructed via the ansatz proposed by N. Berkovits. The nilpotency of the BRST charge leads to a set of constraints for the two spinor fields including pure spinor constraints. A novel non-trivial solution is given for one of the spinor fields which can be written as a sum of two pure spinors., Comment: 23 pages in latex, no figures; discussion added, some typos corrected, more references added

Using the pure spinor formalism for the superstring in an $AdS_5\times S^5$ background, a simple expression is found for half-BPS vertex operators. At large radius, these vertex operators reduce to the usual supergravity vertex operators in a flat background. And at small radius, there is a natural conjecture for generalizing these vertex operators to non-BPS states., Comment: 11 pages, fixed typos

We present an integrable deformation of the $AdS_{5}\times S^{5}$ pure spinor action based on homological perturbation theory. Its equations of motion, Lax connection and BRST symmetry are discussed. The resulting model describes a pure spinor superstring in a generalized supergravity background., Final version. 23 pages. Added references, improved few sentences. Version to be published in JHEP

We consider the kinematics of the locally BPS super-Wilson loop in N=4 super-Yang-Mills with scalar coupling from a twistorial point of view. We find that the kinematics can be described either as supersymmetrized pure spinors or as a point in the product of two super-Grassmannian manifolds G2∣2(4∣4)×G2∣2(4∣4). In this description of the kinematics the scalar–scalar correlation function appearing in the one-loop evaluation of the super-Wilson loop can be neatly written as a sum of four superdeterminants.

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., Comment: 27 pages, typos fixed, presentation clarified, version to appear in PRD

In this article we present bosonic T-dualization in double space of the type II superstring theory in the pure spinor formulation. We use the action with constant background fields obtained from the general case under some physically and mathematically justified assumptions. Unlike Nikolić and Sazdović (EPJ C 77:197, 2017), where we used the first-order theory, in this article fermionic momenta are integrated out. Full T-dualization in double space is represented as a permutation of the initial $$x^\mu $$xμ and T-dual coordinates $$y_\mu $$yμ. Requiring that a T-dual transformation law of the T-dual double coordinate $${}^\star Z^M=(y_\mu ,x^\mu )$$⋆ZM=(yμ,xμ) to be of the same form as for initial one $$Z^M=(x^\mu ,y_\mu )$$ZM=(xμ,yμ), we obtain the form of the T-dual background fields in terms of the initial ones. The advantage of using the action with integrated fermionic momenta is that it gives all T-dual background fields in terms of the initial ones. In the case of the first-order theory Nikolić and Sazdović (2017) a T-dual R-R field strength was obtained out of the double space formalism under additional assumptions.

In this article we consider simultaneous T-dualization of type II superstring action in pure spinor formulation. Simultaneous T-dualization means that we make T-dualization at the same time along some subset of initial coordinates marked by $x^a$. The only imposed assumption stems from the applicability of the Buscher T-dualization procedure - background fields do not depend on dualized directions $x^a$. In this way we obtain the full form of the T-dual background fields and T-dual transformation laws. Because two chiral sectors transform differently, there are two sets of vielbeins and gamma matrices connected by the local Lorentz transformation. Its spinorial representation is the same as in the constant background case. We also found the full expression for T-dual dilaton field., Comment: Comments on dilaton transformation added

In this work, a solution linear in the momentum for the massless constraint $P^{m}P_{m}=0$ is investigated. It is presented in terms of a $SO(2n,\mathbb{C})$ to $U(n)$ decomposition and interpreted in terms of projective pure spinors, which are known to parametrize the $\tfrac{SO(2n)}{U(n)}$ coset. The worldline action is quantized using the BRST formalism and, using the results of Berkovits and Cherkis, the ghost number zero wave function is shown to generate massless solutions for field equations of arbitrary spin. The model can be covariantly expressed by the action recently proposed in $D=10$ by Berkovits, in terms of a twistor-like constraint. However, a thorough account of its gauge symmetries does not lead to a spacetime supersymmetric theory. In order to derive from first principles the superparticle in the pure spinor formalism, a new model is proposed with partial worldline supersymmetry. The gauge algebra is then analyzed within the Batalin-Vilkovisky formalism and the gauge fixed action is finally shown to describe the pure spinor superparticle times a $U(1)$ decoupled sector., Comment: 20 pages. Small clarification added to the published version

In this work we give an explicit construction for the vertex operators of massless states in the pure spinor superstring in a plane wave background. The construction is based on the observation that the full action can be divided in two parts, where the simpler one is based on a smaller coset and closely resembles the gauge fixed Green-Schwarz action., Comment: Latex, 28 pages

The $b$-ghost of the pure spinor formalism in a general curved background is not holomorphic. For such theories, the construction of the string measure requires the knowledge of the action of diffeomorphisms on the BV phase space. We construct such an action for the pure spinor sigma-model in $AdS_5\times S^5$. From the point of view of the BV formalism, this sigma-model belongs to the class of theories where the expansion of the Master Action in antifields terminates at the quadratic order. We show that it can be reduced to a simpler degenerate sigma-model, preserving the AdS symmetries. We construct the action of the algebra of worldsheet vector fields on the BV phase space of this minimalistic sigma-model, and explain how to lift it to the original model., 35pp. LaTeX; v2,3,4: corrections, improved presentation, v5: added Section 5.4 (gluing the BV phase space from charts) and Section 6, v6: published version with minor corrections

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 $D=11$ supergravity., 20 pages. Added reference to paper of Chang, Lin, Wang and Yin

The $D=11$ pure spinor formulation of the superparticle allows a simple realization of covariant quantization, unlike the $D=11$ Brink-Schwarz-like superparticle. We explicitly show the equivalence between the cohomologies of these two models in the context of two different group decompositions: $SO(10,1) \rightarrow SO(1,1)\times SO(9)$ and $SO(10,1) \rightarrow SO(3,1)\times SO(7)$. We also carry out a light-cone analysis of the pure spinor cohomology, and show that it correctly reproduces the $SO(9)$ equations of motion for $D=11$ linearized supergravity., 24 pages. Fixed typos and added footnote

We provide the covariant superspace equations that are sufficient to determine the complete $\theta$ expansion of the vertex operator of the open string massive states with $(mass)^2=1/\alpha'$ 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 $\theta$ expansion of the superfields appearing in the unintegrated vertex to $O(\theta^3)$. Finally, we compute the contribution to a 3-point tree amplitude with the resulting vertex operator upto $O(\theta^3)$ and find its kinematic structure to be identical to the corresponding RNS computation., Comment: 32 pages

Same amplitudes evaluated independently using RNS and pure spinor formalism are expected to agree. While for massless states, this fact has been firmly established, for massive states such an explicit check has been lacking so far. We compute all massless-massless-massive 3-point functions in open supertrings in pure spinor formalism for the first massive states and compare them with the corresponding RNS results. We fix the normalization of the vertex operators of the massive states by comparing same set of 3-point functions for a fixed ordering in the two formalisms. Once fixed, the subsequent 3-point functions for each inequivalent ordering match exactly. This extends the explicit demonstration of equivalence of pure spinor and RNS formalism from massless states to first massive states., Comment: 33 pages

We showed in previous work that for homogeneous Yang-Baxter (YB) deformations of AdS$_5\times$S$^5$, the open string metric and coupling, and as a result the closed string density $e^{-2 \Phi} \sqrt{g}$, remain undeformed. In this work, in addition to extending these results to the deformation associated with the modified CYBE, or $\eta$-deformation, we identify the Page forms as the open string counterpart for RR fields and demonstrate case by case that the non-zero Page forms remain invariant under YB deformations. We give a physical meaning to the Killing vector $I$ of generalized supergravity and show for all YB deformations: 1) $I$ appears as a current for center of mass motion on the worldvolume of a D-branes probing the background, 2) $I$ is equal to the divergence of the noncommutativity parameter, 3) $I$ exhibits "holographic" behavior, where the radial component of $I$ vanishes at the AdS boundary, and 4) in pure spinor formalism $I$ is related to a certain state in the BRST cohomology., Comment: 11 pages, 2 column; v2 references updated; v3 to appear in EPJC

Open supermembrane with a constant three-form flux in the pure spinor formalism is examined. The BRST symmetry of the open supermembrane action leads to non-commutative (NC) M-branes. In addition to the NC M5-brane with a self-dual two-form flux, we find a NC M9-brane with an electric flux and a NC M9-brane with a magnetic flux. The former reduces in the critical electric flux limit to an M2-brane on the M9-brane, while the latter reduces in the strong magnetic flux limit to infinitely many Kaluza-Klein monopoles dissolved into the M9-brane. These NC M-branes are shown to preserve a half of 32 supersymmetries., 17 pages, published version

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 a superfield based on the physical scalars. The formalism should be suited for calculations of correlators of traces of scalar superfields., 13 pp

A $b$-ghost was constructed for the $D=11$ non-minimal pure spinor superparticle by requiring that $\{Q , b\} = T$ where $Q = \Lambda^{\alpha}D_{\alpha} + R^{\alpha}\bar{W}_{\alpha}$ is the usual non-minimal pure spinor BRST operator. As was done for the $D=10$ $b$-ghost, we will show that the $D=11$ $b$-ghost can be simplified by introducing an $SO(10,1)$ fermionic vector $\bar{\Sigma}^{i}$ constructed out of the fermionic spinor $D_{\alpha}$ and pure spinor variables. This simplified version will be shown to satisfy $\{Q, b\} = T$ and $\{b , b\} =$ BRST - trivial., Comment: 47 pages. Added references

The integrated massless vertex operator in an $AdS_5\times S^5$ background in the pure spinor formalism is constructed in terms of superfields., Comment: 22 pages, latex. Fixed discussion on gauge invariant vertex operators

The sectorized description of the (chiral) heterotic string using pure spinors has been misleadingly viewed as an infinite tension string. One evidence for this fact comes from the tree level 3-point graviton amplitude, which we show to contain the usual Einstein term plus a higher curvature contribution. After reintroducing a dimensionful parameter $\ell$ in the theory, we demonstrate that the heterotic model is in fact two-fold, depending on the choice of the supersymmetric sector, and that the spectrum also contains one massive (open string like) multiplet. By taking the $\ell\to\infty$ limit, we finally show that the ambitwistor string is recovered, reproducing the unexpected heterotic state in Mason and Skinner's RNS description., Comment: 30 pages, one figure. Clarifications and references added. Published version

The purpose of this contribution is to initiate the study of integrable deformations for different superstring theory formalisms that manifest the property of (classical) integrability. In this paper we choose the hybrid formalism of the superstring in the background AdS_{2}xS^{2} and explore in detail the most immediate consequences of its lambda-deformation. The resulting action functional corresponds to the lambda-model of the matter part of the fairly more sophisticated pure spinor formalism, which is also known to be classical integrable. In particular, the deformation preserves the integrability and the one-loop conformal invariance of its parent theory, hence being a marginal deformation., LaTex, 37 pages, references added, match published version in Journal of High Energy Physics, 2016

A geometry of superspace corresponding to double field theory is developed, with type II supergravity in D=10 as the main example. The formalism is based on an orthosymplectic extension OSp(d,d|2s) of the continuous T-duality group. Covariance under generalised super-diffeomorphisms is manifest. Ordinary superspace is obtained as a solution of the orthosymplectic section condition. A systematic study of curved superspace Bianchi identities is performed, and a relation to a double pure spinor superfield cohomology is established. A Ramond-Ramond superfield is constructed as an infinite-dimensional orthosymplectic spinor. Such objects in minimal orbits under the OSp supergroup ("pure spinors") define super-sections., Comment: 28 pp., plain tex. v2: refs. added

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 $\Theta^\alpha$ together with a superfield constraint relating $DX^m$ and $D\Theta^\alpha$. The constraint implies that the worldsheet superpartner of $\theta^\alpha$ is a bosonic twistor variable, and different solutions of the constraint give rise to the pure spinor or extended RNS formalisms, as well as a new twistor-string formalism with manifest N=1 worldsheet supersymmetry. These N=1 worldsheet methods generalize in curved Ramond-Ramond backgrounds, and a manifestly N=1 worldsheet supersymmetric action is proposed for the superstring in an $AdS_5\times S^5$ background in terms of the twistor superfields. This $AdS_5\times S^5$ worldsheet action is a remarkably simple fermionic coset model with manifest $PSU(2,2|4)$ symmetry and might be useful for computing $AdS_5\times S^5$ superstring scattering amplitudes., Comment: 30 pages. Corrected typos, added 2 footnotes, and dedicated paper to Mario Tonin

We show that the first-order symmetry operators of twistor spinors can be constructed from conformal Killing-Yano forms in conformally-flat backgrounds. We express the conditions on conformal Killing-Yano forms to obtain mutually commuting symmetry operators of twistor spinors. Conformal superalgebras which consist of conformal Killing vectors and twistor spinors and play important roles in supersymmetric field theories in conformal backgrounds are extended to more general superalgebras by using the graded Lie algebra structure of conformal Killing-Yano forms and the symmetry operators of twistor spinors. The even part of the extended conformal superalgebra corresponds to conformal Killing-Yano forms and the odd part consists of twistor spinors., 16 pages, published version

In this work, some aspects of the ambitwistor pure spinor string are investigated. The $b$ ghost is presented and its main properties are derived in a simple way, very similar to the usual pure spinor $b$ ghost construction. The heterotic case is also addressed with a new proposal for the BRST charge. The BRST cohomology is shown to correctly describe the heterotic supergravity spectrum and a semi-composite $b$ ghost is constructed., 23 pages, reference and clarifications added, typos fixed. Published version

Continuing our exploration of maximally supersymmetric gauge theories (MSYM) deformed by higher dimensional operators, in this paper we consider an off-shell approach based on pure spinor superspace and focus on constructing supersymmetric deformations beyond the first order. In particular, we give a construction of the Batalin-Vilkovisky action of an all-order non-Abelian Born-Infeld deformation of MSYM in the non-minimal pure spinor formalism. We also discuss subtleties in the integration over the pure spinor superspace and the relevance of Berkovits-Nekrasov regularization., Kavli Institute for Theoretical Physics (Graduate Fellowship), Alfred P. Sloan Foundation. Fellowship, Simons Foundation (Simons Investigator Award), National Science Foundation (U.S.) (Award PHY-0847457)

In {\em Phys.\ Lett.} {\bf B 660}, 583 (2008), it was proposed that the D-brane geometry could be produced by open string quantum effects. In an effort to verify the proposal, we consider scattering amplitudes involving {\em massive} open superstrings. The main goal of this paper is to set the ground for two-loop "renormalization" of an oriented open superstring on a D-brane and to strengthen our skill in the pure spinor formulation of a superstring, an effective tool for multi-loop string diagrams. We start by reviewing scattering amplitudes of massless states in the 2D component method of the NSR formulation. A few examples of massive string scattering are worked out. The NSR results are then reproduced in the pure spinor formulation. We compute the amplitudes using the massive vertex operator constructed by Berkovits and Chandia in {\em JHEP} {\bf 0208}, 040 (2002). We point out that it may be possible to discover new Riemann type identities involving Jacobi $\vartheta$-functions by comparing a NSR computation and the corresponding pure spinor computation., Comment: 38 pages, no figure, three-point amplitudes invloving massive fermions added, to appear in NPB

We show that new BRST charges in RNS superstring theory with nonstandard ghost numbers, constructed in our recent work, can be mapped to deformed pure spinor (PS) superstring theories, with the nilpotent pure spinor BRST charge $Q_{PS}=\oint{\lambda^\alpha}d_\alpha$ still retaining its form but with singular operator products between commuting spinor variables $\lambda^\alpha$. Despite the OPE singularities, the pure spinor condition $\lambda\gamma^m\lambda=0$ is still fulfilled in a weak sense, explained in the paper. The operator product singularities correspond to introducing interactions between the pure spinors. We conjecture that the leading singularity orders of the OPE between two interacting pure spinors is related to the ghost number of the corresponding BRST operator in RNS formalism. Namely, it is conjectured that the BRST operators of minimal superconformal ghost numbers $n=1,2,3...$ can be mapped to nilpotent BRST operators in the deformed pure spinor formalism with the OPE of two commuting spinors having a leading singularity order $\lambda(z)\lambda(w)\sim{O}(z-w)^{-2(n^2+6n+1)}$. The conjecture is checked explicitly for the first non-trivial case $n=1$., Comment: 21 pages

All the BRST-invariant operators in pure spinor formalism in $d=10$ can be represented as BRST commutators, such as $V=\lbrace{Q_{brst}},{{\theta_{+}}\over{\lambda_{+}}}V\rbrace$ where $\lambda_{+}$ is the U(5) component of the pure spinor transforming as $1_{5\over2}$. Therefore, in order to secure non-triviality of BRST cohomology in pure spinor string theory, one has to introduce "small Hilbert space" and "small operator algebra" for pure spinors, analogous to those existing in RNS formalism. As any invariant vertex operator in RNS string theory can also represented as a commutator $V=\lbrace{Q_{brst}},LV\rbrace$ where $L=-4c\partial\xi{\xi}e^{-2\phi}$, we show that mapping ${{\theta_{+}}\over{\lambda_{+}}}$ to L leads to identification of the pure spinor variable $\lambda^{\alpha}$ in terms of RNS variables without any additional non-minimal fields. We construct the RNS operator satisfying all the properties of $\lambda^\alpha$ and show that the pure spinor BRST operator $\oint{\lambda^\alpha{d_\alpha}}$ is mapped (up to similarity transformation) to the BRST operator of RNS theory under such a construction. We also observe that BRST cohomology of pure spinor superstring theory contains vertex operators non-trivially coupled to the pure spinor ghost variable. The physical interpretation of these operators remains unclear., Comment: 11 pages, harvmac.tex Typos corrected, reference and appendix added

We consider the covariant quantization of the D=11 massless superparticle (M0-brane) in the spinor moving frame or twistor-like Lorentz harmonics formulation. The action involves the set of 16 constrained 32 component Majorana spinors, the spinor Lorentz harmonics parametrizing (as homogeneous coordinates, modulo gauge symmetries) the celestial sphere S9. There presence allows us to separate covariantly the first and the second class constraints of the model. After taking into account the second class constraints by means of Dirac brackets and after further reducing the first class constraints algebra, the system is described in terms of a simple BRST charge associated to the d=1, n=16 supersymmetry algebra. The study of the cohomology of this BRST operator requires a regularization by complexifying the bosonic ghosts for the kappa-symmetry and further reduction of the regularized cohomology problem to the one for a simpler complex BRST charge which is essentially the pure spinor BRST operator by Berkovits, but with a composite pure spinor. This exhibits a possible origin of the complexity (non-hermiticity) characteristic of the Berkovits pure spinor approach. The simple structure of the nontrivial cohomology of the M0-brane BRST charge finds explanation in the properties that the superparticle action exhibits in the so-called `covariantized light-cone' basis. The the covariant quantization in this basis hints possible hidden symmetries of D=11 supergravity. Besides SO(16), we discuss also some indirect arguments in favor of the possible E8 symmetry., 40 pages, no figures. V2: some references and comments on them added. V.3: misprints corrected. V.4: reference on 1985 preprint by M.J. Duff ([62]) and notice on it added. Published in Nucl. Phys. B

To exhibit the possible origin of the inner complexity of the Berkovits's pure spinor approach, we consider the covariant BRST quantization of the D=11 massless superparticle (M0-brane) in its spinor moving frame or twistor-like Lorentz harmonics formulation. The presence of additional twistor-like variables (spinor harmonics) allows us to separate covariantly the first and the second class constraints. After taking into account the second class constraints by means of Dirac brackets and after further reducing the first class constraints algebra, the dynamical system is described by the cohomology of a simple BRST charge associated to the d=1, n=16 supersymmetry algebra. The calculation of the cohomology of this BRST operator requires a regularization which implies the complexification of the bosonic ghost associated to the kappa-symmetry and further leads to a complex (non-Hermitian) BRST charge which is essentially the `pure spinor' BRST operator by Berkovits, but with a composite pure spinor., Comment: plain LaTeX, 15 pages, no figures. V2- minor changes: misprints corrected, references added, clarifying comments on relation with spinor cohomologies. V3 Shorterned, to appear in Phys.Lett. B

In this article we offer the interpretation of the fermionic T-duality of the type II superstring theory in double space. We generalize the idea of double space doubling the fermionic sector of the superspace. In such doubled space fermionic T-duality is represented as permutation of the fermionic coordinates $\theta^\alpha$ and $\bar\theta^\alpha$ with the corresponding fermionic T-dual ones, $\vartheta_\alpha$ and $\bar\vartheta_\alpha$, respectively. Demanding that T-dual transformation law has the same form as inital one, we obtain the known form of the fermionic T-dual NS-R i R-R background fields. Fermionic T-dual NS-NS background fields are obtained under some assumptions. We conclude that only symmetric part of R-R field strength and symmetric part of its fermionic T-dual contribute to the fermionic T-duality transformation of dilaton field and analyze the dilaton field in fermionic double space. As a model we use the ghost free action of type II superstring in pure spinor formulation in approximation of constant background fields up to the quadratic terms., Comment: Four paragraphs in the Introduction added in order to better motivate the subject, explained the choice of action (detailed derivation)

A simplified pure spinor superstring $b$ ghost in a curved heterotic background was constructed recently. The $b$ ghost is a composite operator and it is not holomorphic. However, it satisfies $\bar\partial b = [ Q , \Omega ]$, where $Q$ is the BRST charge. In this paper, we find a possible $\Omega$., Comment: 21 pages