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.
We present the Y-formalism for the non-minimal pure spinor quantization of superstrings. In the framework of this formalism we compute, at the quantum level, the explicit form of the compound operators involved in the construction of the $b$ ghost, their normal-ordering contributions and the relevant relations among them. We use these results to construct the quantum-mechanical $b$ ghost in the non-minimal pure spinor formalism. Moreover we show that this non-minimal $b$ ghost is cohomologically equivalent to the non-covariant $b$ ghost.
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.
As a check of the first massive integrated vertex operator in the pure spinor formalism constructed in arXiv:1802.04486, we compute the one loop 2-point function of the stable non-BPS massive states in SO(32) heterotic string theory. This allows us to compute the one loop renormalized mass of these states using the pure spinor formalism. Our results are in agreement with the corresponding results obtained by Sen using the RNS formalism. This provides an instance of the equivalence between the RNS and the pure spinor formalism for the massive states at loop level.
A manifestly super-Poincaré covariant formalism for the superstring has recently been constructed using a pure spinor variable. Unlike the covariant Green-Schwarz formalism, this new formalism is easily quantized with a BRST operator and tree-level scattering amplitudes have been evaluated in a manifestly covariant manner. In this paper, the cohomology of the BRST operator in the pure spinor formalism is shown to give the usual light-cone Green-Schwarz spectrum. Although the BRST operator does not directly involve the Virasoro constraint, this constraint emerges after expressing the pure spino…
Using the pure spinor formalism, we compute the tree-level correlation functions for three strings, one closed and two open, in N=1 D=10 superspace. Expanding the superfields in components, the respective terms of the effective action for the type I supergravity are obtained. All terms found agree with the effective action known in the literature. This result gives one more consistency test for the pure spinor formalism.
The discovery of pure spinor formalism makes the computation of superstring scattering amplitudes possible. In this paper, we will illustrate how computer algebra system Cadabra is used in computing the supersymmetric amplitude in pure spinor formalism and provide the source code that computes the tree-level massless 5-gluon amplitude.
All the BRST-invariant operators in pure spinor formalism in $d=10$ can be represented as BRST commutators, such as $V=\lbrace{Q_{brst}},{θ_{+}\over{λ_{+}}}V\rbrace$ where $λ_{+}$ 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\rbr…
After adding an RNS-like fermionic vector ψ^m to the pure spinor formalism, the non-minimal b ghost takes a simple form similar to the pure spinor BRST operator. The N=2 superconformal field theory generated by the b ghost and the BRST current can be interpreted as a "dynamical twisting" of the RNS formalism where the choice of which spin half ψ^m variables are twisted into spin 0 and spin 1 variables is determined by the pure spinor variables that parameterize the coset SO(10)/U(5).
A covariant map between the Ramond-Neveu-Schwarz (RNS) and pure spinor formalisms for the superstring is found which transforms the RNS and pure spinor BRST operators into each other. The key ingredient is a dynamical twisting of the ten spin-half RNS fermions into five spin-one and five spin-zero fermions using bosonic pure spinors that parameterize an SO(10)/U(5) coset. The map relates massless vertex operators in the two formalisms, and gives a new description of Ramond states which does not require spin fields. An argument is proposed for relating the amplitude prescriptions in the two for…
We study to what extent it is possible to generalise Berkovits' pure-spinor construction in d=10 to lower dimensions. Using a suitable definition of a ``pure'' spinor in d=4,6, we propose models analogous to the d=10 pure-spinor superstring in these dimensions. Similar models in d=2,3 are also briefly discussed.
We explicitly compute up to the fifth mass-level the partition function of ten-dimensional pure spinor worldsheet variables including the spin dependence. After adding the contribution from the (x^μ, θ^α, p_α) matter variables, we reproduce the massive superstring spectrum. Even though pure spinor variables are bosonic, the pure spinor partition function contains fermionic states which first appear at the second mass-level. These fermionic states come from functions which are not globally defined in pure spinor space, and are related to the b ghost in the pure spinor formalism. This result cla…
We present the Y-formalism in the pure spinor quantization of superstring theory in detail. Even if the $ω-λ$ OPE is not completely free owing to the presence of the projector reflecting the pure spinor constraint, it is shown that one can construct at the quantum level a conformal field theory in full agreement with the Berkovits' pure spinor formalism. The OPE's of the two formalisms are the same if we define the relevant operators involving $ω$ in a suitable manner. Moreover the Y-formalism at hand is utilized to find the full expression of the covariant, picture raised $b$ antighost $b_B$,…
Following suggestions of Nekrasov and Siegel, a non-minimal set of fields are added to the pure spinor formalism for the superstring. Twisted $\hat c$=3 N=2 generators are then constructed where the pure spinor BRST operator is the fermionic spin-one generator, and the formalism is interpreted as a critical topological string. Three applications of this topological string theory include the super-Poincare covariant computation of multiloop superstring amplitudes without picture-changing operators, the construction of a cubic open superstring field theory without contact-term problems, and a ne…
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.
In this paper we revisit Berkovits' pure spinor formalism in lower dimensions. We are particularly interested in relating a six-dimensional pure spinor action previously constructed in the literature to other superstring formalisms. In order to gain some insight into six-dimensional pure spinors, we first derive their action by gauge-fixing the classical six-dimensional Green-Schwarz action. We then consider a hybrid pure spinor construction in which the spacetime symmetries of six of the ten dimensions are described in pure spinor variables, while the remaining four dimensions are parameteriz…
In order to gain deeper understanding of pure-spinor-based formalisms of superstring, an explicit similarity transformation is constructed which provides operator mapping between the light-cone Green-Schwarz (LCGS) formalism and the extended pure spinor (EPS) formalism, a recently proposed generalization of the Berkovits' formalism in an enlarged space. By applying a systematic procedure developed in our previous work, we first construct an analogous mapping in the bosonic string relating the BRST and the light-cone formulations. This provides sufficient insights and allows us to construct the…
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 subseque…
We study lower-dimensional superstrings in the double-spinor formalism introduced by Aisaka and Kazama. These superstrings can be consistently quantized and are equivalent to the lower-dimensional pure-spinor superstrings proposed by Grassi and Wyllard. The unexpected physical spectrum of the pure-spinor superstrings may thus be regarded as a manifestation of noncriticality. We also discuss how to couple these covariant superstrings to the compactified degrees of freedom described by the N=2 superconformal field theory.
In this paper we study the pure spinor formulation of the superstring in $AdS_5\times S^5$ around point particle solutions of the classical equations of motion. As a particular example we quantize the pure spinor string in the BMN background.
On-shell supergravity vertex operators in an $AdS_5\times S^5$ background are described in the pure spinor formalism by the zero mode cohomology of a BRST operator. After expanding the pure spinor BRST operator in terms of the $AdS_5$ radius variable, this cohomology is computed using N=4 harmonic superspace variables and explicit superfield expressions are obtained for the behavior of supergravity vertex operators near the boundary of $AdS_5$.
The ghost for world-sheet reparametrization invariance is not a fundamental field in the pure spinor formalism. It is written as a combination of pure spinor variables which have conformal dimension two and such that it commutes with the BRST operator to give the world-sheet stress tensor. We show that the ghost variable defined in this way is nilpotent since the OPE of b with itself does not have singularities.
We use the non-minimal pure spinor formalism to compute in a super-Poincare covariant manner the four-point massless one and two-loop open superstring amplitudes, and the gauge anomaly of the six-point one-loop amplitude. All of these amplitudes are expressed as integrals of ten-dimensional superfields in a "pure spinor superspace" which involves five $θ$ coordinates covariantly contracted with three pure spinors. The bosonic contribution to these amplitudes agrees with the standard results, and we demonstrate identities which show how the $t_8$ and $ε_{10}$ tensors naturally emerge from integ…
In a recent paper, the partition function (character) of ten-dimensional pure spinor worldsheet variables was calculated explicitly up to the fifth mass-level. In this letter, we propose a novel application of Pade approximants as a tool for computing the character of pure spinors. We get results up to the twelfth mass-level. This work is a first step towards an explicit construction of the complete pure spinor partition function.
Starting with a classical action where a pure spinor $λ^α$ is only a fundamental and dynamical variable, the pure spinor formalism for superparticle and superstring is derived by following the BRST formalism. In this formalism, not only the string variable $x^m$ but also the space-time spinor $θ^α$ are emerged as the Faddeev-Popov (FP) ghosts of a topological symmetry and its reducible symmetry. This study suggests that the fundamental theory behind the pure spinor formalism of the superstring might be a topological field theory.
The pure spinor formalism for the superstring has recently been used to compute massless four-point two-loop amplitudes in a manifestly super-Poincare covariant manner. In this paper, we show that when all four external states are Neveu-Schwarz, the two-loop amplitude coincides with the RNS result.
In this lectures we outline the construction of pure spinor superstrings. We consider both the open and closed pure spinor superstrings in critical and noncritical dimensions and on flat and curved target spaces with RR flux. We exhibit the integrability properties of pure spinor superstrings on curved backgrounds with RR fluxes.
The character of holomorphic functions on the space of pure spinors in ten, eleven and twelve dimensions is calculated. From this character formula, we derive in a manifestly covariant way various central charges which appear in the pure spinor formalism for the superstring. We also derive in a simple way the zero momentum cohomology of the pure spinor BRST operator for the D=10 and D=11 superparticle.
In a previous paper, the BRST cohomology in the pure spinor formalism of the superstring was shown to coincide with the light-cone Green-Schwarz spectrum by using an SO(8) parameterization of the pure spinor. In this paper, the SO(9,1) Lorentz generators are explicitly constructed using this SO(8) parameterization, proving the Lorentz invariance of the pure spinor BRST cohomology.
An explicit operator mapping in the form of a similarity transformation is constructed between the RNS formalism and an extension of the pure spinor formalism (to be called EPS formalism) recently proposed by the present authors. Due to the enlarged field space of the EPS formalism, where the pure spinor constraints are removed, the mapping is completely well-defined in contrast to the one given previously by Berkovits in the original pure spinor (PS) formalism. This map provides a direct demonstration of the equivalence of the cohomologies of the RNS and the EPS formalisms and is expected to …
The pure spinor formalism for the superstring was recently obtained by gauge-fixing a purely bosonic classical action involving a twistor-like constraint $\partial x^m (γ_mλ)_α=0$ where $λ^α$ is a d=10 pure spinor. This twistor-like constraint replaces the usual Virasoro constraint $\partial x^m \partial x_m =0$, and the Green-Schwarz fermionic spacetime spinor variables $θ^α$ arise as Faddeev-Popov ghosts for this constraint. In this paper, the purely bosonic classical action is simplified by replacing the classical d=10 pure spinor $λ^α$ with a d=10 projective pure spinor. The pure spinor an…
After adding a pair of non-minimal fields and performing a similarity transformation, the BRST operator in the pure spinor formalism is expressed as a conventional-looking BRST operator involving the Virasoro constraint and (b,c) ghosts, together with 12 fermionic constraints. This BRST operator can be obtained by gauge-fixing the Green-Schwarz superstring where the 8 first-class and 8 second-class Green-Schwarz constraints are combined into 12 first-class constraints. Alternatively, the pure spinor BRST operator can be obtained from the RNS formalism by twisting the ten spin-half RNS fermions…
Notes taken by Oscar A. Bedoya of lectures of Nathan Berkovits in June 2009 at the Galileo Galilei Institute School "New Perspectives in String Theory". Topics include the pure spinor superparticle and superstring, tree and loop amplitudes, Ramond-Ramond backgrounds, and open problems.
We construct the integrated vertex operator for the first massive states of open superstrings with $(mass)^2=1/α'$ 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.
This article is based on a talk given at the Memorial Conference for Maximilian Kreuzer at the ESI in Vienna and contains a compact summary of a recent collaboration with P.A. Grassi. A non-linear projection from the space of SO(10) Weyl spinors to the space of pure spinors is presented together with some of its particular properties. This projection can be used to remove the constraints from Berkovits' pure spinor superstring while introducing additional gauge symmetries. This should allow to make transitions to equivalent formulations which might shed light on the origin of the pure spinor g…
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…
We consider pure spinor strings that propagate in the background generated by a sequence of TsT transformations. We use the fact that U(1) isometry variables of TsT-transformed background are related to the isometry variables of the initial background in the universal way that is independent of the details of the background. We will argue that after redefinitions of pure spinors and the fermionic variables we can construct pure spinor action with manifest U(1) isometry. This fact implies that the pure spinor string in TsT-transformed background is described by pure spinor string in the origina…
We describe the pure spinor string in the $AdS_5\times S^5$ using unconstrained matrices first used by Roiban and Siegel for the Green-Schwarz superstring.
It is shown that the pure spinor formulation of the heterotic superstring in a generic gravitational and super Yang-Mills background has vanishing one-loop beta functions.
In a previous work, we have constructed a reparametrization invariant worldsheet action from which one can derive the super-Poincare covariant pure spinor formalism for the superstring at the fully quantum level. The main idea was the doubling of the spinor degrees of freedom in the Green-Schwarz formulation together with the introduction of a new compensating local fermionic symmetry. In this paper, we extend this "double spinor" formalism to the case of the supermembrane in 11 dimensions at the classical level. The basic scheme works in parallel with the string case and we are able to constr…
We construct non-critical pure spinor superstrings in two, four and six dimensions. We find explicitly the map between the RNS variables and the pure spinor ones in the linear dilaton background. The RNS variables map onto a patch of the pure spinor space and the holomorphic top form on the pure spinor space is an essential ingredient of the mapping. A basic feature of the map is the requirement of doubling the superspace, which we analyze in detail. We study the structure of the non-critical pure spinor space, which is different from the ten-dimensional one, and its quantum anomalies. We comp…
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.
The pure spinor formulation of the ten-dimensional superstring leads to manifestly supersymmetric loop amplitudes, expressed as integrals in pure spinor superspace. This paper explores different methods to evaluate these integrals and then uses them to calculate the kinematic factors of the one-loop and two-loop massless four-point amplitudes involving two and four Ramond states.
The SO(32) heterotic superstring on a Calabi-Yau manifold can spontaneously break supersymmetry at one-loop order even when it is unbroken at tree-level. It is known that calculating the supersymmetry-breaking effects in this model gives a relatively accessible test case of the subtleties of superstring perturbation theory in the RNS formalism. In the present paper, we calculate the relevant amplitudes in the pure spinor approach to superstring perturbation theory, and show that the regulator used in computing loop amplitudes in the pure spinor formalism leads to subtleties somewhat analogous …
We report on progresses on the derivation of pure spinor constraints, BRST algebra and BRST invariant sigma models a la pure spinors from the algebraic structure of the FDA underlying supergravity.
In a recent letter we suggested a natural generalization of the flat-space spinor-helicity formalism in four dimensions to anti-de Sitter space. In the present paper we give some technical details that were left implicit previously. For lower-spin fields we also derive potentials associated with the previously found plane wave solutions for field strengths. We then employ these potentials to evaluate some three-point amplitudes. This analysis illustrates a typical computation of an amplitude without internal lines in our formalism.
A Berkovits type action for pure spinors in even dimensions is considered. The equations of motion for pure spinors are investigated by using explicit parameterizations which solve the pure spinor constraints. For general interactions, the equations of motions are shown to be modified from the naive ones. The extra terms contain a particular projector. If the interactions are restricted to the ``ghost number'' u(1) and the Lorentz so(p,q) current couplings, the action has a large ``gauge symmetry''. In this case, in some ``gauges'', the extra terms vanish and the equations of motion for the pu…
Recently, the superstring was covariantly quantized using the BRST-like operator $Q = \oint λ^αd_α$ where $λ^α$ is a pure spinor and $d_α$ are the fermionic Green-Schwarz constraints. By performing a field redefinition and a similarity transformation, this BRST-like operator is mapped to the sum of the Ramond-Neveu-Schwarz BRST operator and $η_0$ ghost. This map is then used to relate physical vertex operators and tree amplitudes in the two formalisms. Furthermore, the map implies the existence of a $b$ ghost in the pure spinor formalism which might be useful for loop amplitude computations.
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.
A $b$-ghost was constructed for the $D=11$ non-minimal pure spinor superparticle by requiring that $\{Q , b\} = T$ where $Q = Λ^αD_α + R^α\bar{W}_α$ 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 $\barΣ^{i}$ constructed out of the fermionic spinor $D_α$ and pure spinor variables. This simplified version will be shown to satisfy $\{Q, b\} = T$ and $\{b , b\} =$ BRST - trivial.
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.
We investigate the complex geometry of D=10 pure spinor space. The Kähler structure and the corresponding metric giving rise to the desired Calabi-Yau property are determined, and an explicit covariant expression for the Laplacian is given. The metric is not that of a cone obtained by embedding pure spinor space in a flat space of unconstrained spinors. Some directions for future studies, concerning regularisation and generalisation to eleven dimensions, are briefly discussed.
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.
We worked out the Batalin-Fradkin-Tyutin (BFT) conversion program of second class constraints to first class constraints in the GS superstring using light cone coordinates. By applying this systematic procedure we were able to obtain a gauge system that is equivalent to the recent model proposed by Berkovits and Marchioro to relate the GS superstring to the pure spinor formalism.
The b ghost in the non minimal pure spinor formalism is not a fundamental field. It is based on a complicated chain of operators and proving its nilpotency is nontrivial. Chandia proved this property in arXiv:1008.1778, but with an assumption on the non-minimal variables that is not valid in general. In this work, the b ghost is demonstrated to be nilpotent without this assumption.
