Pure Spinor BRST Research Articles

Green-Schwarz and pure spinor formulations of chiral strings

Bosonic and RNS chiral strings have been defined from a singular gauge fixing of the respective Polyakov and spinning string actions, enforcing, among other things, the finite nature of their physical spectra. Except for the heterotic case, the tensionless limits of such chiral models have been shown to describe the same field theories predicted by their ambitwistor analogues. In this paper, we study the Green-Schwarz formulation for Type II and heterotic superstrings in a singular gauge. After performing a light-cone gauge analysis, their physical spectra are shown to match those of RNS chiral strings, and their respective tensionless limits are found to describe the same field theories predicted by RNS ambitwistor strings. Their pure spinor counterparts are then introduced by making use of the Oda-Tonin method. In doing so, symmetries hidden in the pure spinor ambitwistor string action become manifest, proposals motivating the sectorized pure spinor BRST charges find simple grounds, and integrated vertex operators emerge naturally.

Simplified D = 11 pure spinor b ghost

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 }{overline{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 {overline{Sigma}}^i constructed out of the fermionic spinor Dα and pure spinor variables. This simplified version will be shown to satisfy {Q, b} = T and {b, b} = BRST - trivial.

Origin of the pure spinor and Green-Schwarz formalisms

The pure spinor formalism for the superstring was recently obtained by gauge-fixing a purely bosonic classical action involving a twistor-like constraint ∂x m (γ m λ) α = 0 where λ α is a d=10 pure spinor. This twistor-like constraint replaces the usual Virasoro constraint ∂x m ∂x m = 0, and the Green-Schwarz fermionic spacetime spinor variables θα 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 and Green-Schwarz formalisms for the superparticle and superstring are then obtained as different gauge-fixings of this purely bosonic classical action, and the Green-Schwarz kappa symmetry is directly related to the pure spinor BRST symmetry. Since a d=10 projective pure spinor parameterizes $$ \frac{\mathrm{SO}(10)}{\mathrm{U}(5)} $$ , this action can be interpreted as a standard ĉ = 5 topological action where one integrates over the $$ \frac{\mathrm{SO}(10)}{\mathrm{U}(5)} $$ choice of complex structure. Finally, a purely bosonic action for the d=11 supermembrane is proposed which reduces upon double-dimensional reduction to the purely bosonic action for the d=10 Type IIA superstring.

Towards one-loop SYM amplitudes from the pure spinor BRST cohomology

In this paper, we outline a method to compute supersymmetric one-loop integrands in ten-dimensional SYM theory. It relies on the constructive interplay between their cubic-graph organization and BRST invariance of the underlying pure spinor superstring description. The five- and six-point amplitudes are presented in a manifestly local form where the kinematic dependence is furnished by BRST-covariant expressions in pure spinor superspace. At five points, the local kinematic numerators are shown to satisfy the BCJ duality between color and kinematics leading to type IIA/B supergravity amplitudes as a byproduct. At six points, the sources of the hexagon anomaly are identified in superspace as systematic obstructions to BRST invariance. Our results are expected to reproduce any integrated SYM amplitude in dimensions .

Multiparticle SYM equations of motion and pure spinor BRST blocks

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 perturbative open and closed superstring, regardless of the loop order. The inherent symmetries of the multiparticle superfields are reproduced by structure constants of the gauge group, hinting a natural appearance of the BCJ-duality between color and kinematics in the fundamentals of super Yang--Mills theory. We present one-loop applications where known scalar cohomology objects are systematically recomputed and a novel vector cohomology particularly relevant to the closed string is constructed for arbitrary multiplicity.

Covariant map between Ramond-Neveu-Schwarz and pure spinor formalisms for the superstring

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 formalisms.

Pure spinor b-ghost in a super-Maxwell background

In the pure spinor formalism for the superstring, the b-ghost is a composite operator satisfying {Q,b}=T where Q is the pure spinor BRST operator and T is the holomorphic stress tensor. The b-ghost is holomorphic in a flat target-space background, but it is not holomorphic in a generic curved target-space background and instead satisfies $\bar\partial b$ = [Q, \Omega] for some \Omega. In this paper, \Omega\ is explicitly constructed for the case of an open superstring in a super-Maxwell background.

Dynamical twisting and the b ghost in the pure spinor formalism

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 $ \frac{1}{2} $ 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).

Harmonic superspace from the AdS 5 × S 5 pure spinor formalism

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 $AdS_5$.

Towards field theory amplitudes from the cohomology of pure spinor superspace

A simple BRST-closed expression for the color-ordered super-Yang-Mills 5-point amplitude at tree-level is proposed in pure spinor superspace and shown to be BRST-equivalent to the field theory limit of the open superstring 5-pt amplitude. It is manifestly cyclic invariant and each one of its five terms can be associated to the five Feynman diagrams which use only cubic vertices. Its form also suggests an empirical method to find superspace expressions in the cohomology of the pure spinor BRST operator for higher-point amplitudes based on their kinematic pole structure. Using this method, Ansaetze for the 6- and 7-point 10D super-Yang-Mills amplitudes which map to their 14 and 42 color-ordered diagrams are conjectured and their 6- and 7-gluon expansions are explicitly computed.

NEW BRST CHARGES IN RNS SUPERSTRING THEORY AND DEFORMED PURE SPINORS

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 QPS = ∮λαdα still retaining its form but with singular operator products between commuting spinor variables λα. Despite the OPE singularities, the pure spinor condition λγmλ = 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 pictures n > 0 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 λ(z)λ(w) ~ O(z-w)-2(n2+6n+1). The conjecture is checked explicitly for the first nontrivial case n = 1.

(NON)TRIVIALITY OF PURE SPINORS AND EXACT PURE SPINOR–RNS MAP

All the BRST-invariant operators in pure spinor formalism in d = 10 can be represented as BRST commutators, such as [Formula: see text] where λ+ is the U(5) component of the pure spinor transforming as [Formula: see text]. Therefore, in order to secure nontriviality 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 = {Q brst , LV} where L = -4c∂ξξe-2ϕ, we show that mapping [Formula: see text] to L leads to identification of the pure spinor variable λα in terms of RNS variables without any additional nonminimal fields. We construct the RNS operator satisfying all the properties of λα and show that the pure spinor BRST operator ∮λαdα is mapped (up to similarity transformation) to the BRST operator of RNS theory under such a construction.

A note on T-dualities in the pure spinor heterotic string

In this note we study the preservation of the classical pure spinor BRST constraints under super T-duality transformations. We also determine the invariance of the one-loop conformal invariance and of the local gauge and Lorentz anomalies under the super T-dualities.

Yang-Mills Chern-Simons corrections from the pure spinor superstring

Nilpotency of the pure spinor BRST operator in a curved background implies superspace equations of motion for the background. By computing one-loop corrections to nilpotency for the heterotic sigma model, the Yang-Mills Chern-Simons corrections to the background are derived.

Explaining the pure spinor formalism for the superstring

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 into five spin-one and five spin-zero fermions, and using the SO(10)/U(5) pure spinor variables to parameterize the different ways of twisting. GSO(−) vertex operators in the pure spinor formalism are constructed using spin fields and picture-changing operators in a manner analogous to Ramond vertex operators in the RNS formalism.

Formula omitted] massless superparticle covariant quantization, pure spinor BRST charge and hidden symmetries

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 v α q − parametrizing (as homogeneous coordinates, modulo gauge symmetries) the celestial sphere S 9 . There presence allows us to separate covariantly the first and the second class constraints of the model. We show that, 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 Q susy 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 κ-symmetry, λ q , and further reduction of the regularized cohomology problem to the one for a simpler complex BRST charge Q ˜ susy . This latter is essentially the pure spinor BRST charge by Berkovits, but with a composite pure spinor constructed from the complex d = 9 spinor with zero norm, λ ˜ q , and the spinorial harmonics v α q − . This exhibits a possible origin of the complexity (non-Hermiticity) characteristic of the Berkovits pure spinor approach. The simple structure of the non-trivial cohomology of the M0-brane BRST charge Q susy finds explanation in the properties that the superparticle action exhibits in the so-called ‘covariantized light-cone’ basis, where the M0-brane action is expressed in terms of κ-symmetry invariant variables. The set of gauge symmetries in this basis reduces to the [ SO ( 1 , 1 ) × SO ( 9 ) ] ▪ K 9 Borel subgroup of SO ( 1 , 10 ) . Imposing their generators as conditions on the superparticle wavefunctions, we arrive at the covariant quantization in terms of physical degrees of freedom which hints possible hidden symmetries of D = 11 supergravity. Besides SO ( 16 ) , which in the twistor like Lorentz harmonic formulation is seen already at the classical level, we discuss also some indirect arguments in favor of the possible E 8 symmetry.

One-loop conformal invariance of the type II pure spinor superstring in a curved background

We compute the one-loop beta function for the Type II superstring using the pure spinor formalism in a generic supergravity background. It is known that the classical pure spinor BRST symmetry puts the background fields on-shell. In this paper we show that the one-loop beta functions vanish as a consequence of the classical BRST symmetry of the action.

Pure spinor formalism as anN= 2 topological string

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 new four-dimensional version of the pure spinor formalism which computes F-terms in the spacetime action.

The Character of Pure Spinors

The character of holomorphic functions on the space of pure spinors in 10, 11 and 12 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

Ten-dimensional supergravity constraints from the pure spinor formalism for the superstring

It has recently been shown that the ten-dimensional superstring can be quantized using the BRST operator Q=∮ λ α d α , where λ α is a pure spinor satisfying λγ m λ=0 and d α is the fermionic supersymmetric derivative. In this paper, the pure spinor version of superstring theory is formulated in a curved supergravity background and it is shown that nilpotency and holomorphicity of the pure spinor BRST operator imply the on-shell superspace constraints of the supergravity background. This is shown to lowest order in α′ for the heterotic and Type II superstrings, thus providing a compact pure spinor version of the ten-dimensional superspace constraints for N=1 Type IIA and Type IIB supergravities. Since quantization is straightforward using the pure spinor version of the superstring, it is expected that these methods can also be used to compute higher-order α′ corrections to the ten-dimensional superspace constraints.