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 ∂xm(γmλ)α = 0 where λα is a d=10 pure spinor. This twistor-like constraint replaces the usual Virasoro constraint ∂xm∂xm = 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 λα 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 SO10U5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \frac{\mathrm{SO}(10)}{\mathrm{U}(5)} $$\end{document}, this action can be interpreted as a standard ĉ = 5 topological action where one integrates over the SO10U5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \frac{\mathrm{SO}(10)}{\mathrm{U}(5)} $$\end{document} 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.


Introduction
In the conventional Ramond-Neveu-Schwarz (RNS) formalism for the superstring, the fermionic worldsheet variables ψ m are d=10 spacetime vectors and d=2 worldsheet spinors. From the spacetime point of view, a d=10 vector representation for fermionic variables is unusual and implies that spacetime supersymmetry acts in a complicated manner mixing ψ m with ghost variables, and that Ramond-Ramond backgrounds cannot be described in a straightforward manner. However, from the worldsheet point of view, their d=2 spinor representation is consistent with spin-statistics and ψ m transforms in a natural manner under worldsheet supersymmetry in the classical RNS worldsheet action.
On the other hand, in the Green-Schwarz (GS) and pure spinor formalisms for the superstring, the fermionic worldsheet variables θ α are d=10 spacetime spinors and d=2 worldsheet scalars. In this case, their d=10 spacetime spinor representation is natural so spacetime supersymmetry acts covariantly and there is no problem in describing Ramond-Ramond backgrounds. However, their d=2 worldsheet scalar representation is in conflict with spin-statistics, and θ α transforms in an unusual manner under the fermionic worldsheet symmetry called kappa symmetry in the GS formalism or BRST symmetry in the pure spinor formalism.
In this paper, a new interpretation for the GS and pure spinor formalisms will be proposed in which the fermionic θ α variables are not classical variables but arise as Faddeev-Popov ghosts from gauge-fixing a twistor-like action. The classical action will be constructed only from bosonic variables which include the usual d=10 spacetime vector variable x m together with a d=10 projective pure spinor variable λ α that plays the role of a JHEP07(2015)091 twistor variable. Since a d=10 projective pure spinor parameterizes SO (10) U (5) , this action can be interpreted as a standardĉ = 5 topological action where one integrates over the SO (10) U (5) choice of complex structure. Different gauge-fixings of this purely bosonic classical action will produce either the pure spinor or GS formalisms, and the GS kappa symmetry will be related in a simple manner to the pure spinor BRST symmetry.
For the superparticle, the spacetime momentum P m will satisfy the twistor-like constraint P m (γ m λ) α = 0 which replaces the mass-shell constraint P m P m = 0, and the fermionic variable θ α is the Faddeev-Popov ghost for this constraint which is a worldline scalar and spacetime spinor. For the Type II superstring, the twistor-like constraints will be (P m + ∂ σ x m )(γ m λ) α = 0 and (P m − ∂ σ x m )(γ m λ)α = 0 (1.1) which replace the Virasoro constraints (P m +∂ σ x m )(P m +∂ σ x m ) = 0 and (P m −∂ σ x m )(P m − ∂ σ x m ) = 0, and the fermionic Faddeev-Popov ghosts θ α and θα for these constraints are worldsheet scalars and d=10 spacetime spinors. Finally, for the supermembrane, the twistor-like constraint will be P M (γ M λ) B + ∂ σ 1 x M ∂ σ 2 x N (γ M N λ) B = 0 which replaces the reparameterization constraints P M P M + det(∂ σ j x M ∂ σ k x M ) = 0 and P M ∂ σ j x M = 0, and the fermionic Faddeev-Popov ghost θ B is a worldvolume scalar and d=11 spacetime spinor. After gauge-fixing the bosonic twistor-like action, spacetime supersymmetry mixes matter and ghost variables in a manner reminiscent of the mixing of matter and ghost variables under spacetime supersymmetry in the RNS formalism. However, unlike the RNS formalism where spacetime supersymmetry acts in a complicated manner, spacetime supersymmetry now acts linearly on the matter and ghost variables and it is straightforward to describe Ramond-Ramond backgrounds.
In previous papers [1] and [2], a similar purely bosonic classical action was used to derive the pure spinor formalism for the superstring by gauge-fixing a twistor-like constraint. However, unlike these previous papers in which the classical variable λ α was a d=10 pure spinor, the classical variable λ α in this paper will be a projective d=10 pure spinor. The scale part of λ α will be a ghost variable coming from gauge-fixing, which will imply that the scale-invariant fermionic variables θ α carry zero ghost number as expected. Furthermore, the gauge-fixing procedure in this paper will be more straightforward than in these previous two papers.
It would be very interesting to generalize the procedure of this paper to curved targetspace backgrounds. For NS-NS backgrounds, the obvious guess is to replace the twistor-like constraints of (1.1) with where E c m (x) is the vierbein satisfying g mn E c m E d n = η cd and c, d = 0 to 9 are tangentspace indices. Constructing a classical twistor-like action for R-R backgrounds is more challenging since the fermionic θ α variables only appear after gauge-fixing.
In section 2 of this paper, the bosonic twistor-like action for the d=10 superparticle will be constructed, and its gauge-fixing to the pure spinor and GS formalisms will be explained. In section 3, this bosonic twistor-like action will be generalized to the d=10 superstring, and its gauge-fixing to the pure spinor and GS formalisms will be explained. And finally in section 4, a twistor-like action for the d=11 supermembrane will be constructed which reduces upon double-dimensional reduction to the twistor-like action for the d=10 Type IIA superstring.

Twistor-like superparticle
Before discussing the d=10 superstring, it will be useful to first discuss the d=10 superparticle and show how its twistor-like action can be interpreted as an action for a topological particle in which one integrates over the choices of complex structure.

Topological particle
The classical action for a d=10 topological particle is whereẋ m = ∂ ∂τ x m , m = 0 to 9 are d=10 vector indices, a, a = 1 to 5 labels a complex split of x m → (x a , x a ) and P m → (P a .P a ), and L a is a Lagrange multiplier imposing the constraint P a = 0. For the action to be real, one should choose spacetime signature (5,5), however, one can easily Wick-rotate to any other d=10 signature. For convenience, we will use Euclidean signature (10, 0) for the rest of this paper which implies that x a is the complex conjugate of x a .
The action of (2.1) is invariant under the gauge transformations δx a = θ a , δL a =θ a (2.2) for arbitary gauge parameters θ a (τ ), and if one gauge-fixes L a = 0 using the standard BRST method, θ a and its conjugate momentum p a are interpreted as fermionic Faddeev-Popov

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ghosts and antighosts associated to this gauge-fixing. The resulting action is where χ = p a L a is the gauge-fixing fermion, Q = θ a P a is the BRST operator which generates the transformation of (2.2), and M a ≡ Qp a is the Nakanishi-Lautrup field whose auxiliary equation of motion is M a = P a . Physical states in the BRST cohomology are easily shown to be anti-holomorphic functions V = f (x a ) which satisfy P a V = 0. Although this topological theory is not SO(10) invariant, one can write the action of (2.1) using ten-dimensional notation as where ξ α for α = 1 to 16 is a fixed d=10 pure spinor which chooses an SO(10) U(5) complex structure. It will now be shown that if one makes the theory SO(10) invariant by integrating over the choice of SO(10) U(5) complex structure, the theory is no longer topological and describes the d=10 super-Maxwell theory coming from quantizing the d=10 superparticle.

Worldline action
To integrate over SO (10) U(5) complex structures, one should replace the fixed ξ α in (2.4) with a projective pure spinor variable λ α (τ ) satisfying the pure spinor condition and which is defined up to the local scale transformation The conjugate momentum to λ α will be called w α and is defined up to the gauge and scale transformations In terms of λ α , the topological constraint P a = 0 can now be expressed covariantly as the twistor-like constraint P m (γ m λ) α = 0, and the corresponding classical worldline action is where ∇λ α =λ α + Aλ α and A is a worldline gauge field transforming as In addition to the local scale invariance of (2.9), (2.8) is also invariant under the local gauge transformations related to the first-class constraint P m (γ m λ) α = 0

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where θ α (τ ) and a(τ ) are arbitrary gauge parameters. 1 Furthermore, one has the local gauge-for-gauge symmetries which transform the gauge parameters as where φ(τ ) is a gauge-for-gauge parameter and χ is the gauge-fixing fermion. The transformation of w α in (2.11) is necessary since δL α of (2.10) transforms under (2.11) as δ ′ (δL α ) = φ∇λ α which is proportional to the equation of motion for w α . So if L α appears in the gauge-fixing fermion χ, the term Qχ in the action will transform under (2.11) as which must be cancelled by a shift of w α → w α − φ δχ δL α .

Pure spinor superparticle
In this section, the pure spinor description of the d=10 superparticle will be obtained by gauge-fixing the twistor-like action of (2.8). The first step is to use the gauge symmetries of (2.10) and the gauge-for-gauge symmetries of (2.11) to gauge-fix L α = 0 and a = 0.
Using the BRST method where the BRST operator Q generates the symmetries of (2.10) and (2.11), this gauge-fixing is accomplished by adding to the classical action S c of (2.8) the BRST-trivial term where θ α and a are fermionic ghosts, φ is a bosonic ghost-for-ghost, p α is a fermionic antighost, β is a bosonic antighost-for-ghost, and M α and N are bosonic and fermionic Nakanishi-Lautrup fields defined by M α ≡ Qp α and N ≡ Qβ. Note that (θ α , p α ) and (φ, β) scale under (2.9) as (2.14) After solving the auxiliary equations of motion of M α and N , the gauge-fixed action of (2.13) is The next step is to gauge-fix the local scale symmetry of (2.9). Although the naive choice would be to gauge-fix one of the 16 components of λ α to be equal to 1, this gaugefixing would break manifest Lorentz invariance and would only be possible if one component 1 In addition to the symmetries of (2.10), (2.8) is also invariant under the transformation δL α = a mn (γmnλ) α for arbitrary gauge parameter a mn . However, this symmetry does not need to be gauge-fixed if one assumes (as will be explained in the following subsection) that the ghost-for-ghost φ is non-vanishing. Including this BRST transformation of L α would modify the gauge-fixed action of the following subsection to include the additonal term dτ pα(γmnλ) α a mn where pα is the antighost. But this additional term is BRST-trivial since it can be expressed as

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of λ α were required to be non-vanishing on the entire worldline. Instead of this gauge-fixing of λ α which would break Lorentz invariance, the local scale symmetry will be gauge-fixed by scaling the ghost-for-ghost variable φ to be equal to 1. This gauge choice manifestly preserves Lorentz invariance, but requires that φ is non-vanishing on the entire worldline.
Since φ carries scale weight −2 and ghost number +2, this gauge-fixing of scale symmetry means that the ghost number of any gauge-fixed operator is shifted by its scale weight. In other words, if an operator O g,s carries ghost number g and scale weight s, the gauge-fixed version of O g,s is the scale-invariant operator φ s 2 O g,s which carries ghostnumber g + s. So after gauge-fixing the scale symmetry, the ghost number of the variables (λ α , w α ) is shifted from (0, 0) to (+1, −1) and the ghost number of the variables (θ α , p α ) is shifted from (+1, −1) to (0, 0). These are the appropriate ghost numbers in the pure spinor formalism for defining physical states. Furthermore, after gauge-fixing φ = 1, the variables A and β of (2.15) satisfy the auxiliary equations of motion A = 0 and β = 1 2 (w α λ α + p α θ α ), and the action of (2.15) reduces to the pure spinor action Finally, the BRST transformations in this gauge reduce to 17) which are the BRST transformations generated by the pure spinor superparticle BRST operator is the worldsheet version of the d=10 superspace derivative D α ≡ ∂ ∂θ α + 1 2 (γ m θ) α ∂ ∂x m . Note that to obtain the pure spinor superparticle action in the usual gauge one needs to add the BRST-trivial term −Qb = 1 2 P m P m to the action of (2.16) where is the pure spinor b ghost and ξ α is any spinor satisfying λ α ξ α = 1. Although it will not be reviewed here, it was explained in [16] how to covariantize the construction of the pure spinor b ghost of (2.21) by adding a non-minimal sector to the pure spinor formalism.

Green-Schwarz superparticle
In this section, the Green-Schwarz description of the d=10 superparticle will be obtained by performing a different gauge-fixing of the twistor-like action of (2.8). One again gauges JHEP07(2015)091 a = 0 by adding the gauge-fixing fermion Q(βa) to the classical action. But instead of gauge-fixing L α = 0 by adding the gauge-fixing term Q(p α L α ), one chooses the gauge-fixing fermion χ so that the BRST variation of w α vanishes. Since this implies that the gauge-fixing fermion χ is chosen to depend on L α as As before, the ghost-for-ghost variable φ will be assumed to be non-vanishing and the scale symmetry of (2.9) will be used to gauge-fix φ = 1. Since λ α carries scale weight +1 and θ α carries scale weight −1, this shifts the ghost-number of λ α from 0 to +1 and shifts the ghost number of θ α from +1 to 0. After this gauge-fixing, the resulting action is where the auxiliary equations of motion A = N − 1 2 (λγ m θ)P m = a = β − 1 2 w α λ α = 0 have been used.
The BRST transformation in this gauge is and one can easily verify that the Noether charge associated with this transformation vanishes. So in this gauge, the global BRST symmetry is a local gauge symmetry which will be related below to the usual GS kappa symmetry.
To relate the action and symmetries of (2.25) and (2.26) with the GS action and symmetries, add to (2.25) the BRST-trivial term to remove the w α dependence where ξ α is a spinor satisfying ξ α λ α = 1 and the 1 2 (wγ m ξ)(λγ mθ ) term in (2.27) is needed to preserve the w α gauge invariance of (2.7). Then shift L α → L α − 1 2 P m (γ m ξ) in (2.25) to obtain the action where S GS is the standard GS superparticle action in the gauge e = 1,

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The role of the second term in (2.28) is to impose the constraint P m (γ m λ) α = 0, which implies that P m P m = 0 and that for some κ α . With this pameterization of λ α , the BRST transformation of (2.26) becomes the standard GS kappa transformation and the transformation of the second term in (2.28) is So the BRST transformation of L α in (2.26), QL α =θ α , replaces the kappa-transformation of the metric, δe = −κ αθ α , in the usual GS action.

Worldsheet action
In this section, we will generalize the superparticle results of the previous section to the superstring and it will be convenient to work in first-order form with x m (τ, σ) and P m (τ, σ) variables for the spacetime position and its conjugate momentum. In addition to these spacetime vector variables, the classical worldsheet variables for the Type II superstring will contain two sets of projective pure spinors, λ α and λα, and their conjugate momenta, w α and wα, satisfying the pure spinor conditions λγ m λ = 0, λγ m λ = 0 (3.1) and related gauge invariances as well as the local scale invariances where Ω(τ, σ) and Ω(τ, σ) are independent local scale parameters and α,α = 1 to 16 are spinor indices of the same chirality for the Type IIB superstring and spinor indices of the opposite chirality for the Type IIA superstring. For the heterotic superstring, only one set of projective pure spinor variables is needed and the right-moving sector of the superstring is the same as in the RNS formalism. The twistor-like constraint P m (γ m λ) α = 0 for the superparticle has the obvious generalization for the superstring (P m + ∂ σ x m )(γ m λ) α = 0 and (P m − ∂ σ x m )(γ m λ) α = 0. (3.4)

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However, since the commutator is nonzero, the additional constraints ∇ σ λ α = 0 and ∇ σ λα = 0 (3.6) will be imposed where and A I andÂ I for I = τ, σ are worldsheet gauge fields which transform as δA I = −Ω −1 ∂ I Ω and δ A I = − Ω −1 ∂ I Ω under the local scale transformations of (3.3). So the classical worldsheet action for the Type II superstring is where the Lagrange multipliers (L α , Lα, K α , Kα) transform under the scale transformations of (3.3) as Note that this action is invariant under worldsheet reparameterizations since the Virasoro constraints

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In addition to the worldsheet reparameterizations and local scale symmetries, the action is invariant under the local symmetries related to the first-class constraints of (3.11), where χ is the gauge-fixing fermion and the variations of the ghosts (c α , cα, θ α , θα, a, a) in the last two lines of (3.15) come from the nonvanishing commutator of (3.5) and from the gauge-for-gauge symmetries related to (2.11). As in (2.12), the shifts in w α and wα proportional to the gauge-fixing fermion χ are needed to cancel terms proportional to the equations of motion ∇ τ λ α and ∇ τ λα in the gauge-for-gauge transformations of (δK α , δL α ) and (δ Kα, δ Lα).

Pure spinor superstring
As in the superparticle, gauge-fixing to the pure spinor formalism is achieved by first using the local symmetries of (3.15) to gauge L α = Lα = 0 and a = a = 0. Using the BRST method, this involves adding to the classical action S c of (3.8) the gauge-fixing fermion where the second line of (3.16) has been included to eliminate (K α , Kα) and (c α , cα) from the action. The resulting gauge-fixed action is

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where (M α , Mα) and (N, N ) are the Nakanishi-Lautrup fields defined by The next step is to use the local scale symmetries to gauge-fix φ = φ = 1, where it will be assumed that these fields are non-vanishing on the entire worldsheet. As in the superparticle, this shifts the ghost number by the scale weight so that after gauge-fixing φ = φ = 1, the pure spinor variables (λ α , w α ) and ( λα, wα) carry ghost number (1, −1) and the fermionic spacetime spinor variables (θ α , p α ) and ( θα, pα) carry ghost number (0, 0).
After solving the auxiliary equations of motion for the variables (A τ , β, M α , L α , N, a) and ( A τ , β, Mα, Lα, N , a), the action reduces to with the BRST transformations These are the usual pure spinor BRST transformations generated by the BRST operator are the worldsheet versions of the N=2 d=10 superspace derivatives D α ≡ ∂ ∂θ α +  (3.19) with the usual pure spinor superstring action in conformal gauge, one needs to add the BRST-trival term − 1

Green-Schwarz superstring
As in the superparticle, gauge-fixing to the GS formalism for the superstring is achieved by gauging a = a = 0 and choosing the L dependence of the gauge-fixing fermion χ such that the BRST variation of w α and wα vanishes where To gauge-fix a = a = 0 and remove the (K α , Kα) dependence from the action, the L-independent terms in χ will be chosen to be The vanishing of Qw α and Q wα therefore implies that the gauge-fixing fermion is

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and the resulting gauge-fixed action is where N = Qβ and N = Q β are Nakanishi-Lautrup fields. After using the scale symmetries to gauge-fix φ = φ = 1 and applying the auxiliary equations of motion for (A τ , β, a, N ) and ( A τ , β, a, N ), the action of (3.29) simplifies to S = dτ dσ P mẋ m + w αλ α + wα˙ λα (3.30) As in the superparticle action, one can relate (3.30) to the GS superstring action in conformal gauge by adding the BRST-trivial term

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and shifting where ξ α and ξα are spinors satisfying ξ α λ α = 1 and ξα λα = 1. One obtains which after integrating out P m is equal to where S GS is the GS action in conformal gauge
As in the superparticle, the role of the second term in (3.34) is to impose the constraints Π m (γ m λ) α = 0 and Π m (γ m λ)α = 0, which implies that Π m Π m = Π m Π m = 0 and that for some κ α and κα. With this parameterization of λ α and λα, the BRST transformations of (3.15) reduce to the usual GS kappa transformations
Using the BRST transformation of the shifted L α and Lα in (3.32),  (4.5) or to the d=11 GS superparticle action with the kappa transformation where λ B = P M (γ M κ) B .

d=11 supermembrane
The purely bosonic classical d=11 supermembrane action will involve the worldvolume variables (x M , P M ) and (λ B , w B ) together with the twistor-like constraint where ∂ j = ∂ σ j for j = 1, 2 and (τ, σ 1 , σ 2 ) are the coordinates of the worldvolume. Using the d=11 γ-matrix identity (γ M ) (BC (γ M N ) DE) = 0, one finds that the commutator of the constraint of (4.8) with itself closes to an algebra if one also imposes the additional constraints ∇ j λ B = 0 and (λγ M N λ)∂ j x M = 0 (4.9) for j = 1, 2 where ∇ j λ B ≡ ∂ j λ B + A j λ B and (A τ , A 1 , A 2 ) is a worldvolume gauge field for the scale symmetry. The twistor-like version of the d=11 supermembrane action will therefore be defined as where L B , K j B and J j M are Lagrange multipliers for the constraints of (4.8) and (4.9). Although it should be possible to verify (4.10) by gauge-fixing it to the pure spinor and GS supermembrane actions of [17] and [18], it will instead be verified here by performing a double-dimensional reduction and comparing with the twistor-like Type IIA superstring action of the previous section.