Pure Spinor Formalism as an N=2 Topological String

Nathan Berkovits (IFT-UNESP; Sao Paulo)

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Pure Spinor Formalism as an N=2 Topological String

High Energy Physics - Theory · Submitted 15 Sep 2005 · 51 references

Pith's verdict: A solid, well-built reorganization of the pure spinor formalism as a critical c-hat=3 N=2 topological string; the loop-amplitude application is the one place where the paper's reach outruns its demonstrated work.

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arXiv IDhep-th/0509120 (version 1)

Pith IDKQDSWTIX, Pith's permanent record of this paper and its review

CitationNathan Berkovits (IFT-UNESP, Sao Paulo). "Pure Spinor Formalism as an N=2 Topological String." arXiv:hep-th/0509120 (2005). https://pith.science/paper/KQDSWTIX

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This is the paper that turns the pure spinor formalism's odd features — the simple BRST operator, the missing (b,c) ghosts, the ghost-number anomaly of −8 — into consequences of a critical topological string. The construction is explicit and checkable: the non-minimal OPEs (3.7) are written down, the b ghost (3.16) is verified to satisfy {Q,b}=T with no self-poles, and the shifted U(1) current J' = wλ − w̄λ̄ is shown to give the right ghost-number anomaly of +3. The tree amplitude (4.9) reduces cleanly to the minimal-formalism result. That is the load-bearing part, and it holds up. The regulator N = exp(−λλ̄ − rθ) is the elegant move: it replaces picture-changing operators with a Gaussian that is manifestly Lorentz covariant, and — crucially for the field theory application — has no kernel, unlike Y²(π/2). That observation is what makes the cubic open superstring field theory action (5.9) actually work, and it is the paper's most useful single idea for anyone working on superstring field theory. The d=4 construction is a conjecture, but a well-motivated one, and the T⁶ check giving the expected ǫ^{ijk} superpotential (6.10) is real evidence, not hand-waving. The citation pattern is honest: Nekrasov and Siegel get credit for the non-minimal fields, [5] for the N=(0,2) resemblance, and the paper independently verifies the algebra rather than leaning on private communications. The soft spot is the loop-amplitude claim. Section 4.2 checks only zero-mode counting — the θ-power in (4.21) and (4.22) — and explicitly defers the Lorentz index contractions to future work. Zero-mode counting fixes the θ-power but not the tensor structure, so a wrong contraction would still pass. The regulator is also restricted to three or fewer b ghosts, i.e. at most two loops, which the authors flag as work in progress with Nekrasov. The abstract's 'multiloop' phrasing slightly outruns the demonstrated two-loop reach. These are acknowledged scope limits, not hidden flaws, and they do not break the construction. This is a paper for anyone in the pure spinor or superstring field theory communities; the b ghost and the regulator are worth knowing. It deserves a serious referee, and I would cite it.

Referee report

Significance

If the construction holds, it explains two long-standing structural features of the pure spinor formalism — the absence of fundamental (b,c) ghosts and the simple form of the BRST operator — by identifying the formalism as a critical topological string, and it does so with an explicit, checkable algebra rather than by analogy. The concrete deliverables are substantial: the OPEs of Eq. (3.7) are written out; the b ghost of Eq. (3.16) is given in closed form; the tree amplitude of Eq. (4.9) is shown explicitly to reduce to the minimal-formalism result, which is the sharpest available check that the non-minimal fields decouple; and the regulator N = exp(−λλ̄ − rθ) is a genuinely Lorentz-covariant replacement for the eleven picture-lowering operators, removing the constant spinors C^α and tensors B^{mn} that made the minimal prescription covariant only up to BRST-trivial surface terms. The observation that N is invertible, so that the cubic action's equation of motion N(QV + VV) = 0 is equivalent to QV + VV = 0, is a clean way around the kernel problem that afflicts the Y²(π/2) cubic RNS action, and the ρ-scaling analysis of Eq. (5.12) shows the regulator dependence can be absorbed by

Major objections (3)

[§4.2, Eqs. (4.21)–(4.22)] The loop-level check that the non-minimal prescription reproduces the minimal (and hence RNS) amplitudes is carried out by zero-mode counting only. Eqs. (4.21) and (4.22) fix the θ-power to (θ)^10 and (θ)^8 respectively, and the author writes that 'it should not be too difficult to verify if the contractions of the Lorentz indices in (4.21) and (4.22) reproduce the appropriate t_8 index contractions.' Zero-mode counting is insensitive to the tensor structure: an incorrect Lorentz contraction in the b ghost or in the integrated vertex operators would pass this test while giving the wrong amplitude. Since the abstract advertises the 'super-Poincare covariant computation of multiloop superstring amplitudes' as one of the three applications, this deferred step is load-bearing for that application. A concrete test is available: contract the indices in (4.21) for the one-loop four-point case and compare with the known minimal-formalism result of Ref. [4] and the two-loop result of Ref. [37]. Either perform this check or state explicitly in §4.2 that the loop amplitudes are verified only at the level of zero-mode counting.
[Abstract and §4.2] The abstract claims 'multiloop' amplitude computation, but §4.2 shows that the regulator method is valid only for three or fewer b ghosts, i.e. at most two loops, because f must diverge slower than λ^{−8−3g} λ̄^{−11} as λλ̄ → 0. The author acknowledges this and states that higher-loop regularization is work in progress with Nekrasov. The mismatch is between the abstract's scope and the demonstrated reach, not a hidden flaw, but it should be corrected: the abstract should say 'one- and two-loop' or otherwise qualify the multiloop claim, so that the reader is not left with the impression that the regularization is established at arbitrary genus.
[§3.2, Eqs. (3.11) and (3.16)] The b ghost is the spin-two generator of the ĉ = 3 N = 2 algebra and is therefore load-bearing for the entire topological-string interpretation, as well as for the amplitude prescription of §4 and the field theory action of §5. The text asserts that {Q_nonmin, b_nonmin} = T_nonmin and that b_nonmin has no poles with itself, but neither computation is displayed. Given the four-term structure with (λλ̄)^{−1} through (λλ̄)^{−4} denominators and the operators G^α, H^{[αβ]}, K^{[αβγ]}, L^{[αβγδ]} defined only by the recursive relations of Eq. (3.13), a reader cannot verify the claim without substantial independent work. A short appendix displaying the pole structure of the individual terms, or at least the cancellation pattern among them, would make the central algebraic claim checkable and would also document the statement that only the antisymmetrized components of H, K, L contribute.

Minor objections (8)

[§3.2, Eq. (3.15)] The text states that 'it will be shown in the appendix that the triple pole of J'_nonmin with T_nonmin is +3', but the appendix (Eq. 7.5) lists the free-field expressions for J^λ, J^r, Φ and T^λ without displaying that triple-pole computation. Since the preservation of the ghost-number anomaly under the shift from J_nonmin to J'_nonmin is one of the two conditions the author identifies as necessary, the one-line check should be added to the appendix.
[§6, Eq. (6.7) and surrounding discussion] The statement that 'the d = 4 mass-shell condition is not imposed by BRST invariance' and that V therefore describes both massive and massless states deserves elaboration, particularly in light of the introduction's remark that Q = λ^a d_a has trivial cohomology in d = 4. It should be made explicit that the nontrivial cohomology requires the G^+_C piece of Eq. (6.4) together with the +1 J_C charge restriction, and how the formalism avoids overcounting states whose d = 4 mass-shell condition is not imposed.
[§4.1, Eq. (4.6)] The measure factor [dr] is stated without derivation. A sentence explaining how the constraint λγ^m r = 0 produces the (ǫT^{−1}) λλ̄λ̄ factor, in analogy with the [dλ] measure of Eq. (4.4), would help the reader follow the counting that leads to the λ^8 λ̄^{11} scaling and hence to the bound on f.
[§4.2, after Eq. (4.12)] The assertion that 'the partition function for the non-zero modes is equal to one since there are an equal number of bosons and fermions at +1 conformal weight' should be spelled out. The non-minimal sector alone pairs eleven bosons (λ̄) against eleven fermions (r), but the minimal sector's λ and w̄ do not obviously pair in the same way, so the counting that yields the stated cancellation is not transparent.
[§5, Eqs. (5.9)–(5.10)] The argument that N(π/2) has no kernel because N^{−1} = exp(λλ̄ + rθ) is well-defined is made at the level of the zero-mode integral of Eq. (5.8). It would strengthen the section to confirm that the invertibility holds as an operator statement on the full off-shell Hilbert space with the midpoint insertion inside the star product, not only on the zero-mode measure.
[§6.2, Eq. (6.13)] The d = 4 b ghost contains a 1/(λλ) pole, and the text argues that only the G^−_C term contributes to a given amplitude by J_C charge conservation, so the pole is harmless and there is no bound on the number of b ghosts. Stating the J_C charge of each term in Eq. (6.13) explicitly would make this argument immediate.
[References [8] and [30]] Both the attribution of the non-minimal BRST operator to Nekrasov and the account of the abandoned Schwarz–Witten construction rest on private communications. The paper does independently verify the quartet decoupling and the OPE algebra, and it would help the reader to say so at the point of attribution, so that the unpublished input is clearly separated from the results established here.
[§5 and References] A careful proofreading pass is warranted: 'singularitites' appears in the closed-string field theory discussion of §5, and Ref. [39] reads 'Phys. Let. B606' where 'Lett.' is intended.

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The paper

The paper claims

The pure spinor superstring is reinterpreted as a critical $N=2$ topological string, once a non-minimal set of extra fields is added, so that the BRST charge becomes the fermionic spin-one generator and a Lorentz-covariant $b$ ghost becomes

The reading

Berkovits asks why the pure spinor formalism for the superstring has such an odd shape: a BRST operator $Q = ∮ λ^α d_α$ built from a constrained bosonic ghost $λ^α$, no fundamental $(b,c)$ ghosts, and a ghost-number anomaly of $-8$ instead of the usual $+3$. His answer is to add a non-minimal set of fields — a second pure spinor $λ̄^α$, a constrained fermionic spinor $r^α$, and their momenta — which leave the cohomology untouched but shift the anomaly to $+3$. With those fields in place, the BRST operator, a Lorentz-invariant $b$ ghost, and the stress tensor assemble into a twisted $ĉ=3$ $N=2$ superconformal algebra, so the whole formalism is a critical topological string in the standard sense. That reinterpretation matters because it replaces the awkward picture-changing operators of the older multiloop prescription with a single smooth Gaussian regulator $N = exp(-λ^αλ̄_α - r^αθ_α)$ on the zero modes, which is manifestly Lorentz covariant and has no kernel. Three payoffs follow: a covariant multiloop amplitude rule, a cubic open superstring field theory action without midpoint singularities, and a four-dimensional version that computes F-terms.

What carries the argument

The load-bearing object is the pair (non-minimal BRST operator, $b$ ghost): $Q = ∮ dz (λ^αd_α + w^αr_α)$ together with $b = s_α∂λ^α + λ^αG_α/(λλ) + λ^αr^βH_{[αβ]}/(λλ)^2 + …$, constructed so that ${Q,b} = T$ with no poles in $b$ with itself and no explicit constant spinors. The Gaussian regulator $N = exp({Q,χ})$ with $χ = -λ^αθ_α$ (extended at genus $g$ by zero-mode terms $½N^I_{mn}S^{mn}_I + J^I_λ S_I$) does the work that picture-changing operators did, and it can be continuously deformed, $N_ρ = exp(-ρ(λλ̄ + rθ))$, so that the $ρ → ∞$ limit recovers the $δ^{11}(λ)$ insertions of the minimal formalism while $ρ → 0$ gives a smooth invertible factor.

Extended reading notes

Core claim

The paper's central claim is that the non-minimal pure spinor formalism is a critical topological string with $ĉ=3$ twisted $N=2$ superconformal symmetry. The generators are the BRST current $λ^αd_α + w^αr_α$ (fermionic, spin one), a $b$ ghost built from $s_α∂λ^α$ plus correction terms with explicit $λ̄$ and $r$ dependence (fermionic, spin two), the stress tensor, and a $U(1)$ current whose triple pole with the stress tensor equals $+3$ — the ghost-number anomaly of the bosonic string. Because the added fields enter only through a quartet, the cohomology is unchanged, so physical states are still the usual pure-spinor states. The practical consequence is a new amplitude prescription: instead of eleven picture-lowering operators $Y_C$ and picture-raising operators $Z_B$ that depend on arbitrary constant spinors and tensors, one inserts $N = exp({Q, -λ^αθ_α}) = exp(-λ^αλ̄_α - r^αθ_α)$ and integrates the zero modes directly. At tree level the author shows explicitly that this reproduces the minimal-formalism answer, $∫ d^{16}θ (ǫ_T^{-1})_{βγδ}^{α_1…α_{11}} θ_{α_1}…θ_{α_{11}} f^{βγδ}(θ)$.

Load-bearing premise

The Gaussian regulator $N = exp(-λλ̄ - rθ)$ is assumed to reproduce the minimal-formalism loop amplitudes, but the zero-mode integral is well-defined only when the integrand grows slower than $λ^{-8-3g}λ̄^{-11}$ as $λλ̄ → 0$, which the author shows restricts the method to three or fewer $b$ ghosts, i.e. at most two loops.

Editorial extensions

If this is right

Multiloop superstring amplitudes can be written in a manifestly Lorentz-covariant way, with picture-changing operators replaced by a zero-mode regulator and no choices of constant spinors or tensors.
A cubic open superstring field theory action $S = ⟨(½VQV + ⅓VVV) N(π/2)⟩$ follows, whose midpoint insertion $N$ has an inverse, so the linearized equation of motion is genuinely $QV = 0$ rather than $N(π/2)QV = 0$.
A four-dimensional version with a two-component pure spinor $λ^a$ gives a $ĉ=3$ critical topological string whose amplitudes compute F-terms of the spacetime action, reproducing for a $T^6$ compactification the expected $∫ d^4x ∫ d^6y ∫ d^2θ ǫ^{ijk} Tr(½Σ_i∂_jΣ_k + ⅓Σ_iΣ_jΣ_k)$.
The $d=4$ prescription reproduces the known $g$-loop count of $2g$ self-dual graviphotons plus chiral Calabi-Yau moduli, matching the hybrid and $ĉ=5$ topological computations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the regulator really is equivalent to the minimal picture-changing prescription in a gauge where the non-minimal fields decouple, then the older Lorentz-non-covariant amplitude computations become a gauge choice inside a covariant framework — which would let one revisit the vanishing theorems and the $R^4$/$∂^4R^4$ coefficients without case-by-case constant-spinor bookkeeping.
The same trick — a regulator with an explicit inverse — may be the right template for closed superstring field theory, where the $b$ ghost's $1/(λλ)^n$ singularities are the obstacle; the paper flags this as open.
In $d=4$ the BRST operator $Q = ∮(λ^a d_a + w^a r_a + G^+_C)$ has trivial cohomology in the minimal sector, so the formalism can only capture a chiral slice of the theory; whether that slice is exactly the F-term sector, rather than merely containing it, is the natural next question and is stated here as a conjecture supported by amplitude matching.
The restriction to at most two loops is a statement about the measure's scaling ($[dλ][dλ̄][dr] → λ^8λ̄^{11}$ versus $f$'s allowed growth), so any alternative zero-mode regulator that softens the $λλ̄ → 0$ behaviour would extend the method to all genus.

Original abstract

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.

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arXiv listing abstract v1. Results and the main claim are underlined; show every role.

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.

background contribution method result interpretation main claim

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Following suggestions of Nekrasov and Siegel, a non-minimal set of fields are added to the pure spinor formalism for the superstring.	contribution 0.54	0.69	0.06	asserted 0.99
s1	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.	interpretation 0.49	0.86	0.48	asserted 1.00
s2	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.	contribution 0.74	0.84	0.41	asserted 1.00

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[51]

N. Marcus, A. Sagnotti and W. Siegel, Ten-dimensional supersymmetric Yang-Mills theory in terms of four-dimensional superﬁelds , Nucl. Phys. B224 (1983) 159. 34

View on Pith 1983

Forward citations

Cited by 18 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Equivalence Proof of Pure Spinor and Ramond-Neveu-Schwarz Superstring Amplitudes
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A new pure spinor amplitude prescription is proven equivalent to the RNS prescription and reduces to the original pure spinor prescription for F-term amplitudes.
The closed-string 3-loop amplitude and S-duality
hep-th 2013-08 conditional novelty 8.0 of 10

The first computation of a genus-three closed superstring amplitude yields a D^6 R^4 coefficient that agrees with the Green-Vanhove S-duality prediction.
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Quantum gravity EFT actions organize higher-curvature corrections into a field-theoretic expansion suppressed by the lightest new mass and a quantum-gravitational expansion suppressed by the species scale.
Vertex operators for the superstring with manifest $d=6$ $\mathcal{N}=1$ supersymmetry
hep-th 2024-12 conditional novelty 7.0 of 10

A manifestly d=6 N=1 supersymmetric vertex operator and amplitude prescription are constructed for the superstring, with BRST invariance implying the SYM equations of motion.
Two-loop superstring five-point amplitudes I: Construction via chiral splitting and pure spinors
hep-th 2020-06 conditional novelty 7.0 of 10

A complete two-loop, five-particle superstring amplitude is assembled from pure-spinor BRST cohomology and chiral splitting, and its low-energy limit reproduces the known two-loop five-point maximal supergravity ampli...

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The Structure of n-Point One-Loop Open Superstring Amplitudes
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The n-point one-loop open superstring integrand is written in closed form, with kinematic and worldsheet bases of dimension equal to an unsigned Stirling number of the first kind, and shown to share its combinatorics ...
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Adding ghost-derivative terms to ambitwistor string vertex operators makes BRST closure equivalent to ∂^n F_nm = 0 and, after two consistent truncations, to the linearized Einstein equation R_mn = 0, with no gauge fix...
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Explicit local superspace numerators are proposed for the two-loop five-point amplitudes of ten-dimensional maximally supersymmetric Yang-Mills and type II supergravity, satisfying color-kinematics duality.
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hep-th 2015-02 conditional novelty 6.0 of 10

A non-minimal pure spinor worldsheet prescription is proposed whose genus-g correlators are conjectured to equal g-loop type II supergravity integrands, with four-point checks at genus one and two.
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Local pure-spinor-superspace representations of the one-loop five- and six-point SYM integrands are constructed, BCJ duality is verified at five points, and the six-point hexagon anomaly is identified as a BRST obstruction.
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A recursive construction generates multiparticle SYM superfields whose BRST variations take a universal deconcatenation form, verified explicitly through rank four and used to build new one-loop cohomology elements.
5 loops in 24/5 dimensions
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Fermionic zero-mode counting in the pure-spinor particle formalism indicates that the derivative-eight curvature interaction is unprotected and produces a seven-loop ultraviolet divergence in four-dimensional N=8 supe...
New Higher-Derivative $R^4$ Theorems
hep-th 2006-09 conditional novelty 6.0 of 10

Using the non-minimal pure spinor formalism, the paper proves that ∂^n R^4 terms in the Type II effective action vanish above n/2 loops for n < 12, and that IIA and IIB contributions coincide for n ≤ 8.
R^4, purified
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The authors derive the complete four-point tree-level effective action of type II superstrings to all orders in alpha prime, and show it equals the tensor product of two copies of the ten-dimensional super Yang-Mills action.
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A canonical Hamiltonian derivation shows that BRST-invariant massless vertex operators of the pure spinor string on AdS5 x S5 obey the linearized on-shell equations of the background.
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How later papers used it

55 later papers cite it

Evidence: substantial. 46 of the citing papers share no author with it, and 13 of those carry a confident label from Pith's citation classifier.

Pith read 35 of the 168 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

5 later papers by other authors are listed below: 5 used its method or tool.

Uses its method or tool 5 of 11 papers with this label

Introduced the non-minimal variables of this paper, entering the action with an energy-momentum tensor and constrained pure and anticommuting spinors.

Notes on the ambitwistor pure spinor string (2016) · 0.81 · paraphrased: the licence does not allow quoting
Used the measure factors described in this paper to integrate over the zero modes of the conformal weight +1 fields.

Superstring Scattering Amplitudes with the Pure Spinor Formalism (2009) · 0.90 · paraphrased: the licence does not allow quoting
Used this paper's regulator, setting its chi on a genus g worldsheet since its pure spinor variables match the superstring's left-moving ones.

Scattering equations, supergravity integrands, and pure spinors (2015) · 0.97 · paraphrased: the licence does not allow quoting
Introduced the non-minimal variables of this paper and extended the formalism into the non-minimal formalism.

Loop amplitudes in maximal supergravity with manifest supersymmetry (2012) · 0.84 · paraphrased: the licence does not allow quoting
Introduced the non-minimal variables defined as in this paper to deal with operators that are not globally defined.

Hilbert space of curved \beta\gamma systems on quadric cones (2008) · 0.93 · paraphrased: the licence does not allow quoting

How the literature uses it

55 arXiv papers cite this in 165 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 118 sentences without a confident label

Contests it 1 paper

From the analysis above we can conclude that for the tree level scattering amplitudes, the naive existence of the homotopy operator [ref] [ref] given by [equation] which by definition satisfies [equation] for $V_1V_2V_3$ unintegrated vertex operators and $U_1...U_N-3$ integrated vertex operators, is not allowed because [equation] is not a global function on $PS∖{0}$.

A New Proposal for the Picture Changing Operators in the Minimal Pure Spinor Formalism · BRST Invariance · contest 0.76

Builds on it 3 papers

We will review and somewhat extend the discussion given in [ref] , which is closely related to the discussion in [ref] for the pure spinor string.

Multi-loop amplitudes in maximally supersymmetric pure spinor field theory · Properties of the amplitude · 0.91
This may be extended to the “non-minimal” formalism (introduced in the string context in [ref] ) and generalised to $=2$ supergravity by mimicking the transition from the open string to the closed string by doubling all the fields in [ref] apart from $X$ and $P$.

5 loops in 24/5 dimensions · Loop amplitudes and pure spinor quantum mechanics · 0.96
In this case, the regulator $ℛ$ should include the genus $g$ zero-modes [ref] of the worldsheet fields, and the prescription [ref] should be modified, possibly with insertions of the $𝒩=2$ generators in a genus-$g$ surface [ref] .

Vertex operators for the superstring with manifest $d=6$ $\mathcal{N}=1$ supersymmetry · Conclusion · 0.80

Uses its method or tool 9 papers

We will derive the conjectured form of the twisted fields and differential starting from the manifestly covariant formulation of eleven-dimensional supergravity [ref] in the pure spinor superfield formalism [ref] .

Maximally twisted eleven-dimensional supergravity · Introduction · 0.89
Given the similarities between the model [ref] and the superstring, we follow [ref] in giving the higher genus amplitude prescription.

Scattering equations, supergravity integrands, and pure spinors · Amplitude prescription · 0.95
As non-minimal pure spinor formalism is a critical topological string, then one can use the bosonic string prescription for computing scattering amplitudes [ref] [ref] .

One-loop Superstring Amplitude From Integrals on Pure Spinors Space · Four point 1-loop massless amplitude · 0.88

Compares with it 2 papers

and, unlike in [ref] , ${^, _, r_}$ are not constrained.

Vertex operators for the superstring with manifest $d=6$ $\mathcal{N}=1$ supersymmetry · Worldsheet variables · 0.92
In contrast with the minimal formalism, in the non-minimal formulation it is possible to compute tree level amplitude with all the vertex operators unintegrated [ref] .

A New Proposal for the Picture Changing Operators in the Minimal Pure Spinor Formalism · Conclusions · 0.89

Cites it as background 21 papers

The most prominent among them is Berkovits' pure spinor formalism [ref] , see [ref] for reviews and lectures.

On the Relation between Hybrid and Pure Spinor String Theory · Introduction · 1.00
The ghost action (left-moving sector) in the non-minimal formalism [ref] of Berkovits' pure spinor string theory is given by [equation] together with the constraints [equation] The variations consistent with these constraints accordingly have to obey [equation] The constraints [ref] generate via the Poisson bracket gauge transformations of the form [equation] where $μ_za,μ̅_z^a$ are some even and $σ_z^a$ are some odd gauge parameters.

On Projections to the Pure Spinor Space · With constrained variables · 0.82
Recall that inverse powers of $λ$ are also required to construct the composite “reparameterization $b$-ghost” that satisfies [ref] [equation] Once inverse powers of $λ$ are allowed, it naively appears that the cohomology of $Q$ becomes trivial due to the relation [equation] Of course, the expression $λ_+^-1θ_+$ is not globally well-defined on $X_10$, but so is the composite $b$-ghost.

Pure Spinor Partition Function and the Massive Superstring Spectrum · Basics and a subtlety · 0.81

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

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