arXiv Vanity renders academic papers from arXiv as responsive web pages so you don’t have to squint at a PDF. Read this paper on arXiv.org.

# Covariant One-Loop Amplitudes in D=11

Lilia Anguelova
Michigan Center for Theoretical Physics, University of Michigan
Ann Arbor, MI 48109-1120, USA,
Pietro Antonio Grassi
C.N. Yang Institute for Theoretical Physics, SUNY at Stony Brook
Stony Brook, NY 11794-3840, USA,
and
DISTA, Università del Piemonte Orientale,
Piazza Amborsoli, 1 15100 Alessandria, Italy
Pierre Vanhove
CEA/DSM/SPhT, URA au CNRS, CEA/Saclay,
F-91191 Gif-sur-Yvette, France
###### Abstract:

We generalize to the eleven-dimensional superparticle Berkovits’ prescription for loop computations in the pure spinor approach to covariant quantization of the superstring. Using these ten- and eleven-dimensional results, we compute covariantly the following one-loop amplitudes: in M-theory; in type II string theory and in type I. We also verify the consistency of the formalism in eleven dimensions by recovering the correct classical action from tree-level amplitudes. As the superparticle is only a first approximation to the supermembrane, we comment on the possibility of extending this construction to the latter. Finally, we elaborate on the relationship between the present BRST language and the spinorial cohomology approach to corrections of the effective action.

pure spinors, superparticle, eleven dimensions
preprint: MCTP-04-46, YITP-04-41
SPHT-T-04/102, NEIP-04-04
hep-th/0408171

## 1 Introduction

Since the discovery of the web of dualities that relate the five 10-dimensional string theories with each other and with 11-dimensional supergravity , it has been realized that perturbative and nonperturbative effects in ten-dimensions are encoded in the effective action of an underlying 11-dimensional theory. In particular, both the fundamental string and the D-brane originate from the 11-dimensional supermembrane. Although the relationship between this latter M-brane and the eleven-dimensonal supergravity fields is what one would expect from a fundamental object and its long-range fields , it has turned out that there is no gap between the massless and massive exitations of the membrane [3, 4]. Nevertheless one can consistently define vertex operators for the massless fields  and study the corresponding scattering amplitudes.111On the other hand, vertex operators, analogous to the massive superstring states, are not expected to exist due to the continuous spectrum of the supermembrane. Instead, the membrane excitations are interpreted as being related to multi-particle (rather than one-particle) states in the Matrix Theory proposal of . In full generality, this is still a task beyond reach. However in the limiting case of the superparticle, i.e. suppressing the transverse fluctuations of the supermembrane, several successful computations have been performed in the light cone gauge. More precisely, the 11d four-graviton scattering was computed to one and two loops in [7, 8] and  respectively. Of course, these supergravity amplitudes are UV divergent but the lack of knowledge about the microscopic degrees of freedom can be compensated by information provided from various dualities. This is how, for example, the one-loop generated eleven-dimensional term was “renormalized” in . The important lesson from the point-particle limit of the membrane is that quantum supergravity in 11d does indeed reproduce the correct form of certain terms in the effective action of M-theory, for example the 11d one-loop term which gives rise to both the tree level and the one-loop terms in the effective superstring action.

However, the light cone gauge, used in the above works, has its drawbacks in that it is not suitable for computing all possible amplitudes in a generic background and also it is less efficient in organizing the calculations, in particular making manifest the cancellations between different contributions, than a covariant formulation would be. The first steps towards a full covariant description were made in , based on the recent progress in the covariant formulation of 10d superstring theory . The idea is to construct a supermembrane action, alternative to , which reduces to the ten-dimensional pure spinor description of the type IIA superstring after double dimensional reduction. This action is invariant under a BRST transformation generated by the same type of operator as the one used in  for the case of superstrings. So again one can define vertex operators via the cohomology of . Using this covariant BRST formulation, the general form of the vertex operators relevant for the supermembrane theory massless excitations, namely the graviton , gravitino and three-form , was obtained in , although some of the superfields in that expression remained undetermined. Clearly, these vertices can be reduced to those of the 11d superparticle by setting to zero the transverse fluctuations of the supermembrane.

Although  gives a tree-level prescription for integration over the zero modes of the pure spinor ghosts and their conjugate momenta it does not derive in general the measure for this integration and so leaves performing loop computations out of reach. The same problem was resolved only very recently for the pure spinor formulation of 10-dimensional string theory . The crucial ingredient was the construction of a composite field playing a role analogous to the one of the NSR -antighost. This is very nontrivial in Berkovits’ formalism as the only field with negative ghost number, , appears always in combination with , i.e. only in expressions with ghost number zero. This problem was overcome in  by introducing picture-changing operators and defining the -ghost via , where is the energy-momentum tensor and is a picture raising operator with ghost-number . As pointed out in that work, the necessity of picture-changing operators should have been anticipated, since the pure spinor ghosts are world-sheet bosons with zero modes similarly to the NSR -ghosts.

Unlike the superstring, the membrane world-volume theory is not conformally invariant. But, as in the string case, it is still reparametrization invariant. Therefore one should expect an analogue of the -ghost in eleven dimensions as well. In this paper we show that the 10-dimensional covariant rules for superstring loop calculations of  can be extended to the case of the 11-dimensional superparticle. In fact, we construct the 11d generalization of the “antighost” field in a form that is also valid for string theory. The advantage of our different, though equivalent to the one in , representation of is that it simplifies the prescription for one loop computations given there. Our construction of covariant one loop amplitudes, as well as the light-cone calculations of , is based on the string-inspired first quantized approach to field theory developed in [13, 14, 7, 9]. Namely, one inserts vertex operators, describing the external fields, on the world-line of the particle circulating in the loop and integrates over that world-line. For that purpose we derive the 11d superparticle vertex operator and show that it is consistent with the general form obtained in .222Recall that, as we already mentioned, many of the superfields in that general form are actually still unknown.

Using the superstring formalism of  and the 11d one of this paper, we compute covariantly several one-loop amplitudes:  the terms in IIA/B string theory and in 11d supergravity and also the supersymmetric in type I (meaning all three cases: four gluons; two gluons and two gluinos; four gluinos), always finding the correct answers. We also show that the tree-level scattering amplitudes in 11d reproduce completely the two-derivative action of eleven-dimensional supergravity . We view our results as a strong confirmation of the viability of the pure spinor approach to covariant quantization.

Finally, we elaborate on the relashionship/equivalence between pure spinor cohomology and the spinorial cohomology developed in [16, 17, 18, 19], recovering the latter from an extended BRST operator similarly to the construction of  that relaxes the pure spinor constraint.

We conclude with some remarks about a possible extension of the present computation to the M-brane.

## 2 From κ-symmetry to BRST symmetry for 11d superparticle

To introduce basic notions, we review in the current section the path leading from the -symmetric superparticle action of  to the action of Berkovits , essentially following . We also explain how the tree-level measure in the pure spinor formalism, introduced initially ad hoc, maps to the light-cone measure.

The Brink-Schwarz super-Poincaré invariant point-particle action is

 Scovariant=∫dτ(PMΠM−12ePMPM), (1)
 ΠM=˙XM−i¯ΘΓM˙Θ,

where ,   parametrizes the superparticle worldline, are Majorana spinors with ,   denotes the worldline einbein and the indices of are lowered by the flat metric. This action is invariant under global supersymmetry,

 δϵΘ=ϵ;δϵXM=i¯ϵΓMΘ;δϵPM=δϵe=0, (2)

and local -symmetry,

 δκΘ=iPMΓMκ;δκXM=i¯ΘΓMδκΘ;δκe=4e˙¯Θκ;δκPM=0. (3)

It is also invariant under local reparametrizations.

Using -symmetry one can eliminate half of the components of . This is easily seen by choosing the light-cone gauge 

 X+=x++p+τ;Γ+Θ=0,

where

 X±=1√2(X0±X10);Γ±=1√2(Γ0±Γ10),{Γ+,Γ−}=1.

The light-cone gauge action used by Brink and Schwarz  is

 Sl.c.=∫dτ(˙XMPM+12˙SαSα−12ePMPM), (4)

where are the sixteen surviving SO(9) components of . In this gauge half of supersymmetry is realized linearly and the other half - nonlinearly:

 δεSα = √p+εα;δϵXM=0;δεPM=δεe=0 (5) δηS = i√p+PMΓMΓ+η;δηXM=−2√p+¯ηΓMΓ+S;δεPM=δεe=0, (6)

where the parameter of the former, , satisfies and the parameter of the latter — . Note also that in order to preserve one has to combine a linear supersymmetry transformation with a -symmetry one.

The wavefunction of the system carries SO(9) vector and fermionic degrees of freedom and can be expanded on the basis of physical states: the metric , 3-form and gravitino (see  for details), where . So the superspace is reduced from 32 original Grassmann coordinates to 16 free ones, , satisfying the Clifford relations . At tree level in string expansion, they can be organized into spinor representations and the tree level measure can be defined by the path integral formula

 ⟨ξa1…ξa8⟩g=0=∫d8ξ8∏i=1ξa1…ξa8=ϵa1…a8. (7)

Accordingly, the states , and are given in terms of representations obtained by acting with on the vacuum defined by with . This agrees with the fact that at tree level only 8 zero modes coming from have to be saturated ( play the role of conjugate momenta, i.e. ). Alternatively, we can write the one loop measure in a completely covariant way by saturating both the zero modes of and those of :

 ⟨Sα1⋯Sα16⟩g=1=∫d16S8∏i=1Sa1⋯Sa16=ϵα1⋯α16. (8)

In order to compare (4) with the pure spinor approach, one adds two new doublets and , of anticommuting and commuting variables respectively, which are not related to the light-cone fermions . Then one defines

 ^dA=pA+12PM[ΓM(θ+1√p+Γ+S)]A, (9)

which satisfies

 {^dA,^dB}=−i2p+PMPM(Γ+)AB .

In deriving this relation we have used that are their own canonical conjugates (i.e., they satisfy ). The first-class constraints 333Notice that is also a first class constraint. are associated with the BRST operator

 Q′=λ′A^dA;λ′Γ+λ′=0 , (10)

and the gauge-fixed action

 Scov=∫dτ(PM˙XM−12ePMPM+˙θApA+12˙SαSα+˙λ′Aw′A). (11)

In order to bring the BRST charge (10) into the form of Berkovits’ BRST charge

we split the SO(10,1) commutting spinor into its SO(9) components with respect to as (see Appendix B for more details)

 λ′A=(λ′α1,λ′α2).

With this decomposition the BRST charge (10) becomes

 Q′=λ′α2d2α+λ′α1^d1α,

where we have used the subspace of SO(9) spanned by , . Since is a null SO(9) spinor preserved by a SO(8) subgroup chosen to act on the indices , there is a similarity transformation  that sends the components onto :

 Q′→e−iSΓ+d/√p+ Q′ eiSΓ+d/√p+ (13)

and the components are re-absorbed by shifting in (9). The similarity transformation has to preserve the null spinor nature of and one component of is not gauged into by (13) but is shifted into an extra component of . As shown in Appendix B the components of are identified with the components of via with , which gives components.

The final result is the BRST invariant action

 Scov=∫dτ(PM˙XM−12PMPM+˙θApA+˙λAwA). (14)

where satisfies the pure spinor constraint .444Note that this definition differs form Cartan’s pure spinors, also used by P. Howe, since the latter obey in addition . Interestingly enough, similar constraints, more precisely , emerge in the quantization of the supermembrane  and the covariant description of D-branes . Another obvious remark about the 11d constraint is that when written in terms of two 10d pure spinors it coincides exactly with the two conditions for the D-brane , which should be the case since the latter is related to the 11d superparticle by dimensional reduction. The BRST operator becomes , which is nilpotent only up to pure spinor constraints and gauge transformations , where is a local gauge parameter. It is easy to check that is invariant under and that also . Notice that on-shell, the only non constant field is . The action is manifestly super-Poincaré invariant. In addition, the ghost field part of the Lorentz generators, , as well as the ghost current are invariant under the gauge transformations of .

The light-cone variables satisfy the zero mode saturation identity (7), which expresses the fact that the superparticle preserves half of the original supersymmetry. We will see now that it gets mapped into the tree level measure

 ⟨(λΓM1θ)⋯(λΓM7θ)(θΓM1⋯M7θ)⟩g=0=1 (15)

in the pure spinor formalism. For that purpose, we work in the space with both species of variables and we define the vertex operator by its expansion in terms of the background fields and . Now, we use the expressions for and in terms of the light-cone -expansion and so obtain a vertex operator of the form . In the same way we have to define a vertex operator (see  and next sections for details). Then one can show that

 ϵa1…a8ξa1…ξa8=⟨U(3)(λ,θ,ξ)U(4)(λ,θ,ξ)⟩g=0, (16)

where the correlator is taken only in the space. This maps the measure for zero modes in the pure spinor approach into the measure factor for light-cone variables at tree level.

Concluding this section, we note that from the above light-cone discussion it follows that any correlator with more than sixteen fermions vanishes. This means that any tree-level correlators , with and positive integers, are zero. In other words, the zero momentum cohomology of (see (12)) is empty for -ghost number bigger than 7, as shown by a direct computation in . The same reasonning can be applied for the ten dimensional open string to infer that the zero momentum cohomology of the corresponding is empty for -ghost number greater than 3.

## 3 Supergravity in 11 dimensions and vertex operators

In this section we analyze the structure of the vertex operators for the particle limit of the eleven-dimensional supermembrane. First we derive them from BRST cohomology and explain their component field expansion. We also point out that they are consistent with the general expression with undetermined coefficients of the integrated covariant vertex of . Then by coupling to a nontrivial background we derive another version of the integrated and unintegrated operators, which will be useful in the rest of the paper.

From now on we use the following conventions in 11d: curved vector indices; curved spinor indices; flat vector indices; flat spinor indices. Also . In 10d we denote by curved vector indices and by curved spinor indices. Finally run over the spatial directions of the membrane and is a derivative w.r.t. the M2 worldvolume time.

### 3.1 Vertex Operators and N=1,d=11 Supergravity

The vertex operator for the 11d superparticle at ghost number zero can be viewed as a deformation of the quantum action expanded around flat background. It emerges as the first non-trivial contribution

 Sdef=Scov+∫dτV(0)+O(V), (17)

where denotes higher terms in the expansion. In order that the action is still invariant under BRST symmetry, one has , where is a ghost number one vertex operator. The latter satisfies by consistency.

Acting with the BRST operator on a generic polynomial of the fields and , one finds that

and

 U(1)=λAEArEMsPMδrs, (19)

where  ,   and are superfields which satisfy the supergravity field equations. We also note that (18) is easily seen to be consistent with the light-cone vertices of .

The vertex operator is defined up to a gauge transformation generated by total derivative terms, i.e. . Correspondently, the vertex is defined up to BRST exact terms of the form . For the case at hand, the gauge transformations are given by

At leading order in the expansion their parameters are the following:

 ΩM=ξM+O(θ),     ΩA=ϵA+O(θ),     Ω[MN]=ω[MN]+O(θ), (21)

where is the one of infinitesimal diffeomorphisms, of infinitesimal supersymmetry transformations and of Lorentz gauge transformations (i.e., rotations).

The superfields and , contained in the vertex operator , are comprised from the fields of the usual 11d supergravity multiplet namely the graviton , gravitino and 3-form . However, the 11d superparticle does not couple directly to the potential , but only feels its field-strength . As a result, the tensor gauge transformations cannot be seen among the transformation rules given in (20).

The expansion of the superfields , up to order in terms of the on-shell supergravity fields is the following [29, 30]:

 EMr = eMr+2¯θΓrψM+(¯θUr|NPQLMθ)GNPQL+14¯θΓrstθωM[st]+O(θ3) EMa = ψMa+14ωM[rs](Γrsθ)a+(TMNRSTθ)aGNRST+O(θ3) EAr = −(¯θΓr)A+O(θ3) H[rs]M = ω[rs]M+O(θ2) (22)

with

 TMNRST = 1288(ΓMNRST−8ηM[NΓRST]); (23) UR|MNPQL = 1288(5ηR[MΓNPQL]+8ΓR[MNPηQ]L). (24)

We note in passing that there are several different ways of obtaining the above expansions. The most well-known one is the gauge completion method used in . A much easier alternative is the following: one can use the field equations and Bianchi identities to derive recursion relations for the component fields in the expansion of the supergravity superfields similarly to the case of 10-dimensional SYM . This procedure was applied recently to ten- and eleven- dimensional supergravity in  and  respectively.555We should mention yet another way of calculating the component fields in the expansion of superfields, namely the normal coordinates approach advocated for example in .

The superspace description of 11d supergravity, obtained by starting from the ghost number zero vertex operator , coincides with the superspace description given in [22, 23]. However, as shown in [24, 16], one can also use a formulation based on the constraint of the spinorial component of the four form . This constraint can be translated into a BRST closedness condition by considering another type of vertex operator . The superfield is defined up to gauge transformations . By requiring that is BRST closed, one can easily derive the condition 

 D(AABCD)=ΓM(ABACD)M, (25)

where is an auxiliary superfield.666These equations are readily solved by imposing the Wess-Zumino gauge . This form of the unintegrated vertex operator is related to the integrated one for the supermembrane   , where

 +AΣΓΛΞ∂IZΣ∂JZΓ∂KZΛ∂LZΞϵIJϵKL+YΣBA∂IZΣwA∂JλBϵIJ. (26)

By setting to zero the transverse fluctuations of the membrane , one reduces (26) to the first line, which is indeed of the same form as the vertex given in (18).

### 3.2 Coupling of the action to a background

Recall that in curved space we denote by capital greek letters superspace curved indices in 11d and by latin capital letters the tangent space indices. So and are the supersymmetric conjugate momenta to . The superparticle action can be written as

 Scurved=∫dτ(PMEΩM∂ZΩ+dAEΩA∂ZΩ−12ηMNPMPN+ (27)
 +wA(∂λA+∂ZΣΩΣ,[MN](ΓMN)BAλB)).

As always the connection couples to the ghost with a spin coupling. Clearly the equations of motion for the fields and are changed.

In addition, the action (27) is no longer BRST invariant. This reflects itself in the lack of nilpotency of the BRST operator

 Q=λA(dA+EAΣ∂ZΣ) (28)

in curved space. The non-invariance of the action implies that . Imposing

 ˙Q=0,         {Q,Q}=0 (29)

and using the pure spinor conditions, one ends up with the usual supergravity constraints on the supertorsion and field strenght :

 T    ABC=T   NAM=0,    TMAB=(ΓM)AB, (30)
 GABCD=GABCM=0,    GABMN=(ΓMN)AB.

For later use we write down another version of the vertex operators (18) and (19),

obtained after integrating out in (27). Notice that the integrated vertex operator, expressed in this form, resembles the integrated vertex of the closed superstring.

## 4 Tree-level amplitudes

### 4.1 Path integral definition for the tree-level measure

It is convenient to rewrite (15) in a functional integral form for later comparison with one-loop amplitudes. In order to be able to integrate over the 23 independent pure spinor components, we have to define a suitable measure which respects the pure spinor constraint and by gauge fixing the zero modes of and (at three level and for flat worldvolume with marked points, there are no zero mode for and ). So we introduce the Lagrange multiplier and its BRST partner , where , as well as their respective conjugate momenta and . Now the BRST operator changes to . We also introduce the constant gauge fixing parameters with the help of which the tree level functional integral becomes

 ⟨∏iU(ni)i(λ,X,θ)⟩g=0 = ∫d11Xd11Pd32θd23χd23η[dλA1…dλA23] (ϵ32T−1)(B1…B7)[A1…A23]7∏j=1∂∂λBj(e[Qnew,ηICIAλA])∏iU(ni)i(λ,X,θ),

where and is a unique tensor constructed in terms of Lorentz invariants. Integrating over and , one can replace the exponential with the picture lowering operator . This expression is manifestly independent of since the vertex operators , as well as the measure, are BRST invariant. In the following section we will see that the measure is indeed consistent with the pure spinor constraint. Computing the derivatives in the above expression, we end up with

 ⟨∏iU(ni)i(λ,X,θ)⟩g=0 = ∫d11Xd11Pd32θd23χd23η[dλA1∧⋯∧dλA23] (ϵ32T−1)(B1…B7)[A1…A23]7∏j=1(CIBjχI)(e[Qnew,ηICIAλA])∏iU(ni)i(λ,X,θ).

Since the amplitude is independent of the gauge fixing parameters , an easy way to compute it is to average over those parameters with a Gaussian-type measure such that .

### 4.2 Measure of integration for the pure spinor λ

In this section we construct the bosonic ghost measure . The measure for the canonically conjugate momentum (recall that ) will be constructed in section 5.1 when discussing one-loop amplitudes.

The pure spinor condition

 λΓMλ=0 (34)

is a set of first class constraints and for constructing the measure of integration we use the same method as in . Namely, from the zero mode prescription at tree-level (15) we construct a tensor , antisymmetric in the indices and traceless symetric in , defined by

 1 = ⟨(λΓm1θ)⋯(λΓm7θ)(θΓm1⋯m7θ)⟩ (35) =: λB1⋯λB7θA1⋯θA9T((B1⋯B7))[A1⋯A9]

and write

 [d23λ][α1⋯α23] := dλα1∧⋯∧dλα23=[Dλ]+16(ϵ32T)[α1⋯α23]((β1⋯β7))λβ1⋯λβ7 (36)

with the totally antisymmetric tensor and as scalar measure factor with ghost-charge .

The inverse tensor is given by

 (T−1)((α1⋯α7))[ρ1⋯ρ9] = ⟨λα1⋯λα7θρ1⋯θρ9⟩ = Π((α1⋯α7))β1⋯β7(ΓM1)β1[ρ1⋯(ΓM7)β7ρ7](ΓM1⋯M7)ρ8ρ9],

where is the projector on symmetric -traceless 7-tensors:

 Π((α1⋯α7))β1⋯β7 = δ(α1β1⋯δα7)β7−2174(ΓM1)(α1α2δα3(β1⋯δα7)β5(ΓM1)β6β7) + 21592(ΓM1)(α1α2(ΓM2)α3α4δα5(β1⋯δα7)β3(ΓM2)β4β5)(ΓM1)β6β7) − 2120128(ΓM1)(α1α2⋯(ΓM3)α5α6δα7)(β1(ΓM3)β2β3⋯(ΓM1)β6β7).

Before concluding this subsection we make a few remarks to stress the differences between the eleven-dimensional case studied in the present paper and the ten-dimensional setup.

We can parametrize the pure spinor by first splitting the 32 of SO(11) as of SO(10), i.e. , and then decomposing each SO(10) spinor into U(5) representations and as

 λαL=(λ+,λa,λab),λαR=(λ−,λa,λab),a=1,⋯,5

In this language the pure spinor constraint reads

 λ+λ−+λaλa+12λabλab = 0 λ+λa+18ϵabcdeλbcλde = −λbλab λ−λa+18ϵabcdeλbcλde = −λbλab (39)

where the first equation comes from and the other two from . We solve these equations as

 5!λ[aλbcλde] = 8ϵabcdeλ+(λ+λ−+12λabλab) 5!λ[aλbcλde] = 8ϵabcdeλ−(λ+λ−+12λabλab). (40)

This determines completely all components of and . The solution is very similar to the one of the ten-dimensional pure spinor constraint, where the 5 is solved in terms of the of U(5) as (see ). But there is a crucial difference since the 5 of is solved in terms of the 1 and 10 of and . This gives a solution with complex parameters. The SU(5) decomposition of an SO(11) pure spinor has an extra complex parameter , which leaves equations (39) and (4.2) invariant under the rescaling

 (λL,λR)→(ρλL,ρ−1λR). (41)

This symmetry is related to the constraint (i.e. ; this condition can be solved by a generic choice of and up to a scale gauge symmetry), which states that an eleven dimensional pure spinor is not just the square of two ten dimensional pure spinors. It is now clear that the measure on can also be written in the following form:

 d23λA=dρ∧(dλ+∧dλab)∧(dλ−∧dλab). (42)

### 4.3 Tree-level action from scattering amplitudes

In this subsection we will reproduce the effective action for eleven dimensional supergravity up to four-fermi terms from tree-level amplitude computations. The supergravity action up to this order in the fermion fields reads 

 S=12ℓ9P∫d11x[eR+e¯ΨMΓMNP^DNΨP+12G4∧∗G4+16C3∧G4∧G4], (43)

where is the supercovariant derivative (again up to fermion bilinears), involving the field strength. The tensor is defined in (23).

The zero momentum cohomology of contains a non zero -ghost number 3 vertex operator whose components are comprised of the supergravity fields and their derivatives and a -ghost number 4 vertex operator containing the antifields. The relevant components of are :

 U(3)C3 = (λΓMθ)(λΓNθ)(λΓKθ)CMNKeik⋅X
{"ft_lang_label":"__label__en","ft_lang_prob":0.91142493,"math_prob":0.96623844,"size":23199,"snap":"2020-45-2020-50","text_gpt3_token_len":5032,"char_repetition_ratio":0.15641302,"word_repetition_ratio":0.013119143,"special_character_ratio":0.212035,"punctuation_ratio":0.087448806,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98444897,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-12-05T22:19:33Z\",\"WARC-Record-ID\":\"<urn:uuid:1175d229-d235-4d84-a7e2-2fbbb4a245ff>\",\"Content-Length\":\"1049561\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:e7658435-bb81-4a0e-bd2c-c8b0d9f8c3b7>\",\"WARC-Concurrent-To\":\"<urn:uuid:dd7b73eb-0065-4731-bc4f-3953d389ef59>\",\"WARC-IP-Address\":\"104.28.20.249\",\"WARC-Target-URI\":\"https://www.arxiv-vanity.com/papers/hep-th/0408171/\",\"WARC-Payload-Digest\":\"sha1:NCAL5ZISGTMSQODWCDUVAFTSEB4MXJD3\",\"WARC-Block-Digest\":\"sha1:EL6HIIEEV646HSSRIGQACFVZSH6MV4VV\",\"WARC-Truncated\":\"length\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-50/CC-MAIN-2020-50_segments_1606141750841.83_warc_CC-MAIN-20201205211729-20201206001729-00234.warc.gz\"}"}