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ScienceDirect

Nuclear Physics B 907 (2016) 542–571

www.elsevier.com/locate/nuclphysb

On the construction of integrated vertex in the pure 

spinor formalism in curved background

Andrei Mikhailov 1

Instituto de Física Teórica, Universidade Estadual Paulista, R. Dr. Bento Teobaldo Ferraz 271, Bloco II, Barra Funda, 
CEP:01140-070, São Paulo, Brazil

Received 25 June 2015; received in revised form 26 February 2016; accepted 4 April 2016

Available online 7 April 2016

Editor: Herman Verlinde

Abstract

We have previously described a way of describing the relation between unintegrated and integrated vertex 
operators in AdS5 × S5 which uses the interpretation of the BRST cohomology as a Lie algebra cohomol-
ogy and integrability properties of the AdS background. Here we clarify some details of that description, 
and develop a similar approach for an arbitrary curved background with nondegenerate RR bispinor. For an 
arbitrary curved background, the sigma-model is not integrable. However, we argue that a similar construc-
tion still works using an infinite-dimensional Lie algebroid.
© 2016 Published by Elsevier B.V. This is an open access article under the CC BY license 
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

1. Introduction

The construction of the worldsheet sigma-model for the Type II superstring in the pure spinor 
formalism is a fundamental problem. It was more or less solved in [1]. However, we feel that 
some better understanding is possible. First of all, the sigma-model suggested in [1] is techni-
cally very special, and it is not clear why this is the most general solution. In particular, the 
formulation depends crucially on a special choice of fields. Indeed, the theory is not invariant un-

E-mail address: andrei@ift.unesp.br.
1 On leave from Institute for Theoretical and Experimental Physics, ul. Bol. Cheremushkinskaya, 25, Moscow 117259, 

Russia.
http://dx.doi.org/10.1016/j.nuclphysb.2016.04.004
0550-3213/© 2016 Published by Elsevier B.V. This is an open access article under the CC BY license 
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

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A. Mikhailov / Nuclear Physics B 907 (2016) 542–571 543
der field redefinitions mixing matter fields with ghosts. It would be desirable to have an axiomatic 
formulation of the sigma-model. Something along these lines:

• A sigma-model with two nilpotent symmetries, QL and QR , such that the current of QL is 
holomorphic and the current of QR is antiholomorphic, and there are symmetries U(1)L and 
U(1)R , such that QL and QR are appropriately charged under them.

However, we feel that this is not enough; the axiomatics sketched above is probably too weak, 
although it is enough to correctly describe small deformations of the flat space. The correct 
axiomatics should somehow encode the singularity of the pure spinor cone.

Also, we believe that the worldsheet sigma-model should be formulated as a problem in 
cohomological perturbation theory. A small neighborhood of each point in space–time can be 
approximated by flat space:

S =
∫

dτ+dτ− (
∂+Xμ∂−Xμ + p+∂−θL + p−∂+θR + w+∂−λL + w−∂+λR

)
(1)

with BRST symmetries:

QL = λα
L

(
∂

∂θα
L

+ �m
αβθ

β
L

∂

∂xm

)
+ (. . .)

∂

∂w+
(2)

QR = λα̂
R

(
∂

∂θ α̂
R

+ �m

α̂β̂
θ

β̂
R

∂

∂xm

)
+ (. . .)

∂

∂w−
(3)

Then we say that a general background is obtained by the deformation of the action accompanied 
by some deformation2 of QL and QR . The infinitesimal deformations at the linearized level 
are well known to correspond to the linearized SUGRA waves. They were classified in [3]. 
However, it was shown also in [3] that there is a potential obstacle to extending the deformations 
beyond the linearized level. The obstacle is a nonzero cohomology group, namely the ghost 
number three vertex operators. Without doubt, the obstacle actually vanishes (there is a nonzero 
cohomology group, but the actual class vanishes). This, however, is not well understood. As we 
explained in [3], one way to prove the vanishing of the obstacle is to consider the action of the 
b-ghost in cohomology. The formalism that would allow to do such calculation has not yet been 
fully developed. The definition of the b-ghost requires including the non-minimal fields which 
makes the lack of axiomatic formulation even more acute. And the b-ghost is nonpolynomial, 
opening the possibility that at some order the deformed action will also become non-polynomial.3

Such questions should be addressed together with the problem of axiomatic formulation of the 
worldsheet theory.

When we study the pure spinor formalism as a cohomological perturbation theory, one im-
portant technical aspect is the relation between integrated and unintegrated vertex operators. The 
deformation of the action is described by integrated vertices:

S → S +
∫

dτ+dτ− U (4)

2 As we have shown in [2], the very leading effect will be actually the deformation of QL and QR leaving S unde-
formed; this corresponds to the linear dilaton.

3 We have no doubt that this does not happen, it is just that we don’t know how to see this using the cohomological 
perturbation theory.



544 A. Mikhailov / Nuclear Physics B 907 (2016) 542–571
It is very important, that such deformations are in one-to-one correspondence with the uninte-
grated vertices, which correspond to the cohomology of QL +QR . One of the goals of this paper 
is to better understand the correspondence between integrated and unintegrated vertices.

In [4–6] we have studied the relation between the pure spinor cohomology in AdS5 × S5 and 
the Lie algebra cohomology, and argued that it is useful for understanding the relation between 
the integrated and unintegrated vertices. The pure spinor cohomology is the cohomology of the 
operator QBRST acting on the space of functions F(g, λL, λR):

(QBRSTF)(g,λL,λR) =
(
λα

LLα + λα̂
RLα̂

)
F(g,λLλR) (5)

Here g ∈ G = PSU(2, 2|4) and Lα , Lα̂ are left shifts by some generators of psu(2, 2|4). 
We introduced an infinite-dimensional Lie superalgebra Ltot, and shown that the cohomol-
ogy of QBRST is equivalent to some cohomology of Ltot. Unintegrated vertices of the physi-
cal states correspond to the elements of the second cohomology group. Moreover, there is a 
Lax pair J+, J− taking values in Ltot. Given a cohomology class represented by a cocycle 
ψ : 
2Ltot → Fun(G), the corresponding integrated vertex is ψ(J+, J−). This construction 
uses special properties of AdS5 × S5.

Here we will describe a similar construction for an arbitrary curved spacetime with the non-
degenerate Ramond–Ramond field strength.4 Instead of the superalgebra Ltot we will use some 
super Lie algebroid. We will conjecture that the cohomology of this algebroid is equal to the 
BRST cohomology, i.e. unintegrated vertex operators. Moreover, there seems to be an analogue 
of a Lax pair, which allows to construct integrated vertices. However, this Lax pair takes values 
in the sections of a Lie algebroid (instead of a fixed Lie algebra), and presumably does not lead 
to integrability.

Better understanding of the integrated vertices could also help to explain the consistency of 
the higher orders of the deformation of the action (1). Superficially, this problem looks similar 
to the PBW theorem of quadratic-linear algebras which (coincidentally?) is also useful in the 
construction of the integrated vertex.

In eleven dimensional SUGRA, the analogous problem is the membrane worldsheet the-
ory [7]. However, it appears more difficult than string worldsheet theory. But unintegrated 
vertices are more or less understood. Constructing integrated vertex operators is very close to 
understanding the worldsheet theory. Maybe some methods which we are developing here could 
be useful.

Summary of results Section 2 provides a streamlined proof of previously obtained results for 
the pure spinor superstring in AdS5 × S5. Sections 3 and 4 provide a new point of view on the 
Type II pure spinor superspace, based on the study of the algebra generated by the covariant 
derivatives. This is somewhat similar to the Yang–Mills algebra, but quite different in details. 
Understanding this algebra requires careful analysis of the space–time gauge symmetries. The 
relations become degenerate in the case of zero Ramond–Ramond field; we feel that our approach 
could be particularly useful in the case of strong RR field. Actually, in Section 4.6 we give a 
new definition of the Ramond–Ramond fields which, in our opinion, is geometrically the most 
transparent one. (The problem of “reading” the RR field from the superspace action turns out to 
be nontrivial; we feel that Eqs. (176) and (177) provide the nicest solution.) As an immediate 

4 Some elements of our construction become degenerate if the Ramond–Ramond field strength is zero. We do use the 
inverse of the RR bispinor Pαα̂ in Section 4. See the discussion of the flat space limit in [6].



A. Mikhailov / Nuclear Physics B 907 (2016) 542–571 545
application, we provide a relation between integrated and unintegrated vertex operators, which 
is a nontrivial generalization of the similar relation in AdS5 × S5. Unfortunately (or perhaps 
fortunately) this requires the introduction of some infinite-dimensional algebroid, which could 
be an interesting object in itself. There is even an algebroid analogue of a Lax pair, but it is not 
clear in what sense it leads to integrability.

Plan of the paper

• In Section 2 we give a streamlined review of [4–6], also simplifying some of the proofs in 
those references

• Section 3 develops a different point of view on the formalism of [1]; our approach empha-
sizes the similarity between the constraints of the Type IIB SUGRA and the constraints of 
the supersymmetric Yang–Mills theory

• In Section 4 we study the worldsheet currents. We construct an object resembling the Lax 
pair of the AdS theory, but using an algebroid instead of a Lie algebra. We conjecture that 
the cohomology of this algebroid corresponds to integrated vertex operators

2. Brief review of the case of AdS5 × S5

Here we will briefly review the relation between the unintegrated and integrated vertices de-
scribed in [4–6]. Both types of vertices are obtained from the same relative cocycle of some 
infinite-dimensional Lie superalgebra which we call Ltot. In this sense, the relative Lie algebra 
cohomology of Ltot provides the unified description of both types of vertex operators.

2.1. Definition of Ltot and the PBW theorem

The infinite-dimensional superalgebra Ltot is defined in [5] by “gluing together” two copies 
of the Yang–Mills algebra which we call LL and LR , in the following way. The LL is generated 
by letters ∇L

α , and LR is generated by ∇R
α̂

, satisfying the super-Yang–Mills constraints:

{∇L
α , ∇L

β } = �m
αβAL

m , {∇R
α̂

, ∇R

β̂
} = �m

α̂β̂
AR

m (6)

(The existence of such AL and AR are the constraints.) All we need to do is to explain how ∇L
α

anticommutes with ∇R
α̂

. For that we add a copy of the finite-dimensional algebra g0̄ = so(1, 4) ⊕
so(5) with the generators denoted t0[mn]. We impose the commutation relations:

{∇L
α , ∇R

α̂
} = fαα̂

[mn]t0[mn] (7)

[t0[mn], ∇L
α ] = f[mn]αβ∇L

β (8)

[t0[mn], ∇R
α̂

] = f[mn]α̂ β̂∇R

β̂
(9)

[t0[mn], t0[pq]] = f[mn][pq][rs]t0[rs] (10)

where the coefficients f••• are the structure constants of g = psu(2, 2|4).
One can consider the Lie algebra generated by the letters ∇L

α , ∇R
α̂

, t0[mn] with the above re-
lations, or the associative algebra generated by them. The associative algebra is the same as the 
universal enveloping ULtot.



546 A. Mikhailov / Nuclear Physics B 907 (2016) 542–571
The algebra ULtot is an example of a quadratic-linear algebra. It is a filtered algebra; FpULtot
consists of those elements which can be represented by sums of products of ≤ p letters. For 
example AL

m ∈ F 2ULtot.
One can also define a homogeneous quadratic algebra qULtot as an associative algebra 

generated by the letters ∇L
α , ∇R

α̂
, t0[mn] with the relations (6) and {∇L

α , ∇R

β̂
} = [t0[mn], ∇L

α ] =
[t0[mn], ∇R

α̂
] = [t0[mn], t0[pq]] = 0. The algebra qULtot is graded; grpULtot consists of those ele-

ments which consist of p letters.

Theorem 1 (PBW).

grpULtot = grp(qULtot) (11)

Proof. Uses the fact that qULtot is a Koszul quadratic algebra. We will give a proof following 
Section 3.6.8 of [8]. We will need some standard language, which we will now review. Let V
be the vector space generated by the letters ∇L

α , ∇R
α̂

, t0[mn]. Consider the subspace R ⊂ V ⊗ V

generated by the following elements (the relations of qULtot):

t0[mn] ⊗ t0[pq] − t0[mn] ⊗ t0[pq] (12)

t0[mn] ⊗ ∇L
α − ∇L

α ⊗ t0[mn] (13)

t0[mn] ⊗ ∇R
α̂

− ∇R
α̂

⊗ t0[mn] (14)

∇L
α ⊗ ∇R

β̂
+ ∇R

β̂
⊗ ∇L

α (15)

(�m1···m5)
αβ∇L

α ⊗ ∇L
β (16)

(�m1···m5)
α̂β̂∇R

α̂
⊗ ∇R

β̂
(17)

Notice that qULtot can be defined as the factorspace of the tensor algebra (=free algebra) T V

modulo the ideal generated by R.
The dual coalgebra UL¡

tot is defined as the following subspace of T V :

UL¡
tot = C ⊕ V ⊕ R ⊕

∞⊕
p=3

p−2⋂
q=0

(V ⊗q ⊗ R ⊗ V ⊗(p−q−2)) (18)

The coalgebra structure is induced from the standard coalgebra structure of the tensor product:

�(a ⊗ b ⊗ c ⊗ · · · ) = a|(b ⊗ c ⊗ · · · ) + (a ⊗ b)|(c ⊗ · · · ) + (a ⊗ b ⊗ c)|(· · · ) + . . . (19)

Explanation of notations For any coalgebra C, the coproduct � acts from C to C ⊗ C. In 
our case, it so happens that C is itself defined as a tensor product. In this case it is common to 
use the notation C|C instead of C ⊗ C, just to avoid confusion. The spaces C|C| · · · |C form the 
so-called cobar complex, because there is a natural differential:

d(x|y|z| · · · ) = �(x)|y|z| · · · − x|�(y)|z| · · · + x|y|�(z)| · · · − . . . (20)

The nilpotence of this differential is equivalent to the co-associativity of �. This complex is 
denoted �(C). As a vector space �(C) is:

�(C) =
∞⊕

C⊗p (21)

p=0



A. Mikhailov / Nuclear Physics B 907 (2016) 542–571 547
It is naturally an algebra, just a tensor (=free) algebra over C:

�(C) = T C (22)

Also notice that d respects the multiplication: d(X|Y) = d(X)|Y − (−1)rk(X)X|dY . This means 
that �(C) is a differential algebra. Let us consider the cohomology of d .

Observation 1.

H 0
d (�(UL¡

tot)) = qULtot (23)

Proof. This is obvious from the definitions.
So far the definition of UL¡

tot only used the homogeneous relations of qULtot, it is really 
(qULtot)

¡ rather than UL¡
tot. We have to somehow take into account the nonzero right hand sides 

of (7), (8), (9), (10). This is done by supplying UL¡
tot with a differential d1, which is defined as 

follows5:

d1(a ⊗ b ⊗ c ⊗ · · · ) = ((d1(a ⊗ b)) ⊗ c ⊗ · · · ) − (a ⊗ (d1(b ⊗ c)) ⊗ · · · ) + . . . (24)

d1

(
t0[kl] ⊗ t0[mn] − t0[mn] ⊗ t0[kl]

)
= f[kl][mn][pq]t0[pq] (25)

d1

(
t0[mn] ⊗ ∇L

α − ∇L
α ⊗ t0[mn]

)
= f[mn]αβ∇L

β (26)

d1

(
t0[mn] ⊗ ∇R

α̂
− ∇R

α̂
⊗ t0[mn]

)
= f[mn]α̂ β̂∇R

β̂
(27)

d1

(
∇L

α ⊗ ∇R

β̂
+ ∇R

β̂
⊗ ∇L

α

)
= f

αβ̂
[mn]t0[mn] (28)

d1

(
(�m1···m5)

αβ∇L
α ⊗ ∇L

β

)
= 0 (29)

d1

(
(�m1···m5)

α̂β̂∇R
α̂

⊗ ∇R

β̂

)
= 0 (30)

The verification of the nilpotence of d1 is equivalent to the verification of the Jacobi identity of 
Ltot in filtration ≤ 3. There are the following cases to verify:

d2
1

(
�αβ

m1...m5
∇L

α ∧ ∇L
β ∧ ∇R

α̂

)
= 0 (31)

d2
1

(
�αβ

m1...m5
∇L

α ∧ ∇L
β ∧ t0[mn]

)
= 0 (32)

d2
1

(
∇L

α ∧ ∇R

β̂
∧ t0[mn]

)
= 0 (33)

d2
1

(
∇L

α ∧ t0[mn] ∧ t0[pq]
)

= 0 (34)

d2
1

(
t0[mn] ∧ t0[pq] ∧ t0[rs]

)
= 0 (35)

and similar equations with L ↔ R. Eq. (35) is the Jacobi identity for g0̄. Eq. (34) says that the 
spinor representation is a representation of g0̄. Eq. (33) is one of the Jacobi identities of the 
psl(4|4):

5 Notice that the signs do not depend on whether a, b, c, . . . are “odd” or “even”, and in fact we do not use such words 
at this point. The notion of “odd” or “even” elements only becomes useful when we say that our quadratic-linear algebra 
is in fact a universal enveloping of a super-Lie algebra.



548 A. Mikhailov / Nuclear Physics B 907 (2016) 542–571
f
αβ̂

[pq]f[pq][mn][rs] = fαγ̂
[rs]f

β̂[mn]
γ̂ + fα[mn]γ f

γ β̂
[rs] (36)

Eq. (32) is derived as follows. After first time applying d1 we get:

d1

(
�αβ

m1...m5
∇L

α ∧ ∇L
β ∧ t0[mn]

)
=

= �αβ
m1...m5

(
fα[mn]γ ∇L

γ ∧ ∇L
β + fβ[mn]γ ∇L

α ∧ ∇L
γ

)
=

= [�mn,�m1...m5]αβ∇L
α ∧ ∇L

β (37)

Since [�mn, �m1...m5] is a five-form, the second application of d1 results in zero. Eq. (31) is 
derived as follows:

d2
1

(
�αβ

m1...m5
∇L

α ∧ ∇L
β ∧ ∇R

α̂

)
=

= 2�αβ
m1...m5

fβα̂
[mn]fα[mn]γ ∇L

γ =
= −�αβ

m1...m5
fβα

kfα̂k
γ ∇L

γ = 0 (38)

where we have taken into account that fβα
k = �βα

k and therefore the contraction with �αβ
m1...m5

is zero. �
Observation 2.

H 0
d+d1

(�(UL¡
tot)) = ULtot (39)

Proof. This is also obvious from the definitions.
Notice that until now we have not done anything nontrivial, just developed a language. But 

now we are ready to proceed with the proof of the PBW theorem (11). Before the proof, we prob-
ably have to explain why the statement is nontrivial. Let us consider, for example, the following 
element of Ltot:

X = [{∇L
α ,∇L

β },∇L
γ ] (40)

This expression can be represented by the following element of V |V |V ⊂ �(UL¡
tot):

X = (∇L
α |∇L

β + ∇L
β |∇L

α )|∇L
γ − ∇L

γ |(∇L
α |∇L

β + ∇L
β |∇L

α ) (41)

The question is, how do we know that this element is nonzero? Maybe one can prove that it 
is zero, using the relations of Ltot? We know however that it is nonzero as an element of LL. 
(We are not going to prove it now; in fact this particular expression corresponds to the field 
strength superfield.) The LL is a homogeneous quadratic algebra. We want to prove that X it 
is also nonzero as an element of Ltot, an inhomogeneous (quadratic-linear) algebra. The danger 
is that maybe there is some element Y0, for example in R|V |V ⊂ �(UL¡

tot), such that dY0 = 0
and d1Y0 = X. This would imply that (d + d1)Y0 = X and therefore X is actually zero as an 
element of Ltot. Or, perhaps there are Y0 ∈ V |V |V |R and Y1 ∈ V |V |R such that d1Y1 = X and 
dY1 = −d1Y0 and dY0 = 0; then again (d + d1)(Y0 + Y1) = X and therefore X is zero. We deal 
with such fears in the following manner. Suppose X = (d + d1)Y and Y0 be the highest bar-order 
term of Y (the term with the highest number of |). Then dY0 = 0. Because qLtot is Koszul,6 this 

6 The Koszul property implies that the cohomology group corresponding to d-closed Y0 modulo d-exact Y0 vanishes, 
see Section 3.6.8 of [8] for details. The algebra of functions of ten-dimensional spinors satisfying (λ�mλ) = 0 is Koszul 



A. Mikhailov / Nuclear Physics B 907 (2016) 542–571 549
implies that Y0 = dZ0. We therefore have (d + d1)(Y − (d + d1)Z0) = X, and the bar-order of 
Y − (d + d1)Z0 is one less than the bar-order of Y . We repeat this until the bar-order of Y is 
equal to the bar-order of X. Now (d + d1)Y = X implies that the leading order term in X is zero 
in qULtot. This contradicts the assumption and completes the proof of the PBW theorem. �
Theorem 2. As a linear space

Ltot = LL ⊕LR ⊕ g0̄ (42)

Proof. The Lie superalgebra Ltot can be considered a subspace of ULtot, consisting of those 
elements which can be represented as nested commutators. Then (42) follows from the PBW 
theorem. �
Comment A physical interpretation of qULtot could be the flat space limit of ULtot.

2.2. BRST complex

2.2.1. The structure of the dual coalgebra
Besides the PBW theorem, the Koszulity also implies that the complex (ULtot)

¡ ⊗ ULtot is 
acyclic. Notice that (ULtot)

¡ has the following structure. As a linear space:

(ULtot)
¡ = (ULL)¡ ⊗ (ULR)¡ ⊗ 
g0̄ (43)

Any element ω ∈ (ULtot)
¡ can be presented as a linear sum of the expressions of the “decom-

posable” elements of the form:

ω = Fα1···αp β̂1···β̂q [m1n1]···[mrnr ]

∇L
α1

⊗ · · · ⊗ ∇L
αp

⊗ ∇R

β̂1
⊗ · · · ⊗ ∇R

β̂q
⊗ t0[m1n1] ⊗ · · · ⊗ t0[mrnr ] +

+ . . . (44)

where Fα1···αp β̂1···β̂q [m1n1]···[mrnr ] is symmetric in α1, . . . , αp and in β̂1, . . . , β̂q and antisymmet-
ric in [m1n1], . . . , [mrnr ] and satisfies:

�k
α1α2

Fα1···αp β̂1···β̂q [m1n1]···[mrnr ] = 0 (45)

�k

β̂1β̂2
Fα1···αp β̂1···β̂q [m1n1]···[mrnr ] = 0 (46)

and . . . in (44) stand for the terms which are obtained from the first term by permutations of the 
tensor product, which are needed so that the resulting expression belong to the exterior product 
of p + q + r copies of the linear superspace generated by ∇L, ∇R and t0. For example, when 
p = 2 and r = 1, we get:

ω = Fα1α2[mn] (∇L
α1

⊗ ∇L
α2

⊗ t0[mn] − ∇L
α1

⊗ t0[mn] ⊗ ∇L
α2

+ t0[mn] ⊗ ∇L
α1

⊗ ∇L
α2

)
(47)

The action of d1 on this ω is:

d1 ω = Fα1α2[mn]f[mn]α1
α∇L

α ⊗ ∇L
α2

(48)

by the results of [9]. The SYM algebras LL and LR are both Koszul as quadratic duals to the Koszul algebra of pure 
spinors. The algebra qULtot is the commutative product of ULL , ULR and 
g0̄ , and therefore is Koszul by the Corol-

lary 1.2 in Chapter 3 of [10] (where the commutative product is denoted ⊗q=1).



550 A. Mikhailov / Nuclear Physics B 907 (2016) 542–571
2.2.2. Computing Hp(ULtot , V ) as ExtULtot(C, V )

We have seen that the complex:

. . . → (ULtot)
¡
2 ⊗ ULtot → (ULtot)

¡
1 ⊗ ULtot → ULtot → C → 0 (49)

provides a free resolution of C as a ULtot-module. Therefore, for any representation V of ULtot, 
we can compute the cohomology group:

Hp(ULtot , V ) = ExtULtot(C,V ) (50)

as the cohomology of the complex HomULtot

(
(ULtot)

¡
p ⊗ ULtot , V

)
:

0 → V → HomC

(
(ULtot)

¡
1 , V

)
→ HomC

(
(ULtot)

¡
2 , V

)
→ . . . (51)

We will now interpret this complex in terms of ghosts. An element of (ULtot)
¡ is the sum of 

expressions of the form (44). Notice that the dual space 
(
(ULtot)

¡)′ is the space of functions of 

commuting variables cα
L, cα̂

R satisfying the pure spinor constraints cα
L�m

αβc
β
L = cα̂

R�m

α̂β̂
c
β̂
R = 0 and 

anticommuting variables c[mn]
0 . In this language the BRST operator becomes

QBRST = c
[mn]
0 ρ(t0[mn]) + f α

β[mn]cβ
Lc

[mn]
0

∂

∂cα
L

+ f α̂
β̂[mn]c

β̂
Rc

[mn]
0

∂

∂cα̂
R

+

+ 1

2
f [mn][m1n1][m2n2]c

[m1n1]
0 c

[m2n2]
0

∂

∂c
[mn]
0

+

+ cα
Lρ(∇L

α ) + cα̂
Rρ(∇R

α̂
) +

+ f [mn]
αα̂cα

Lcα̂
R

∂

∂c
[mn]
0

(52)

Notice that the ghosts corresponding to g0̄ are non-abelian while the ghosts cα
L and cα̂

R are pure 
spinors. We will therefore call (51) the “mixed complex”: it is the pure spinor BRST complex 
coupled with the Serre–Hochschild complex of the finite-dimensional Lie algebra g0̄.

2.2.3. Decoupling of the c0-ghosts
In the mixed complex (51) Let us consider the decreasing filtration by the power of cα

L plus 
the power of cα̂

R . The leading term is the cohomology of g0̄ with values in the functions of 
(x, c0, cL, cR). Let us restrict ourselves to those vertex operators which are polynomial functions 
of x, c0, cL, cR . The space of such operators splits as an infinite sum of finite-dimensional 
representations of g0̄. Then the cohomology sits on the functions which do not depend on c0 and 
are invariant under the action of g0̄. The resulting complex is the physical BRST complex for the 
unintegrated vertex operators in AdS5 × S5:

In the Serre–Hochschild complex of Ltot Similarly, the decoupling of the c0 ghosts in the 
Serre–Hochschild complex of Ltot leads to the relative cohomology:

Hp(Ltot , (Ug)′) = Hp(Ltot , g0̄ ; (Ug)′) (53)

This establishes the relation between the BRST cohomology and the relative Lie algebra coho-
mology [4].



A. Mikhailov / Nuclear Physics B 907 (2016) 542–571 551
2.2.4. Cohomology of the ideal
Consider the ideal I ⊂ Ltot such that:

Ltot/I = g (54)

By the Shapiro’s theorem:

Hp(Ltot , (Ug)′) = Hp(I) (55)

This helps to identify various supergravity field strengths [4].

2.3. Integrated vertex

2.3.1. Generalized Lax operator
Consider a classical string solution in AdS5 × S5, i.e. a field configuration in the worldsheet 

sigma-model solving the classical equations of motion. It was shown in [5,6] that one can con-
struct the Lax pair:

L+ =
(

∂

∂τ+ + J
[mn]
0+ t0[mn]

)
+ Jα

3+∇L
α + Jm

2+AL
m + (J1+)αWα

L +

+ λα
LwL

β+
(
{∇L

α , W
β
L } − fα

β [mn]t0[mn]
)

(56)

L− =
(

∂

∂τ− + J
[mn]
0− t0[mn]

)
+ J α̇

1−∇R
α̇ + Jm

2−AR
m + (J3−)α̇W α̇

R +

+ λα̇
RwR

β̇−
(
{∇R

α̇ , W
β̇
R } − fα̇

β̇ [mn]t0[mn]
)

(57)

where J± and λ, w are worldsheet fields and t0, ∇ , A, W generators of Ltot, satisfying the zero 
curvature equations:

[L+,L−] = 0 (58)

and having simple BRST transformation laws:

QBRST L± =
[
L± ,

(
λα

L∇L
α + λα̇

R∇R
α̇

)]
(59)

We will denote J± the connections in L±:

L± = ∂

∂τ± + J± (60)

2.3.2. Bicomplex d + QBRST
Let us denote: J = J+dτ+ + J−dτ− – an Ltot-valued one-form on the worldsheet. For the 

purpose of calculations, it is convenient to assume that dτ+ and dτ− anticommute with the 
worldsheet fields θ :

dτ+θα = − θαdτ+ (61)

dτ−θα = − θαdτ− (62)

We also introduce arbitrarily many anticommuting parameters ε1, ε2, . . . , which anticommute 
among themselves, with θ , and with dτ±. With these notations, we have:



552 A. Mikhailov / Nuclear Physics B 907 (2016) 542–571
(ε1d + ε1QBRST)
(
ε2dτ jJj − ε2λ

)
= (63)

= − 1

2

[
ε1dτ iJi − ε1λ , ε2dτ iJi − ε2λ

]
(64)

Schematically:

ε1(d + QBRST) ε2(J − λ) = −1

2
[ ε1(J − λ) , ε2(J − λ) ] (65)

Also:

ε1(d + QBRST)g = −ε1π(J − λ) g (66)

Given an n-cochain ψ ∈ Cn(Ltot, g0̄; (Ug)′), let us consider an inhomogeneous form ev(ψ) on 
the worldsheet which can schematically be defined by the following formula:

evε1,...,εn(ψ) = ψ (ε1(J − λ) ⊗ ε2(J − λ) ⊗ . . .) (g) (67)

This schematic notation is deciphered as follows. Notice that ψ is a function of the type:


nLtot → [Ug → C] (68)

We first evaluate it on ε1(J − λ) ⊗ ε2(J − λ) ⊗ . . . , which gives us a function of the type 
Ug → C. We then evaluate it on a “group element”g ∈ Ug. The “group elements” are defined as 
expressions of the form g = eξ where ξ ∈ g. Being infinite series, they strictly speaking do not 
belong to Ug. This rises the question of convergence, which we will ignore.

Then we observe:

ε1(d + QBRST) evε2,...,εn+1(ψ) = 1

n + 1
evε1,...,εn+1(QLieψ) (69)

The derivation of this formula, besides (65) and (66), also uses the fact that ψ is a relative
cocycle, and therefore:

ψ ({ λ ,λ } ⊗ . . .) = ψ
(

2λα
Lλα̂

Rfαα̂
[mn]t0[mn] ⊗ . . .

)
= 0 (70)

In our case ψ is a 2-cocycle. Therefore:

(d + QBRST)evε1,ε2(ψ) = 0 (71)

This means that the ghost number two part of evε1,ε2(ψ) is an unintegrated vertex, and the ghost 
number zero part of evε1,ε2(ψ) is an integrated vertex.

Therefore our construction provides one way of thinking about the relation between uninte-
grated and integrated vertices.

3. General curved superspace

The pure spinor description of the Type IIB SUGRA emphasizes the local Lorentz symmetry 
of the supergravity theory. More specifically, the Type IIB superstring combines left and right 
sectors, and there are two copies of the local Lorentz group.

We will now describe some structure on the superspace, which we call “SUGRA data”. We 
first describe it as an abstract geometrical structure, and then explain how it emerges in the 
sigma-model.



A. Mikhailov / Nuclear Physics B 907 (2016) 542–571 553
3.1. Weyl superspace

The formulation of the pure spinor sigma-model in [1] uses the so-called Weyl superspace. 
In this formalism, besides the local Lorentz symmetry, there are also two copies of the local 
dilatation symmetries:

ĥ = ĥL ⊕ ĥR = spin(1,9)L ⊕ RL ⊕ spin(1,9)R ⊕ RR (72)

Let SL ⊕ SR denote the spinor representation of ĥ. We will also denote Ĥ the Lie group corre-
sponding to ĥ. To summarize:

Ĥ = Spin(1,9)L × R×
L × Spin(1,9)R × R×

R (73)

ĥ = Lie(H) (74)

SL ⊕ SR = spinor representation of H (75)

We will now start describing the SUGRA data.
Let M be a 10|32-dimensional supermanifold, the super-space–time.

The first part of the SUGRA data is:

• a distribution SL ⊕ SR ⊂ T M

• for every point x ∈ M , an orbit of the action of Ĥ on some linear map D : SL ⊕ SR →
SL(x) ⊕ SR(x) (the action of h ∈ Ĥ is D 
→ D ◦ h); notice that the map D itself does not 
enter into the SUGRA data, only its orbit (with the action of Ĥ on it)

Let M̂
π−→ M be the principal bundle over M whose fiber over a point x ∈ M is that orbit. In 

other words, a point of M̂ is a pair (x, D). Let π denote the natural projection:

π : M̂ → M

π(x,D) = x (76)

More explicitly, any linear map D is of the form:

D(sL + sR) = sα
LEL

α + sα̂
RER

α̂
(77)

EL
α ∈ SL(x) (78)

ER
α̂

∈ SR(x) (79)

Sometimes we will simply write Eα and Eα̂ instead of EL
α and ER

α̂
.

Let Vect(M̂) = �(T M̂) denote the infinite-dimensional space of all vector fields on M̂.

The second part of the SUGRA data is a map

D : SL ⊕ SR → Vect(M̂) (80)

D(sL + sR) = sα
LDL

α + sα̂
RDR

α̂
(81)

satisfying the following properties:



554 A. Mikhailov / Nuclear Physics B 907 (2016) 542–571
• D commutes with the action of Ĥ
• D is “fixed modulo VectM̂/M” in the following sense: for any point (x, D) ∈ M let π(x) be 

the natural projection T(x,D)M̂ → TxM , then

π(x)
(
(D(sL + sR))(x,D)

)
=D(sL + sR) (82)

(in other words, only the vertical component of D is non-obvious; the projection to T M is 
tautological)

• SUGRA constraints:

{D(sL + sR) , D(sL + sR)} =
= (sL�msL)AL

m + (sR�msR)AR
m +

+ RLL
αβ sα

Ls
β
L + RRR

α̇β̇
sα̇
Rs

β̇
R + RLR

αβ̇
sα
Ls

β̇
R (83)

where:
– AL

m and AR
m are some sections of T M̂ and

– RLL
αβ , RRR

α̇β̇
and RLR

αβ̇
some sections of T M̂/M (i.e. vertical vector fields); they are essen-

tially “curvatures”
Notice that satisfying the SUGRA constraints does depend on the vertical component of D.

Moreover:

• there is an equivalence relation, which we will describe in Section 3.2.4

3.2. Relation to the formalism of [1]

3.2.1. SUGRA constraints, oversimplified
Let M be the super-space–time. In supergravity, M comes equipped with the distribution 

S ⊂ T M . The SUGRA constraints are conditions on the Frobenius form of S , which go roughly 
speaking as follows. We choose some vector fields ∇α , α ∈ {1, . . . , dimS} tangent to S and say 
that:

{∇α,∇β} = �m
αβAm mod S (84)

where Am are some other vector fields. (The point of the constraint being that the RHS is pro-
portional to �m

αβ .) It is important to remember that when we write such conditions, we need to 
fix a basis of S , i.e. a set of ∇α . If we choose some linear combination:

∇′
α = Xβ

α∇β (85)

then, generally speaking, ∇′
α will not satisfy the constraint (84). If we want ∇′

α to satisfy the 
constraint, we should require that X ∈ so(1, 9) ⊕ R – an antisymmetric matrix plus a scalar. 
This means that S actually comes with an additional structure, namely an orbit of the action 
of SO(1, 9) × R× on some linear map D : S → S where S is the spinor representation of 
so(1, 9) ⊕ R. As we said in Section 3.1, the map D itself does not enter into the SUGRA data, 
only its orbit (with the action of SO(1, 9) × R× on it). Given a point x ∈ M , and an orbit of 
SO(1, 9) × R× in S(x), we can choose a point D in this orbit, then choose any set of vector 
fields ∇α such that ∇α(x) = Dα , and verify Eq. (84).



A. Mikhailov / Nuclear Physics B 907 (2016) 542–571 555
This means that it is useful instead of M to consider M̂ , which is the SO(1, 9) × R×-bundle 
over M whose fiber over x ∈ M is that SO(1, 9) × R×-orbit in HomC(S, S(x)) which we 
should have received as part of our SUGRA data. It is natural to think that the matter fields 
live in M̂ rather than M , except that the fiber is a gauge degree of freedom. The fiber can be 
gauged away because, as we said, the map D ∈ HomC(S, S(x)) itself does not enter into the 
SUGRA data, only its orbit. This is how the AdS5 × S5 sigma-model is formulated [11]. In 
that case M is PSU(2, 2|4)/(SO(1, 4) × SO(5)) and M̂ is PSU(2, 2|4). The sigma model has 
the SO(1, 4) × SO(5) gauge symmetry which gauges away the fiber. It is SO(1, 4) × SO(5)

rather than SO(1, 9) × R× because in that particular case some of the gauge symmetry can be 
canonically fixed.

In the sigma-model we couple matter fields with the ghosts λ which belong to the pure spinor 
cone C ⊂ S. As D ∈ HomC(S, S) can be thought of as linear functions from S to S , it make sense 
to apply it to λ ∈ S. The resulting vector field D(λ) describes the action of the BRST operator on 
the matter fields:

Qmatter =D(λ) (86)

3.2.2. Sigma-model
The target space of the sigma-model is M̂ , but as we explained there is a gauge symmetry 

which reduces M̂ → M . The action, copied from [11], is:

S = 1

2πα′

∫
d2z

(
1

2

(
GMN(Z) + BMN(Z)

)
∂ZM∂ZN + Eα

M(Z)dα∂ZM +

+ Eα̂
M(Z)d̃α̂∂ZM + �Mα

β(Z)λαwL
β ∂ZM + �̂Mα̂

β̂ (Z)λ̃α̂wR

β̂
∂ZM +

+ P αβ̂(Z)dαd̃
β̂

+ Cβγ̂
α (Z)λαwL

β d̃γ̂ + Ĉ
β̂γ

α̂
(Z)λ̃α̂wR

β̂
dγ +

+ S
βδ̂

αγ̂
(Z)λαwL

β λ̃γ̂ wR

δ̂
+ 1

2
α′�(Z)r + wL

α+∂−λα
L + wR

α̂−∂+λα̂
R

)
(87)

In a generic background, one can integrate out d , d̃ and get a simpler-looking action. It is postu-
lated that the field d should be the same as the density of the BRST charge. This is, essentially, 
a restriction on the choice of fields. Notice that the form of the Lagrangian (87) is not invariant 
under the field redefinitions, specifically under those redefinitions which mix the ghosts λ with 
the matter fields ZM . (And this, in our opinion, is a defect of the formalism in its current form.)

The phase space of this sigma-model will be denoted X . It can be identified with the moduli 
space of all classical solutions:

X = the space of classical solutions of the string σ -model

We can also consider the space of off-shell field configurations:

XOS = the space of off-shell field configurations

3.2.3. From pure spinor Q to SUGRA constraints
We just said that the target space of the sigma-model is M̂ . This, however, is not the full truth, 

because there are also ghosts. With ghosts, the target space is a cone in the associated vector 
bundle of the principal bundle M̂ corresponding to the spinor representation of Ĥ :

Target space with ghosts = M̂ × ˆ (CL × CR) (88)

H



556 A. Mikhailov / Nuclear Physics B 907 (2016) 542–571
where CL is the pure spinor cone in SL and CR the pure spinor cone in SR . The BRST operator 
of the sigma-model is a nilpotent odd vector field:

Q ∈ Vect(M̂ ×
Ĥ

(CL × CR)) (89)

Generally speaking, consider a coset space X/G, where the action of G on X is free and tran-
sitive. Then vector fields on X/G can be described as follows. Let us start by considering the 
subalgebra A ⊂ Vect(X) which consists of those vector fields which are invariant under the ac-
tion of G, i.e. g∗v = v for any g ∈ G (a.k.a “Atiyah algebroid”). One can check that the vertical 
vector fields (those which are tangent to the orbits of G) are an ideal I ⊂ A. The factoralgebra is 
isomorphic to the algebra of vector fields on X/G:

Vect(X/G) � A/I (90)

Let us see how this description works in the particular case:

X = M̂ × (CL × CR) , G = Ĥ and X/G = M̂ ×
Ĥ

(CL × CR)

Let us fix some lift

Lift : M̂ ×
Ĥ

(CL × CR) → M̂ × (CL × CR) (91)

Consider Lift∗Q – a vector field on the image of Lift. Notice that (Lift∗Q)2 = 0, but don’t forget 
that Lift∗Q is not a vector field on the whole M̂ × (CL × CR), but only on a submanifold – the 
image of Lift. However, we can define an Ĥ -invariant vector field on the whole M̂ × (CL ×CR)

using the fact that M̂ × (CL ×CR) is foliated by the translations of the image of Lift by elements 
of Ĥ :

M̂ × (CL × CR) =
⋃
h∈Ĥ

h(im(Lift)) (92)

This means that we can extend Lift∗Q to the whole M̂ × (CL × CR) in an h-invariant matter, 
simply by translating. In other words, let Q↑ be the nilpotent vector field on M̂ × (CL × CR)

such that:

Q↑|im(Lift) = Lift∗Q (93)

for any h ∈ Ĥ : h∗Q↑ = Q↑ (94)

A different choice of Lift will result in another Q↑, but the difference in Q↑ will be in adding a 
vertical vector field, i.e. and element of I . This is precisely (90).

Notice that the vertical component of Q↑ is Ĥ -invariant. Moreover, CL × CR is an orbit of 
the action of Ĥ on SL × SR . In other words, any pair of pure spinors (λL, λR) can be obtained 
from a fixed pair (λ(0)

L , λ(0)
R ) by the action of some element h ∈ Ĥ . Therefore exists a vertical 

vector field ω such that the following vector field on M̂ × (CL × CR):

Q̂ = Q↑ + ω (95)

acts trivially on CL × CR . In other words, Q̂ is a vector field on M̂ .
To clarify the construction, let us describe it in coordinates. A point of M̂ × (CL × CR) is 

described in coordinates as follows: (Z, (EL
α ), (ER

α̂
), λL, λR). A point of M̂ ×

Ĥ
(CL × CR) is 

described in the same way but with the equivalence relation:

(Z, (EL), (ER), λL,λR) ∼ (Z, ((h−1)α
′

α EL′), ((h−1)α̂
′
ER′), hLλL,hRλR) (96)
α α̂ L α R α̂ α̂



A. Mikhailov / Nuclear Physics B 907 (2016) 542–571 557
Our Lift is essentially gauge fixing. It is described by specifying the functions:

EL
α = EL0

α (Z) and ER
α̂

= ER0
α̂

(Z) (97)

The way it works, for every point in M̂ ×
Ĥ

(CL ×CR), to calculate its lift we use the equivalence 
relations (96) to bring its coordinates (Z, (EL

α ), (ER
α̂
), λL, λR) to the form satisfying (97). The 

resulting (Z, (EL0
α (Z)), (ER0

α̂
(Z)), λnew

L , λnew
R ) specifies a point in M̂ × (CL × CR), which is the 

lift. The lift of the BRST field Lift∗Q is of the form:

Lift∗QL = λα
L

(
EL0M

α (Z)
∂

∂ZM
+ Xα

β
γ (Z)λ

γ

L

∂

∂λ
β
L

)
(98)

We must stress that Lift∗Q is only defined on the image of Lift. To extend this vector field to the 
whole M̂ × (CL × CR), we must relax the gauge fixing (97). We observe that any EL

α and ER
α̂

can be presented in the form:

EL
α = (gL)α

′
α EL0

α′ (Z) and ER
α̂

= (gR)α̂
′

α̂
ER0

α̂′ (Z) (99)

Let us use (Z, gL, gR, λL, λR) as coordinates on M̂ × (CL × CR). Then we have:

Q
↑
L = λα

L

(
(gL)α

′
α EL0M

α′ (Z)
∂

∂ZM
+ (gL)α

′
α Xα′β

′
γ ′(Z)((gL)−1)

β

β ′(gL)γ
′

γ λ
γ

L

∂

∂λ
β
L

)
(100)

Finally:

Q̂L = λα
L

(
(gL)α

′
α EL0M

α′ (Z)
∂

∂ZM
+ (gL)α

′
α Xα′β

′
γ ′(Z)(gL)

γ ′
δ

∂

∂(gL)
β ′
δ

)
(101)

– a vector field on M̂ .
Notice that Q̂L depends linearly on λL. Therefore, Q̂ defines sixteen vector fields DL

α :

Q̂L = λα
LDL

α (102)

These are the vector fields which were postulated in Section 3.1.

Ambiguity However, the definition of Q̂L, and therefore of DL
α , contains an ambiguity. It is 

possible to add to Q̂L a vertical vector field:

Q̂L,new = Q̂L + λα
LωL

α (103)

such that λα
LωL

α ∈ St(λL) ⊂ ĥL. This corresponds to the “shift gauge transformations” of [1]. We 
will now describe such ωα .

3.2.4. Shift gauge transformations
Let us modify DL

α by adding to it a vector field in T M̂/M (i.e. tangent to the fiber) of the form 
(cf. Eq. (61) of [1]):

(ωL
α )βγ = (�n�m)βγ �m

α•h•n
L (104)

The characteristic property of such ωL
α is that λα

LωL
α ∈ St(λL) ⊂ ĥL; in other words:

λα (ωL)βλ
γ = 0 (105)
L α γ L



558 A. Mikhailov / Nuclear Physics B 907 (2016) 542–571
Similarly, we can modify DR
α̂

by adding to it some ωR
α̂

defined in a similar way; we stress that ωL

takes values in ĥL and ωR takes values in ĥR . Obviously, these “shift transformations” depend 
on two parameters: hαn

L and hα̂n
R . In terms of Section 3.1 this modifies D:

Dnew(sL + sR) = D(sL + sR) + sα
LωL

α + sα̂
RωR

α̂
(106)

3.2.5. SUGRA fields
The action (87) involves various SUGRA fields, which are either sections of associated vector 

bundles over M , or connections on them. They enter the action through their pullback on the 
string worldsheet.

Sections For example, P αα̂ is a section of M̂ ×
Ĥ

(SL ⊗ SR) 
π→ M . Such sections can be in-

terpreted as Ĥ -invariant maps7 M̂ → SL ⊗ SR . From this point of view we consider P αα̂ as a 
function P αα̂(Z, (EL

β ), (ER
β̂
)) such that:

P αα̂(Z, ((hL)
β ′
β EL

β ′), ((hL)
β̂ ′
β̂

ER
β̂ ′)) = (h−1

L )αα′(h−1
R )α̂

α̂′P α′α̂′
(Z, (EL

β ), (ER
β̂
)) (107)

Connections Connections are needed to define the kinetic terms for the ghost fields. A con-
nection on the associated vector bundle M̂ ×

Ĥ
(SL ⊗ SR) 

π→ M is constructed from a connec-

tion on the principal bundle M̂
π→ M . We will now remind how this works. For any vector 

field ξ ∈ Vect(M), a connection in the principal bundle defines a lift ξ ′ ∈ Vect(M̂), which 
is Ĥ -invariant in the sense that for any χ ∈ ĥ the corresponding vector field v(χ) commutes 
with ξ ′:

[v(χ), ξ ′] = 0 for any χ ∈ ĥ (108)

For any representation ρ : ĥ → End(V ), sections of the associated bundle M̂ ×
Ĥ

V
π→ M can 

be understood as maps σ : M̂ → V , invariant under Ĥ in the following sense:

Lv(χ)σ = ρ(χ)σ for any χ ∈ ĥ (109)

where L is the Lie derivative. Eq. (108) implies that for any σ satisfying (109), Lξ ′σ also satisfies 
(109). This means that the lift ξ 
→ ξ ′ consistently defines the action of ξ on the sections of the 
associated vector bundle.

Let us explain how a connection in the principal bundle M̂ → M defines a kinetic term for the 
ghosts. Consider the ghost λL; in the flat space limit it is a left-moving field. In the general curved 
space, the kinetic term for λL should involve the derivative ∂−λL. A point of the target space is 
(Z, (EL

α ), (ER
α̂
), λL, λR). The worldsheet is foliated by the characteristics. Let us consider the 

right-moving characteristic τ+ = const. It is parametrized by the τ−:

(Z(τ−), (EL
α (τ−)), (ER

α̂
(τ−)), λL(τ−), λR(τ−)) (110)

Let us choose a representative so that 
(

dZ(τ−)
dτ− , (

dEL
α (τ−)

dτ− ), (
dER

α̂
(τ−)

dτ− )

)
is a horizontal vector, in 

the sense defined by the principal bundle connection in M̂. Then the kinetic term is:

7 Indeed, every such map defines σ : M → M̂ × ˆ (SL ⊗ SR) such that π ◦ σ = id.

H



A. Mikhailov / Nuclear Physics B 907 (2016) 542–571 559
∫
dτ+dτ−

(
wL+ ,

dλL

dτ−

)
(111)

where wL+ is the conjugate momentum to λL.

3.3. Lorentz superspace

There is a way to canonically fix R×
L × R×

R . In this paper we will use the variant of the 
formalism which has R×

L × R×
R fixed. For us the gauge algebra is:

h = hL ⊕ hR = spin(1,9)L ⊕ spin(1,9)R (112)

This version of the formalism is called “Lorentz superspace”. We will now review how the 
Lorentz superspace is derived, as much as we understand.

Consider the sigma-model (87) and let us integrate out dα and d̃
β̂

. It turns out that it is always 
possible to choose the gauge so that the coupling to the ghosts is only through the traceless
currents8:

(wL+�mnλL) and (wR−�mnλR) (113)

The u(1) combinations (wL+λL) and (wR−λR) appear only in the kinetic terms (wL+∂−λL) and 
(wR−∂+λR). This fixes the gauge from ĥL ⊕ ĥR to hL ⊕ hR . In the language of the present paper 
this simply means that we can use a slightly simpler M̂ . A point of this simplified M̂ is a point 
x ∈ M and a point in the orbit of H in S(x) ⊂ TxM ; the simplification is in replacing the orbit of 
Ĥ = Spin(1, 9)L × R×

L × Spin(1, 9)R × R×
R with the orbit of H = Spin(1, 9)L × Spin(1, 9)R . 

As in Section 3.2.3, we can still trade the BRST operator for an H -invariant vector field on M̂. 
This statement is somewhat nontrivial, because what if the BRST operator Q involves a rescaling 
of λL and λR? Let us consider the action of QR on λL:

QRλα
L = λα̂

RXα̂
α
βλ

β
L (114)

In particular, the QR variation of the kinetic term wL
α+∂−λα

L gives the term wL
α+Xα̂

α
βλ

β
L∂−λα̂

R

which has nothing to cancel unless if Xα̂ is traceless, i.e. if Xα̂
α
α �= 0. (In this case it is canceled

by the variation of the connection on which wL
α+∂−λα

L depends, implicitly in our language.) Now 
consider the action of QL on λL:

QLλα
L = λα

LXα
β
γ λ

γ

L (115)

Now it is even meaningless to ask if Xα is traceless or not, because Xα is only defined by (115)
up to a shift transformation of Section 3.2.4. We therefore use these shift transformations to 
remove the trace of Xα . Then we have to remember that when we work in the Lorentz superspace 
formalism, the shift transformations have their parameter restricted to:

�n
α•h•n

L = 0 (116)

8 N. Berkovits, private communication; notice that we semiautomatically arrived at this gauge in our study of linearized 
excitations of AdS5 × S5 in [6].



560 A. Mikhailov / Nuclear Physics B 907 (2016) 542–571
4. Worldsheet currents, quadratic-linear algebroid

Let us consider the algebroid A over M̂ freely generated by D satisfying Eq. (83) where 
AL

m and AR
m are free and RLL

αβ , RRR
α̇β̇

and RLR
αβ̇

are same sections of �(T M̂/M) as in Eq. (83). 
The definition of A is a direct generalization of the definition of Ltot in [4]. We “leave alone” 
the vertical generators RLL

αβ , RRR
α̇β̇

and RLR
αβ̇

in the sense that their commutation relations are 

postulated as the commutation relation in �(T M̂/M). But we consider DL
α and DR

α̂
and AL

m and 
AR

m as free generators modulo the relations (83).

(Open question: Does A satisfy a PBW theorem?)

From now on we will use letters DL
α and DR

α̂
to denote the generators of the algebroid. The vector 

fields defined in Eq. (81) will now be interpreted as the corresponding values of the anchor and 
therefore denoted a(DL

α ) and a(DR
α̂
) (instead of simply DL

α and DR
α̂

):

DL
α , DR

α̂
, AL

m, AR
m : generators of the algebroid A

a(DL
α ), a(DR

α̂
), a(AL

m), a(AR
m) : vector fields on M̂

We introduce a bidirectional filtration on A in the following sense. For n > 0, we will say that 
ξ ∈ A≤n if ξ can be represented as a nested supercommutator of ≤ n generators DL

α . For n < 0, 
we will say that ξ ∈ A≥n if ξ can be represented as a nested supercommutator of ≤ |n| genera-
tors DR

α̂
. Notice that the expression containing nested supercommutators of both DL

α and DR
α̂

can 
be reduced to expressions containing either all DL

α or all DR
α̂

.

4.1. Basic consequences of the defining relations

We observe that the basic commutation relations of (83) imply the existence of Wα
L such that:

[ DL
α , AL

m ] = �mαβWβ
L mod A≤1 (117)

Furthermore, notice the existence of F[mn] such that:

{ DL
α , Wβ

L } = (�mn)βαF[mn] mod A≤2 (118)

Indeed:

�mβ(γ {DL
α) , Wβ

L} = {DL
(α , [DL

γ ) , AL
m]} = �n

αγ [AL
n , AL

m] mod A≤2 (119)

and

10{DL
α , Wα

L} = �
γα
m �m

αβ{DL
γ , Wβ

L} = �
γα
m {Dγ , [Dα,Am]} = 0 mod A≤2 (120)

This implies the existence of F[mn].

4.2. Worldsheet currents

Remember that the string worldsheet is spanned by the left-moving characteristics τ− =
const. Consider an observer moving along a characteristic with the constant velocity τ̇+ = 1. 
The velocity vector can be decomposed via the worldsheet currents:

∂+ZM = J̃ LM
0+ + J̃ RM

0+ + J̃ α+aM(DL
α ) + �̃m+aM(AL

m) + ψ̃α+aM(Wα
L) (121)



A. Mikhailov / Nuclear Physics B 907 (2016) 542–571 561
Here we used the abbreviation:

J̃ LM
0+ = J̃

L[mn]
0+ aM(tL0[mn]) (122)

where t0[mn] are generators of hL.

Notice that the “currents” J̃ L[mn]
0+ , Ĩ L

0+, J̃ α+, �̃m+, ψ̃α+ are local functions on the phase space. 
We will denote the space of such functions Loc(X ):

X = phase space

Loc(X ) = the space of local functions on X
At the same time, a(tL0), a(DL

α ), a(AL
m) and a(Wα

L) are vector fields on M̂ . Notice that a 
function f (Z) on M̂ and a point (τ+, τ−) on the worldsheet define a function on X , namely 
f (Z(τ+, τ−)). In this sense, we should think of ∂+Z as an element of the space:

V = Loc(X ) ⊗Fun(M̂) Vect(M̂) (123)

This is not an algebroid over X , because generally speaking there is no way to lift a vector field 
on M̂ to a vector field on the phase space. But this is possible if the vector field generates a 
symmetry of the sigma-model. When two elements X ∈ V and Y ∈ V both correspond to some 
symmetry of the sigma-model, then it is possible to define the commutator [X, Y ]. Another way 
of turning V into an algebroid is to go off-shell, i.e. replace the X with the space of off-shell 
configurations XOS.

4.3. Tautological Lax pair

4.3.1. The case of AdS5 × S5

Consider the sigma model of the classical string in AdS5 × S5. It is classically integrable. 
There is a Lax pair, which depends on the spectral parameter z. At some particular value of z, 
the Lax pair becomes tautological, the zero curvature equations being just the Maurer–Cartan 
equation for the worldsheet currents. We will now briefly review how this goes.

The current is J = −dgg−1. For any representation of g with generators ta , it is straightfor-
ward to verify the Maurer–Cartan equation:[

∂

∂τ+ + J a+ta ,
∂

∂τ− + J b−tb

]
= 0 (124)

We will need a slight variation of this construction. Let g̃ be the Lie superalgebra obtained from 
g by changing the sign of the anticommutators (all the commutators are the same, but all the 
anticommutators have the opposite sign). The left regular representation of g̃ on the space of 
functions on the group manifold of G is defined as follows:

(L(ξ)f )(g) = d

dt

∣∣∣∣
t=0

f (e−tξ g) (125)

This means that:

∂f (g(τ+, τ−))

∂τ± − (L(J )f )(g(τ+, τ−)) = 0 (126)

We get9:

9 Notice the difference in sign between (126) and (127).



562 A. Mikhailov / Nuclear Physics B 907 (2016) 542–571
[
∂

∂τ+ + L(J+) ,
∂

∂τ− + L(J−)

]
= 0 (127)

Eq. (127) is almost the particular case of (124) corresponding to the left regular representation. 
The only difference is that the left regular representation, as we defined it, is the representation 
of g̃ and not g. But at the same time, notice that the odd–odd terms in L(J+) are of the form: 
(−∂+θα + . . .) 

(
∂

∂θα + . . .
)

where (−∂+θα + . . .) is Jα+ and 
(

∂
∂θα + . . .

) = t̃α is the corresponding 
generator of g̃, let us call it t̃α . Notice that t̃α anti-commutes with Jα+, while in (124) by definition 
J a commute with ta . In spite of this subtlety, the two definitions are actually equivalent. Given a 
Lax pair in the sense of (127), let us replace every term of the form Jαt̃α with Jαt̃α(−)F . Notice 
that t̃α(−)F are the generators of some representation of g (which should also be called “left 
regular”), and also that t̃α(−)F commutes with Jα . Therefore we obtained the Lax pair in the 
sense of (124).

We can interpret the operator ∂
∂τ± + L(J±) in the following way. Consider the space XOS of 

all field configurations (off-shell) in the classical sigma-model. Let Loc(XOS) denote the space of 
all local functions on XOS. Let us consider the space:

Fun(M̂) ⊗Fun(M̂) Loc(XOS) (128)

This is, obviously, the same as Loc(XOS). Let us, however, define the action of the Lax operator 
on this space, as follows: ∂

∂τ± acts only on Loc(XOS) and L(J±) acts only on Fun(M̂). Our point 
here is that this action is correctly defined. For example, the action of ∂

∂τ+ + αL(J+) with α �= 1
would not be correctly defined on (128), because it would act differently on f ⊗ φ and 1 ⊗ f φ.

4.3.2. General case
Consider the velocity of the coordinate function ZM̂ pulled back on the string worldsheet:

∂ZM̂(τ+, τ−)

∂τ+ = J̃
L[mn]

0+ aM̂(tL0[mn]) + J̃
R[mn]

0+ aM̂(tR0[mn]) +
+ J̃ α+aM̂(DL

α ) + �̃m+aM̂(AL
m) + ψ̃α+aM̂(Wα

L) (129)

We write ZM̂ instead of simply ZM , to stress that the coordinates include also the fiber. In 
the AdS5 × S5 language, ZM̂ would parametrize PSU(2, 2|4) rather than AdS. The terms 
J̃

L[mn]
0+ aM̂(tL0[mn]) and J̃ R[mn]

0+ aM̂(tR0[mn]) are vertical (along the fiber).

Then 
[

∂
∂τ+ , ∂

∂τ−
]
ZM̂ = 0 leads to the tautological zero curvature equation:

∂

∂τ+
(
J̃ LM

0− + J̃ RM
0− + J̃ α−a(DL

α ) + �̃m−a(AL
m) + ψ̃α−a(Wα

L)
)

−

− ∂

∂τ−
(
J̃ LM

0+ + J̃ RM
0+ + J̃ α+a(DL

α ) + �̃m+a(AL
m) + ψ̃α+a(Wα

L)
)

+
+

[
J̃ LM

0+ + J̃ RM
0+ + J̃ α+a(DL

α ) + �̃m+a(AL
m) + ψ̃α+a(Wα

L) ,

J̃ LM
0− + J̃ RM

0− + J̃ α−a(DL
α ) + �̃m−a(AL

m) + ψ̃α−a(Wα
L)

]
= 0 (130)

In this formula ∂
∂τ+ in the first line and ∂

∂τ− in the second line only act on the currents J̃ , �̃, 
ψ̃ and do not act on a(D), a(A), a(W). The commutator is the commutator of the vector fields, 
e.g.: [

�̃m+a(AL
m) , �̃n−a(AL

n )
]

= �̃m+�̃n−
[
a(AL

m) , a(AL
n )

]
(131)



A. Mikhailov / Nuclear Physics B 907 (2016) 542–571 563
Consequences of [QL, ∂+] = 0 Let us consider the BRST variation:

εQLZM̂ = ελα
LaM̂(DL

α ) (132)

We have two vector fields on the phase space, QL and ∂
∂τ+ . They commute:

(εQLJ̃
L[mn]
0+ )a(tL[mn]) + (εQLJ̃

R[mn]
0+ )a(tR[mn]) +

+ (εQLJ̃ α+)a(DL
α ) + (εQL�̃m+)a(AL

m) + (εQLψ̃α+)a(Wα
L) −

− ∂+(ελ
β
L)a(DL

β )+
+ J̃

L[mn]
0+ a

(
ελ

β
L[DL

β , t0
L[mn]]

)
+ J̃

R[mn]
0+ a

(
ελ

β
L[DL

β , t0
R[mn]]

)
+

+ J̃ α+a
(
ελ

β
L{DL

β , DL
α }

)
+

+ �̃m+a
(
ελ

β
L[DL

β , AL
m]

)
+

+ ψ̃α+a
(
ελ

β
L{DL

β , Wα
L}

)
= 0 (133)

In particular, we can say something about QLψ̃α+. Let us define the superfield Cα
βγ (Z) by the 

following formula:

{ a(DL
β ) , a(Wα

L) } = Cα
β γ a(Wγ

L) mod A[0,2] (134)

where mod A[0,2] stands for a linear combination of a(DL
α ), a(AL

m), a(tL[mn]) and a(tR[mn]). From 
(133) we read:

(εQLψ̃α+)a
(
Wα

L

) + a
(
[ελγ

LDL
γ , ψ̃α+Wα

L]
)

+ a
(
[ελγ

LDL
γ , �̃m+Am

L ]
)

∈ A[0,2] (135)

and therefore:

QLψ̃α+ − (ψ̃+CαλL) + λ
γ

L�̃m+�m
γα = 0 (136)

Similarly we have:

QL�̃m+ + J̃ α+�m
αβλ

β
L + ψ̃α+Fα

β
mλ

β
L = 0 (137)

with some Fαm
β originating from [ελγ

LDL
γ , ψ̃α+Wα

L]. The nilpotence of QL implies:

Q2
Lψ̃α+ = − (ψ̃+(QLCα)λL) + ((ψ̃+CλL)CαλL) − (λLQL(�̃m+)�m)α =

= − Rα1α2
α′
α ψ̃α′+ (138)

Happily, the only part of QL�̃m+ which gives a nonvanishing contribution is proportional to ψ̃α+; 
let us extract its coefficient:

λ
α1
L λ

α2
L a(DL

α1
)Cβ

α2α
=

= λ
α1
L λ

α2
L C

β
α1δ

Cδ
α2α

+ λ
α1
L λ

α2
L Rα1α2

β
α + λ

α1
L λ

α2
L Fβ

α1
m�m

α2α
(139)



564 A. Mikhailov / Nuclear Physics B 907 (2016) 542–571
4.4. Identification of ψ̃α+

We will now show that ψ̃α+ can be identified as the matter part of the BRST charge density.
Remember that wL+ is the momentum conjugate to λL – see Eq. (87). Let us define10 ψα+ and 

dα+ as follows:

ψα+ =
(
ψ̃α+ − wL◦+C◦•αλ•

L

)
(140)

dα+ = QL(wL
α+) (141)

It follows:

QLdα+ = −λ
α1
L λ

α2
L Rα1α2

β
αwL

β+ mod (_)m+�m
αγ λ

γ

L (142)

Here “mod (_)m+�m
αγ λ

γ

L” means “up to adding um+�m
αγ λ

γ

L with some arbitrary um+”. Notice 
that λα

Ldα+ is the left BRST current. This follows from the fundamental property of the formal-
ism: the U(1)L charge of the left BRST current is +1. We have:

QLψα+ = QLψ̃α+ − (d◦+C◦•αλ•
L) − (w◦+(QC◦•α)λ•

L) =
= (ψ̃�+C�•αλ•

L) − λ•
L�̃m+�m•α − (d�+C�•αλ•

L) − (w�+(QC�•α)λ•
L) =

= ((w�+C�◦•λ◦
L)C•�αλ�

L) − (w�+(QC�•α)λ•
L) −

− λ•
L�̃m+�m•α + ((ψ+ − d+)CαλL) (143)

It follows from Eq. (139) that:

((w+CλL)CαλL) − (w(QCα)λL) = −wβ+λ
α1
L λ

α2
L Rα1α2

β
α mod (_)m+�m

αγ λ
γ

L (144)

Therefore:

QLψα+ = −λ
α1
L λ

α2
L Rα1α2

β
αwL

β+ + ((ψ◦+ − d◦+)C◦•αλ•
L) mod (_)m+�m

αγ λ
γ

L (145)

Let us denote ζα+ = ψα+ − dα+ mod (_)m�m
αγ λ

γ

L.

Theorem 3.

ζα+ = 0 mod (_)m+�m
α•λ•

L (146)

Proof. Comparing Eqs. (145) and (142) we get:

QLζα+ = ζ◦+C◦•αλ•
L mod (_)m+�m

α•λ•
L (147)

(the same equation as (136)).
It follows from the analysis of Eq. (147) in the flat space limit that any ζα+ satisfying (147) is 

of the form:

ζα+ = φ
(
dα+ + C◦•αw◦+λ•

L

) + B◦•αw◦+λ•
L (148)

10 One could define dα+ through the density of the BRST charge QL , which is λα
L
dα+ . Such a definition would 

only specify dα+ up to an addition of the terms of the form Xm�m
αβλ

β
L

and Xklm�klm
αβ λ

β
L

. It is possible to reduce this 
ambiguity by defining dα+ from Eq. (141); this leaves only the ambiguity of the form Xm�m

αβλ
β
L

. In this sense, dα+ is 
“better-defined” than one might think.



A. Mikhailov / Nuclear Physics B 907 (2016) 542–571 565
where φ = φ(Z) and Bβ
γα = B

β
γα(Z) are some functions. Indeed, in the flat space limit, in the 

neighborhood of any point of M , if θ and λ scale as R−1/2 and x as R−1 and w± as R−3/2, 
then ζα+ should be of the order R−3/2; this means that the coefficients of J̃+ and �̃+ in ζα+
are zero. The leading term in the flat space expansion of ζα+ is then of the form φβ

αdβ+, and its 
BRST variation is in the leading order (QLφ

β
α )dβ+ + φ

β
α �m

βγ �m+λ
γ

L. Therefore the vanishing of 

the leading term in Eq. (147) up to (_)m+�m
α•λ• implies that φβ

α is proportional to δβ
α .

Notice that Eq. (147) is satisfied when φ = const and Bβ
γα = 0. When φ is not constant, the 

vanishing of the coefficient of d+ in (147) implies:

Bβ
γα = δβ

αDL
γ φ − 1

2
�m

αγ �β•
m DL• φ (149)

and the vanishing of the coefficient of w+λLλL implies:

λ•
Lλ•

L

(
DL• Bβ•α + C◦•αBβ•◦

)
wβ+ = (_)m�m

α•λ•
L (150)

This is equivalent to the following equation being satisfied for any pure spinor λL:

λ•
Lλ•

Lλ•
L

(
DL• Bβ•• + C◦••Bβ•◦

)
= 0 (151)

Substitution of (149) gives:

Cβ••λ•
Lλ•

Lλ•
LDL• φ = 0 (152)

This implies that either Cβ••λ•
Lλ•

L = 0 for any λL, which is generally speaking not the case, or 
λ•

LDL• φ = 0, which implies that φ = const. In the case of AdS5 × S5 we know ζα+ is zero. 
Therefore φ = 0 and Eq. (146) follows. �

This means that in terms of the sigma-model (87):

ψ̃α+ = P −1
αα̂

Eα̂
M∂+ZM (153)

dα+ = ψ̃α+ − C
γ
βαλβwγ+ mod (_)m+�m

αβλ
β
L (154)

4.5. Identification of Cβ
αγ

Let us consider the SUGRA superfields Cβα̂
γ and Pαα̂ defined in Eq. (87). (Notice that we use 

the same letter C as for Cβ
αγ , but with a different set of indices; we hope this will not lead to 

confusion.) Eq. (154) implies that:

Cα
β

γ̂ P −1
γ̂ γ

= −Cα
βγ (155)

In particular, this implies that in the Lorentz superspace formalism (112):

Cα
α

γ̂ = 0 (156)

(This is not stated in [1].) One difference of our approach with [1] is that we do not require that 
T α

βγ = 0. In fact, it is difficult to define T α
βγ in our language.

We will now confirm this by comparing the “shift” gauge transformations defined in Eq. (61) 
of [1]. They correspond to the following variation of ∇L

α :



566 A. Mikhailov / Nuclear Physics B 907 (2016) 542–571
δh∇L
α = ωα (157)

ωα
β
γ = (�k

α•h•n)(�β◦
n �k◦γ ) (158)

where hαn is a gauge parameter. In the Lorentz superspace formalism (112) the shift parameter 
satisfies:

�n
αβhβn = 0 (159)

Let us determine the transformation of ψα+ and Cα
βγ .

{δh∇L
(α1

,∇L
α2)

} = −(�k
(α1|•h

•n)∇L◦ (�◦•
n �k

•|α2)
) = (160)

= 1

2
�p

α1α2
hn��p�•�•α

n ∇L
α (161)

In other words:

δhAp = hn��p�•�•◦
n ∇L◦ (162)

[∇α , δhAp] = −hn��p�•�•◦
n �k◦αAk =

= hn��n�•�•◦
p �k◦αAk − 2hp•�k•αAk =

= −hn��n�•�•◦
k �p◦αAk − 2hp•�k•αAk + 2hn•�n•αAp (163)

[δh∇α , Ap]�p
α1α2

= −2ωα
•
(α1

Ap�
p

α2)• = (164)

= −2(�k
α◦h◦n)(�•�

n �k
�(α1

)Ap�
p

α2)• = (165)

= 2(�k
α◦h◦n)�n

α1α2
Ak − 4(�k

α◦h◦(k)�p)
α1α2

Ap (166)

where we used the gamma-matrix identity:

�
p

(α1|•�
•�
n �k

�|α2)
= (�p�n�

k)(α1α2) = (�(p�n�
k))(α1α2) = −δpk�n

α1α2
+ 2δn(k�p)

α1α2
(167)

and therefore:

[δh∇α , Ap] = 2�k
α◦h◦pAk − 2(�k

α◦h◦k)Ap − 2(�p
α◦h◦k)Ak (168)

This implies:

δhWα
L = −hn•�n•◦�◦α

k Ak − 2hαkAk (169)

{∇L
β , δhWα

L} = hn•�n•◦�◦α
k [∇L

β ,Ak] + 2hαk[∇L
β ,Ak] = (170)

= hn•�n•◦�◦α
k �k

βγ Wγ

L + 2hαk�k
βγ Wγ

L (171)

At the same time:

{δh∇L
β ,Wα

L} = (hn◦�k◦β)(�α•
n �k•γ )Wγ

L − 4hk◦�k◦βWα
L (172)

where the term −4hk◦�k◦βWα
L corresponds to the trace part of ω. Therefore:

δh{∇L
β ,Wα

L} = hn•�n•◦�◦α
k �k

βγ Wγ

L + 2hαk�k
βγ Wγ

L +
+ (hn•�k•β)(�α◦

n �k◦γ )Wγ

L − 4hk◦�k◦βWα
L (173)



A. Mikhailov / Nuclear Physics B 907 (2016) 542–571 567
This implies that11:

δhC
α
βγ = hn•�n•◦�◦α

k �k
βγ + 2hnα�n

βγ +
+ hn•�k•β�α◦

n �k◦γ − 4hn•�n•βδα
γ (174)

Therefore:

δhC
α
βγ + δh(C

α
β

α̂P −1
α̂γ

) = hn•�n•◦�◦α
k �k

βγ + 2hnα�n
βγ +

+ 2hn•�k
•(β|�

α◦
n �k

◦|γ ) − 4hn•�n•βδα
γ =

= − 4hn•�n•βδα
γ mod (_)m�m

αβλ
β
L (175)

Given (159), this is in agreement with (155).

4.6. Ramond–Ramond fields

The Ramond–Ramond bispinor P αα̂ must have a similar interpretation. Let us expand DR
α̂

in 
terms of Wα

L, Am
L , DL

α and T M̂/M . We will get:

DR
α̂

= Pα̂αWα
L + . . . (176)

where . . . stand for the terms proportional to Am
L and DL

α and T M̂/M . (Again, the coefficients 
of Am

L and DL
α are not defined unambiguously, but the coefficient of Wα

L is well-defined.) We 
conjecture that Pα̂α is the inverse of the P αα̂ of (87), however we do not have a proof. Similarly:

DL
α = Pαα̂Wα̂

R + . . . (177)

This is probably the most concise definition of the Ramond–Ramond bispinor in the framework 
of the pure spinor sigma-model.

4.7. Weighing anchor

Consider a linear map κ :

κ : T M̂ → A (178)

such that:

im κ = A[0,3] (179)

a ◦ κ = id : T M̂ → T M̂ (180)

Notice that the following operator:

a⊥ = id − κ ◦ a (181)

is the projection to ker(a) along A[0,3].
Let us unapply the anchor from the RHS of (121):

L̃+ = ∂+ + J̃
L[mn]
0+ t0

L[mn] + J̃
R[mn]
0+ t0

R[mn] + J̃ α+DL
α + �̃m+AL

m + ψ̃α+Wα
L (182)

11 As a consistency check, δhCα
αγ = 0 and δ(�klmn)

β
αCα = 0.
βγ



568 A. Mikhailov / Nuclear Physics B 907 (2016) 542–571
Notice that:

(QL + QR)2L̃+ = 0 (183)

Indeed, let us for example look at the λLλL part:

Q2
L = Q2

L + 1

2
λα

Lλ
β
L{DL

α , DL
β } = Q2

L + 1

2
λα

Lλ
β
LRαβ (184)

This implies that the calculation of the action of Q2
L on ̃L+ does not lead out of A[0,3]. Therefore 

the calculation is the same as it would be under the anchor, and the result is zero.
Also notice that:

a
(
(QL + QR) L̃+

) = 0 (185)

However it is not true that (QL + QR)L̃+ = 0; we will therefore correct ̃L+ by adding to it some 
expression with zero anchor.

4.8. Correction ̃L+ → L+

4.8.1. General theory
Let us consider deforming:

L̃+ 
→ L̃+ + �L+ (186)

where �L+ does not contain the derivative ∂+ and is an anchorless element of A such that:

(QL + QR)2�L+ = 0 (187)

We also require that �L+ be h-invariant. Let us denote Y+ the linear space of all expressions 
X+ ∈ A satisfying the following properties:

1. X+ is h-invariant
2. X+ has conformal dimension (1, 0)

3. (QL + QR)2X+ = 0

By definition �L+ belongs to Y+.

Observation 3. The cohomology of the operator QL + QR acting in Y+ is zero.

Proof. Let us prove that the cohomology of QL is zero. First of all let us prove this statement 
in flat space. In flat space the algebroid is homogeneous, it is defined by the same relations as 
qULtot. Given an expression annihilated by QL, let us consider the term with the lowest number 
of the letters ∇ . It is QL-closed. Since the cohomology of QL in the expressions of the conformal 
dimension (1, 0) is trivial, this means that this lowest order term is exact. This completes the 
proof that the cohomology of QL is zero in flat space.

In a general curved space, let us use the near-flat-space expansion (see [2] for details). For 
an element φ ∈ A let us define its degree deg(φ) so that deg(θ) = deg(λ) = 1, deg(w) = 3, 
deg(x) = 2 and deg(DL) = deg(DR) = −1. The proof follows from the following observations:

• deg((QL + QR)φ) ≥ deg(φ)

• the action in the associated graded space is the same as in flat space,
• we have just proven that the cohomology in flat space is zero. �



A. Mikhailov / Nuclear Physics B 907 (2016) 542–571 569
Observation 3 implies the existence of such a Y+ ∈ Y+ that:

(QL + QR)(L̃+ + Y+) = 0 (188)

We therefore denote:

L+ = L̃+ + Y+ (189)

4.8.2. Explicit construction
(If the reader is not familiar with the construction of the Lax operator for AdS5 × S5 [12] we 

would recommend to first look at [13].)
We will now show that the leading term of Y+ is in degree four, i.e. L+ is of the form:

L+ = ∂+ + J
L[mn]
0+ t0

L[mn] + J
R[mn]
0+ t0

R[mn] +
+ Jα+DL

α + �m+AL
m + ψα+Wα

L + λα
LwL

β+P
α′β
αβ ′ { DL

α′ , Wβ ′
L } (190)

where P αβ
γ δ is the projector on the zero-form plus two-form. In other words,

Y+ = λα
LwL

β+P
α′β
αβ ′ a⊥{ DL

α′ , Wβ ′
L } (191)

where a⊥ is defined in (181). We have to verify that Y+ satisfies (188). First of all, notice that 
ψα+ in Eq. (190) is by its definition12 the same as ψα+ defined in Eq. (140). This implies that 
QLL+ falls into A[0,3]. But at the same time, the anchor of QLL+ is zero. This implies that 
QLL+ = 0.

Observation 4.

QRL+ = 0 (192)

Proof. Unfortunately we did not manage to prove it directly, but we have an indirect argument. 
Consider the action of QR on L+. Let us look at the leading term (which is in A4):

QR{λγ

LDL
γ , wL

α+Wα
L} = {λγ

LDL
γ , (QRwL

α+)Wα
L} mod A≤3 (193)

The direct examination of the action shows that QRwL
α+ = 0. (If QRwL

α+ were nonzero, the 
variation of the kinetic term w+∂−λL would result in the term with the structure λRwL+∂−λL

which would have nothing to cancel with.) Therefore QRL+ falls into A[−1,3]. Since the anchor 
is automatically zero, it remains to prove that QRL+ actually falls into A[0,3]. Let us look at the 
component of QRL+ in grading −1. It is of the form:

X+ = φα̂+DR
α̂

(194)

We know that QLQRL+ = 0. This implies that QLφα̂+ = 0. We conclude that φα̂+ has confor-
mal dimension (1, 0), ghost number (0, 1) and is QL-closed. But there are not such operators, 
therefore φα̂+ = 0. �

12 We also use the fact that the projector Pα′β
′ only affects things in A[0,2] – see Section 4.1.
αβ



570 A. Mikhailov / Nuclear Physics B 907 (2016) 542–571
4.8.3. Zero curvature
Theorem 4.

[L+,L−] = 0 (195)

Unfortunately we did not manage to prove it directly, but we have an indirect argument. We 
know that [L+, L−] is a dimension (1, 1) operator with components in A[−4,4], annihilated by 
both QL and QR . Let us consider the highest component:

X = [L+,L−]mod A≤3 (196)

We know that X is of the conformal dimension (1, 1) and ghost number zero. It follows 
that QRX = 0. But there are no operators with such properties (as can be seen from the 
flat space limit). Therefore the components of [L+, L−] span A−4,3. Then we can consider 
[L+, L−] mod A≤2 and so on.

4.9. Relation between integrated and unintegrated vertex

There must be some analogue of the Koszul duality for algebroids, which should imply that 
the cohomology of the BRST operator λα

La(DL
α ) + λα̂

Ra(DR
α̂
) is equivalent to the Lie algebroid 

cohomology of A. Let us define J+ and J− from the Lax pair:

L± = ∂

∂τ± + J± (197)

Then, given a 2-cocycle ψ representing the Lie algebroid cohomology, we can construct the 
corresponding integrated vertex as in [6]:

U = ψ(J+,J−) (198)

Moreover, the Koszul duality must also imply the consistency of the definition of the algebroid 
A (PBW).

We leave the details for future work.

Is A an overkill? Notice that J± only requires a small part of the A; indeed, J+ belongs to 
A[0,4] and J− belongs to A[−4,0]. This suggests that our definition of A is quite an overkill.

Acknowledgements

We are grateful to N. Berkovits, R. Heluani and M. Movshev for explanations and useful/criti-
cal comments. The work was partially supported by the RFBR Grant 15-01-99504 “String theory 
and integrable systems”.

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	On the construction of integrated vertex in the pure spinor formalism in curved background
	1 Introduction
	2 Brief review of the case of AdS5xS5
	2.1 Definition of Ltot and the PBW theorem
	2.2 BRST complex
	2.2.1 The structure of the dual coalgebra
	2.2.2 Computing Hp(ULtot, V) as ExtULtot(C,V)
	2.2.3 Decoupling of the c0-ghosts
	2.2.4 Cohomology of the ideal

	2.3 Integrated vertex
	2.3.1 Generalized Lax operator
	2.3.2 Bicomplex d + Q BRST


	3 General curved superspace
	3.1 Weyl superspace
	3.2 Relation to the formalism of [1]
	3.2.1 SUGRA constraints, oversimplified
	3.2.2 Sigma-model
	3.2.3 From pure spinor Q to SUGRA constraints
	3.2.4 Shift gauge transformations
	3.2.5 SUGRA fields

	3.3 Lorentz superspace

	4 Worldsheet currents, quadratic-linear algebroid
	4.1 Basic consequences of the defining relations
	4.2 Worldsheet currents
	4.3 Tautological Lax pair
	4.3.1 The case of AdS5xS5
	4.3.2 General case

	4.4 Identification of ψ̃α+
	4.5 Identification of Cβαγ
	4.6 Ramond-Ramond fields
	4.7 Weighing anchor
	4.8 Correction L̃+->L+
	4.8.1 General theory
	4.8.2 Explicit construction
	4.8.3 Zero curvature

	4.9 Relation between integrated and unintegrated vertex

	Acknowledgements
	References