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Published for SISSA by Springer

Received: August 7, 2017

Revised: April 23, 2018

Accepted: July 15, 2018

Published: July 25, 2018

A minimalistic pure spinor sigma-model in AdS

Andrei Mikhailov1

Instituto de F́ısica Teórica, Universidade Estadual Paulista,

R. Dr. Bento Teobaldo Ferraz 271, Bloco II — Barra Funda,

CEP:01140-070, São Paulo, Brasil

E-mail: a.mkhlv@gmail.com

Abstract: The b-ghost of the pure spinor formalism in a general curved background is

not holomorphic. For such theories, the construction of the string measure requires the

knowledge of the action of diffeomorphisms on the BV phase space. We construct such

an action for the pure spinor sigma-model in AdS5 × S5. From the point of view of the

BV formalism, this sigma-model belongs to the class of theories where the expansion of

the Master Action in antifields terminates at the quadratic order. We show that it can

be reduced to a simpler degenerate sigma-model, preserving the AdS symmetries. We

construct the action of the algebra of worldsheet vector fields on the BV phase space of

this minimalistic sigma-model, and explain how to lift it to the original model.

Keywords: BRST Quantization, Gauge Symmetry, Superstrings and Heterotic Strings,

Topological Strings

ArXiv ePrint: 1706.08158

1On leave from Institute for Theoretical and Experimental Physics, ul. Bol. Cheremushkinskaya, 25,

Moscow 117259, Russia

Open Access, c© The Authors.

Article funded by SCOAP3.
https://doi.org/10.1007/JHEP07(2018)155

mailto:a.mkhlv@gmail.com
https://arxiv.org/abs/1706.08158
https://doi.org/10.1007/JHEP07(2018)155


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Contents

1 Introduction 2

2 Master Actions quadratic-linear in antifields 3

3 Pure spinor superstring in AdS5 × S5 7

3.1 Notations 7

3.2 Standard action 9

3.3 New action 9

3.4 The b-ghost 10

3.5 Gauge fixing SO(1, 4)× SO(5) 10

3.6 In BV language 11

4 Action of diffeomorphisms 12

4.1 Formulation of the problem 12

4.2 Subspaces associated to a pair of pure spinors 13

4.3 Construction of Φξ 14

5 Regularization 15

5.1 Adding more fields 15

5.2 A canonical transformation 16

5.3 Constraint surface and its conormal bundle 17

5.4 Gluing charts 18

6 Taking apart the AdS sigma model 21

7 Generalization 22

8 Open problems 23

A The projector 23

A.1 Definition 23

A.2 Matrix language 24

A.3 Explicit formula for the projector 25

A.4 Properties of P13 and P31 26

A.5 Subspaces of g associated to pure spinors 26

B BRST variation of the b-tensor 27

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1 Introduction

The b-ghost of the pure spinor formalism in a general curved background is only holo-

morphic up to a Q-exact expression [1]. The construction of the string measure for such

theories was suggested in [2, 3]. It requires the knowledge of the action of the group of

worldsheet diffeomorphisms on the BV phase space. For a vector field ξ on the worldsheet

(= infinitesimal diffeomorphism) let Φξ be the BV Hamiltonian generating the action of ξ

on the BV phase space. Then, the string measure is, schematically:

exp (SBV + σ + ΦF ) (1.1)

where:

• SBV is the worldsheet Master Action

• σ is the generating function of the variations of the Lagrangian submanifold (for the

standard choice of the family, this is just the usual
∫
µzz̄bzz + µz̄zbz̄z̄)

• F is the curvature of the connection on the equivalence class of worldsheet theories,

considered as a principal bundle over the space of theories modulo diffeomorphisms

It is not completely trivial to construct Φξ for the pure spinor superstring in AdS. One

of the complications is the somewhat unusual form of the pure spinor part of the action.

Schematically:

Sλw =

∫
wL+(∂− +A−)λL + wR−(∂+ +A+)λR + SwL+λLwR−λR (1.2)

where S is a linear combination of Ramond-Ramond field strengths. Notice that the conju-

gate momenta wL and wR only enter through their (1, 0) and (0, 1) component, respectively.

We can try to integrate out w, ending up with a “standard” kinetic term for ghosts:

(∂− +A−)λL (∂+ +A+)λR
SλLλR

(1.3)

Notice that S landed in the denominator. It would seem that the theory depends quite

irregularly on the Ramond-Ramond field, but this is not true. All physics sits at λ = 0,

and the wλwλ term is in some sense subleading.

In this paper we will show, closely following [4, 5], that the pure spinor terms (1.2)

can actually be removed by reduction to a smaller BV phase space, keeping intact all the

symmetries of AdS5 × S5. The resulting action is degenerate, and therefore can not be

immediately used for quantization. On the other hand, it is simpler than the original action.

In particular, the action of worldsheet diffeomorphisms in this reduced BV phase space is

rather transparent, although the explicit expression eq. (4.25) is somewhat involved. We

then explain how to lift this action to an action on some quantizable theory which is

basically the same as the original pure spinor sigma-model of [7].

For the case of flat spacetime, the formal expressions are somewhat more complicated.

The construction of the action of diffeomorphisms is a work in progress with Renann

Lipinski [6].

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Formal application of BV formalism. Here, as in [3], we formally apply the for-

malism of odd symplectic manifolds in the infinite-dimensional case (the field space of

two-dimensional sigma-models). This should be proven in perturbation theory, but in this

paper we restrict ourselves with purely formal manipulations. We believe that supersym-

metry will play crucial role in controlling quantum anomalies; therefore it is important

that our constructions preserve supersymmetries (see section 4.1).

Plan of the paper. We begin in section 2 with the general discussion of the reduction

procedure when a BV Master Action is a quadratic-linear functional of antifields. In

section 3 we apply this to the case of pure spinor superstring in AdS5 × S5. In sections 4

we construct the action of diffeomorphisms in the minimalistic sigma-model. Then in

section 5 we construct the action of diffeomorphisms on the BV phase space of the non-

degenerate theory, which is essentially equivalent (quasiisomorphic) to the original sigma-

model. Sections 6 and 7 contain summary and generalizations, and section 8 open problems.

2 Master Actions quadratic-linear in antifields

Suppose that the BV phase space is an odd cotangent bundle, i.e. is of the form ΠT ∗N

for some supermanifold N (the “field space”). If φa are coordinates on N , then φ?a are

coordinates on ΠT ∗N , and “Π” means that the statistics of φ?a is opposite to the statistics

of φa. There is an odd Poisson bracket (the “BV bracket”):

{φ?a, φb} = δba (2.1)

This bracket is geometrically well-defined, in a sense that the bracket of two functions

{F,G} is actually independent of how the coordinates φa on N are choosen. Equivalently,

there is an odd symplectic form1 (which, as any differential form, can be considered a

function on ΠT (ΠT ∗N)):

ωBV =
∑
a

(−1)ādφa dφ?a (2.2)

(As a slight overuse of Einstein notations, we will omit the summation sign Σa in such

cases.) Suppose that the Master Action is of the form:

SBV = Scl(φ) +Qa(φ)φ?a +
1

2
φ?aπ

ab(φ)φ?b (2.3)

(writing φ?aπ
ab(φ)φ?b rather than πab(φ)φ?aφ

?
b simplifies some signs later).

We will assume that SBV satisfies the classical Master Equation :

{SBV, SBV} = 0 (2.4)

If N is purely even, we can think of functions on ΠT ∗N as polyvector fields on N . For

example, Qa(φ)φ?a corresponds to the vector field Qa(φ) ∂
∂φa , and 1

2π
ab(φ)φ?aφ

?
b corresponds

to a Poisson bivector πab(φ) ∂
∂φa ∧

∂
∂φb

. The odd Poisson bracket { , } corresponds to the

Schouten bracket of polyvector fields.

1http://andreimikhailov.com/math/bv/BV-formalism/Odd symplectic manifolds.html

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If N is a super -manifold, then this polyvector picture does not seem to be very illumi-

nating. However, one can still apply the intuition of Hamiltonian mechanics. The linear

function Q = Qa(φ)φ?a still defines a vector field; the derivative of a function f ∈ C∞(N)

along it is: {Qa(φ)φ?a , f(φ)} = Qa∂af . The quadratic function π = 1
2φ

?
aπ

ab(φ)φ?b still

defines a map from functions on N to vector fields on N :

f 7→ {π, f} (2.5)

The Master Equation (2.4) implies, order by order in expansion in φ?:

{Q,Scl} = 0 (2.6)

{Q,Q}+ 2{π, Scl} = 0 (2.7)

{Q, π} = 0 (2.8)

{π, π} = 0 (2.9)

It follows from eq. (2.9) that vector fields of the form {π, f}, f ∈ C∞(N), form a closed

subalgebra in the algebra of vector fields. They are all tangent to a family of submanifolds

of N which can be called “symplectic leaves of π”. As a slight abuse of notations, the

letter Q will denote both the BRST transformation Qa∂a and the function Qaφ?a on ΠT ∗N .

Eq. (2.7) says that generally speaking the BRST operator Q is only nilpotent on-shell [8].

We will show that under some conditions, this theory can be reduced to a simpler

theory which has BRST operator nilpotent off-shell (and therefore its Master Action has

no quadratic terms φ?φ?).

The case when π is non-degenerate. Let us first consider the case when the Poisson

bivector πab is nondegenerate. Eq. (2.8) implies that an odd function ψ ∈ Fun(N) locally

exists, such that Q = {π, ψ}. Suppose that ψ is also defined globally. Let us consider the

canonical transformation of the Darboux coordinates generated by ψ:

(φ, φ?) → (φ̃, φ̃?)

φa = φ̃a (2.10)

φ?a = φ̃?a +
∂

∂φ̃a
ψ(φ̃)

More geometrically: φ̃ and φ̃? (functions on ΠT ∗N) are pullbacks of φ and φ? by the flux

of the Hamiltonian vector field {ψ, } by the time 1. (The flux integrates to eqs. (2.10)

because ψ only depends on φ, and therefore the velocity of φ? is φ?-independent.)

In the new coordinates:

S = S̃cl +
1

2
φ?aπ

ab(φ)φ?b (2.11)

where S̃cl = Scl +
1

2
∂aψπ

ab∂bψ (2.12)

The φ?-linear term is gone! The Master Equation implies that {S̃cl, π} = 0. Since we

assumed that π is nondegenerate, this implies:

S̃cl = const. (2.13)

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The case of degenerate π. We are actually interested in the case when π is degenerate.

Let P ⊂ TN be the distribution tangent to symplectic leaves of π:

P = imπ ⊂ TN (2.14)

This distribution is integrable because π satisfies the Jacobi identity. We also assume that

Q is transverse to P:

Q /∈ P (2.15)

Let us also consider the distribution P +Q which is generated by elements of P and by Q.

Eqs. (2.8) and (2.9) imply that P +Q is also integrable. Let us assume the existence of a

2-form2 ω on each integrable surface3 of P +Q and a function ψ ∈ Fun(N) which satisfy:

πωπ = π (2.16)

ωπω = ω (2.17)

dω|P+Q = 0 (2.18)

(ιQω − dψ)|P = 0 (2.19)

where πωπ and ωπω are defined as follows:

φ?a (πωπ)ab φ?b = φ?aπ
aa′ωa′b′π

b′bφ?b (2.20)

dφa (ωπω)ab dφ
b = dφaωaa′π

a′b′ωb′bdφ
b (2.21)

Existence of ψ satisfying eq. (2.19) locally follows from eqs. (2.16) and (2.18), because they

imply d(ιQω)|P = 0. But we also require this ψ to be a globally well-defined function on

N . Contracting ιQω − dψ with πω we find that:

Q− {π, ψ} ∈ ker (ω|P+Q) (2.22)

Let us define the new odd vector field:

Q̃ = Q− {π, ψ} (2.23)

Eq. (2.18) implies that ker (ω|P+Q) is an integrable distribution inside an integral surface

of P + Q. Therefore eq. (2.22) implies that Q̃2 is proportional to Q̃, i.e. there exists a

function ζ such that: Q̃2 = ζQ̃. In fact ζ = 0, since Q̃2 ∈ P and Q̃ /∈ P . We conclude:

Q̃2 = 0 (2.24)

Let us consider the canonical transformation (2.10) of Darboux coordinates generated by

ψ. With these new Darboux coordinates:

SBV = Scl −
1

2
ω(Q,Q) + (Q− {π, ψ})a φ̃?a +

1

2
φ̃?aπ

abφ̃?b (2.25)

2This ω is even; it should not be confused with the odd symplectic form of ΠT ∗N .
3It is enough to define ω on each integrable surface of P + Q; it does not have to be defined on the

whole N .

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Notice that the new “classical action”:

S̃cl = Scl −
1

2
ω(Q,Q) (2.26)

is automatically constant on symplectic leaves of π. Also, it follows that Q̃ consistently

defines an odd nilpotent vector field on the moduli space of symplectic leaves of π. These

facts follow from {SBV, SBV} = 0. To summarize:

SBV = S̃BV +
1

2
φ̃?aπ

ab(φ)φ̃?b (2.27)

where S̃BV = S̃cl(χ) + Q̃(χ)mχ?m (2.28)

where χ is coordinates on the space of symplectic leaves of π. We therefore constructed a

new, simpler theory, on the space of symplectic leaves of π.

This theory can be interpreted as the result of integrating out4 some antifields. More

precisely, let us define a submanifold N0 ⊂ N by picking one point from each symplectic

leaf. Fibers of the odd conormal bundle5,6 ΠT ∗N0 are isotropic submanifolds in ΠT ∗N ,

and we can integrate them out as described in [3]. In this paper the coordinates in these

fibers will be called w? (and integrated out).

Oversimplified example. We will now illustrate the relation by a toy sigma-model (we

will actually run the procedure “in reverse”). Let Σ be a two-dimensional worldsheet. Let

us start with:

SBV = Scl +

∫
Σ
λθ? (2.29)

where Scl does not depend neither on the fermionic field θa nor on the bosonic field λa. (It

depends on some other fields φµ.) We postulate the odd symplectic form so that our fields

are Darboux coordinates7 [3], as in eq. (2.2):

ωBV =

∫
Σ
dλ?dλ− dθ?dθ +

∑
µ∈{ other

fields}
(−1)µ̄dφ?µdφ

µ (2.30)

This action is highly degenerate; the path integral
∫

[dλ][dθ][dφ]eScl(φ) is undefined (infinity

from integrating over λ times zero from integrating over θ). To regularize ∞× 0, let us

introduce a new field-antifield pair w,w?, where w is a bosonic 1-form on the worldsheet

and w? is a fermionic 1-form on the worldsheet:

w =w+dz + w−dz (2.31)

w? =w?+dz + w?−dz (2.32)

4http://andreimikhailov.com/math/bv/transfer/Partial Integration.html
5http://andreimikhailov.com/math/bv/BRST-formalism/Family of Lagrangian submanifolds.html

#(part. .Conormal bundle)
6The fiber of the conormal bundle of N0 ⊂ N at the point φ ∈ N0 consists of those elements of T ∗φN

which vanish on TφN0 ⊂ TφN .
7http://andreimikhailov.com/math/bv/BV-formalism/Odd symplectic manifolds.html

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http://andreimikhailov.com/math/bv/BRST-formalism/Family_of_Lagrangian_submanifolds.html#(part._.Conormal_bundle)
http://andreimikhailov.com/math/bv/BRST-formalism/Family_of_Lagrangian_submanifolds.html#(part._.Conormal_bundle)
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The total odd symplectic form is postulated as follows:

ωtot
BV = ωBV +

∫
Σ
dw? ∧ dw (2.33)

(where d is the field space differential, not the worldsheet differential). Let us add (w?)2

to the BV action:

SBV = Scl +

∫
λθ? +

∫
w? ∧ w? (2.34)

(Notice that this
∫
w? ∧ w? does not involve the worldsheet metric.) This corresponds to:

ω =

∫
Σ
dw ∧ dw (2.35)

(again, d is the field space differential, not the worldsheet differential). In this case P is

the subspace of the tangent space generated by ∂
∂w , and Q is generated by λ ∂

∂θ . Then, shift

the Lagrangian submanifold by a gauge fermion:

Ψ =

∫
Σ
w ∧ dθ (2.36)

This results in the new classical action:

Snew
cl =Scl +

∫
Σ
w ∧ dλ+

∫
Σ
dθ ∧ dθ (2.37)

SBV =Snew
cl +

∫
λθ? +

∫
dθ w? +

∫
w? ∧ w? (2.38)

Qnew =λ
∂

∂θ
+ dθ

∂

∂w
(2.39)

Here we have run the procedure of section 2 “in reverse”. That is, eq. (2.37) is an example

of the Scl of eq. (2.3), and eq. (2.34) is an example of the “split” eq. (2.11). Notice that

π is degenerate, as it does not involve ∂
∂θ and ∂

∂λ . Because of that, the Scl of eq. (2.34)

is not constant as in eq. (2.26), but just independent of w. The vector field {π,Ψ} is the

dθ ∂
∂w -part of Qnew, as in eq. (2.22).

This is, still, not a quantizable action (the kinetic term for θ is a total derivative). One

particular way of choosing a Lagrangian submanifold leading to quantizable action is to

treat w+ and w− asymmetrically (pick a worldsheet complex structure), see section on A-

model in AKSZ [9] and section 5.3 of this paper. This requires more than one flavour of w.

3 Pure spinor superstring in AdS5 × S5

3.1 Notations

We follow the notations in [10]. The superconformal algebra g = psu(2, 2|4) has Z4-

grading:

g = g0̄ + g1̄ + g2̄ + g3̄ (3.1)

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Bars over subindices are to remind that they are mod 4. Geometrically, g2̄ can be identified

with the tangent space to the bosonic AdS5 × S5, which is the direct sum of the tangent

space to AdS5 and the tangent space to S5:

T (AdS5 × S5) = T (AdS5)⊕ T (S5) (3.2)

Therefore elements of g2̄ are vectors from this tangent space. We can also consider the

tangent space to the full superspace M :

M = super(AdS5 × S5) =
PSU(2, 2|4)

SO(1, 4)× SO(5)
(3.3)

T

(
PSU(2, 2|4)

SO(1, 4)× SO(5)

)
= g1̄ ⊕ g2̄ ⊕ g3̄ (3.4)

this is a direct sum of three vector bundles. We parametrize a point in M by g ∈ PSU(2, 2|4)

modulo the equivalence relation:

g ' hg for all h ∈ SO(1, 4)× SO(5) (3.5)

We are identifying representations of g0̄ = Lie(SO(1, 4)× SO(5)), such as g1̄, g2̄, g3̄, with

the corresponding vector bundles over the coset space (3.3). In fact, the worldsheet field

λL takes values in the fibers of g3̄ and λR takes values in the fibers of g1̄. The pure spinor

conditions define the cones CL and CR:

CL : {λL, λL} = 0 (3.6)

CR : {λR, λR} = 0 (3.7)

Here { , } denotes the anticommutator (the Lie superalgebra operation) of elements of

g. It should not be confused with neither the odd Poisson bracket, nor the even Poisson

bracket corresponding to πab of section 2. Again, we identify CL and CR as bundles over

super-AdS. (They are not vector bundles, because their fibers are cones and not linear

spaces.) We will denote:

PS AdS5 × S5 =
CL × CR × PSU(2, 2|4)

SO(1, 4)× SO(5)
(3.8)

where the prefix PS on the l.h.s. stands for “Pure spinors” (and on the r.h.s. for “Projective”

and “Special”).

In appendix A we construct PSU(2, 2|4)-invariant surjective maps of bundles (“projec-

tors”):

P31 : (g3̄ × CL)→ TCL (3.9)

P13 : (g1̄ × CR)→ TCR (3.10)

They are rational functions of λL and λR.

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3.2 Standard action

The action of the AdS sigma-model has the following form [7]:

S0 =

∫
dz dz̄ Str

(
1

2
J2+J2− +

3

4
J1+J3− +

1

4
J3+J1− (3.11)

+ w1+D0−λ3 + w3−D0+λ1 −N0+N0−

)
where Jn are the gn̄-components of J = −dgg−1 = J+dz + J−dz̄. We write λ3 instead of

λL and λ1 instead of λR, just to highlight the Z4-grading.(And also because neither λL
is strictly speaking left-moving, nor is λR right-moving.) The covariant derivative D0± is

defined as follows:

D0± = ∂± + [J0±, ] (3.12)

Since λ3 and λ1 both satisfy the pure spinor constraints, the corresponding conjugate

momenta are defined up to “gauge transformations”:

δv2w1+ = [v2+, λ3] (3.13)

δu2w3− = [u2−, λ1] (3.14)

where v2 and u2 are arbitrary sections of the pullback to the worldsheet of g2̄. The BRST

transformations are defined up to gauge transformations corresponding to the equivalence

relation (3.5). It is possible to fix this ambiguity8 so that:

QλL = QλR = 0 (3.15)

Qg = (λL + λR)g (3.16)

Qw1+ = − J1+ , Qw3− = −J3− (3.17)

The first line in eq. (3.11) is by itself not BRST invariant. Modulo total derivatives, its

BRST variation is:

Q

∫
dτ dσ Str

(
1

2
J2+J2− +

3

4
J1+J3− +

1

4
J3+J1−

)
=

∫
dτ dσ Str (−D0+λ1 J3− −D0−λ3 J1+) (3.18)

This cancels with the BRST variation of the second line in eq. (3.11).

3.3 New action

On the other hand, we observe that:

Q STr (J1+P31J3−) = STr (−D0+λ1 J3− −D0−λ3 J1+) (3.19)

Notice that the projector drops out on the r.h.s. because D0±λ is automatically tangent to

the cone. Comparing this to (3.18) we see that the following expression:

S′0 =

∫
dτ dσ STr

(
1

2
J2+J2− +

3

4
J1+J3− +

1

4
J3+J1− − J1+P31J3−

)
(3.20)

is BRST invariant. It does not contain neither derivatives of pure spinors, nor their conju-

gate momenta.

8http://andreimikhailov.com/slides/talk Perimeter/LiftOfQ.html

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3.4 The b-ghost

We define:

b++ =
STr

(
({J3+, λ3} − {J1+, λ1})J2+

)
STr(λ3λ1)

=
Tr (({J3+, λ3} − {J1+, λ1})J2+)

STr(λ3λ1)
(3.21)

b−− = same but with + replaced with − (3.22)

(See appendix A for notations. We use the fact that Str(A2B2) = Str(A2B2Σ) =

Tr(A2B2).) These expressions satisfy (appendix B):

Qb++ = T++ and Qb−− = T−− (3.23)

where T++ = Str

(
1

2
J2+J2+ + J1+(1−P31)J3+

)
T−− = Str

(
1

2
J2−J2− + J1−(1−P31)J3−

)
Notice that:

S′0 = S′′0 +QB (3.24)

wher B =

∫
dτdσ

Tr
(
({J3+, λ3} − {J1+, λ1})J2− + (+↔ −)

)
STr(λ3λ1)

(3.25)

S′′0 =

∫
STr (J1 ∧ (1−P31)J3 − J1 ∧P31J3) (3.26)

and S′′0 is diffeomorphism-invariant (and therefore degenerate!). The BRST invariance of

S′′0 can be verified explicitly as follows:

QS′′0 =

∫
STr

(
[λ3, J2] ∧ J3 − [λ1, J2] ∧ J1 −D0λ1 ∧ J3 +D0λ3 ∧ J1

)
=

∫
d STr(λ3J1 − λ1J3) = 0 (3.27)

3.5 Gauge fixing SO(1, 4) × SO(5)

Consider the action of the BRST operator given by Eq. (3.16) on g. It is nilpotent only up

to the g0-gauge transformation by {λ3, λ1}. We have so far worked on the factorspace by

gauge transformations. This means that we think of the group element g and pure spinors

λ as defined only modulo the gauge transformation:

(g, λ) ' (hg, hλh−1) (3.28)

It turns out that the action of these gauge transformations on the BV phase space is

somewhat nontrivial, see section 5.4. We will now just fix the gauge, postponing the

discussion of gauge transformations to section 5.4. Let us parametrize the group element

g ∈ PSU(2, 2|4) by u, x, θ:

g = euex+θ (3.29)

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where u ∈ g0, x ∈ g2 and θ ∈ g3 + g1, and impose the following gauge fixing condition:

u = 0 (3.30)

Since eq. (3.30) does not contain derivatives, this gauge is “ghostless”, the Faddeev-Popov

procedure is not needed.9 In this gauge fixed formalism, the BRST operator includes the

gauge fixing term (cp. eqs. (3.15), (3.16), (3.17)):

Qg = (λ3 + λ1 +A0)g (3.31)

Qλ3 = [A0, λ3] , Qλ1 = [A0, λ1] (3.32)

Qw1+ = − J1+ + [A0, w1+] , Qw3− = − J3− + [A0, w3−] (3.33)

where A0 ∈ g0̄ is some function of θ, λ and x, defined by eqs. (3.31) and (3.30); schemat-

ically A0 = {θL, λ1} + {θR, λ3} + . . . This A0 is usually called “the compensating gauge

transformation”. It automatically satisfies:

QA0 = −{λ3, λ1}+
1

2
[A0, A0] (3.34)

Gauge fixing is only possible locally in AdS5 × S5. In order for our constructions to work

globally, we will cover AdS5 × S5 with patches and gauge-fix over each patch. Then we

have to glue overlapping patches. We will explain how to do this in section 5.4.

3.6 In BV language

We will now show that the difference between the original action and the action (3.20)

can be interpreted in the BV formalism as a particular case of the construction outlined in

section 2.

The BRST symmetry of the pure spinor superstring in AdS5 × S5 is nilpotent only

on-shell. More precisely, the only deviation from the nilpotence arises when we act on the

conjugate momenta of the pure spinors:

Q2w1+ =
δS0

δw3−
(3.35)

Q2w3− =
δS0

δw1+
(3.36)

(while the action of Q2 on the matter fields is zero even off-shell). This means that the BV

Master Action contains a term quadratic in the antifields:

SBV = S0 +

∫
(QZi)Z?i +

∫
(Qλ)λ? +

∫
(Qw)w? +

∫
Str

(
w?1+w

?
3−
)

(3.37)

In this formula Z and Z? stand for matter fields (x and θ) and their antifields, and S0 is

given by eq. (3.11). The matter fields Z are essentially x and θ where J = −dgg−1 with

g = ex+θ, x ∈ g2, θ ∈ g3 ⊕ g1:

Z = x and θ (3.38)

9The Faddeev-Popov procedure in such cases leads to ghost action of the form
∫
f(φ)c̄c where f(φ) is

some function of the fields. Integration out c and c̄ leads to local expressions (in fact, proportional to δ(0))

which are absorbed by counterterms. Similar topics were discussed in [11, 12].

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Their BRST transformation QZi is read from eq. (3.31). We observe that the action is of

the same type as described in section 2. The Poisson bivector is:

π =

∫
Str

(
∂

∂w1+
∧ ∂

∂w3−

)
(3.39)

The 2-form ω discussed in section 2 can be choosen as follows:

ω =

∫
Str (dw1+ ∧P31dw3−) (3.40)

The projector P31 is needed to make ω invariant with respect to the gauge transforma-

tions (3.13) and (3.14). We take the following generating function ψ satisfying eq. (2.19):

ψ =

∫
Str (w1+P31J3− + w3−P13J1+ + w1+[A0, w3−]) (3.41)

The new “classical action” S̃cl is given by Eq. (3.20). (We will provide more details for a

slightly more general calculation in section 5.) It is, indeed, constant along the symplectic

leaves of π, as the fields w± are not present in this new Lagrangian at all. The new BV

action is:

S̃BV =

∫
dτ dσ Str

(
1

2
J2+J2− +

3

4
J1+J3− +

1

4
J3+J1− − J1+P31J3−

+
∑

Z∈{x,θ,λ}

(QZ)Z?

)
(3.42)

where Zi runs over θ, x, λ and the action of Q on Zi is the same as it was in the original

σ-model. The new BV phase space is smaller, it only contains θ, x, λ, θ?, x?, λ?. The BRST

operator is now nilpotent off-shell; the dependence of the BV action on the antifields is

linear. The fields λL|R enter only through their combination invariant under local rescalings

(they enter through P31). This in particular implies that the BRST symmetry Q is now a

local symmetry.

Of course, the new action (3.20) is degenerate.

4 Action of diffeomorphisms

4.1 Formulation of the problem

Let L?2 be the BV Hamiltonian generating the left shift by elements of g2̄; if f is any

function of g, then:

{Str(A2L
?
2) , f}BV (g) =

d

dt

∣∣∣∣
t=0

f
(
etA2g

)
(4.1)

The L?0, L?1 and L?3 are defined similarly. In particular:

{S̃BV, } =

∫
Str (λ3L

?
1 + λ1L

?
3) (4.2)

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With these notations, when X and Y are two even elements of g,{∫
Str(XL?),

∫
Str(Y L?)

}
BV

= −
∫

Str([X,Y ]L?) (4.3)

(Even elements are generators of g2 and g0, and also the generators of g3 and g1 multiplied

by a Grassmann odd parameter.)

The infinitesimal action of diffeomorphisms is generated by the following BV Hamil-

tonian Vξ:

Vξ =

∫
Str
(

(ιξD0λ3)λ?1 + (ιξD0λ1)λ?3 − (ιξJ3)L?1 − (ιξJ1)L?3 − (ιξJ2)L?2

)
(4.4)

where D0λ = dλ+ [J0, λ] (4.5)

In this section we will construct Φξ such that:

Vξ = {S̃BV,Φξ}BV (4.6)

It is very easy to construct such Φξ if we don’t care about the global symmetries of

AdS5 × S5. (Something like Φξ = θα

λαVξ.) But we will construct a Φξ invariant under

the supersymmetries of AdS5 × S5, i.e. invariant under the right shifts of g. We believe

that such an invariant construction has better chance of satisfying the equivariance condi-

tions of [2, 3] at the quantum level, because supersymmetries restrict quantum corrections.

In particular, the equivariance condition must require that the Φξ correspond, in some

sense, to a primary operator.

Comment on gauge transformations. In this section we discuss vector fields on the

factorspace PS AdS defined by eq. (3.8). They are the same as SO(1, 4)×SO(5)-invariant

vector fields on CL × CR × PSU(2, 2|4) modulo SO(1, 4) × SO(5)-invariant vertical vector

fields. All the formulas here are modulo vertical SO(1, 4)× SO(5)-invariant vector fields.

4.2 Subspaces associated to a pair of pure spinors

We use the notations of section A.5. For X3 ∈ [g2L, λ1] and X1 ∈ [g2R, λ3], let T2(X1+X2)

denote the map:

T2 : [λ1,g2L]⊕ [λ3,g2R] −→ g2L ⊕ g2R (4.7)

T2([λ1, v2L] + [λ3, v2R]) = v2L + v2R (4.8)

(This is a direct sum of two completely independent linear maps.) For a pair I3 ⊕ I1 ∈
T⊥CR ⊕ T⊥CL we decompose:10

I3 ⊕ I1 = (Isplit
3 ⊕ Isplit

1 ) + (Iker
3 ⊕ Iker

1 ) + (Icoker
3 ⊕ Icoker

1 ) (4.9)

where Isplit
3 ⊕ Isplit

1 ∈ [λ1,g2L]⊕ [λ3,g2R] (4.10)

Iker
3 ⊕ Iker

1 ∈ ker
[
T⊥CR ⊕ T⊥CL

(+)◦({λ3, }⊕{λ1, })−→ g2

]
(4.11)

Icoker
3 ⊕ Icoker

1 ∈ coker
[
g2
{λ3, }+{λ1, }−→ T⊥CR ⊕ T⊥CL

]
(4.12)

10For example, Isplit3 denotes the component of I3 which belongs to [λ1,g2L]; the label “split” is because

we could not invent any better notation.

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where we must use a special representative of the cokernel:

Iker
3 =

1

2

Tr(λ3I3 − λ1I1)

Tr ([λ1, λ3]STL)2 [λ1, [λ3, λ1]STL] (4.13)

Iker
1 =

1

2

Tr(λ3I3 − λ1I1)

Tr ([λ1, λ3]STL)2 [λ3, [λ3, λ1]STL] (4.14)

Icoker
3 =

1

2

Tr(λ3I3 + λ1I1)

Tr ([λ1, λ3]STL)2 [λ1, [λ3, λ1]STL] (4.15)

Icoker
1 = − 1

2

Tr(λ3I3 + λ1I1)

Tr ([λ1, λ3]STL)2 [λ3, [λ3, λ1]STL] (4.16)

Similarly, any I2 ∈ g2 (assumed to be both TL and STL) can be decomposed:

I2 = IL
2 + IR

2 + Iker
2 + Icoker

2 (4.17)

where IL
2 ∈ g2L (4.18)

IR
2 ∈ g2R (4.19)

Iker
2 ∈ ker

[
g2
{λ3, }+{λ1, }−→ T⊥CL ⊕ T⊥CR

]
(4.20)

Icoker
2 ∈ coker

[
T⊥CR ⊕ T⊥CL

(+)◦({λ3, }⊕{λ1, })−→ g2

]
(4.21)

Explicitly:

Iker
2 =

Tr(I2[λ3, λ1])

Tr([λ3, λ1]STL)2
[λ3, λ1]TL (4.22)

Icoker
2 =

Tr(I2[λ3, λ1])

Tr([λ3, λ1]STL)2
[λ3, λ1]STL (4.23)

4.3 Construction of Φξ

The generating function Vξ of the infinitesimal worldsheet diffeomorphisms (= vector fields)

ξ = ξτ∂τ + ξσ∂σ, given by eq. (4.4), is BV-exact:

Vξ = {S̃BV,Φξ} (4.24)

Φξ = −
∫

Str
(

(P31ιξJ3)λ?1 + (P13ιξJ1)λ?3

+
(
T2

(
ιξJ

split
3 + ιξJ

split
1

)
+A[λ3, λ1]STL

)
L?2

+ B ([λ1, [λ3, λ1]STL]L?3 − [λ3, [λ3, λ1]STL]L?1)
)

(4.25)

where ιξJ = − ξα∂αgg−1

A =
1

2

Tr (λ3(1−P31)ιξJ3 − λ1(1−P13)ιξJ1)

Tr([λ3, λ1]STL)2

B =
STr([λ3, λ1]ιξJ2)

STr(λ3λ1) Tr([λ3, λ1]STL)2
(4.26)

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The coefficients A and B satisfy:

(QA)[λ3, λ1]STL = ιξJ
coker
2 (4.27)

A ([λ1, [λ3, λ1]STL] + [λ3, [λ3, λ1]STL]) = ιξJ
ker
3 + ιξJ

ker
1 (4.28)

(QB) ([λ1, [λ3, λ1]STL]− [λ3, [λ3, λ1]STL]) = ιξJ
coker
1 + ιξJ

coker
3 (4.29)

B ({λ3, [λ1, [λ3, λ1]STL]} − {λ1, [λ3, [λ3, λ1]STL]}) = ιξJ
ker
2 (4.30)

Eq. (4.24) follows from:

ιξJ = P31ιξJ3 + P13ιξJ1

+ ιξJ
split
3 + ιξJ

split
1 + ιξJ

ker
3 + ιξJ

ker
1

+ (QB) ([λ1, [λ3, λ1]STL]− [λ3, [λ3, λ1]STL])

+ ιξJ
split
2 + ιξJ

ker
2 + (QA)[λ3, λ1]STL (4.31)

Some useful identities.

STr ([λ3, [λ3, λ1]STL] (1−P31)ιξJ3) = − STr(λ3λ1)

2
Tr (λ3(1−P31)ιξJ3) (4.32)

{λ1, [λ3, [λ3, λ1]STL]} = − {λ3, [λ1, [λ3, λ1]STL]}

=
1

2
[λ1, λ3]TLStr(λ3λ1) +

1

8

(
(Str(λ3λ1))2 − 2Tr[λ1, λ3]2

)
1

=
1

2
[λ1, λ3]TLStr(λ3λ1)− 1

4
Tr([λ3, λ1]STL)2 1 (4.33)

STr
(

[λ1, [λ3, λ1]STL] [λ3, [λ3, λ1]STL]
)

= − 1

2
Str(λ3λ1)Str

(
[λ1, λ3]STL[λ1, λ3]TL

)
= −1

2
Str(λ3λ1)Tr([λ3, λ1]STL)2 (4.34)

Notice that we have Tr([λ3, λ1]STL)2 in denominators. At the same time:

STr([λ3, λ1]STL)2 = STr([λ3, λ1])2 = 0 (4.35)

5 Regularization

The “minimalistic action” (3.42) cannot be regularized in a way that would preserve the

symmetries of AdS5×S5; it is impossible to choose a PSU(2, 2|4)-invariant Lagrangian sub-

manifold so that the restriction of the Master Action of eq. (3.42) to it be non-degenerate.

Let us therefore return to the original action of eqs. (3.11), (3.37), but in a way preserving

the worldsheet diffeomorphisms. The construction is somewhat similar to the description

of the topological A-model in [9].

5.1 Adding more fields

Add a pair of bosonic 1-form fields ω3 and ω1, taking values in g3 and g1, respectively, and

their antifields ω?1 and ω?3, also 1-forms:

ωBV =

∫
STr (dω?3 ∧ dω1 + dω?1 ∧ dω3) (5.1)

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(where d is the differential in the field space, not on the worldsheet!). In other words, for

any “test 1-forms” f1 and g3:{∫
Str f1 ∧ ω?3 ,

∫
Str ω1 ∧ g3

}
=

∫
Str f1 ∧ g3 (5.2){∫

Str g3 ∧ ω1 ,

∫
Str ω?3 ∧ f1

}
=

∫
Str g3 ∧ f1 (5.3)

We define the BV Master Action as follows:

S̃+
BV = S̃BV +

∫
STr(ω?3 ∧ ω?1) (5.4)

and the BV Hamiltonian for the action of diffeomorphisms as follows:

V̂ξ = {S̃+
BV , Φ̂ξ}BV (5.5)

Φ̂ξ = Φξ +

∫
STr(ω3 ∧ Lξω1) (5.6)

where Lξ is the Lie derivative.

The expression
∫

STr(δω3 ∧ δω1) defines a symplectic structure on the space of 1-forms

with values in godd. The expression
∫

STr(ω?3∧ω?1) is the corresponding Poisson bivector.11

The Lie derivative preserves this (even) symplectic structure, and
∫

STr(ω3 ∧ Lξω1) is the

corresponding Hamiltonian.

5.2 A canonical transformation

Let us do the canonical transformation by a flux of the following odd Hamiltonian:

Ψ(0) =

∫
STr [A0, ω3] ∧ ω1 = −

∫
STr [A0, ω1] ∧ ω3 (5.7)

This is the Hamiltonian of [A0, ] in the same sense as
∫

STr(ω3∧Lξω1) is the Hamiltonian

of Lξ; we again use the same procedure of passing from eq. (2.3) to eq. (2.25), actually

in reverse.

The effect of the flux of Ψ(0) on the BV Master Action S̃+
BV of eq. (5.4) is:

S̃+
BV = S̃BV +

∫
STr ω?3 ∧ ω?1

becomes S̃′BV = S̃BV +

∫
STr

[(
{λ3, λ1} −

1

2
[A0, A0]

)
, ω3

]
∧ ω1

+

∫
STr(ω?3 + [A0, ω3]) ∧ (ω?1 − [A0, ω1])

= S̃+
BV +

∫
STr {λ3, λ1}{ω3,∧ω1} (5.8)

+

∫
STr

(
[A0, ω3] ∧ ω?1 + [A0, ω1] ∧ ω?3

)
11http://andreimikhailov.com/math/bv/omega/Duistermaat-Heckman formula.html

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Notice that the terms of the form A2ω2 cancelled. This is automatic, because such terms

would contradict the Master Equation (the bracket
{
ω?ω?, A2ω2

}
would have nothing to

cancel against).

The purpose of this canonical transformation was, essentially, to introduce the com-

pensator term [A,ω] into the action of Q on ω, cp. eq. (3.31). We will discuss this in a

more general context in section 5.4.

We are now ready to construct the Lagrangian submanifold.

5.3 Constraint surface and its conormal bundle

The configuration space X of this new theory is parametrized by g, λ3,λ1,ω3± and ω1±.

Let us consider a subspace Y ⊂ X defined by the constraints:

(1−P13)ω1+ = 0

(1−P31)ω3− = 0 (5.9)

ω1− = 0

ω3+ = 0

Consider the odd conormal bundle12 ΠT⊥Y of Y ⊂ X in the BV phase space ΠT ∗X.

As any conormal bundle, this is a Lagrangian submanifold. The restriction of S̃′BV on

this Lagrangian submanifold is still degenerate. But let us deform it by the following

generating function:

Ψ =

∫
STr

(
ω3−P13J1+ + ω1+P31J3−

)
(5.10)

The restriction of S̃′BV to this deformed Lagrangian submanifold is equal to:∫
STr

(
1

2
J2+J2− +

3

4
J1+J3− +

1

4
J3+J1−

+ w1+D0−λ3 + w3−D0+λ1 +N0+N0− + ω?3+ω
?
1− + ω?3−ω

?
1+

)
(5.11)

where N0+ = {w1+, λ3}, N0− = {w3−, λ1},

w1+ = P13ω1+ and w3− = P31ω3− (5.12)

Notice that the terms:∫
STr

(
[A0, λ3]λ?3 + [A0, λ1]λ?1

+ [A0, ω3+]ω?1− + [A0, ω3−]ω?1+ + [A0, ω1+]ω?3− + [A0, ω1−]ω?3+

)
(5.13)

vanish on T⊥Y . Indeed, the vector field:

[A0, λ3]
∂

∂λ3
+ [A0, λ1]

∂

∂λ1

+ [A0, ω3+]
∂

∂ω3+
+ [A0, ω3−]

∂

∂ω3−
+ [A0, ω1+]

∂

∂ω1+
+ [A0, ω1−]

∂

∂ω1−
(5.14)

12http://andreimikhailov.com/math/bv/BRST-formalism/Family of Lagrangian submanifolds.html

#(part. .Conormal bundle)

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is tangent to the constraint surface (5.9); the conormal bundle, by definition, consists of

those one-forms which vanish on such vectors. The term ω?3+ω
?
1− + ω?3−ω

?
1+ computes the

contrubution to the action from the fiber ΠT⊥Y . The coordinates of the fiber enter without

derivatives, and decouple.

We therefore return to the original AdS5 × S5 action of eq. (3.11).

But now we understand how the worldsheet diffeomorphisms act, at the level of the

BV phase space.

5.4 Gluing charts

In our construction we used a lift of AdS5 × S5 to PSU(2, 2|4) (section 3.5). This is only

possible locally. Therefore, we have to explain how to glue together overlapping patches.

This is a particular case of a general construction, which we will now describe.

The idea: is to build a theory which is locally (on every patch of AdS5 × S5) a direct

product of two theories S(φ) and S(w):

Stot = S(φ) + S(w) = Scl(φ) +Qµ(φ)φ?µ +
1

2
w?a(ω

−1)abw?b (5.15)

but transition functions between overlapping patches mix φ and w.

Consider the following data, consisting of two parts. The first part is a Lie group H and

a principal H-bundle E with base B. Suppose that B comes with a nilpotent vector field

Q ∈ Vect(B) and a Q-invariant action Scl ∈ Fun(B). Then SB(φ, φ?) = Scl(φ) +Qµ(φ)φ?µ
satisfies the Master Equation on the BV phase space ΠT ∗B. The second part of the data

is a symplectic vector space W which is a representation of H. This means that W is

equipped with an even H-invariant symplectic form ω.

Let us cover B with charts {Ui|i ∈ I} and trivialize E over each chart:

p−1(Ui) ' Ui ×H (5.16)

At the intersection Ui ∩ Uj we identify (φ, hi) ∈ Ui ×H with (φ, hj) ∈ Uj ×H if

hj = uji(φ)hi (5.17)

All this comes from E
H→ B. We will now construct a new odd symplectic manifold, which

is locally ΠT ∗Uj ×ΠT ∗W , with some transition functions, which we will now describe.

Technical assumption: in this section we assume that all w are bosons, and that H is

a “classical” (i.e. not super) Lie group. This is enough for our considerations.

Transition functions: let h be the Lie algebra of H. For each α ∈ Map(B,h) consider

the following BV Hamiltonian:

χα = {Stot, Fα} (5.18)

where Fα = − 1

2
wbρ∗(α(φ))ab ωac w

c (5.19)

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Here ρ∗ is the representation of the Lie algebra corresponding to the representation ρ of

the group, and ω is the symplectic form of W . Eq. (5.19) defines Fα as the Hamiltonian of

the infinitesimal action of α on w, i.e. the “usual” (even) moment map. (Here we use our

assumption that ω is H-invariant.) The explicit formula for χα is:

χα = ρ∗(α(φ))abw
b w?a −

1

2
wbρ∗(Qα(φ))abωacw

c (5.20)

Notice that:

{χα1 , Fα2} = −F[α1,α2] (5.21)

The flux of the BV-Hamiltonian vector field {χα, } is a canonical transformation, and

eq. (5.18) implies that this canonical transformation is a symmetry of Stot. This canonical

transformation does not touch φµ, it only acts on φ?, w, w?. We identify (φ, φ?i , wi, w
?
i )

on chart U(i) with (φ, φ?j , wj , w
?
j ) on chart U(j) when (φ?j , wj , w

?
j ) is the flux of (φ?i , wi, w

?
i )

by the time 1 along the vector field {χαji , } where αji is the log of uji, i.e. uji = eαji .

Explicitly:

waj = ρ (uji)
a
b w

b
i (5.22)

w?ja = ρ
(
u−1
ji

)b
a
w?ib − ωab Qρ (uji)

b
c w

c
i (5.23)

φ?jµ = φ?iµ − w?jaρ∗

uji ←
∂

∂φµ
u−1
ji

a

b

wbj −
1

2
wajωab

∂

∂φµ
ρ∗

(
Qujiu

−1
ji

)b
c
wcj (5.24)

These gluing rules are consistent on triple intersections because of eq. (5.21).

Lagrangian submanifold. Eqs. (5.23) and (5.24) look somewhat unusual. In particu-

lar, the “standard” Lagrangian submanifold13 φ? = w? = 0 is not well-defined, because it

is incompatible with our transition functions. One simple example of a well-defined La-

grangian submanifold is w = φ? = 0. We will now give another example, which repairs the

ill-defined w? = φ? = 0.

The construction requires a choice of a connection in the principal bundle E
H→ B.

To specify a connection, we choose on every chart Ui some h-valued 1-form Aiµ, with the

following identifications on the intersection Ui ∩ Uj :

∂

∂φµ
+Ajµ(φ) = uji(φ)

(
∂

∂φµ
+Aiµ(φ)

)
(uji(φ))−1 (5.25)

and in particular:

Qρ(uji)
a
b + ρ∗(Q

µAjµ)acρ (uji)
c
b − ρ(uji)

a
cρ∗(Q

µAiµ)cb = 0 (5.26)

13In BV formalism, there is no such thing as the standard Lagrangian submanifold. We invented this

notion to denote the one where all antifields (w.r.to some Darboux coordinates) are zero. This is often a

useful starting point to construct Lagrangian submanifolds.

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On every chart, let us pass to a new set of Darboux coordinates, by doing the canonical

transformation with the following gauge fermion:

Ψi =
1

2
wai ωab Q

µ(φ)ρ∗(Aiµ(φ))bc w
c
i (5.27)

Notice that Ψi does not depend on antifields; therefore this canonical transformation is

just a shift:

w̃ai = wai (5.28)

w̃?ia = w?ia − ωab Q
ν(φ)ρ∗(Aiν(φ))bc w

c
i (5.29)

φ̃?iµ = φ?iµ −
1

2

∂

∂φµ

[
wai ωab Q

ν(φ)ρ∗(Aiν(φ))bc w
c
i

]
(5.30)

This canonical transformation does not preserve SBV, therefore the expression for the

action will be different in different charts, see eq. (5.8). In particular, it will contain the

term w̃?Qµρ∗(Aiµ)w̃, which means that the action of the BRST operator on w̃ involves the

connection. On the other hand, the transition functions simplify:

w̃aj = ρ (uji(φ))ab w̃
b
i (5.31)

w̃?ja = ρ
(
uji(φ)−1

)b
a
w̃?ib (5.32)

φ̃?jµ = φ̃?iµ − w̃?icρ
(
uji(φ)−1

)c
a

ρ (uji(φ))ab

←
∂

∂φµ

 w̃bi (5.33)

These are the usual transition functions of the odd cotangent bundle ΠT ∗W, where W is

the vector bundle with the fiber W , associated to the principal vector bundle E
H→ B.

In particular, the “standard” Lagrangian submanifold w̃? = φ̃? = 0 is compatible with

gluing. The corresponding BRST operator is defined by the part of the BV action linear

in the antifields:

QBRST = Qµ
∂

∂φµ
+Qνρ∗(Aν)ab w̃

b ∂

∂w̃a
(5.34)

After this canonical transformation of eqs. (5.28), (5.29) and (5.30), the new Scl is such

that this QBRST is nilpotent on-shell.

Gluing together Φξ: let us consider the relation between the functions Φ̂ξ defined by

eq. (5.6) on two overlapping charts. It is enough to consider the case of infinitesimal

transition function, i.e. uji = 1 + εαji, where ε is infinitesimally small. With Fα defined in

eq. (5.19), the difference between Φ̂ξ on two coordinate charts is:

δjiΦ̂ξ = {{Stot, Fαji}, Φ̂ξ} = −{{Stot, Φ̂ξ}, Fαji}+ {Stot, {Fαji , Φ̂ξ}} (5.35)

The first term on the r.h.s. is zero:

{{Stot, Φ̂ξ}, Fαji} = 0 (5.36)

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since Fα is diffeomorphism-invariant. Let us study the second term. We have:

{Fαji , Φ̂ξ} = {Fαji ,Φξ} =
1

2
waωabΦ

µ
ξ

∂

∂φµ
(αji)

b
cw

c

= − 1

2

(
waωabΦ

µ
ξ (δjiAµ − [αji, Aµ])bcw

c
)

= − FΦµξ δjiAµ
+

1

2
waωabΦ

µ
ξ [αji, Aµ]bcw

c

= − FΦµξ δjiAµ
− {{Stot, Fαji}, FΦµAµ} (5.37)

where Aµ is any connection, transforming as in eq. (5.25). Therefore the following

expression:

Φ̂′ξ = Φ̂ξ + {Stot, FΦµξAµ
} (5.38)

is consistent on intersections of patches.

The correcting term {Stot, FΦµξAµ
} is the infinitesimal gauge transformation (see

eqs. (5.18) and (5.19)) with the parameter Φµ
ξAµ.

Back to AdS5 ×S5: in our case B is the pure spinor bundle over super-AdS5×S5; the

coordinates φ are functions from the worldsheet to PS AdS5 × S5 (defined in eq. (3.8)).

The total space E is the space of maps from the worldsheet to CL × CR × PSU(2, 2|4).

Notice that CL × CR × PSU(2, 2|4) is a principal H-bundle over PS AdS5 × S5. It has a

natural PSU(2, 2|4)-invariant connection, which for every tangent vector:

(λ̇L, λ̇R, ġ) ∈ T (CL × CR × PSU(2, 2|4)) (5.39)

declares its vertical component to be (ġg−1)0̄, i.e. the projection of ġ on the denominator

of (3.3) using the Killing metric. This defines, pointwise, the connection on the space

of maps.

It is natural to use this connection as Aµ in eq. (5.38).

Notice that we do not need a connection to write the BV Master Action (eq. (5.4)). But

the connection is needed to construct Φ̂′ξ (and also in our construction of the Lagrangian

submanifold).

6 Taking apart the AdS sigma model

The standard action given by eq. (3.11) depends on the worldsheet complex structure and

is polynomial in the pure spinor variables. In the BV formalism, it corresponds to a specific

choice of the Lagrangian submanifold. We can change the action to a physically equivalent

one, by adding BRST quartets and/or deforming the Lagrangian submanifold. We can ask

ourselves, what is the simplest formulation of the theory, in the BV language, preserving

the symmetries of AdS5 × S5? (Of course, the notion of “being the simplest” is somewhat

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subjective.) In this paper we gave an example of such a “minimalistic” formulation:

SBV = S(g,λ) + S(ω)

=

∫
STr (J1 ∧ (1−P31)J3 − J1 ∧P31J3)

+

∫
STr (λ3L

?
1 + λ1L

?
3) +

∫
1

2
STr (ω?3 ∧ ω?1) (6.1)

Here L? are the BV Hamiltonians of the left shift, eq. (4.1). The relation of eq. (6.1) to

the original BV action (3.37) is through adding BRST quartet (section 5) and canonical

transformations (eqs. (3.24), (5.7), (5.10)). Subjectively, eq. (6.1) is the simplest way of

presenting the worldsheet Master Action for AdS5 × S5.

The Master Action (6.1) does not depend on the worldsheet metric. The dependence

on the worldsheet metric (through the complex structure) comes later when we choose the

Lagrangian submanifold.

The way eq. (6.1) is written, it seems that w is completely decoupled from g and λ.

But the transition functions on overlapping charts, described in section 5.4, do mix the

two sets of fields.

The Master Action (6.1) is non-polynomial in λ, because of P31.

7 Generalization

Consider a sigma-model whose target space is some supermanifold X . Suppose that X is

equipped with a nilpotent odd vector field Q ∈ Vect(X ), generating a gauge symmetry of

the sigma-model. In minimalistic sigma-models the BRST operator is just an odd nilpotent

vector field on the target space.

This means that the field configuration X(σ, τ) has the same action as eε(σ,τ)QX(σ, τ)

for an arbitrary odd gauge parameter function ε on the worldsheet:

S[X] = S[eεQX] (7.1)

Locally and away from the fixed points of Q this implies that one of the target space

fermionic coordinates completely decouples from the action (the action does not depend

on it). In case of pure spinor sigma-model, this gauge symmetry does not account for all

degeneracy of the action. All directions in the θ space tangent to the pure spinor cones are

degenerate directions of the quadratic part of the action.

Let us add an additional scalar field on the worldsheet Λ(σ, τ) and consider the fol-

lowing solution of the Master Equation:

SBV = S +

∫
ΛQA(X)X?

A (7.2)

In the pure spinor case X is parametrized by g ∈ PSU(2, 2|4) and λL, λR modulo rescaling

(i.e. projective pure spinors).

In Type II pure spinor theory, there are actually two anticommuting BRST symmetries,

QL and QR, and the term in SBV linear in antifields is∫
ΛLQ

A
L(X)X?

A + ΛRQ
A
R(X)X?

A (7.3)

The action S is given by eq. (3.20). Such a theory requires regularization.

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The minimalistic sigma-model action is written in terms of the target space metric G

and the B-field B. For example, the action of eq. (3.20) corresponds to:

G = Str

(
1

2
J2J2 + J1(1−P31)J3

)
(7.4)

B = Str

(
1

2
J1 ∧ J3 − J1 ∧P31J3

)
(7.5)

The existence of the b-ghost is equivalent to the metric being the Lie derivative along Q of

some symmetric tensor b:

G = LQb (7.6)

where LQ is the Lie derivative along the vector field Q. In our case (appendix B):

b =
Tr (({J3, λ3} − {J1, λ1})J2)

STr(λ3λ1)
(7.7)

As in section 3.4, the part of the action involving the target space metric G is BRST exact.

8 Open problems

We did not verify that Φ̂′ξ of eq. (5.38) satisfies the conditions14 formulated in [2, 3]. In

particular, we may hope for {Φ̂′ξ, Φ̂′ξ} = 0, but more complicated scenarios are also possible.

We believe that the invariance of our construction under the symmetries of AdS5 × S5 is

important to satisfy those conditions at the quantum level.

We did not explicitly calculate the restriction of the Φ̂′ξ to the standard family of

Lagrangian submanifolds, corresponding to the integration over the space of metrics. It

can probably be expressed in terms of O where ∂b = QO as calculated in [1]. In any case, it

is most likely nonzero, and therefore the string measure of eq. (1.1) is not just the product

of Beltrami differentials, but involves also the curvature terms Φ̂′F .

Acknowledgments

We want to thank Nathan Berkovits, Henrique Flores and Renann Lipinski for discussions

and comments. This work was supported in part by the FAPESP grant 2014/18634-9

“Dualidade Gravitac, ão/Teoria de Gauge” and in part by the RFBR grant 15-01-99504

“String theory and integrable systems”.

A The projector

A.1 Definition

Let πA and πS denote the projectors:

πA : T (AdS5 × S5)→ T (AdS5) projector along T (S5) (A.1)

πS : T (AdS5 × S5)→ T (S5) projector along T (AdS5) (A.2)

πA(v) + πS(v) = v (A.3)

14http://andreimikhailov.com/math/bv/omega/Equivariant Form.html

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For any vector v, we will denote by v the difference of its AdS5 and S5 components:

v
def
= πA(v)− πS(v) (A.4)

The projector P13 : g1 → g1 was defined in [10] as follows:

P13A1 = A1 + [S2, λ3] (A.5)

{λ1 , P13A1} = 0 (A.6)

where S2 ∈ g2 is adjusted to satisfy (A.6). In fact P13 is the projection to the tangent

space TCR along the space T⊥CL which is orthogonal to TCL with respect to the metric

defined by Str:

(1−P13)A1 ∈ T⊥CL (A.7)

In other words, for generic λ3 and λ1 we have an exact sequence:

0 −→ T⊥CL
i−→ g1

P13−→ TCR −→ 0 (A.8)

In section A.3 we will give an explicit formula for P13 following [1].

A.2 Matrix language

It turns out that computations can often be streamlined by thinking about elements of g

literally as 4|4-matrices. In fact g is a factorspace of sl(4|4) modulo a subspace generated

by the unit matrix. Therefore, when talking about a matrix corresponding to an element

of g, we have to explain every time how we choose a representative. The Z4 grading

of psl(4|4) can be extended to sl(4|4); the unit matrix has grade two. Therefore, the

ambiguity of adding a unit matrix only arises for representing elements of g2. To deal with

this problem, we introduce some notations. Given a matrix X of grade two, we denote by

XTL the corresponding traceless matrix:

XTL = X − Tr(X)

8
1 (A.9)

(The subscript “TL” is an abbreviation for “traceless”.) Also, it is often useful to consider

4|4-matrices with nonzero supertrace. Such matrices do not correspond to any elements of

g. For a 4|4-matrix Y we define:

YSTL = Y − STr(Y )

8
Σ (A.10)

where Σ = diag(1, 1, 1, 1,−1,−1,−1,−1) (A.11)

In particular:

(YTL)STL = (YSTL)TL = Y − Tr(Y )

8
1− STr(Y )

8
Σ (A.12)

We also define, for any even matrix Y :

Y = Y Σ = ΣY (A.13)

This definition agrees with eq. (A.4).

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A.3 Explicit formula for the projector

In fact S2 is given by the following expression:

S2 =
2

Str(λ1λ3)
{λ1, A1}STL (A.14)

Notice that {λ1, A1}STL is actually both super-traceless and traceless; it is the same as

{λ1, A1}TL (with the overline extending over “TL”). We have to prove that the S2 defined

this way satisfies (A.6). Indeed, we have:

[S2, λ3] =
2

Str(λ1λ3)

[
{λ1, A1}STL , λ3

]
(A.15)

and we have to prove eq. (A.6). We have:{
λ1 ,

[
{λ1, A1}STL , λ3

]}
= {λ1 , [Σ{λ1, A1}TL , λ3]}

= − Σ [λ1 , {{λ1, A1}TL, λ3}]
= − Σ {[λ1, λ3] , {λ1, A1}TL} (A.16)

Both {λ1, A1} and [λ1, λ3] have Z4-grading two. Let us use:

[λ1, λ3] =
1

4
Str(λ1λ3)Σ + [λ1, λ3]STL (A.17)

For all grade 2̄ matrices A2 and B2 such that TrA2 = TrB2 = STrA2 = StrB2 = 0 the

following identity holds:15

{A2, B2} = A2B2 +B2A2 =
1

4
(Str(A2B2)Σ + Tr(A2B2)1) (A.18)

Therefore: {
λ1 ,

[
{λ1, A1} , λ3

]}
mod 1 = −1

2
Str(λ1λ3){λ1, A1}TL (A.19)

(where “mod1” means “modulo the center of psl(4|4)”, i.e. up to a multiple of the unit

matrix). This proves (A.6).

The central part of {λ1 , [{λ1, A1} , λ3]} is generally speaking nonzero:

Tr
{
λ1 ,

[
{λ1, A1} , λ3

]}
= 2Tr

(
λ1

[
{λ1, A1} , λ3

])
= − 2Tr ([λ1, λ3]STLΣ{λ1, A1})
= − 2STr ([λ1, λ3]STL {λ1, A1}) (A.20)

In Γ-matrix notations, [λ1, λ3]STL is (λ1,Γ
m
λ3) and {λ1, A1} is (λ1,Γ

mA1).

15http://andreimikhailov.com/math/pure-spinor-formalism/AdS5xS5/Symmetries.html

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Let us define (cp. eq. (A.14)):

S2/1 : g1 → g2

S2/1A1 =
2

Str(λ1λ3)
{λ1, A1}STL (A.21)

S2/3 : g3 → g2

S2/3A3 =
2

Str(λ3λ1)
{λ3, A3}STL (A.22)

so that:

P13A1 = A1 + [S2/1A1, λ3] (A.23)

P31A3 = A3 + [S2/3A3, λ1] (A.24)

A.4 Properties of P13 and P31

It follows from the definition, that for any v2 ∈ g2̄ we have

P13[v2, λ3] = 0 (A.25)

Let us verify this explicitly using the definition (A.5) with the explicit expression for S2

given by (A.14). We have:

P13[v2, λ3] = [v2, λ3] +
2

Str(λ1λ3)

[
{λ1, [v2, λ3]} , λ3

]
(A.26)

Consider the expression [ {λ1, [v2, λ3]} , λ3 ]:[
{λ1, [v2, λ3]} , λ3

]
= [ Σ{λ1, [v2, λ3]} , λ3 ] (A.27)

= − [ Σ{[λ1, λ3], v2} , λ3 ] + [ Σ[{λ1, v2}, λ3] , λ3 ] (A.28)

Let us consider the first expression on the r.h.s. of (A.27). Using (A.17) we rewrite:

− [ Σ{[λ1, λ3], v2} , λ3 ] =− 1

4
Str(λ1λ3)[Σ{Σ, v2} , λ3] = −1

2
Str(λ1λ3) [v2, λ3] (A.29)

This cancels with the first term on the r.h.s. of (A.26). And the second expression on the

r.h.s. of (A.27) is zero:

[ Σ[{λ1, v2}, λ3] , λ3] = [ {Σ{λ1, v2}, λ3} , λ3 ] = 0 (A.30)

A.5 Subspaces of g associated to pure spinors

Consider the decomposition:

g2 = g2L ⊕ g2R ⊕C[λ3, λ1]STL ⊕C[λ3, λ1]TL (A.31)

Here g2L is a 4-dimensional subspace Tr-orthogonal to C[λ3, λ1]TL and commuting with

λ3, and g2R is Tr-orthogonal to C[λ3, λ1]TL and commuting with λ1.

Similarly we can refine T⊥CR and T⊥CL:

g3 ⊃ T⊥CR = [g2L, λ1]⊕C[[λ3, λ1]STL, λ1] (A.32)

g1 ⊃ T⊥CL = [g2R, λ3]⊕C[[λ3, λ1]STL, λ3] (A.33)

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B BRST variation of the b-tensor

Here we will prove:

(QL +QR)
Tr{J1, λ1}J2

Str(λ3λ1)
= −Str

(
1

4
J2J2 +

1

2
J1(1−P31)J3

)
(B.1)

(Remember that Tr . . . = Str(. . .Σ).) In fact, only QL contributes; the action of QR
is zero:

QR Str ({J1, λ1}J2Σ) = −Str ({J1, λ1}{J1, λ1}TL) = 0 (B.2)

because J1 is a fermion. Let us compute the action of QL:

QL Str ({J1, λ1}J2Σ) = − Str (({[J2, λ3], λ1}J2 + {J1, λ1}{J3, λ3}TL) Σ)

= − Tr ({J2, [J2, λ3]}λ1) + Str
(
{J1, λ1}STL{J3, λ3}

)
= − 1

4
Str(λ3λ1)Str(J2

2 ) + Str
(
J3

[
{J1, λ1}STL, λ3

])
= − 1

4
Str(λ3λ1)Str(J2

2 )− 1

2
Str(λ3λ1)Str (J3(1−P13)J1) (B.3)

Open Access. This article is distributed under the terms of the Creative Commons

Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in

any medium, provided the original author(s) and source are credited.

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	Introduction
	Master Actions quadratic-linear in antifields
	Pure spinor superstring in AdS(5) x S**5
	Notations
	Standard action
	New action
	The b-ghost
	Gauge fixing SO(1,4) x SO(5)
	In BV language

	Action of diffeomorphisms
	Formulation of the problem
	Subspaces associated to a pair of pure spinors
	Construction of Phi(xi)

	Regularization
	Adding more fields
	A canonical transformation
	Constraint surface and its conormal bundle
	Gluing charts

	Taking apart the AdS sigma model
	Generalization
	Open problems
	The projector
	Definition
	Matrix language
	Explicit formula for the projector
	Properties of P(13) and P(31)
	Subspaces of g associated to pure spinors

	BRST variation of the b-tensor