Master symmetry in the AdS 5 × S 5 pure spinor string
Master symmetry in the
Open Access 0 1 2 4
c The Authors. 0 1 2 4
0 Texas A&M University
1 Diagonal Las Torres 2640, Pen ̃alol ́en , Santiago , Chile
2 Facultad de Ingenier ́ıa y Ciencias, Universidad Adolfo Ib ́an ̃ez
3 Departamento de Ciencias F ́ısicas, Universidad Andres Bello
4 Sazie 2212 , Santiago , Chile
We lift the set of classical non-local symmetries recently studied by Klose, Loebbert, and Mu¨nkler in the context of Z2 cosets to the pure spinor description of the superstring in the AdS5 × S5 background.
AdS-CFT Correspondence; Conformal Field Models in String Theory
pure spinor string
1 Introduction 2 3 4
Pure spinor string in AdS5 × S
The flat current
Conclusions and prospects
Finding and studying integrable structures in the context of the AdS/CFT correspondence
has been one of the most active areas of research in high energy physics. The theories
on both sides of the conjecture enjoy a large number of symmetries that make it possible
to obtain impressive results and checks of the conjecture. Although it lacks more recent
updates, a good review with an extensive list of references is . A more recent development
not covered in  is the the Quantum Spectral Curve method [2, 3]. For some of its
applications, including higher loop computations, see [4–9].
In the famous work of Bena, Polchinski, and Roiban , it was shown that the
GreenSchwarz superstring in AdS5 × S5  has an infinite set of classical conserved currents.
The existence of an analogous set of currents in the context of the AdS5 × S5 pure spinor
superstring was demonstrated in reference . Since this string is a generalization of the
usual Z2 coset to a super-coset with Z4 symmetry, the ability to lift this symmetry to the
super-coset is non-trivial. In this note, we go one step further and show that the pure spinor
Loebbert, and Mu¨nkler . This symmetry complements the Yangian symmetry, acting as
a raising operator on the classical Yangian charges. The master symmetry is not essentially
new, however it provides a unifying picture containing all local and non-local symmetries
of a coset model. In particular, it is interesting that the conserved charge associated with
to be of practical use, for example, in applications to the study of supersymmetry Wilson
loops in the AdS5 × S5 super-coset.
The work presented here extends this structure to its super-analogue, specifically, to
the Z4 super-coset description of the AdS5 × S5 pure spinor string. In a sense the ghosts
present in the pure spinor string make the Z4 symmetry manifest with the ghosts’ Lorentz
current playing the role of a gauge covariant current with vanishing Z4 charge.
The classical and quantum integrability of the string in this background has been
explored much more for the GS string (see e.g. ) than for the pure spinor version. Some
interesting results concerning the classical and quantum integrability in the pure spinor
formalism are given in references [15–18]. A possible application of integrability techniques
to the quantum pure spinor string is to study its worldsheet dilatation operator . It
has been shown that semi-classical computations in the pure spinor string give the same
results as the GS string for a set of classical solutions [20, 21], but very little is known
about solutions dual to Wilson loops. This is an interesting line of research to which the
results presented in this work may have suitable applications.
This paper is organized as follows: in section 2, we give a short review of the pure
spinor string in AdS5 × S5 including its flat current using a notation that will be useful in
the subsequent sections. In section 3, we extend the master symmetry discussed in  to
the pure spinor string. In section 4, we derive how the existence of the first Yangian charge
is a consequence of the master symmetry and the global psu(2, 2|4) symmetry. We then
give a general derivation of all higher non-local and non-abelian charges the superstring
has. We conclude the paper and discuss directions for future research in section 5.
Pure spinor string in AdS5 × S
The pure spinor string in the AdS5 × S5 background is described in terms of the
superg = Li3=0 gi with the projections satisfying
h := g0
m := M gi.
J = g−1dg = K + A,
dJ + J ∧ J = 0
∇ = d + [A, · ]. The Maurer-Cartan identity
will also decompose into four independent identities along each gi.
current and its dz component can be understood from context.
The Killing form Str(·) also respects this symmetry in the sense that
[gi, gj ] ⊂ gi+j mod 4.
Str(gigj ) 6= 0 iff
i + j = 0 mod 4.
This is a Z4 generalization of the usual Z2 symmetry present in any symmetric space and,
in particular, in the bosonic coset construction. For comparison and general convenience,
we will define
They suffer the gauge transformations
where A and B are any two local bosonic elements of g2. We will also define
which are the Lorentz generators for the ghosts. Note that they have zero Z4 charge. The
pure spinor condition implies
In addition to the geometric part, the pure spinor string is defined with pure spinor
ghosts and their conjugate momenta. These are invariant under global PSU(2, 2|4)
transformations. The ghosts are fermionic elements of the algebra
This is the coset generalization of the pure spinor condition in flat space. The momenta
conjugate to the pure spinor variables are denoted
Having all the ingredients, we can write the pure spinor action [22–24]
1 Z d2z Str K1K¯3 + 2K2K¯2 + 3K3K¯1 − 4N N¯ + 4ω∇¯ λ + 4ω¯∇λ¯ .
The geometric part of this action is the standard kinetic term of a coset model plus a
SWZ = −
1 Z d2z Str K1K¯3 − K3K¯1 .
This particular coefficient of the Wess-Zumino term is fundamental for BRST symmetry
and integrability [12, 16].
By construction, the action has global PSU(2, 2|4) invariance and local SO(1, 4) ×
SO(5) invariance. Global transformations act on g by left multiplication and the local
transformations act on g by right multiplication. The current J is invariant under the
global symmetry. On the other hand, K tranforms in the adjoint representation of h if
momenta transform in the adjoint representation of h as well.
The next fundamental symmetry is BRST invariance defined by2
δg = g(λ + λ¯), δλ = 0, δλ¯ = 0, δω = −K3, δω¯ = −K¯1.
2These transformations are nilpotent only up to local SO(1, 4) × SO(5) transformations and equations
of motion. There are ways to fix both these issues [25–28], however, they will not be needed here.
The conserved current associated with BRST symmetry is given by
the charges defined by
after using the equations of motion which will be discussed below. This fact means that
also generate symmetries for any two independent holomorphic and anti-holomorphic
functions ǫ(z) and ǫ¯(z¯). In this case the BRST transformations above generalize to3
We now compute the current associated with the global PSU(2, 2|4) symmetry. The
Noether method. The left invariant currents transform as
When inserting this transformation into the action, we can drop the restriction on the
subspaces since the transformations will always come together with a dual algebra element
inside a supertrace. The action transforms as
1 Z d2z Str g−1∂ΩgK¯3 + g−1∂¯ΩgK1 + · · · + 4g−1∂ΩgN¯ + 4g−1∂¯ΩgN ,
from which we read off the Noether current
These equations are calculated by varying the action (2.12) with respect to a variation
4 decomposition, the
Maurer-Cartan identity for J , and
∇N¯ − ∇¯ N − 2[N, N¯ ] = 0,
3It may seem surprising that the BRST invariance in the pure spinor superstring implies a much larger
symmetry than the usual BRST symmetry in field theory. However, we should remember that the pure
spinor BRST should also imply Virasoro symmetry which is an infinite-dimensional chiral symmetry.
we can separate (2.22) into eight equations of motion. To derive this last equation we use
the equations of motion for the ghosts coming from (2.12):
Then, the action can be written as
and the components of the Noether current can be written as
1 K¯ Σ(K) + ω∇¯ λ + ω¯∇λ¯ − N N¯ ,
jz = gΣ(K + N )g−1, jz¯ = gΣ(K¯ + N¯ )g−1.
For the supertrace, we have
it does not preserve the Lie bracket).
The flat current
In contrast to the Noether current of the bosonic cosets, the conserved current (2.21) of
the Z4 super-coset is not flat. Instead, it was shown in reference  that the pure spinor
string in AdS5 × S5 has a family of flat currents depending on a complex parameter µ :
Lµ = lµ dz + ¯lµ dz¯,
The current is flat
as a consequence of the equations of motion.
dLµ + Lµ ∧ Lµ = 0
and lµ = g e−µ Σ − 1 (J¯ + N¯ )g−1.
The existence of this current is remarkable given that, as just mentioned, the conserved
current of a Z4 super-coset is generally not flat. In particular, there is no value of µ for
which the flat current (2.30) reduces to (2.21). However, note that
L′0 = ∗j,
the first charge generated by the flat current Lµ is the conserved charge of the psu(2, 2|4)
algebra. This can be seen from the monodromy matrix4
The charge of the psu(2, 2|4) algebra
M (µ ) = P exp
is the coefficient of the first power of µ in the expansion of M (µ ).
Maurer-Cartan current by
Following Klose, Loebbert, and Mu¨nkler , we can define a flat deformation of the
+ A¯ + e−3µ K¯1 + e−2µ K¯2 + e−µ K¯3 + (e−4µ − 1)N¯ dz¯. (3.1)
can be written as
Lµ = eµ Σ(J + N ) − N dz + e−µ Σ(J¯ + N¯ ) − N¯ dz¯.
(It is actually more straightforward to verify that Lµ satifies a flatness condition.)
A deformation gµ of the coset element g can be defined by the differential
equadgµ (z, z¯) = gµ (z, z¯)Lµ
with gµ (z0, z¯0) = g(z0, z¯0).
Here, (z0, z¯0) is any reference point on the worldsheet needed to fix an “initial condition”.
This equation is well-defined since Lµ is flat. Consequently, this deformation of g is only
defined on-shell. An ansatz to solve it is
4For simplicity and to avoid global subtleties, we will assume the worldsheet is infinite and open. This
is not essential to any of the results discussed below which are classical. It would be interesting to check
that finite size effects do not spoil our conclusions, but we expect they do not as Berkovits has shown that
the Yangian symmetries are preserved even when such quantum corrections are taken into account .
Again, this equation is well defined since Lµ is flat. We can expand this differential equation
of motion are satisfied. The solution is given by
With this, we are finally in the position to define the “master symmetry” 
(dzjz − dz¯jz¯).
This is a non-local transformation acting on the currents as
δˆJ = g−1dχ(1)g = δˆK2 + δˆK1 + δˆK3 + δˆA = g−1(jzdz − jz¯dz¯)g =
= (K1 + 2K2 + 3K3 + 4N ) dz − K¯3 + 2K¯2 + 3K¯1 + 4N¯ dz¯
Up until this point, the master symmetry has been discussed at the level of geometry.
Since the structure under discussion is on-shell, we will consider N and N¯ as fundamental
will define the extension of the master symmetry to act on them as
With these transformations, it is immediate to verify that (3.9) is a symmetry of
the equations of motion (2.22), turning it into a Maurer-Cartan identity together with
symmetries of a Z2 coset. The analogous statement for the pure spinor string for any value
of µ will be proved at the end of this paper. The main difference here, apart from the
presence of fermionic terms, is that the gauge field A transforms into the ghost current N ,
thereby mixing matter and ghosts.
Since this symmetry is only defined on-shell, discussing it at the level of the action
is potentially meaningless. Nevertheless, we follow reference  and try to use Noether
procedure to calculate the current associated to the master symmetry anyway. We include
a local parameter ǫ(z, z¯) in the transformation of g
The currents transform as
δˆJ = dǫg−1χ(1)g + ǫΣ (K + N ) dz − ǫΣ K¯ + N¯ dz¯.
Inserting this transformation into the action and only collecting terms depending on dǫ,
1 Z d2z Str ∂ǫχ(1)jz¯ + ∂¯ǫχ(1)jz .
Using the same normalization as (2.21), we read off the conserved current
Note that by (3.6),
Using this, we can perform the integral to find the conserved charge
use that the psu(2, 2|4) charge is given by
∗ J (0) =
∗J (0) =
is the Casimir of the psu(2, 2|4) algebra, as in .
In principle we could find the higher scalar charges associated with higher powers of
in (3.14) suggests that the scalar current containing all higher master symmetry charges
element of the algebra, it is not conserved, and its µ 2 coefficient does not match the Casimir
with respect to µ of the flatness condition (2.33). Then, the complete tower of nonlocal
charges can be defined by
The zeroth power of µ in the expansion gives the Casimir C(0), and it is easy to show that
δˆj = j(1) = g(K1 + 4K2 + 9K3 + 16N )g−1dz
which is the first non-local current given by the monodromy matrix (2.35) constructed
from the flat current (2.30). Schematically, this current generates a transformation on the
use the Noether method again to see if we can obtain the non-local current as the Noether
current associated with this transformation. However, carrying this out, we obtain only
the last term in (4.1). As mentioned previously, it is not surprising that this time we could
not obtain the desired result since this is an on-shell symmetry. In the case of the principal
chiral model it is possible to interpret these non-local currents as Noether currents [32, 33],
but it is not clear that we can use the same method here.
We could obtain the higher non-local currents by successive applications of the master
The non-local current associated with the global symmetry and all higher Yangian charges
is calculated by replacing g → gµ (3.3), (3.4) in the definition of the Noether current (2.21)
and defining a finite µ deformation of ghost currents as
N → Nµ = e4µ N,
N¯ → N¯µ = e−4µ N¯ ,
from which we can see that the master symmetry (3.10) corresponds to the first power of
the µ deformation. With these definitions we have
Jµ = gµ ((Kµ )1 + 2(Kµ )2 + 3(Kµ )3 + 4Nµ )gµ−1dz
+ gµ (3(K¯µ )1 + 2(K¯µ )2 + (K¯µ )3 + 4N¯µ )gµ−1dz¯
= χµ g(eµ K1 + 2e2µ K2 + 3e3µ K3 + 4e4µ N )g−1χu−1dz
+ χµ g(3e−3µ K¯1 + 2e−2µ K¯2 + e−µ K¯3 + 4e−4µ N¯ )g−1χu−1dz¯,
identify this current as
− ∂¯lµ′ + lµ , −¯lµ′
∂¯Jµ = χµ g[∇¯ + K¯ , 2e2µ K2 + eµ K1 + 3e3µ K3 + 4e4µ N ]g−1
+ [¯lµ , g(2e2µ K2 + eµ K1 + 3e3µ K3 + 4e4µ N )g−1] χµ−1
∂J¯µ = χµ g[∇ + K, 2e−2µ K¯2 + 3e−3µ K¯1 + e−µ K¯3 + 4e−4µ N¯ ]g−1
+ [lµ , g(2e−2µ K¯2 + 3e−3µ K¯1 + e−µ K¯3 + 4e−4µ N¯ )g−1] χµ−1
So, the conservation of Jµ is simply the first derivative with respect to µ of the flatness
condition (2.33) of Lµ :
∂¯Jµ + ∂J¯µ = χµ ∂¯lµ′ − ∂¯lµ′ + ¯lµ , lµ′
This relation proves that if g is a solution, then the deformation gµ is also a solution. The
current Jµ contains a whole tower of non-local conserved currents of the model, starting
with the global psu(2, 2|4) current (2.4). It is easily checked that
where j(1) is the first Yangian current. To prove the higher µ powers are all higher Yangian
Jµ = j + µ j(1) + · · · ,
After some manipulations, one can show that
To get to final result let us now compute the derivative with respect to µ of Jµ
χµ−1δˆJµ χµ = ∗Lµ′′ + [χµ−1δˆχµ , ∗Lµ′ ] + [χ(1), Lµ′ ].
χµ−1Jµ′ χµ = ∗Lµ′′ + [χµ−1χµ′ , ∗Lµ′ ].
If we subtract both equations we have
χµ−1 δˆJµ − Jµ′ χµ = [χµ−1δˆχµ − χµ−1χµ′ , ∗Lµ′ ] + [χ(1), ∗Lµ′ ].
Let us call φµ := χµ−1δˆχµ − χµ−1χµ′ and note that φ0 = −χ(1). Using that
it can be shown that the non-local Casimir (3.20) satisfies
Conclusions and prospects
We have shown that the classical pure spinor string in the AdS5 × S5 background has
the full set of classical non-local symmetries extending those recently studied by Klose,
Loebbert and Mu¨nkler in the context of Z2 cosets . We find that the inclusion of ghosts
in a sense makes the Z4 symmetry manifest, and all non-local symmetries can be lifted to
the super-coset PSU(2, 2|4)/SO(1, 4) × SO(5).
An immediate extension of the results of this paper is to derive the analog for the
GreenSchwarz superstring. That can be done by erasing the ghosts and imposing an appropriate
gauge choice. Classical solutions of the pure spinor string should preserve BRST symmetry
which means the BRST charge should vanish when evaluated on the solution. If we do
not set the ghosts to zero, this means that the currents K3 and K¯1 should vanish. In
reference , it was shown that the pure spinor flat current is equivalent to the one in the
Green-Schwarz formalism  in this gauge. We expect that the GS string enjoys all of
the symmetries discussed in the present work.
It would be interesting to apply the results of this paper to supersymmetric Wilson
loops in AdS5 as in reference . However, it is as yet not known how to study such
classical solutions in the pure spinor formalism. There is hope such a task can be done,
since it was shown by explicit computations in references [20, 21] that the semi-classical
quantization of the pure spinor string is equivalent to the Green-Schwarz string in a certain
class of solutions. In  it was argued that the equivalence holds for any physical solution.
With these results in mind, it is likely that one can extend the results of, for example,
references [35, 36] to the pure spinor string.
A more speculative line of research is the relevance of the master symmetry in the
quantum theory. Since the Yangian currents are still conserved at quantum level  it is
possible that there is some quantum version of the master symmetry. However we cannot
say if it will provide any additional help in achieving an exact solution of the model.
The work of Oc and Bcv is partially supported by FONDECYT grant number 1151409.
Bcv also has partial support from CONICYT grant number DPI20140115. Wdl3 is
supported by NSF grants PHY-1214333 and PHY-1521099.
This article is distributed under the terms of the Creative Commons
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any medium, provided the original author(s) and source are credited.
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