#### Yang-Baxter deformations beyond coset spaces (a slick way to do TsT)

Revised: May
Yang-Baxter deformations beyond coset spaces (a slick way to do TsT)
I. Bakhmatov 0 1 4 5 7 8 9 10
E. O Colgain 0 2 4 5 7 8 9 10
M.M. Sheikh-Jabbari 3 4 5 7 8 9 10
H. Yavartanoo 4 5 6 7 8 9 10
0 Asia Paci c Center for Theoretical Physics
1 Institute of Physics, Kazan Federal University
2 Physics Department , Postech
3 School of Physics, Institute for Research in Fundamental Sciences , IPM
4 Chinese Academy of Sciences , Beijing 100190 , China
5 P. O. Box 19395-5531, Tehran , Iran
6 State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics
7 geometries , such as AdS
8 Postech , Pohang 37673 , Korea
9 Pohang 37673 , Korea
10 Kremlevskaya 16a , 420111, Kazan , Russia
Yang-Baxter string sigma-models provide a systematic way to deform coset shown that the Yang-Baxter deformation in target space is simply an open-closed string map that can be de ned for any geometry, not just coset spaces. Given a geometry with an isometry group and a bivector that is assumed to be a linear combination of antisymmetric products of Killing vectors, we show the equations of motion of (generalized) supergravity reduce to the Classical Yang-Baxter Equation associated with the isometry group, proving the statement made in [1]. These results bring us closer to the proof of the \YB solution generating technique" for (generalized) supergravity advertised in [1] and in particular provide an economical way to perform TsT transformations.
Classical Theories of Gravity; Integrable Field Theories; Supergravity Models
1 Introduction
Preliminaries Perturbative analysis 2 3
4
3.1
3.2
4.1
4.2
4.3
Examples 5
Discussion
Review of generalized supergravity
expansion
Flat spacetime
Bianchi III
Lunin-Maldacena-Frolov
A Consistency of the generalized supergravity eld equations B
Details of the perturbative analysis
B.1 Zeroth order
B.2 First order
B.3 Second order B.4 Third order equations
through, we now understand noncommutative [7{9] and marginal deformations [10, 11] of
AdS/CFT geometries in a new light: they are part of a larger family of integrable
deformations of AdSp
Sp geometries, where the deformation is given by an r-matrix solution
to the Classical Yang-Baxter Equation (CYBE). One exciting outcome of this research line
has been the observation that the deformed geometries may not be consistent string theory
backgrounds in the usual sense; they are not solutions of usual supergravity.
Nonetheless, it has been noted that they are solutions to the generalized supergravity equations
of motion [12, 13], which di er from usual supergravity through an additional Killing
vector I. Furthermore, exploring the connection to TsT transformations [14, 15], established
in [16], it was conjectured [17], and later proved [18, 19] that homogeneous Yang-Baxter
deformations are equivalent to non-Abelian duality transformations.1
1See [20{80] for a sample of related developments.
{ 1 {
A subsequent simpler proposal surfaced in [84{86], where it was shown that the
closedopen string map, which is an extension of the map between open string and closed string
frames initially introduced by Seiberg & Witten [87] (see also [88]), undoes all known
Yang-Baxter deformations. The obvious implication of this nding is that Yang-Baxter
models are really the open-closed string map in disguise. While this observation may have
various bearings for open or closed string theories on these backgrounds, in this work we
focus on the backgrounds themselves and explore a potentially powerful solution generating
technique for (generalized) supergravity. For backgrounds where the Killing vector I is not
sourced, our methods constitute a supergravity solution generating technique.
More concretely, given a spacetime metric G
with an isometry group and
an
de ned through the map:
applied to explicit coset and non-coset geometries alike.2 In this paper, we move beyond
examples and towards a general proof of the statements in [
1
].
Before proceeding, let us brie y take stock. The map (1.1) may look like a
straightforward generalisation of the Yang-Baxter -model to general spacetimes, but there is a
palpable di erence in philosophy. In the traditional Yang-Baxter
-model narrative of
Klimcik [3, 4], the r-matrix solution to the CYBE is an input and this is the magic
ingredient that guarantees integrability of the deformed coset -model. Here, we relax this input
and adopt the milder assumption that
is bi-Killing. Recalling that CYBE is an algebraic
equation on a given Lie algebra, this bi-Killing structure is well justi ed. To \geometrise"
the CYBE, it is hence natural to consider Killing vectors and the isometry algebra of a given
geometry. The coe cients of the bi-Killing antisymmetric bivector
are then a constant
skew-symmetric matrix r. As noted for explicit examples in [
1
], the dynamical equations
of motion (EOMs) of generalized supergravity then reduce to the purely algebraic CYBE
on r. Thus, the CYBE becomes the output. Moreover, the connection to integrability is
severed, since it is clear that even for non-integrable geometries3 the map (1.1) exists. Our
observation ultimately means that supergravity can be exploited to classify solutions to the
CYBE, providing a striking application of physics to a mathematics problem. Conversely,
our analysis provides a solution generating technique for (generalized) supergravity. We
start from any solution, construct a
from solutions to the CYBE associated with the
2In a series of papers [89{91], the same map has been embedded in DFT, where
= , but with a
3As remarked earlier for the geometry T 1;1 [92].
isometries of the background, then use this data and the above map (1.1) to construct the
deformed background, which is a solution to generalized supergravity. This method as we
demonstrate later provides a more economical way to perform TsT transformations.
As mentioned, the purpose of this current manuscript is to substantiate the claims
of [
1
] by moving beyond examples to generic spacetimes.
When working with solution
generating techniques in supergravity, it is the case that once one nails the NS sector,
the transformation of the RR sector can be pieced together.4 For this reason, we focus
purely on the NS sector of generalized supergravity, while extension to the full generalized
supergravity with inclusion of the RR sector is just an added technicality. Furthermore,
since the map is only de ned for geometries with vanishing NSNS two-form, we are forced to
restrict ourselves to geometries that are supported by a scalar dilaton , which guarantees
that they are not only Ricci- at but can be curved. While it is easy to invert (1.1) for
explicit solutions, such as AdS2
S2 and the Schwarzschild black hole [
1
], for arbitrary G
and
extracting g and B, so that one can check the EOMs, is challenging.
To overcome this di culty, we work perturbatively in the deformation parameter
about an arbitrary background G, which may be supported by a scalar dilaton
. This
allows us to expand g and B in
and substitute the expressions directly into the EOMs
of generalized supergravity [12]. However, it turns out that little progress can be made for
generic
, so we are forced to also assume that it is bi-Killing,
where Ki denote Killing vectors of the background and the constant coe cients are
skewsymmetric, rij =
rji. Doing so, we arrive at a number of results, which we have checked to
third order in . At rst order, we are able to prove that the Killing vector I of generalized
supergravity is the divergence of the bivector,
= rij Ki Kj ;
I = r
;
(1.2)
(1.3)
thus providing a proof of a relation identi ed earlier in [84, 85]. Since
is bi-Killing, it
should be noted that I is Killing by construction. Previously this relation was motivated
by the
-gauge symmetry of the NSNS two-form, B ! B + d , where
is an arbitrary
one-form [86]. At second order, we con rm that the dilaton and Einstein equation reduce
to the CYBE. Imposing the CYBE we nd that the third order equations are trivially
satis ed, which is consistent with the claim of [
1
] that the EOMs are equivalent to the
CYBE once
is bi-Killing. For explicit solutions, it is possible to go further and check
our claim to all orders.
The structure of this paper runs as follows. In section 2 we introduce the bi-Killing
CYBE. We also explain how the dilaton transforms. In section 3 we study the map (1.1)
perturbatively to third order in
, in the process proving (1.3) and demonstrating that
the CYBE emerges from the Einstein and dilaton EOM at second order, as well as the
4For T-duality and Yang-Baxter -models, it is well-known that the frame rotation manifests itself in a
Lorentz transformation on the ux bispinor [91, 93{95].
{ 3 {
B- eld EOM at third order. In section 4, we provide deformations of at spacetime,
Bianchi spacetimes and provide an example that includes the RR sector. In particular,
we con rm that the Lunin-Maldacena-Frolov geometries [10, 11] can be easily recovered
using the methods outlined in [
1
]. Thus, Yang-Baxter deformations provide a smart way to
perform TsT transformations and there is no need to resort to T-duality transformations.
For sadomasochists, gory details can be found in the appendix.
2
Preliminaries
In this section, we provide a setting for later calculations. We start with a description of
the bi-Killing structure of the bivector
in the open-closed string map (1.1). We recall
that we are considering generic spacetime metrics G
with an isometry group. From the
Killing vectors Ki, one can construct an antisymmetric product of Killing vectors,
=
rij
2
(Ki Kj
Ki Kj ) = rij Ki Kj ;
before antisymmetrising,
[
r
] = Ki Kj Kk (cl1l2irjl1 rkl2 + cl1l2
kril1 rjl2 + cl1l2j rkl1 ril2 ) = 0:
{ 4 {
where rij is a skew-symmetric matrix, rij =
rji, with constant coe cients. The above
ansatz is motivated by the fact that we are exploring possible connections between the
CYBE over the isometry algebra of a given solution and a class of deformations. Being
Killing vectors associated to an isometry group, Ki satisfy the commutation relation
[Ki; Kj ] = cij kKk;
where cij k denote the structure constants. One may recast this relation in terms of
components as,
Ki r K
j
Kj r K
i = cij kK ;
k
where it makes no di erence if one replaces the covariant derivatives with partial
derivatives, since the Christo el symbols cancel.
We recall that the map (1.1) appears in the string theory literature in the context of
noncommutativity in string theory [87], where
is the noncommutativity (NC) parameter.
In the open string setting, the endpoints of the open string parametrised by X coordinates
satisfy a commutation relation
[X ; X ] = i
(X):
For the above algebra to be consistent, the NC parameter should satisfy the Jacobi identity
r
+
r
+
r
= 0:
Using the bi-Killing structure of
, it is an easy exercise to show that the Jacobi
identity is equivalent to the CYBE. First we consider
r
= rij rkl Ki Kl Kj r K
k + Ki Kk Kj r K
l ;
(2.1)
(2.2)
(2.3)
(2.4)
(2.5)
(2.6)
(2.7)
Modulo the Killing vectors, the right hand side is the homogeneous CYBE5
See [89{91] for a similarly explicit derivation of the relation between the CYBE and the
vanishing of R- ux, essentially the Jacobi identity above. We also note that the left hand
side of (2.5) is also known as the Schouten bracket in the context of double eld theory
and general O(d; d) string theory compacti cations [89].
At this point, it is an opportune time to recall that r-matrix solutions to the CYBE
take the form
HJEP06(218)
cl1l2
[irjl1 rk]l2 = 0:
r =
2
1 rij Ti ^ Tj ;
(2.8)
(2.9)
where Ti are elements of the Lie algebra, Ti 2 g. It should now be clear that the bi-Killing
mimics the r-matrix. In other words, one can assume that
is the r-matrix
written in the basis of Killing vectors. This relation has been observed for all Yang-Baxter
deformations, even for r-matrix solutions to the modi ed CYBE [85] (see appendix B). We
note that for G=H coset spaces the Killing vectors are basically the same as the generators
of G which also provide a complete basis for expanding any tensor. In this sense, the
bi-Killing structure allows for
to have all possible components. Of course, the spacetime
dependence of
is still not xed by the bi-Killing assumption.
At this stage, we have introduced the bivector
, which plays a central role in our
map (1.1), and explained its bi-Killing structure. We have further demonstrated a
connection between the interpretation of
as an NC parameter, which is required to satisfy
the Jacobi identity, and its role as an r-matrix solution to the CYBE.
We recall that
NC deformations of eld theories are intimately connected to Drinfeld twists [96] of Lie
algebras, where the twist element is precisely an r-matrix solution to the CYBE [97{99].6
For this reason, it is expected that the Jacobi identity is the CYBE when the bivector
is
bi-Killing. Of course, it is more careful to state that the CYBE implies the Jacobi identity
since there may be solutions to the Jacobi identity that are not bi-Killing.
Moving on, we will now address the relation between the Killing vector I of generalized
supergravity [12] and the bivector
(1.3). In [85] it was checked that this relation holds for
a large class of solutions to generalized supergravity and we will prove it is an outcome of
the generalized supergravity EOMs in the next section. Here, using the bi-Killing structure,
we motivate this relation in a simple way. A short calculation reveals that
I = rij r (Ki Kj ) = rij Ki r K
j =
2
1 rij cij kK ;
k
(2.10)
where we have used the fact that rij is antisymmetric and the commutation relation (2.3).
Therefore, by construction I is a linear combination of Killing vectors with constant
coe cients and is hence guaranteed to be Killing. As a further check, we note that when
5We note that since (2.7) should hold for all points on spacetime, then it can be satis ed only if the
constant, spacetime independent piece vanishes.
6We thank Anca Tureanu for a discussion on this point.
{ 5 {
I = 0, so that the solution corresponds to a solution of usual supergravity, we recover the
unimodularity condition of [95],
In summary, given the fact that the relation (1.3) holds for a large number of explicit
solutions [85], it can be explained for D-brane geometries [86] and that it recovers a result
in the independent literature [95], this should put any doubts about the validity of (1.3) to
rest. That being said, we have yet to identify the Killing vector I with the Killing vector
appearing in the EOMs of generalized supergravity. This we will do in the next section.
Before proceeding to the next section, where we will study the EOMs of generalized
supergravity, it is important to address the transformation of the dilaton. At this stage, given
the original metric G, our map (1.1) and (1.3), the deformed metric g, NSNS two-form B
and Killing vector I are completely determined in terms of . As proposed originally in [86],
the usual T-duality density e 2 pg (valid for both Abelian and non-Abelian T-duality) is
invariant. With this assumption, given the metric G and scalar , the transformed dilaton
is
=
+
log
g
G
:
1
4
complete prescription including the RR sector can be found in [
1
].
3
Perturbative analysis
In this section, we will extract the CYBE from the EOMs of generalized supergravity. As
stated earlier, we restrict our attention to the NS sector on the basis that repeating the
calculations for the RR sector will not o er new insights. Indeed, since we are working
perturbatively, yet ultimately interested in exact solutions, we will fall short of our goal
of establishing the map (1.1) and dilaton transformation (2.12) as a solution generating
technique. Instead, we will expand in
around a generic background and enumerate the
conditions that should hold through third order in
so that a solution can exist. We will
see that all conditions, including the CYBE, follow once one assumes that
is bi-Killing.
As we have seen, the bi-Killing vector
is essentially the r-matrix. Since the CYBE is
quadratic in components of the r-matrix, it is reasonable to expect that the CYBE emerges
from the EOMs of generalized supergravity at second order in
. For this reason, in this
section, we expand our map (1.1) to second order in
in the EOMs. At leading order,
we identify conditions that are satis ed once
is bi-Killing and I is a Killing vector, a
fact that is guaranteed by the relation (1.3). At second order, we nd from the B- eld
EOM that the Lie derivative of
with respect to I must vanish, LI
= 0, while from the
Einstein and dilaton EOMs we recover the CYBE. Details of the calculations can be found
in the appendix.
Let us begin by recalling the EOMs of generalized supergravity [12],
denote the covariant derivative and curvature of the deformed solution
g , we have used the trace of the Einstein equation to eliminate R^ in (3.2), and we have
X
Throughout the remainder of this work, we will refer to the equations (3.1), (3.2) and (3.3)
as the NSNS two-form (B- eld) EOM, the dilaton EOM and the Einstein equation,
respectively. To derive these expressions [12] it has been assumed that I is a Killing vector. Here,
one can drop that assumption as this condition appears from the EOMs at leading order,
thus providing a further consistency condition on the work presented in [12]. In other
words, it is enough to assume the above equations. It is worth noting also that setting
I = 0, we recover the EOMs of usual supergravity.
We remark that the generalized gravity EOMs, similarly to the supergravity EOMs,
are closely related to the string theory
-model. One may start from a -model obtained
from a generic non-Abelian T-duality over a usual consistent string worldsheet theory. As
a result of non-Abelian T-duality, the worldsheet anomaly cancelation does not lead to
supergravity equation, but rather the generalized supergravity EOM [82] (see also [83]
for earlier work and [12, 19, 48] for addition of the RR- elds). From the perspective
of the
-model, the Killing vector I appears to be the trace of the structure constants
of the non-semisimple group on which one T-dualises [100]. In contrast to the original
treatment of generalized supergravity [12], where an explicit solution and T-duality on
a non-isometric direction were used to motivate the EOMs, or [13] where -symmetry is
assumed, the derivation [100] from the T-dual -model of [82] is purely bosonic and does
not assume fermions.7
Check of consistency of the EOMs.
Regardless of their -model roots, one can ask
if the generalized supergravity EOMs provide a consistent set of di erential equations. For
the set of equations (3.1), (3.2), (3.3), this amounts to checking if the Bianchi identity
r^ (R^
12 R^g ) = 0 holds for any on-shell con guration. As the detailed calculations of
7The analysis presented in [100] is restricted to the NS sector.
{ 7 {
the appendix demonstrates, straightforward but tedious analysis, reveals that this identity
is satis ed i the one-form X
has the form (3.4) for an arbitrary Killing one-form
eld
I . That is, (3.4) is also an outcome of the set of generalized supergravity EOMs and need
not be put in by hand. We also comment that while this consistency check is the necessary
condition for the EOMs to come from a di eomorphism invariant action, it is not su cient;
generalized supergravity is described by its EOMs and it is not known whether this theory
has an action.
where all indices are raised and lowered with respect to the background metric G .
Zeroth order.
At this order the B- eld equation is trivial and the other two equations
read as
tively yield
R
where the curvature is computed using background metric G .
First order.
At rst order the dilaton and Einstein equations (3.2) and (3.3),
respecI r
= 0;
which just con rm I as a Killing vector for the background solution, speci ed by G ; .
The NSNS two-form EOM (3.1) using the bi-Killing structure of
(1.2), after
straightforward algebra and using Killing vector identities (see appendix for details), yields
r (r
I ) = 0 =) I = r
The constant part may be dropped using the fact that we want I = 0 at zeroth order when
= 0. We hence recover (1.3) as a consequence of the rst order EOMs.
Second order.
The NSNS two-form EOM, once we use the rst order results, takes a
very simple form:
LI
= diI
+ iI d
= 0;
which essentially tells us that I is not only a Killing vector of the original geometry but
also remains Killing in the deformed geometry.
{ 8 {
We next consider the dilaton and Einstein equations at second order. To work these
out, one should note that the covariant derivatives appearing in the EOMs are with respect
to the metric g
and hence one should expand the Christo el symbols too,
1
2
G
r^ X
= r X
(r^
2 + r^
2
^
r
2 )X +
;
(3.10)
where r
denotes covariant derivative with respect to the metric G
and
stand for
higher powers of
. The Riemann curvature then receives even power corrections due to
the correction to the Christo el connection. The H2 terms in the EOM also contribute to
the second and all even powers.
After lengthy calculations, the dilaton equation of motion takes the form
CYBE (2.8). The Einstein equation can also be massaged and brought to the form
1
2 (r Ki Kj K
k + r Ki Kj K ) cl1l2irjl1 rkl2 + cl1l2
k
kril1 rjl2 + cl1l2j rkl1 ril2
= 0; (3.12)
where we have used symmetries of the curvature terms and Killing identities (see appendix
for more details). This again, gives the CYBE (2.8).
Third order. To work out equations at the third order, we recall (3.5) and that g
and
have even powers of
while the NSNS two-form has odd powers and hence X
has all powers from zero to three. Therefore, only the X-terms in the dilaton and Einstein
equations contribute to third order. One may show that these equations become an identity
once we use the fact that I is a Killing vector, namely, LI
= LI G
= LI
= 0.
The only non-trivial equation at third order is hence the NSNS two-form equation.
Again, after lengthy but straightforward analysis, one nds that this equation upon using
I being a Killing yields the CYBE.
Higher orders.
From (3.5) one can readily see the following structure: for even powers
of
the NSNS two-form EOM is satis ed trivially if I is a Killing vector, while the dilaton
and Einstein equations are non-trivial. Conversely, for odd powers of
, the dilaton and
Einstein equations are readily satis ed once we assume I is Killing. Given our analysis
above, we expect the dilaton and Einstein equations at even powers, and the B- eld EOM
at odd powers yield the CYBE. It is, of course, desirable to provide such an analysis
and give an all-orders proof for our proposed \YB solution generating technique", but we
leave this to future work. It is clear that unless one can work by induction, perturbative
expansions are not a means to provide such a proof.
4
Examples
In this section, we provide examples of generalised Yang-Baxter deformations in a bid to
get the reader better acquainted with the solution generating technique outlined in [
1
]. We
{ 9 {
focus on two examples that fall outside the usual examples studied via the Yang-Baxter
-model, before presenting a more familiar example with an RR sector.
Let us consider at spacetime in three dimensions 3D. One may imagine that this is trivial
compared to deformations of AdS spacetimes, but it turns out that generalising the
YangBaxter -model to at spacetime is complicated by the fact that the bilinear of the coset
Poincare group is degenerate [101]. As a result, our analysis here, simple though it may
be, is novel.
Consider the metric,
ds2 =
dt2 + dx2 + dy2:
Since we will initially study the CYBE, we identify the isometry group of the spacetime.
Flat spacetime is a maximally symmetric space and for this reason it permits six Killing
vectors in 3D. Let us label the Killing vectors as follows:
and record the non-zero commutation relations:
[T1; T4] = T2;
[T1; T5] = T3; [T2; T4] = T1; [T2; T6] = T3;
[T3; T5] = T1;
are constant coe cients. We have deliberately picked the Lorentz
generators, as once combined with translations one can easily generate more involved r-matrices
through inner automorphisms of the algebra. As we shall see, inner automorphisms
correspond to coordinate transformations in the geometry. Identifying r45 =
; r56 =
, etc, we
can substitute them into the CYBE (2.8), to identify a single constraint on the coe cients:
2 =
of generalized supergravity. To see this, we rst recast the r-matrix as , using
where we have replaced the generators Ti by their Killing vector representation. Having
done so, one can easily read o the components of ,
tx =
y ;
ty = x ;
xy = t ;
(4.1)
(4.2)
(4.3)
(4.4)
(4.5)
(4.7)
HJEP06(218)
=
t + y
x. One can determine the corresponding Killing
vector from (1.3),
(4.8)
At this stage, one generates a deformed supergravity solution from (1.1) and (2.12),
It is interesting to look at the symmetries preserved by the deformation. Obviously, there is
one Killing vector I that is a linear combination of the Lorentz transformations generated by
T4; T5 and T6. Plugging the deformed geometry, along with I, into the EOMs of generalized
supergravity, one quickly con rms that a deformation exists provided (4.5) holds, in line
with our expectations. It is worth noting that one can easily generate more complicated
solutions by shifting t; x and y by constants, since translation symmetries are broken.
The point of this example is to demonstrate the EOMs are equivalent to the CYBE.
However, if the focus is on inequivalent r-matrix solutions to the CYBE, we note that
r-matrices related through inner automorphisms of the algebra are equivalent. Therefore,
by applying an inner automorphism to (4.4), we can bring it to the simpler form,
r = (T4 ^ T5 + T5 ^ T6):
To see that the r-matrices are equivalent, note that one can generate an r-matrix
satisfying (4.5) through the inner automorphism e T6 Xe
T6 , where X 2 fT4; T5; T6g and
=
cos ;
=
sin . The inner automorphism of the algebra corresponds to a
rotation by angle
in the (x; y)-plane.
4.2
The previous example involved a deformation of at spacetime. In a bid to consider
spacetimes that are not Ricci- at, let us consider the following Bianchi III spacetime,
where we have de ned the functions
and Maurer-Cartan one-forms:
ds2 =
A solution exists provided the constants satisfy the condition:
The Killing vectors are and they satisfy the commutation relation:
(4.14)
(4.15)
(4.16)
(4.17)
(4.18)
(4.19)
(4.20)
(4.21)
Given that we only have three Killing vectors, the most general r-matrix one can consider is
It is easy to check that this is a solution to the CYBE (2.8) provided
= 0. Here, the
term corresponds to a naive TsT transformation in the (y; z)-directions, both of which are
Killing. Since the TsT deformation is of less interest, we will henceforth consider
= 0.
It remains now to check that the EOMs agree with the CYBE and that valid solutions
exist when either
6= 0 or 6= 0. As before, we extract the components of ,
xy = ;
yz =
z;
zx = ;
and identify the corresponding Killing vector,
I =
When
= 0, it can be checked that the deformed geometry
is a solution to usual supergravity. Since we have encountered a solution to usual
supergravity, this deformation can be interpreted as a TsT transformation with respect to the
shift symmetries generated by T1 and T2, respectively. It should be noted that both of these
Killing vectors commute and the r-matrix is Abelian, so it is a TsT transformation [16].
On the other hand, setting = 0, we encounter the geometry g dx dx =
[1 + e2x 2a2a2] dx ^ dz;
log[1 + e2x 2a21a23]:
1
[1 + e2x 2a2a2]
1 3
a21dx2 + a32e2xdz2 ;
It is straightforward to check that the EOMs of generalized supergravity are satis ed. This
deformation is of Jordanian type.
4.3
As promised we give one example of a geometry with an RR sector simply to illustrate
the utility of the methods outlined in [
1
]. While it is easy to consider a new example, and
we invite readers to do so, this risks distracting the reader from our main message. For
this reason, we nd it instructive to study an example familiar to all. The key take-home
message is that one can now perform a complicated series of TsT transformations in the
NS sector by simply inverting a matrix, while the transformation of the RR sector follows
from a knowledge of the bivector
and the Page-forms [102] of the original geometry, as
discussed in [85, 86]. While we do not provide a proof of the transformation of the RR
sector, it can be checked case by case that it works and it is expected to work since all
information about the deformation has to be encoded in
. Our prescription for the RR
sector transformation, which we review below is an elegant way to generate new
uxes
solely based on a knowledge of .
We focus on a well-known geometry that can be generated through a series of TsT
transformations, namely the Lunin-Maldacena-Frolov geometries [10, 11]. As the reader
will observe, while the deformation is traditionally de ned in terms of a series of
Tduality transformations, using our prescription this is a single matrix inversion: there
is no need to return to the Buscher T-duality rules. We begin by recalling the original
undeformed geometry:
G 1 +
2
ds2 = R
2
ds2(AdS5) + X(dri2 + ri2d i2) ;
F5 = 4R4 vol(AdS5) + vol(S5) ;
3
i=1
!
where we have introduced the constrained coordinates ri,
r1 = cos ;
r2 = sin cos ;
r3 = sin sin :
To nd the deformed background using our method, it is easiest to work with the
constrained coordinates. In terms of these coordinates the matrix to be inverted to get g; B,
may be written as
0 R 2
= BBB
B
B
B
B
C ;
C
C
C
A
where we have labeled the columns and rows r1; 1; r2; 2, etc. Inverting this matrix, while
rede ning ^i = R2 i, we get the following metric and NSNS two-form:
(4.22)
(4.23)
(4.24)
3
i=1
ds2 = R2 4ds2(AdS5) + X(dri2 + Gri2d i2) + Gr12r22r32
3
i=1
X ^id i
!23
5 ;
R2G ^3r12r22d 1 ^ d 2 + ^1r22r32d 2 ^ d 3 + ^2r32r12d 3 ^ d 1 ;
G 1 = 1 + ^32r12r22 + ^12r22r32 + ^22r32r12:
solution. The dilaton is read o from the T-duality invariant e 2 p
g, leading to
It is easy to check that this is, up to a sign in the NSNS two-form, the usual form of the
e
2 = G;
and it can be checked that I = 0, so we nd a bona de supergravity solution, as expected.
Before illustrating how the RR sector transforms, let us review the logic. Since the
Page charges are quantised, we do not expect them to change under the deformation since i
are continuous deformation parameters. Therefore, the Page ve-form should be invariant.
We note that from the perspective of AdS/CFT, this invariance is very natural as the Page
charges, which arise from integrating the Page forms over compact cycles, are related to
the ranks of the gauge groups. Using the invariance of the Page form, we can get the
lower dimension Page forms by simply contracting the bivector
and its products into the
invariant Page forms. This procedure works for all the geometries we have considered, so
we expect it to work in this setting too, and we will quickly con rm it does.
To extract the RR sector, we de ne the Page forms in terms of the usual RR
eld
strengths,
Q1 = F~1;
Q3 = F~3
B ^ F~1;
Q5 = F~5
B ^ F~3;
where we have added tildes to distinguish the deformed RR sector from the original RR
sector. We have also ipped the sign of the B- eld relative to [
1
] to make our conventions
consistent with [10, 11]. As explained above, we now use the fact that Q5 is invariant,
which implies it is the same as the original ve-form
ux,
Q5 = F5 = 4R4 vol(AdS5) + vol(S5) ;
since there was no NSNS two-form in the beginning.
Our prescription [
1
] now demands that we contract in
and its products to get the
lower-dimensional Page forms. It should be noted that all products of
vanish when
contracted into forms, so we only need to contract
to nd the Page three-form Q3 with
the Page one-form being trivially zero. As a result, we have
~
F3 1 2 3 = Q3 1 2 3 =
Q5
Following this procedure, we get
F~3 = 4R2 sin3
cos sin cos d
^ d ^
3
i=1
X ^id i:
Now that we have B and F~3, we can read o F~5 from (4.28). The result is
F~5 = 4R4 vol(AdS5) + Gvol(S5) :
(4.26)
(4.27)
(4.28)
(4.29)
(4.30)
(4.31)
(4.32)
Up to signs, our expressions for the RR sector agree with [10, 11]. We emphasise again that
there was no need to perform any T-duality transformation or use results in the literature
detailing how the RR sector transforms [91, 95]. It is much quicker to get the RR sector
using invariance of the Page forms and descent through
contractions, as was also checked
earlier in [85, 86].
5
Discussion
In this work, we focused on the generalized supergravity EOMs and analysed what they
imply on solutions obtained from deformations generated through the open-closed string
map, and in this way, substantiated the claims of our earlier letter [
1
]. Assuming the
bivector
to be a generic linear combination of anti-symmetric products of Killing vectors,
we imposed the EOMs of generalized supergravity and studied the equations perturbatively
in . Our analysis revealed:
HJEP06(218)
1. The consistency (integrability condition) of generalized supergravity EOMs implies
the form of the X
eld appearing in these equations in terms of I and other elds.
2. The I vector appearing in the generalized supergravity must be a Killing vector of
the deformed background and is also the divergence of . This generalises the earlier
unimodularity condition of [95], which is recovered when I = 0. It also proves the
observation made through a long list of examples [84{86, 91, 100].
3. Most importantly, the CYBE comes out as a result of the EOMs and not as an input.
We have hence provided strong evidence that the CYBE and our open-closed map can be
used beyond coset or maximally symmetric spaces. We have then checked the \YB solution
generating technique" proposal in various examples. Here we showed that the EOMs are
automatically satis ed at the third order in
once the CYBE is imposed. However, based
on explicit examples, either presented earlier in [
1
], or eshed out in section 4, it should
be clear that this statement is true for all orders. That being said, the proof of the YB
solution generating technique is still outstanding.
Since we were largely working perturbatively, but ultimately are interested in
generating exact solutions to supergravity, we opted not to address the RR sector. Admittedly, if
one is only working perturbatively, there is little motivation to do so. Our experience with
many examples [84{86, 91, 100] indicates that the addition of the RR sector should largely
be a technical issue and would just con rm the results we have enumerated above. The
only new feature we expect to appear with the addition of the RR sector is the possibility
of obtaining the modi ed CYBE; note that as we showed, the NS sector yields only the
homogeneous CYBE. To obtain the modi ed CYBE within our framework, in [
1
] it was
proposed to make an extra constant shift in the dilaton. However, given that the only
known deformed geometries based on the modi ed CYBE are deformations of AdSp
geometries, we can use our approach to study deformations of Minkowski vacua supported
Sp
by RR
ux. We will report on this elsewhere.
Our method, open-closed map plus solutions to CYBE, can be used as a very handy and
simple solution generating proposal, as outlined in [
1
]. To demonstrate this, we reworked
the Lunin-Maldacena-Frolov geometries to highlight the economy of the approach. Our
method provides clearly a smarter way to perform TsT transformations, rather than going
through the standard Buscher T-duality procedure.
As a nal remark, we point out that in this work we focused on original backgrounds
without any B- eld; the B- elds that appear in the solutions are all generated through .
There are, however, interesting geometries, such as AdS3
S3
T 4, which are supported
by H- ux (brie y commented on in [85, 91]). For this example, the matrix g + B is
singular and cannot be inverted. In short, our method does not work. Nevertheless, one
can consider the more general framework of O(d; d) and -transformations [103] (also [80]),
which include both non-Abelian T-duality and Yang-Baxter deformations as special cases.8
Acknowledgments
We thank T. Araujo, O . Kelekci, J. Sakamoto, Y. Sakatani and S. van Tongeren for
discussion on related topics. I. B. is partially supported by the Russian Government program for
the competitive growth of Kazan Federal University. E. O C. thanks Cafe Dudart,
Sangamdong, Seoul for refuge. M.M. Sh-J. is partially supported by the grants from ICTP NT-04,
INSF grant No 950124. The work of HY is supported in part by National Natural Science
Foundation of China, Project 11675244.
A
Consistency of the generalized supergravity eld equations
To check the consistency of equations (3.1){(3.3), we rst rewrite the Einstein equation (3.3)
1
2
H2g
81 r^ H2
=)
81 r^ H2
H
H
H
X):
(A.1)
R^) = 0, we get
X = 0: (A.2)
r^ r^ X
= r^ r^
X + X R^ ;
r^ 2X
We also record the following,
r^ r^ X
= r^ r^ X
+ X R^
1
4
Taking the derivative of the above equation and noting that r^ (R^
r^ H
+
H
r^ H
r^ 2X
r^ r^ X
+ r^ r^
= H
= H
= H
8We thank Y. Sakatani and J. Sakamoto for correspondence on this issue.
so we infer the relation
H
r^ H
=
Using the above identity and the B- eld and Einstein equations (3.1), (3.3), we get
1
2
Using the dilaton equation (3.2) to replace r H2 and the Einstein equation to replace
+ 2X r^ X +
(r^ X
r^ X ) + r^ r^ X
r^ X2
r^ 2X
= 0; (A.9)
r^ 2X + r^ r^ X
(r^ X
r^ X ) + 2X r^ X
r^ X2 = 0:
and
This can be written as
r^ f
=
H
1
2
so we arrive at
H
1
2
H
1
2
H
+
1
2
H
f
r^ I
1
2
H
(A.6)
(A.7)
(A.10)
(A.11)
(A.12)
(A.13)
= 0;
(A.15)
One can now decompose the one-form X as an exact form, a Killing form and an extra
part normal to the Killing form
X
+ I +
;
f
We note that, when B = 0 we get I = 0 and arrive back to the usual supergravity equations
with X
= z H
, where z is an H-independent vector
eld. Replacing this ansatz in (A.11) , we nd
r^ 2I
r^ r^ I
r^ z
z r^ H
1
2
H
r^ I +
1
2
H
+ z H
H
2X r^ I + 2X r^ I + 2X z H
= 0: (A.14)
Using the equations of motion again, we get
r^ 2I
r^ r^ I
r^ z
2z X H
2z r^ X +2z r^ X
r^ I
1
2
+
r^ I +2z R +2z r^ X +2z r^ X
2X r^ I +2X r^ I +2z X H
and after some further simpli cations we get
2(z
I )R
H
r^ (z
I ) + 4z r^ X
+ 4X r^ I = 0:
Next, noting that we arrive at
LI X
= I r^ X
+ X r^ I = 0;
This is an identity when z = I. This along with the fact that when B = 0, the Killing
vector I vanishes, implies
=
B
I . It should be noted that in writing this solution
we have absorbed a total derivative in .
B
Details of the perturbative analysis
In this section, we provide some details of the results quoted in the text.
Some useful Killing identities. In our perturbative analysis we have heavily used
Killing vectors and their properties. So we start with some useful identities. Given a set
of Killing vectors K ,
i
there is a well-known identity,
r Ki + r Ki = 0;
r r K
= R
K ;
)
r2K
=
R
K :
Assuming the bi-Killing structure for
(2.1), one can readily show:
H
= r
+ r
+ r
= 2rij r Ki Kj + r Ki Kj + r Ki Kj ;
(B.3)
where the spacetime indices are lowered and raised by the metric G .
Perturbative expansion. Expanding the metric g , B- eld and dilaton for small ,
we get
2(z
I )R
H
r^ (z
I ) + 4(z
I )r^ X
(A.18)
HJEP06(218)
(A.16)
(A.17)
(B.1)
(B.2)
(B.4)
g
B
= G
=
=
+
+
1
4
Plugging the above expressions directly into the EOMs, we can also expand them for small
. We now detail the information extracted at each order from the EOMs.
Zeroth order
At zeroth order in , equations (3.1){(3.3) become
r X
r X
= 0 =) r r
r r
R
+ r X + r X
= 0 =) R
X
2X2 = 0 =) r
2
2r
r
where we remark that the rst equation is trivial, whereas the second and third are simply
the EOMs satis ed by the original undeformed solution, in line with expectations.
HJEP06(218)
B.2
First order
At rst order the linear terms in
give the following contribution to the EOMs,
(B.5)
(B.6)
(B.7)
(B.8)
(B.9)
(B.10)
(B.11)
(B.12)
(B.13)
(B.14)
Now we assume that
is bi-Killing (1.2) and use the identities (B.2)
1
2 r H
(r
)H
r I
4r
I = 0
r I + r I = 0;
r I + r I = 0;
=)
I r
= 0:
r
K = 0;
K
r
= 0;
1
2 r H
the second term is
where the latter is valid for any eld
. This allows us to write H = dB in terms of the
components of
as
H
= rij Kj r Ki + Ki r Kj + Ki r Kj + Kj r Ki + Ki r Kj + Kj r Ki ;
= 2rij Ki r Kj + Ki r Kj + Ki r Kj :
Now each term in equation (B.8) can be expanded as follows: the rst term is
= rij 2r Kj r Ki
2Kj r Ki r
2Ki r Kj r
+ R
Kj Ki ;
r
H
=
2rij (Kj r Ki r
+ Ki r Kj r
);
so that equation (B.8) can be further simpli ed,
rij 2r Kj r Ki + R
Kj Ki
Equation (B.9) implies that I is a Killing vector of the original metric G
and
therefore (B.10) is automatically satis ed for the scalar eld .
H
I H
+ B
I H
:
Therefore, the B- eld equation (3.1) becomes,
H
+ B
I H
and also
and
as follows,
=
=
r^ I +r^ I +(r^ B
r^ I +r^ I +I H
+r^ B
;
r^ B )I +(B
+r^ B )I ;
Second order
Before trying to expand and solve the second order equations, it would be useful to simplify
the EOMs using what we have found from the zeroth and the rst order equations. Using
the fact that I is a Killing vector we nd
LI g = r^ I + r^ I = 0;
LI B = I r^ B
+ B
r^ I + B
r^ I = 0:
Using these expressions, we get
HJEP06(218)
r^ X +r^ X = g
r^ I +g
r^ I +(r^ B
r^ I
B
r^ I );
To expand covariant derivatives we note that ^bac can be expand in
and gives
=
+ C ; where C
=
G (r
2 + r
2
r
2 ) + O( 4); (B.22)
^
R
= R
+ D
= R
+ r C
r C :
It is clear from (B.21) that there is no second order term in the B- eld equation.
The Einstein equations. The quadratic term in the Einstein equation can be organised
D
H
H
+ r (
I ) + r (
I ) + 2C
(B.24)
where D denotes the second order terms in the Ricci tensor expansion, de ned in (B.23).
The rst term can be expand as follows
D
=
1
2 r r
= rijrmn 2Kj KmKnr r Ki
(B.16)
(B.17)
(B.18)
(B.19)
(B.20)
(B.21)
(B.23)
(B.25)
The second term gives
4H H = 14(r
1
= rijrmn KjKn r Kir Km +KjKn r Kir Km
+r
+r )(r
+r
+r );
+KjKnr Kir Km
KjKn r Kir Km
+ 1KjKnr Kir Km +( $ ):
2
(B.26)
The third and the fourth terms together can be simpli ed as follows
+
r r
+( $ ); (B.27)
=rijrmn KiKmr Kjr Kn+KmKnr Kjr Ki +KmKn Ki Kj R
+( $ );
and nally the last part of the Einstein equations give
2C @ =2rijrmn KiKjKm r Kn +KiKjKm r Kn +Ki Kj Knr Km
+Ki Kj Kmr Kn+KiKj Knr Km +KiKjKm r Kn @ ;
=rijrmn 2KiKjKm Kn R +Ki Kj KmKnR +( $ ):
Adding all these terms together we get
rijrmn 2Kj KmKnr r Ki Kj KmKnr2Ki KjKm Kn r r Ki
KjKn r Kir Km KjKn r Kir Km
KjKnr Kir Km
+KjKn r Kir Km
1KjKnr Kir Km +KiKmr Kjr Kn
2
+KmKnr Kjr Ki +KmKn Ki Kj R
+2KiKjKm Kn R
+Ki Kj KmKnR
+( $ )=0:
(B.28)
(B.29)
After simpli cation, this can be written as
rijrmn Ki Kmr Kjr Kn Ki Kmr Knr Kj +Km Kjr Knr Ki
Km Knr Kjr Ki +KjKi r Km r Kn KnKi r Km r Kj
Ki KmKnR
Kj +Ki KmKjR
Kn +( $ )=0;
(B.30)
which upon further permutation of the indices, takes the neat form,
which is identically satis ed one the CYBE holds.
(KjKl r Ki +KiKjKl r ) rjprlq cpqi+riprjq cpql+rlpriq cpqj =0; (B.31)
Dilaton equation.
Having discussed the more involved Einstein equation, we focus on
the dilaton equation at second order. It takes the form,
+
1
12
H2 + r (
I ) + C
2I I = 0:
(B.32)
The second and the last term can be together simpli ed as
r (
I ) 2I I = r
r r
2r
r
;
(B.33)
= rijrmn Ki Km
r Kj r Kn + Ki Kjr Km
r Kn + Ki KjKm
r r Kn
:
Using equation (B.10) and the fact that
is a bi-Killing, it is easily con rmed that
= 0. The remaining term with
can be simpli ed as
HJEP06(218)
(B.34)
(B.35)
(B.36)
(B.39)
C
where 2 is de ned as
As a result, we get
)2 + r (
)
2
r
2 =
:
1
2 r (
= rijrmnKi Km Knr2Kj :
in terms of
and K. The rst
term in the dilaton equation can hence be simpli ed as
1
12
H2 =
1
12 (r
+ r
+ r
+ r
+ r
);
= rijrmn Ki Km
r Kj r Kn + 2KnKi r Kj r Km :
(B.37)
Putting everything together we get
rijrmn Ki Km
r Kj r Kn +Ki Kjr Km
r Kn +Ki KjKm
r r Kn
Ki Km Knr2Kj +Ki Km
r Kj r Kn +2KnKi r Kj r Km
= 0: (B.38)
Further simplifying the above expression we get
Ki Km
r Kj r Kn + KnKjr Ki r Km + KmKjr Ki r Kn
2Ki Kj r Knr Km
Kj Knr Km
r Ki + Ki Kj Km
r r Kn
+ Ki KjKm
r r Kn + Ki Kj Km
r r Kn
Ki KmKnr r Kj
+ Ki KmKj r r Kn
rijrmn = 0:
Now using (B.2) and the Bianchi identity for the Riemann tensor, we can factorise the
above expression as
Ki Kl
r Km riprlqcpq m + rmpriqcpq l + rlprmqcpq i
+rijrmnKi Kj Km Kn (R
+ R
+ R
) = 0:
(B.40)
(B.41)
This expression vanishes identically for any curved background once the CYBE is satis ed.
B.4
Third order equations
To test that nothing funny happens at the higher order, we study the EOMs to third order
in
. We will see in this order that the dilaton and Einstein equation are somewhat trivial,
while the B- eld EOM encapsulates information of the CYBE.
Dilaton equations at third order. In order to expand the EOMs, it is useful to
HJEP06(218)
+ g I
+ r
+
r
+
2 r
+ O( 4): (B.42)
note that
Therefore, we have
where we de ne
satis ed identically.
be the following:
ially satis ed.
(B.43)
(B.44)
(B.45)
where we have used (B.16). By expanding H up to cubic order in , we nd
H
= H(1) + H(3) +
= (r
+ r
+ r
) + (r
3 + r
3 + r
3 ) + O( 5);
3 =
:
We note that the rst term in the dilaton equation (3.2) does not have any term cubic in
. Therefore, the dilaton equation reduces to,
C
r
2
r
r
+ 2r
r
= 0:
(B.46)
Using (B.22) and noting that I is a Killing vector, we nd that the above equation is
Einstein equation at third order. At cubic order, the Einstein equation turns out to
X = g X = r
+ r
r
C
r
+ ( $ ) = 0:
(B.47)
Using (B.22) and noting again that I is Killing, we nd that above equation is also
triv
B- eld equation.
We now arrive at a non-trivial equation. We expand the B- eld
equation (3.1), or equivalently (B.21), and keep the cubic terms in
. The rst term
in (B.21) gives
r^ H
1
2
Using (B.22), the covariant derivative can be expanded as
G
C
)r^ H
:
(B.48)
r^ H
= r H(1) + r H(3)
C
H(1)
C
H(1)
H(1) + O( 5);
(B.49)
where H(1) and H(3) are de ned in (B.44). Therefore the cubic terms in
in (B.48) are
1
2 r H(3)
The second term in equation (B.21) takes the form
r H(1)
G (C
H(1) + C
H(1) + C
H(1) ):
(B.50)
H(3) =
3 +
r
(B.51)
where to write down the above equations we have used (B.16). Further, using the
following identities:
we can simplify (B.51) as follows
r r
+
r r
(B.52)
1
2
(B.53)
(B.54)
(B.55)
1
2
H(3) =
( 3 R
3 R ):
The third term in (B.21) takes the form
B
I H
+ r
+ r
)r
Adding all the above terms, massaging them and using bi-Killing structure of , the
B- eld equation at third order can be written as
K[i Kkr Km] cl1l2mril1 rkl2 + cl1l2krml1 ril2 + cl1l2irkl1 rml2
= 0;
where [
] denotes anti-symmetrization with respect to all indices. The above equation is
satis ed identically once the CYBE holds.
Open Access.
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any medium, provided the original author(s) and source are credited.
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