.

.

. .

... .

.

.

*Squashed group manifolds in String Theory*

*exact solutions, brane realization, classical integrability*

Domenico Orlando 数物連携宇宙研究機構

Institute for the Physics and Mathematics of the Universe

National Taiwan University 6 May 2011

Based on:[arXiv:1003.0712, arXiv:1011.1771, arXiv:1104.0738]and work in progress Collaboration with:

I.Kawaguchi (Kyoto),S.Reffert (IPMU), L.Uruchurtu (Imperial College), K.Yoshida (Kyoto)

. .

. .

*Outline*

. .* ^{1}*.

*Why are we here?*

. .* ^{2}*.

*The main idea*. .

*.*

^{3}*D–brane construction*

. .* ^{4}*.

*Integrability of the Principal Chiral Model*. .

*.*

^{5}*Heterotic construction (or, the S–dual story)*. .

*.*

^{6}*Conclusion*

. .

. .

*Outline*

. .* ^{1}*.

*Why are we here?*

. .* ^{2}*.

*The main idea*. .

*.*

^{3}*D–brane construction*

. .* ^{4}*.

*Integrability of the Principal Chiral Model*. .

*.*

^{5}*Heterotic construction (or, the S–dual story)*. .

*.*

^{6}*Conclusion*

. .

. .

*Squashed geometries*

String theory provides a dual description of the same physics: worldsheet andtarget space

In very few cases we can access both and learn from both sides

Most of these cases (ﬂat spacetime, group manifolds, plane waves) have a high degree of symmetry

More symmetry meansmore structure(and simpler analysis)

Can we break part of this symmetry, at the same time preserving the “nice”

structures?

. .

. .

*Three dimensional gravity*

Three–dimensional gravityprovides a simple laboratory for quantum gravity There are no propagating gravitons

There are non-trivial solutions (BTZ black holes) Pure gravity solutions are alwayslocally AdS3

Anti–de Sitter spaces appear innear-horizon geometriesof various D–brane conﬁgurations

AdS3appears as anexact string theory background(Wess–Zumino–Witten model)

. .

. .

*TMG*

Three–dimensional AdS spaces also appear as solutions of topologically massive gravity(TMG)

*S**tmg**(g) =* 1
*16πG*

∫

d^{3}*x*√

*−g*
(

*R +* 2
*ℓ*^{2}

)

*+ μ S**CS*

for μ*̸= 1 the anti–de Sitter solution is unstable, but there are*two warped
solutionswith metric of the type

*ds*^{2}[WAdS_{3}*] = R*^{2}
[

dω^{2}*− cosh*^{2}ω dτ^{2}+ 1
cosh^{2}Θ_{w}

(dβ + sinh ω dτ)2]

where Θ* _{w}* is a deformation parameter:

for Θ* _{w}*= 0 this is AdS

_{3}for Θ

_{w}*→ ∞ this is AdS*2

*× S*

^{1}

. .

. .

*Today’s talk*

Geometryof squashed groups in general (and AdS3 in particular) T–dualityacts on principal ﬁbrations

Type IIsolutions with squashed AdS_{3}*and S*^{3}
Integrability(without RR ﬁelds)

ExactCFTwith squashed group geometry

. .

. .

*Outline*

. .* ^{1}*.

*Why are we here?*

. .* ^{2}*.

*The main idea*. .

*.*

^{3}*D–brane construction*

. .* ^{4}*.

*Integrability of the Principal Chiral Model*. .

*.*

^{5}*Heterotic construction (or, the S–dual story)*. .

*.*

^{6}*Conclusion*

. .

. .

*The Geometry – Group Manifold*

Consider agroup manifold*G (e.g. AdS*_{3}*or S*^{3})

*There is a bi-invariant metric such that the isometry group is G× G*
*The metric can be written in terms of the currents J** _{a}*that generate half of
the symmetry

*ds*^{2}*[G] =*

*dim G*∑

*a=1*

*J*_{a}*⊗ J*_{a}

*If H⊂ G is compact, G is the total space of a*Hopf ﬁbration*over G/H:*

*H* *−−−−→ G*

y
*G/H*
*Today H = U(1) and G/H = S*^{2}*, AdS*_{2}.

. .

. .

*The Geometry – Squashed group*

*SqG is a deformation of G described by the*same currents
*The symmetry group is G(R) × U(1) (only left-invariance)*

*The metric can be written in terms of the same J*_{a}*currents. Fix dim G = 3*
*ds*^{2}*[SqG] = J*_{1}*⊗ J*_{1}*+ J*_{2}*⊗ J*_{2}+ 1

cosh^{2}Θ_{w}*J*_{3}*⊗ J*_{3}
SqG is thebase space for a ﬁbration*that has G× S*^{1} total space

*S*^{1}*× S*^{1} *−−−−→ G × S*^{1}

y
*G/U(1)*

choose embedding

*−−−−−−−−−−→*

*S*^{1} *−−−−→ G × S*^{1}

y
SqG
*the embedding of S*^{1}*,→ S*^{1}*× S*^{1} is described by Θ*w*

. .

. .

*T-duality to “undo” ﬁbrations*

ConstructNLSM on squashed groups via T-dualityfrom principal chiral
*models with group manifold target space G*

We are not considering here conformal models: we start with the metric
*only (a B-ﬁeld will appear)*

*Main observation: if the space has a S*^{1} ﬁbration structure
*S*^{1} *−−−−→ M*

y
*N*

T-duality along the ﬁber will “undo” the ﬁbration and give adirect product
*M = N*e *× S*^{1}

. .

. .

*T-duality on the worldsheet*

*If there is a S*^{1}ﬁbration, the action can be written as
*S[u*^{i}*, z] =*

∫

Σ

*G*_{ij}*(u) du*^{i}*∧ ∗du** ^{j}*+(

*dz + f*_{i}*(u) du** ^{i}*)

*∧ ∗*(

*dz + f*_{j}*(u) du** ^{j}*)

*,*

which is to say, the metric has a block form
( *G*_{ij}*(u) + f*_{i}*(u)f*_{j}*(u)* *f*_{i}*(u)*

*f*_{i}*(u)* 1

)
*.*

*we want to T-dualize the S*^{1}*described by z.*

. .

. .

*T-duality on the worldsheet*

*introduce a gauge ﬁeld A and a Lagrange multiplier*
*S[u*^{i}*, A,ez] =*

∫

Σ*G**ij**(u) du*^{i}*∧ ∗du** ^{j}*+(

*A + f*_{i}*(u) du** ^{i}*)

*∧ ∗*(

*A + f*_{i}*(u) du** ^{i}*)

*− 2ezdA .*

The EOM for*ez give*

*dA = 0* *⇒ A = dz ,*
which leads back to the original action

*The EOM for A gives*

*∗dez = A + f*_{i}*(u) du*^{i}*= dz + f*_{i}*(u) du*^{i}*.*

. .

. .

*T-duality on the worldsheet*

The resulting action describes adirect product metricplus a*B ﬁeld*
*S[u*^{i}*,ez] =*

∫

Σ*G**ij**(u) du*^{i}*∧ ∗du** ^{j}*+ d

*ez ∧ ∗dez − 2 dez ∧ f*

*i*

*(u) du*

*as promised*

^{i}*S*^{1} *−−−−→ M*

y
*N*

*−−−−→*

T-duality

*M = N*e *× S*^{1}

. .

. .

*Squashed groups*

start with

U(1) *−−−−→ G × U(1)*

y SqG

the ﬁber is a linear combination of the U(1) and one direction in the
*Cartan of G*

The metric on SqG is

*ds*^{2}*[SqG] = ds*^{2}*[G] + tanh*^{2}*Θ j*_{C}*⊗ j*_{C}*,*
whereΘ measures the combinationof the two U(1):

for Θ*→ 0, SqG = G*

for Θ*→ ∞, SqG = G/U(1) × U(1).*

. .

. .

*Outline*

. .* ^{1}*.

*Why are we here?*

. .* ^{2}*.

*The main idea*. .

*.*

^{3}*D–brane construction*

* ^{4}*.

*Integrability of the Principal Chiral Model*. .

*.*

^{5}*Heterotic construction (or, the S–dual story)*. .

*.*

^{6}*Conclusion*

. .

. .

*Initial setup*

Consider the superposition of a*D*_{1}*/D*_{5}system with amagnetic monopole
and aplane wave

*This is the T–dual of the D = 4 extremal dyonic black string.*

The ﬁeld content is
*ds*^{2}*= H*^{1/2}_{1} *H*^{1/2}_{5}

(

*H*^{−1}_{1} *H*^{−1}_{5}
(

*du dv + K du*^{2}
)

*+ H*^{−1}_{5}
(

*dy*^{2}_{1}*+ . . . dy*^{2}_{4}
)

+
*+ V** ^{−1}*(

*dψ + A*_{i}*dx** ^{i}*)2

*+ V*
(

*dx*^{2}_{1}*+ . . . dx*^{2}_{3}
))

*e*^{2φ}*= H*^{−1}_{1} *H*_{5}*,* *F*_{[3]}*= H*^{−1}_{1} *dt∧ du ∧ dv − B**i**dx*^{i}*∧ dψ*
*where H*_{1}*(x), H*_{5}*(x), K(x), V(x), A*_{i}*(x), B*_{i}*(x) are harmonic functions of the*
*transverse coordinates x*_{i}*, i = 1, 2, 3 and*

*dB =− ∗ dH*5*,* *dA =− ∗ dV*

. .

. .

*Initial setup*

Consider thenear–horizon limit
*ds*^{2}*= Q*_{m}*Q*^{1/2}_{1} *Q*^{1/2}_{5}

(*− dτ*^{2}+ dω^{2}*+ Q** _{w}*dσ

^{2}

*+ 2Q*

_{w}*sinh ω dσ dτ )*

^{1/2}*+ Q*_{m}*Q*^{1/2}_{1} *Q*^{1/2}_{5}
(

dθ^{2}+ dφ^{2}+ dψ^{2}+ 2 cos θ dψ dφ
)

*+ Q*^{1/2}_{1} *Q*^{−1/2}_{5}
(

*dy*^{2}_{1}+*· · · + dy*^{2}_{4})
*,*

*F*_{[3]}*= Q*_{m}*Q*^{1/2}_{1} *Q*^{1/2}_{5} (

cosh ω dτ*∧ dω ∧ dσ + sin θ dφ ∧ dψ ∧ dθ*)
*.*
The geometry isAdS3*× S*^{3}*× T*^{4}

*The radii are ﬁxed by the monopole charge Q** _{m}*(quantized)
The plane wave charge appears in the AdS

_{3}part

. .

. .

*Hopf–T–duality*

We want to use the fact that

*S*^{1} *−−−−→ AdS*3*× S*^{1}

y WAdS3

Single out a AdS_{3}*× S*^{1} part from the ten-dimensional geometry
ImplementHopf–T–duality.

*If the geometry is the total space for a S*^{1} ﬁbration and there are only
Ramond–Ramond ﬁelds, the*T–dual along the ﬁber has geometry B× S*^{1}.

*S*^{1} *−−−−→ E*

y
*B*

+ RR ﬁelds *−−−→*^{T–dual} (
*B× S*^{1})

+ RR and NS ﬁelds

. .

. .

*Hopf–T–duality*

Starting from AdS_{3}*× S*^{1}:
*ds*^{2}*= R*^{2}

(*− dτ*^{2}+ dω^{2}*+ Q** _{w}*dσ

^{2}+ 2√

*Q** _{w}*sinh ω dσ dτ
)

+

√
*Q*_{1}
*Q*_{5}*dy*^{2}_{1}
*F*_{[3]}*= R*^{2}cosh ω dτ*∧ dω ∧ dσ*

T–duality
we obtain the metric we want: WAdS_{3}*× S*^{1}
*ds*^{2}*= R*^{2}

[

dω^{2}*− cosh*^{2}ω dτ^{2}+ 1
cosh^{2}Θ_{w}

(dβ + sinh ω dτ)_{2}]

+ dζ^{2}_{w}

*F*_{4}= *R*^{2}

cosh^{2}Θ* _{w}*cosh ω dω

*∧ dτ ∧ dβ ∧ dζ*

*w*

*,*

*F*2

*= R tanh Θ*

*w*cosh ω dω

*∧ dτ ,*

*H*3*= R tanh Θ** _{w}*cosh ω dω

*∧ dτ ∧ dζ*

*w*

*.*

. .

. .

*Observations*

Having started from a pure RR background, the geometry isglobally WAdS3. We can get orbifolds by adding NS ﬂuxes in the initial background Theparametersof the solution are understoodin terms of charges:

*R*^{2} *= Q** _{m}*√

*Q*_{1}*Q*_{5} sinh^{2}Θ_{w}*= 4Q*_{w}*Q*_{m}*Q*_{5}

the quantization of the deformation corresponds to the quantization of the linear combination of the ﬁbers.

*The group SL*_{2}(R) has three different types of generators. Each can be
chosen for this construction and lead to different geometries.

The same construction can be used to describe other squashed groups
*(simplest case: squashed S*^{3})

. .

. .

*Outline*

. .* ^{1}*.

*Why are we here?*

. .* ^{2}*.

*The main idea*. .

*.*

^{3}*D–brane construction*

* ^{4}*.

*Integrability of the Principal Chiral Model*. .

*.*

^{5}*Heterotic construction (or, the S–dual story)*. .

*.*

^{6}*Conclusion*

. .

. .

*Currents and EOM*

Consider the*PCM for a group G*
*S =−*1

2

∫

Σ*Tr[dg(x, t)∧ ∗dg*^{−1}*(x, t)] ,*
*where g is a map from the worldsheet to the group g : Σ→ G*
Theequations of motionare

d*∗(g*^{−1}*dg) = d∗(dg g*^{−1}*) = 0 .*
these are the conservation laws for twocurrents

*j = g*^{−1}*dg ,* *j =− dg g*^{−1}*.*

the currents areﬂat*and thus fulﬁll the Maurer–Cartan (MC) equations:*

*dj + j∧ j = 0 ,* *dj + j∧ j = 0 .*
Conservation and ﬂatness are the reasons for the integrability.

. .

. .

*Lax current*

We will consider only the left currents. The right side works in the same way Introduce theone-parameter familyof currents

*J(x, t; ζ) =−* ζ

1*− ζ*^{2}*(ζ j(x, t) +∗j(x, t))*
where ζ*∈ C is the spectral parameter.*

*The ﬂatness of J is an equation for the components (J*_{x}*, J** _{t}*), the so-calledLax
equations:

*∂*_{t}*J*_{x}*− ∂**x**J*_{t}*+ [J*_{t}*, J** _{x}*] = 0
this is a Lax Pair.

. .

. .

*Lax current*

*The ﬂatness of J and J implies both the EOM and the MC equations.*

Conversely, imposing the EOM and MC equations results in the ﬂatness of the currents.

This can be easily veriﬁed by observing that
*dJ(ζ) + J(ζ)∧ J(ζ) =* ζ

ζ^{2}*− 1*(d*∗j + ζ (dj + j ∧ j)) .*

We started with a conserved current. Its ﬂatness implies theexistence of a one-parameter family of ﬂat currents.

*Algebraically we passed from j∈ g to the*loop algebra*J∈ g ⊗ C[ζ, ζ** ^{−1}*]
We literally haveinﬁnitely more currents(after Fourier transform).

. .

. .

*Wilson line*

We have constructed inﬁnite currents. Where are theconserved charges?

*Since J(ζ) is ﬂat we can introduce a*Wilson lineas the path-ordered
exponential

*W(x, t|x*0*, t*0; ζ) = P
{

exp [∫

*C:(x*0*,t*_{0})*→(x,t)**J(ξ, τ; ζ)*
]}

*,*

and

*J(x, t; ζ) = W*^{−1}*(x, t; ζ) dW(x, t; ζ) .*
*This generalizes the relation j = g*^{−1}*dg to the loop algebra.*

For spin chain experts: this is the transfer matrix.

. .

. .

*Wilson loop and conserved charges*

We can now deﬁne aone-parameter family of conserved charges:

*Q(t; ζ) = W(∞, t| − ∞, t; ζ) = P*
{

exp
[∫ _{∞}

*−∞**J*_{x}*(x, t; ζ) dx*
]}

*.*
*note that Q goes “all around” the worldsheet. This is a*Wilson loop
Key point: using theLax equationsand with appropriate BC, the
*one-parameter charge Q(t; ζ) is conserved*

d

*dtQ(t; ζ) = 0*

Expand on ζ and ﬁnd an*inﬁnite set of conserved charges Q*_{n}*Q(t; ζ) = 1 +*

∑*∞*
*n=0*

ζ^{n+1}*Q*^{(n)}*(t) .*
for which

d

*dtQ*^{(n)}*(t) = 0,* *∀ n = 0, 1, . . . .*

. .

. .

*Integrability of the PCM (in short)*

The principal chiral model has twoconserved currentscorresponding to the
*G× G symmetry*of the action

These currents are alsoﬂat

Out of these one can construct twoone-parameter families of ﬂat currents The Wilson loops of these currents are time-independent

The Fourier development givesinﬁnite conserved charges

These charges close under an inﬁnite-dimensional algebra (Yangianor afﬁne)
the g*⊕ g symmetry of the action is the zero mode of thebg ⊕ bg*symmetry of
the equations of motion.

. .

. .

*Integrability and squashed groups*

How much of this structure remainsafter T–duality?

We have alinear transformation of the current*components J(ζ)7→eJ(ζ)*
that leaves the (on-shell) ﬂatness conditions invariant:

*deJ+eJ∧eJ= 0 ,*

*concretely we deﬁne T–dual Lax currents eJ(ζ) by imposing the condition*

*∗dez = dz + f*_{i}*(u) du*^{i}*.*

Flatness is the key. This is preserved: the system isstill integrable.

The condition is not local (mix time and space derivatives). The resulting charges are all non-localanddo dot correspond to isometries.

. .

. .

*A technical remark*

The current that we use for T–duality does not commute with the others
*some of the components of J depend explicitly on z, when we have an*
*equation for dz.*

We need to perform agauge transformation
*J*^{′}*= h*^{−1}*J h + h*^{−1}*dh ,*

after the transformation, the new current has a zero–mode (in the ζ expansion)

*eJ*^{′}*(ζ) = h*^{−1}*dh− Λ(ζ)h*^{−1}*jh*

*dz=**∗dez−f**i**(u) du*^{i}*= eJ*^{′(0)}*− Λ(ζ)ej.*

*for the experts: in the hierarchies we will have to covariantize w.r.t. eJ** ^{′(0)}*.

. .

. .

*The simplest example: Squashed three-sphere*

*eJ** ^{′1}*(ζ) =

*−ıΛ(ζ) sin θ dφ ,*

*eJ*

*(ζ) =*

^{′2}*−ıΛ(ζ) dθ ,*

*eJ*

*(ζ) =*

^{′3}*−ı*

[

(1 + Λ(ζ))

(dα + cos θ dφ

cosh^{2}Θ + tanh Θ*∗dez*
)

*− cos θ dφ*
]

*,*
*eJ** ^{′4}*(ζ) =

*−ı tanh ΘΛ(ζ)*(

*∗dez − tanh Θ*(

dα + cos θ dφ))
*,*

and

e*J*^{1}(ζ) =*−ıΛ(ζ)*[ 1

cosh^{2}Θcos φ sin θ dα*− sin φ dθ + tanh*^{2}Θ cos φ sin θ cos θ dφ + tanh Θ cos φ sin θ∗dez]
*,eJ*^{2}*(ζ) = ıΛ(ζ)*[ 1

cosh^{2}Θsin φ sin θ dα + cos φ dθ*− tanh*^{2}Θ sin φ sin θ cos θ dφ + tanh Θ sin φ sin θ*∗dez*]
*,*
e*J*^{3}*(ζ) = ıΛ(ζ)*

[ 1

cosh^{2}Θcos θ dα +(

1*− tanh*^{2}Θ cos^{2}θ)

dφ + tanh Θ cos θ*∗dez*
]

*,*
e*J*^{4}*(ζ) = ı tanh ΘΛ(ζ)*(

*∗dez − tanh Θ*(

dα + cos θ dφ))
*.*

we recoverSU(2)*× SU(2) × U(1)*currents even if the isometry is
SU(2)*× U(1) × U(1)*

this is promoted to afﬁne when looking at thenon-local charges

. .

. .

*Symmetries of the squashed group model*

The action with squashed group target space is obtained via T–duality T–dualitypreserves the integrable structureof the PCM

Thefull*bg ⊕ bg*symmetry is preserved

Onlypart of the zero-modes are realized as isometriesg*⊕ u(1)*
The other zero modes are non-local

Adding RR ﬂuxesdoes not change the overall picture:

NS and R sectors are separated under T–duality We already know (from the previous section) the expressions for the RR ﬁelds

The PCM + RR ﬁelds is integrable and has an inﬁnite symmetry

This inﬁnite symmetry will be preserved

. .

. .

*Outline*

. .* ^{1}*.

*Why are we here?*

. .* ^{2}*.

*The main idea*. .

*.*

^{3}*D–brane construction*

* ^{4}*.

*Integrability of the Principal Chiral Model*. .

*.*

^{5}*Heterotic construction (or, the S–dual story)*. .

*.*

^{6}*Conclusion*

. .

. .

### WAdS

_{3}

*in Heterotic strings*

AdS3is the target space for aWess-Zumino-Witten model The action can be written as

*S[g] =* *k*
16π

∫

Σd^{2}*z⟨g*^{−1}*∂g, g*^{−1}*∂g*¯ *⟩ +* *k*
24π

∫

*M*

*⟨g*^{−1}*dg, [g*^{−1}*dg, g*^{−1}*dg]⟩*

*where g∈ SL*2(*R), ∂M = Σ*
What is interesting for us is:

The model describes string propagating on AdS_{3}target space
*ds*^{2}*= k (J*_{1}*⊗ J*_{1}*+ J*_{2}*⊗ J*_{2}*+ J*_{3}*⊗ J*_{3})

The effective action at all orders in α^{′}*is the classical action after a shift in k*

. .

. .

*Marginal deformations*

WZW models admittruly marginal deformations(all orders in α* ^{′}*).

*The operator must have weight (1, 1). So it is written as a product of two*
currents

*O*λ= λ

∫

d^{2}*z*∑

*a,b*

*J*_{a}*(z)¯J** _{b}*(¯

*z) .*

*The deformation S = S**WZW*+*O*λ is truly marginal if the currents commute
among each other

The resulting geometry preserves all the symmetries that commute with these currents.

. .

. .

### WAdS

_{3}

*as a marginal deformation*

*Remember the presence of an external U(1). Consider the*WZW model
*for the group SL*_{2}(*R) × U(1)*

Now add the deformation

*O*λ = λ

∫

d^{2}*z J*_{3}*(z)¯I(¯z)*

*where J*_{3} *is the hyperbolic left current in SL*_{2}(*R) and¯Iis the right current in*
*the U(1) component*

The deformation is exact for all values of λ

*The resulting geometry has symmetry U(1)× SL*2(*R) (the J*_{3}current used
*for the deformation and the right-moving initial SL*_{2}(R))

The metric reads

*ds*^{2}*= J*_{1}*⊗ J*_{1}*+ J*_{2}*⊗ J*_{2}+
(

1*− 2λ*^{2})
*J*_{3}*⊗ J*_{3}

. .

. .

*Outline*

. .* ^{1}*.

*Why are we here?*

. .* ^{2}*.

*The main idea*. .

*.*

^{3}*D–brane construction*

* ^{4}*.

*Integrability of the Principal Chiral Model*. .

*.*

^{5}*Heterotic construction (or, the S–dual story)*. .

*.*

^{6}*Conclusion*

. .

. .

*Summary*

Squashed group manifolds have met renewed attention during the last years Topologically massive gravity

Schrödinger spacetimes

They can be understood as natural deformations of group manifolds Using the Hopf ﬁbration structure we can constructtype II backgrounds.

Using the Lie algebra structure we can construct exactheterotic backgrounds.

Using both structures we can prove their classical integrability

. .

. .

*The end*

.

... .

.

.