Squashed group manifolds in String Theory exact solutions, brane realization, classical integrability

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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)

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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

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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

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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?

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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)

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TMG

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

Stmg(g) = 1 16πG

∫

d³x√

−g (

R + 2 ℓ²

)

+ μ SCS

for μ̸= 1 the anti–de Sitter solution is unstable, but there aretwo warped solutionswith metric of the type

ds²[WAdS₃] = R² [

dω²− cosh²ω dτ²+ 1 cosh²Θ_w

(dβ + sinh ω dτ)2]

where Θ_w is a deformation parameter:

for Θ_w= 0 this is AdS₃ for Θ_w→ ∞ this is AdS2× S¹

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Today’s talk

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

Type IIsolutions with squashed AdS₃and S³ Integrability(without RR fields)

ExactCFTwith squashed group geometry

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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

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The Geometry – Group Manifold

Consider agroup manifoldG (e.g. AdS₃or S³)

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_athat generate half of the symmetry

ds²[G] =

dim G∑

a=1

J_a⊗ J_a

If H⊂ G is compact, G is the total space of aHopf fibrationover G/H:

H −−−−→ G

 y G/H Today H = U(1) and G/H = S², AdS₂.

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The Geometry – Squashed group

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

The metric can be written in terms of the same J_acurrents. Fix dim G = 3 ds²[SqG] = J₁⊗ J₁+ J₂⊗ J₂+ 1

cosh²Θ_wJ₃⊗ J₃ SqG is thebase space for a fibrationthat has G× S¹ total space

S¹× S¹ −−−−→ G × S¹

 y G/U(1)

choose embedding

−−−−−−−−−−→

S¹ −−−−→ G × S¹

 y SqG the embedding of S¹,→ S¹× S¹ is described by Θw

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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 S¹ fibration structure S¹ −−−−→ M

 y N

T-duality along the fiber will “undo” the fibration and give adirect product M = Ne × S¹

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T-duality on the worldsheet

If there is a S¹fibration, the action can be written as S[uⁱ, z] =

∫

Σ

G_ij(u) duⁱ∧ ∗du^j+(

dz + f_i(u) duⁱ)

∧ ∗(

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 S¹described by z.

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T-duality on the worldsheet

introduce a gauge field A and a Lagrange multiplier S[uⁱ, A,ez] =

∫

ΣGij(u) duⁱ∧ ∗du^j+(

A + f_i(u) duⁱ)

∧ ∗(

A + f_i(u) duⁱ)

− 2ezdA .

The EOM forez give

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

The EOM for A gives

∗dez = A + f_i(u) duⁱ= dz + f_i(u) duⁱ.

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T-duality on the worldsheet

The resulting action describes adirect product metricplus aB field S[uⁱ,ez] =

∫

ΣGij(u) duⁱ∧ ∗du^j+ dez ∧ ∗dez − 2 dez ∧ fi(u) duⁱ as promised

S¹ −−−−→ M

 y N

−−−−→

T-duality

M = Ne × S¹

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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

ds²[SqG] = ds²[G] + tanh²Θ 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).

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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

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Initial setup

Consider the superposition of aD₁/D₅system with amagnetic monopole and aplane wave

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

The field content is ds²= H^1/2₁ H^1/2₅

(

H⁻¹₁ H⁻¹₅ (

du dv + K du² )

+ H⁻¹₅ (

dy²₁+ . . . dy²₄ )

+ + V⁻¹(

dψ + A_idxⁱ)2

+ V (

dx²₁+ . . . dx²₃ ))

e^2φ= H⁻¹₁ H₅, F_[3]= H⁻¹₁ dt∧ du ∧ dv − Bidxⁱ∧ dψ where H₁(x), H₅(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 =− ∗ dH5, dA =− ∗ dV

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Initial setup

Consider thenear–horizon limit ds²= Q_mQ^1/2₁ Q^1/2₅

(− dτ²+ dω²+ Q_wdσ²+ 2Q_w^1/2sinh ω dσ dτ )

+ Q_mQ^1/2₁ Q^1/2₅ (

dθ²+ dφ²+ dψ²+ 2 cos θ dψ dφ )

+ Q^1/2₁ Q^−1/2₅ (

dy²₁+· · · + dy²₄) ,

F_[3]= Q_mQ^1/2₁ Q^1/2₅ (

cosh ω dτ∧ dω ∧ dσ + sin θ dφ ∧ dψ ∧ dθ) . The geometry isAdS3× S³× T⁴

The radii are fixed by the monopole charge Q_m(quantized) The plane wave charge appears in the AdS₃part

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Hopf–T–duality

We want to use the fact that

S¹ −−−−→ AdS3× S¹

 y WAdS3

Single out a AdS₃× S¹ part from the ten-dimensional geometry ImplementHopf–T–duality.

If the geometry is the total space for a S¹ fibration and there are only Ramond–Ramond fields, theT–dual along the fiber has geometry B× S¹.

S¹ −−−−→ E

 y B

+ RR fields −−−→^T–dual ( B× S¹)

+ RR and NS fields

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Hopf–T–duality

Starting from AdS₃× S¹: ds²= R²

(− dτ²+ dω²+ Q_wdσ²+ 2√

Q_wsinh ω dσ dτ )

+

√ Q₁ Q₅dy²₁ F_[3]= R²cosh ω dτ∧ dω ∧ dσ

T–duality we obtain the metric we want: WAdS₃× S¹ ds²= R²

[

dω²− cosh²ω dτ²+ 1 cosh²Θ_w

(dβ + sinh ω dτ)₂]

+ dζ²_w

F₄= R²

cosh²Θ_wcosh ω dω∧ dτ ∧ dβ ∧ dζw, F2= R tanh Θwcosh ω dω∧ dτ ,

H3= R tanh Θ_wcosh ω dω∧ dτ ∧ dζw.

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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:

R² = Q_m√

Q₁Q₅ sinh²Θ_w = 4Q_wQ_mQ₅

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

The group SL₂(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 S³)

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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

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Currents and EOM

Consider thePCM for a group G S =−1

2

∫

ΣTr[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⁻¹dg) = d∗(dg g⁻¹) = 0 . these are the conservation laws for twocurrents

j = g⁻¹dg , j =− dg g⁻¹.

the currents areflatand thus fulfill the Maurer–Cartan (MC) equations:

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

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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− ζ²(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:

∂_tJ_x− ∂xJ_t+ [J_t, J_x] = 0 this is a Lax Pair.

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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

ζ²− 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 theloop algebraJ∈ g ⊗ C[ζ, ζ⁻¹] We literally haveinfinitely more currents(after Fourier transform).

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Wilson line

We have constructed infinite currents. Where are theconserved charges?

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

W(x, t|x0, t0; ζ) = P {

exp [∫

C:(x0,t₀)→(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⁻¹dg to the loop algebra.

For spin chain experts: this is the transfer matrix.

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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 aWilson 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 aninfinite set of conserved charges Q_n Q(t; ζ) = 1 +

∑∞ n=0

ζⁿ⁺¹Q⁽ⁿ⁾(t) . for which

d

dtQ⁽ⁿ⁾(t) = 0, ∀ n = 0, 1, . . . .

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Integrability of the PCM (in short)

The principal chiral model has twoconserved currentscorresponding to the G× G symmetryof 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 ⊕ bgsymmetry of the equations of motion.

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Integrability and squashed groups

How much of this structure remainsafter T–duality?

We have alinear transformation of the currentcomponents 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) duⁱ.

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.

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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⁻¹J h + h⁻¹dh ,

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

eJ^′(ζ) = h⁻¹dh− Λ(ζ)h⁻¹jh

dz=∗dez−fi(u) duⁱ = eJ^′(0)− Λ(ζ)ej.

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

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The simplest example: Squashed three-sphere

eJ^′1(ζ) =−ıΛ(ζ) sin θ dφ , eJ^′2(ζ) =−ıΛ(ζ) dθ , eJ^′3(ζ) =−ı

[

(1 + Λ(ζ))

(dα + cos θ dφ

cosh²Θ + tanh Θ∗dez )

− cos θ dφ ]

, eJ^′4(ζ) =−ı tanh ΘΛ(ζ)(

∗dez − tanh Θ(

dα + cos θ dφ)) ,

and

eJ¹(ζ) =−ıΛ(ζ)[ 1

cosh²Θcos φ sin θ dα− sin φ dθ + tanh²Θ cos φ sin θ cos θ dφ + tanh Θ cos φ sin θ∗dez] ,eJ²(ζ) = ıΛ(ζ)[ 1

cosh²Θsin φ sin θ dα + cos φ dθ− tanh²Θ sin φ sin θ cos θ dφ + tanh Θ sin φ sin θ∗dez] , eJ³(ζ) = ıΛ(ζ)

[ 1

cosh²Θcos θ dα +(

1− tanh²Θ cos²θ)

dφ + tanh Θ cos θ∗dez ]

, eJ⁴(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

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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

Thefullbg ⊕ bgsymmetry 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

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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

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WAdS

in Heterotic strings

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

S[g] = k 16π

∫

Σd²z⟨g⁻¹∂g, g⁻¹∂g¯ ⟩ + k 24π

∫

M

⟨g⁻¹dg, [g⁻¹dg, g⁻¹dg]⟩

where g∈ SL2(R), ∂M = Σ What is interesting for us is:

The model describes string propagating on AdS₃target space ds²= k (J₁⊗ J₁+ J₂⊗ J₂+ J₃⊗ J₃)

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

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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

Oλ= λ

∫

d²z∑

a,b

J_a(z)¯J_b(¯z) .

The deformation S = SWZW+Oλ is truly marginal if the currents commute among each other

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

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WAdS

as a marginal deformation

Remember the presence of an external U(1). Consider theWZW model for the group SL₂(R) × U(1)

Now add the deformation

Oλ = λ

∫

d²z J₃(z)¯I(¯z)

where J₃ is the hyperbolic left current in SL₂(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)× SL2(R) (the J₃current used for the deformation and the right-moving initial SL₂(R))

The metric reads

ds²= J₁⊗ J₁+ J₂⊗ J₂+ (

1− 2λ²) J₃⊗ J₃

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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

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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

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The end

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Thank you

for your attention