Silva

Geometry and topology of LLM backgrounds from gauge theory Guillermo Silva IFLP-Departamento de F´ısica, UNLP

PRD 76 (2007), H.Y.Chen, D.Correa, G.A.S. [hep-th/0703068]

Outline AdS/CFT correspondence 1 2

BPS LLM solutions of IIB supergravity

1 2

BPS states of N = 4 SYM

Single traces probing

1 2

BPS states of SYM

↔ Probe strings on LLM geometries

AdS/CFT Correspondence

[Maldacena 97]

N = 4 SYM in d = 4 IIB string theory ←→ G = U (N ) gauge group on AdS5 × S5 R2 , gs 2πα0

2 (gYM , N ) →λ = gYM N, N1

These two theories are conjectured to be equivalent. The parameters are related as:  λ =

2

R α0

2

λ = gs 4πN Rank(G) = F5 flux over S5 The conjecture gives a concrete realization of an old idea suggested by ’t Hooft.

Large N Limit

[’t Hooft 74]

:

• Take the large N → ∞ limit with λ ≡ gY2 M N kept fixed. (This amounts to take gY M → 0 at the same time.) • N12 controls genus expansion. The leading contribution comes from planar diagrams. • N → ∞ implies gs = 0. Strings do not interact with one another (free strings). • Strings nevertheless fluctuate. The problem on the string side is essentially QM. • The fluctuation of the string worldsheet is governed by the effective string tension Tef f =

R2 2πα0

=

√ λ 2π .

The large N limit conserves one parameter: λ

Computability regimes in ’t Hooft limit: • λ1 Weakly coupled gauge theory perturbative regime (keep only planar diagrams). AdS/CFT

⇒

perturbative gauge theory describes free fluctuating strings.

• λ1 Classical (rigid) string theory (Tef f  1). Low energy effective action is IIB supergravity. AdS/CFT

⇒

perturbative supergravity describes strongly coupled gauge theory.

AdS/CFT is a strong/weak coupling duality ⇒ hard to prove

AdS/CFT Dictionary:

[Maldacena 97] [Guser, Klebanov, Polyakov 98] [Witten 98]

• Both theories have the same Hilbert space: • Global symmmetries in the CFT

↔

|ψiSYM

↔

|ψiAdS

Isometries of AdS5 × S 5

The full symmetry group is PSU(2,2|4) with bosonic subgroup: SO(4, 2) × d=4 Conformal group d = 10 AdS5 isometry group

SO(6) R symmetry group S5 isometry group

States on H carry quantum numbers corresponding to PSU(2,2|4). — Cartan PSU(2, 2|4)

=

(s+ , s− ) SO(3, 1)

×

∆ SO(1, 1)

×

(r1 , r2 , r3 ) SO(6)R

• SYM gauge theory is defined on R × S 3 ↔ Conf. boundary of AdS5 × S 5 time on SYM IIB time of AdS ≡ on R × S3 in global coordinates Equivalently, DSYM ≡ Hstring The spectrum of both operators should coincide, but, how to check?

Keep in mind:

∆ scale dimension in R4

R-charges

↔

E energy in global AdS coordinate

↔

J momentum comp. along S 5

LLM geometries All

1 2

[Lin, Lunin, Maldacena 04]

BPS IIB supergravity solutions with SO(4) × SO(4) × R isometry

˜ 23 ds2 = −h−2(dt + Vadxa)2 + h2(dy 2 + dx2a) + yeGdΩ23 + ye−GdΩ

LLM geometries All

1 2

[Lin, Lunin, Maldacena 04]

BPS IIB supergravity solutions with SO(4) × SO(4) × R isometry

˜ 23 ds2 = −h−2(dt + Vadxa)2 + h2(dy 2 + dx2a) + yeGdΩ23 + ye−GdΩ Functions Va, h, G are determined in terms of ρ(x1, x2, y), satisfying  ∂i∂iρ(x1, x2, y) + y∂y

∂y ρ(x1, x2, y) y

 =0

LLM geometries All

1 2

[Lin, Lunin, Maldacena 04]

BPS IIB supergravity solutions with SO(4) × SO(4) × R isometry

˜ 23 ds2 = −h−2(dt + Vadxa)2 + h2(dy 2 + dx2a) + yeGdΩ23 + ye−GdΩ Functions Va, h, G are determined in terms of ρ(x1, x2, y), satisfying  ∂i∂iρ(x1, x2, y) + y∂y

∂y ρ(x1, x2, y) y

with boundary data given on 2d LLM plane y = 0.

 =0

LLM geometries All

1 2

[Lin, Lunin, Maldacena 04]

BPS IIB supergravity solutions with SO(4) × SO(4) × R isometry

˜ 23 ds2 = −h−2(dt + Vadxa)2 + h2(dy 2 + dx2a) + yeGdΩ23 + ye−GdΩ Functions Va, h, G are determined in terms of ρ(x1, x2, y), satisfying  ∂i∂iρ(x1, x2, y) + y∂y

∂y ρ(x1, x2, y) y

with boundary data given on 2d LLM plane y = 0. Regular geometries demand ρ(x1, x2, 0) = ±1/2

 =0

LLM geometries All

1 2

[Lin, Lunin, Maldacena 04]

BPS IIB supergravity solutions with SO(4) × SO(4) × R isometry

˜ 23 ds2 = −h−2(dt + Vadxa)2 + h2(dy 2 + dx2a) + yeGdΩ23 + ye−GdΩ Functions Va, h, G are determined in terms of ρ(x1, x2, y), satisfying  ∂i∂iρ(x1, x2, y) + y∂y

∂y ρ(x1, x2, y) y

with boundary data given on 2d LLM plane y = 0. Regular geometries demand ρ(x1, x2, 0) = ±1/2 Any two-coloring of the plane represents a solution.

 =0

The topology of the droplet picture determines the topology of the solution. On y = 0 plane either S 3 or S˜3 collapse to zero radius: ρ = −1/2 ρ = 1/2

→ →

S 3 shrinks to zero S˜3 shrinks to zero

The topology of the droplet picture determines the topology of the solution. On y = 0 plane either S 3 or S˜3 collapse to zero radius: ρ = −1/2 ρ = 1/2

→ →

S 3 shrinks to zero S˜3 shrinks to zero

Boundary conditions given by a disk generate AdS5 × S 5.

The topology of the droplet picture determines the topology of the solution. On y = 0 plane either S 3 or S˜3 collapse to zero radius: ρ = −1/2 ρ = 1/2

→ →

S 3 shrinks to zero S˜3 shrinks to zero

Boundary conditions given by a disk generate AdS5 × S 5. Any static point on the boundary of the droplet corresponds to a null geodesic. Zooming around an edge → Penrose limit

The topology of the droplet picture determines the topology of the solution. On y = 0 plane either S 3 or S˜3 collapse to zero radius: ρ = −1/2 ρ = 1/2

→ →

S 3 shrinks to zero S˜3 shrinks to zero

Boundary conditions given by a disk generate AdS5 × S 5. Any static point on the boundary of the droplet corresponds to a null geodesic. Zooming around an edge → Penrose limit Half filled plane generates the maximally supersymetric plane wave.

Matrix model for

1 2

BPS states

[Corley, Jevicki, Ramgoolam 01] [Berenstein 04]

N = 4 SYM is defined on R × S 3. N = 4 SYM field content is: X, Y, Z, Aµ, fermions in the adjoint of U (N ).

Matrix model for

1 2

BPS states

[Corley, Jevicki, Ramgoolam 01] [Berenstein 04]

N = 4 SYM is defined on R × S 3. N = 4 SYM field content is: X, Y, Z, Aµ, fermions in the adjoint of U (N ). General

1 2

BPS operators of N = 4 SYM:

Matrix model for

1 2

BPS states

[Corley, Jevicki, Ramgoolam 01] [Berenstein 04]

N = 4 SYM is defined on R × S 3. N = 4 SYM field content is: X, Y, Z, Aµ, fermions in the adjoint of U (N ). General

1 2

BPS operators of N = 4 SYM:

• Constructed out from a single complex scalar say Z O = O(Z) ⇒ preserve SO(4) ⊂ SO(6)R . Call J the U (1) ⊂ SO(6)R generator corresponding to Z → eiαZ.

Matrix model for

1 2

BPS states

[Corley, Jevicki, Ramgoolam 01] [Berenstein 04]

N = 4 SYM is defined on R × S 3. N = 4 SYM field content is: X, Y, Z, Aµ, fermions in the adjoint of U (N ). General

1 2

BPS operators of N = 4 SYM:

• Constructed out from a single complex scalar say Z O = O(Z) ⇒ preserve SO(4) ⊂ SO(6)R . Call J the U (1) ⊂ SO(6)R generator corresponding to Z → eiαZ. • Only s-waves should be kept in the harmonic expansion of Z on S 3 Z = Z(t) ⇒ preserve SO(4) ⊂ SO(4, 2) .

Matrix model for

1 2

BPS states

[Corley, Jevicki, Ramgoolam 01] [Berenstein 04]

N = 4 SYM is defined on R × S 3. N = 4 SYM field content is: X, Y, Z, Aµ, fermions in the adjoint of U (N ). General

1 2

BPS operators of N = 4 SYM:

• Constructed out from a single complex scalar say Z O = O(Z) ⇒ preserve SO(4) ⊂ SO(6)R . Call J the U (1) ⊂ SO(6)R generator corresponding to Z → eiαZ. • Only s-waves should be kept in the harmonic expansion of Z on S 3 Z = Z(t) ⇒ preserve SO(4) ⊂ SO(4, 2) . • (D − J) O = 0 generates a translation invariance R.

Matrix model for

1 2

BPS states

[Corley, Jevicki, Ramgoolam 01] [Berenstein 04]

N = 4 SYM is defined on R × S 3. N = 4 SYM field content is: X, Y, Z, Aµ, fermions in the adjoint of U (N ). General

1 2

BPS operators of N = 4 SYM:

• Constructed out from a single complex scalar say Z O = O(Z) ⇒ preserve SO(4) ⊂ SO(6)R . Call J the U (1) ⊂ SO(6)R generator corresponding to Z → eiαZ. • Only s-waves should be kept in the harmonic expansion of Z on S 3 Z = Z(t) ⇒ preserve SO(4) ⊂ SO(4, 2) . • (D − J) O = 0 generates a translation invariance R. ¯ = 0. • Z is normal matrix: [Z, Z]

Matrix model for

1 2

BPS states

[Corley, Jevicki, Ramgoolam 01] [Berenstein 04]

N = 4 SYM is defined on R × S 3. N = 4 SYM field content is: X, Y, Z, Aµ, fermions in the adjoint of U (N ). General

1 2

BPS operators of N = 4 SYM:

• Constructed out from a single complex scalar say Z O = O(Z) ⇒ preserve SO(4) ⊂ SO(6)R . Call J the U (1) ⊂ SO(6)R generator corresponding to Z → eiαZ. • Only s-waves should be kept in the harmonic expansion of Z on S 3 Z = Z(t) ⇒ preserve SO(4) ⊂ SO(4, 2) . • (D − J) O = 0 generates a translation invariance R. ¯ = 0. • Z is normal matrix: [Z, Z] 1 2

BPS states preserve a SO(4) × SO(4) × R symmetry

The reduced action for the N × N normal matrix Z(t) is Z   1 ˙ 2 − |Z|2 S= dt tr |Z| 2

The reduced action for the N × N normal matrix Z(t) is Z   1 ˙ 2 − |Z|2 S= dt tr |Z| 2 A normal matrix can be diagonalized: Z = U DU −1 with U ∈ U (N ). U factors out for U (N ) gauge invariant states. Dynamics of states reduces to the dynamics of the N eigenvalues of Z.

The reduced action for the N × N normal matrix Z(t) is Z   1 ˙ 2 − |Z|2 S= dt tr |Z| 2 A normal matrix can be diagonalized: Z = U DU −1 with U ∈ U (N ). U factors out for U (N ) gauge invariant states. Dynamics of states reduces to the dynamics of the N eigenvalues of Z. 1 2

BPS states reduces to N particles in a harmonic oscillator potential

The reduced action for the N × N normal matrix Z(t) is Z   1 ˙ 2 − |Z|2 S= dt tr |Z| 2 A normal matrix can be diagonalized: Z = U DU −1 with U ∈ U (N ). U factors out for U (N ) gauge invariant states. Dynamics of states reduces to the dynamics of the N eigenvalues of Z. 1 2

BPS states reduces to N particles in a harmonic oscillator potential

Residual permutation of eigenvalues would imply symmetric wave functions, but, the change of variables Zji → (zi, θα) entails a Jacobian µ2 Y i,j

dZji

=µ

2

Y i

dzi

Y α

dθα

2

µ =

Q

i<j |zi

− zj |2

(a.k.a. Vandermonde det.)

The Jacobian enters in the Hamiltonian through the Laplacian   X Y 1 −2 ∂ 2 ∂ 2 2 H= −µ |zi − zj |2 µ + |zi| µ = 2 i ∂ z¯i ∂zi i<j

The Jacobian enters in the Hamiltonian through the Laplacian   X Y 1 −2 ∂ 2 ∂ 2 2 H= −µ |zi − zj |2 µ + |zi| µ = 2 i ∂ z¯i ∂zi i<j Eigenstates of H are given in terms Ω(Z) (holomorphic in Z) as X tr Ω(Z) ψΩ(z1, · · · , zN ) = e exp(− ziz¯i/2) i

The Jacobian enters in the Hamiltonian through the Laplacian   X Y 1 −2 ∂ 2 ∂ 2 2 H= −µ |zi − zj |2 µ + |zi| µ = 2 i ∂ z¯i ∂zi i<j Eigenstates of H are given in terms Ω(Z) (holomorphic in Z) as X tr Ω(Z) ψΩ(z1, · · · , zN ) = e exp(− ziz¯i/2) i

Each Ω characterizes a

1 2

BPS state

The Jacobian enters in the Hamiltonian through the Laplacian   X Y 1 −2 ∂ 2 ∂ 2 2 H= −µ |zi − zj |2 µ + |zi| µ = 2 i ∂ z¯i ∂zi i<j Eigenstates of H are given in terms Ω(Z) (holomorphic in Z) as X tr Ω(Z) ψΩ(z1, · · · , zN ) = e exp(− ziz¯i/2) i

Each Ω characterizes a

1 2

BPS state

Normalization of ψΩ ↔ Partition function of random normal matrix model Z Z X X 2 2 2 2 d zi µ |ψΩ| = d zi exp(− W (zi, z¯i) + 2 log |zi − zj |) i

2

W (z, z¯) = −|z| + Ω(z) + Ω(z)

i<j

µ2 amounts to a repulsive interaction zi become fermions

Large N limit: sums are approximated by integrals X

Z (...) →

d2z ρ(z) (...)

[Berenstein 05]

Z N=

i

introducing the the eigenvalue density distribution ρ(z).

d2z ρ(z)

Large N limit: sums are approximated by integrals X

Z (...) →

[Berenstein 05]

Z

d2z ρ(z) (...)

N=

d2z ρ(z)

i

introducing the the eigenvalue density distribution ρ(z). Minimizing the exponential argument give eom for the most likely distribution |z|2 − Ω(z) − Ω(z) = 2 Acting with ∇2 = 2∂z ∂z¯ one obtains:

Z

d2z 0ρ(z 0) log(|z − z 0|) ρ(z) = π1 .

Large N limit: sums are approximated by integrals X

Z (...) →

[Berenstein 05]

Z

d2z ρ(z) (...)

N=

d2z ρ(z)

i

introducing the the eigenvalue density distribution ρ(z). Minimizing the exponential argument give eom for the most likely distribution |z|2 − Ω(z) − Ω(z) = 2 Acting with ∇2 = 2∂z ∂z¯ one obtains:

Z

d2z 0ρ(z 0) log(|z − z 0|) ρ(z) = π1 .

Eigenvalues distribute uniformly in a droplet. The droplet shape depends on Ω(z). Standard matrix model techniques can be used to determine the shape in a few paradigmatic examples:

ρ=0 ρ=

1 π

Ω(z) = 0 ψ0(z1, · · · , zN ) = exp(−

X

ziz¯i/2)

i

Eigenvalues distribute uniformly in a circular droplet.

Ω(z) = 0 ψ0(z1, · · · , zN ) = exp(−

X

ziz¯i/2)

i

Eigenvalues distribute uniformly in a circular droplet.

Ω(z) = M log(z)

[Berenstein 05]

M

ψ(z1, · · · , zN ) = det(Z)

exp(−

X

ziz¯i/2)

i

Eigenvalues distribute uniformly in an annular droplet.

Ω(z) = 0 ψ0(z1, · · · , zN ) = exp(−

X

ziz¯i/2)

i

Eigenvalues distribute uniformly in a circular droplet.

Ω(z) = M log(z)

[Berenstein 05]

M

ψ(z1, · · · , zN ) = det(Z)

exp(−

X

ziz¯i/2)

i

Eigenvalues distribute uniformly in an annular droplet. Ω(z) = 2 z 2

[V´azquez 06]

ψ(z1, · · · , zN ) = e

tr(Z 2 )/2

exp(−

X

ziz¯i/2)

i

Eigenvalues distribute uniformly in an elliptical droplet.

What did we do with this set-up?

[V´azquez 06] [Chen, Correa, Silva 07]

• Evidence showing the correspondence |ψiSYM ↔ LLM geometry, for BPS states seems too qualitative (a common droplet picture).

What did we do with this set-up?

[V´azquez 06] [Chen, Correa, Silva 07]

• Evidence showing the correspondence |ψiSYM ↔ LLM geometry, for BPS states seems too qualitative (a common droplet picture). Idea: Placing a single trace on top of the BPS operator render it non-BPS. This should correspond to a light string probing the LLM geometry, e.g. O = tr(ZY ZZ...Y Y Z) · OBPS

OBPS ↔ backreacted AdS5 × S 5 tr (. . .) ↔ Closed string excitation

Extrapolating the 1-loop anomalous dimension analysis to strong coupling (`a la BMN) may reassure the correspondence (quantitative check).

What did we do with this set-up?

[V´azquez 06] [Chen, Correa, Silva 07]

• Evidence showing the correspondence |ψiSYM ↔ LLM geometry, for BPS states seems too qualitative (a common droplet picture). Idea: Placing a single trace on top of the BPS operator render it non-BPS. This should correspond to a light string probing the LLM geometry, e.g. O = tr(ZY ZZ...Y Y Z) · OBPS

OBPS ↔ backreacted AdS5 × S 5 tr (. . .) ↔ Closed string excitation

Extrapolating the 1-loop anomalous dimension analysis to strong coupling (`a la BMN) may reassure the correspondence (quantitative check). As a by product we shown that some geometric and topological features of the dual LLM solutions pop up from gauge theory computations!

Scale dimensions analysis in the CFT

[Minahan, Zarembo 02] [Beisert, Kristjansen, Staudacher 03]

Dilatation operator: Can be computed from two-point correlation functions Γab : mixing matrix of 1 † hOa(x)Ob(0)i = 2∆ (δab − 2Γab ln(Λ|x|) + . . .) anomalous dimensions |x| 0 • At 1 and higher-loop operators with the same classical dimension mix.

Scale dimensions analysis in the CFT

[Minahan, Zarembo 02] [Beisert, Kristjansen, Staudacher 03]

Dilatation operator: Can be computed from two-point correlation functions Γab : mixing matrix of 1 † hOa(x)Ob(0)i = 2∆ (δab − 2Γab ln(Λ|x|) + . . .) anomalous dimensions |x| 0 • At 1 and higher-loop operators with the same classical dimension mix. • Diagonalizing Γ one obtains the perturbative correction in λ to ∆0.

Scale dimensions analysis in the CFT

[Minahan, Zarembo 02] [Beisert, Kristjansen, Staudacher 03]

Dilatation operator: Can be computed from two-point correlation functions Γab : mixing matrix of 1 † hOa(x)Ob(0)i = 2∆ (δab − 2Γab ln(Λ|x|) + . . .) anomalous dimensions |x| 0 • At 1 and higher-loop operators with the same classical dimension mix. • Diagonalizing Γ one obtains the perturbative correction in λ to ∆0. Dilatation operator on the SU (2) sector: Z and Y in O(x) ⇒ two R charges

Scale dimensions analysis in the CFT

[Minahan, Zarembo 02] [Beisert, Kristjansen, Staudacher 03]

Dilatation operator: Can be computed from two-point correlation functions Γab : mixing matrix of 1 † hOa(x)Ob(0)i = 2∆ (δab − 2Γab ln(Λ|x|) + . . .) anomalous dimensions |x| 0 • At 1 and higher-loop operators with the same classical dimension mix. • Diagonalizing Γ one obtains the perturbative correction in λ to ∆0. Dilatation operator on the SU (2) sector: Z and Y in O(x) ⇒ two R charges The action of the dilatation operator can be computed order by order in λ D = D0 + λD1 + O(λ2)

D0 = tr(Z∂Z + Y ∂Y ) D1 = N −1tr([Z, Y][∂Z, ∂Y ])

D0 gives the classical scale dimension and D1 is a mixing matrix.

Labelling: single trace operators on top of det as states of a bosonic lattice Anullar distribution :

tr(Y Z n1 Y Z n2 . . . Y Z nL ) det Z M

Y ’s define a L sites lattice and Z’s are boson occupancy

↔ |n1, n2, . . . , nLi [Berenstein, Correa, V´azquez 05]

.

Labelling: single trace operators on top of det as states of a bosonic lattice Anullar distribution :

tr(Y Z n1 Y Z n2 . . . Y Z nL ) det Z M

↔ |n1, n2, . . . , nLi

Y ’s define a L sites lattice and Z’s are boson occupancy [Berenstein, Correa, V´azquez 05]. The action of ∂Z from D1 when hitting the determinant and proper account of indices imply that integers in probe trace can be negative!

Labelling: single trace operators on top of det as states of a bosonic lattice Anullar distribution :

tr(Y Z n1 Y Z n2 . . . Y Z nL ) det Z M

↔ |n1, n2, . . . , nLi

Y ’s define a L sites lattice and Z’s are boson occupancy [Berenstein, Correa, V´azquez 05]. The action of ∂Z from D1 when hitting the determinant and proper account of indices imply that integers in probe trace can be negative! • The action of D1 in the large N limit is closed on the set above and takes the form [Chen, Correa, Silva 07] D1 ∼ 2

L X l=1

a ˆ†l a ˆl −

L−1 X l=1

(ˆ a†l a ˆl+1 + a ˆl a ˆ†l+1)

Labelling: single trace operators on top of det as states of a bosonic lattice Anullar distribution :

tr(Y Z n1 Y Z n2 . . . Y Z nL ) det Z M

↔ |n1, n2, . . . , nLi

Y ’s define a L sites lattice and Z’s are boson occupancy [Berenstein, Correa, V´azquez 05]. The action of ∂Z from D1 when hitting the determinant and proper account of indices imply that integers in probe trace can be negative! • The action of D1 in the large N limit is closed on the set above and takes the form [Chen, Correa, Silva 07] D1 ∼ 2

L X l=1

a ˆ†l a ˆl −

L−1 X

(ˆ a†l a ˆl+1 + a ˆl a ˆ†l+1)

l=1

Where shift operators a ˆi , a ˆ†i lower and raise the occupation at each site with  √ M 1 + γ |ni − 1i if ni > 0 † √ γ= a ˆi |nii = |ni + 1i a ˆi|ni = γ |ni − 1i if ni ≤ 0 N

Labelling: single trace operators on top of det as states of a bosonic lattice Anullar distribution :

tr(Y Z n1 Y Z n2 . . . Y Z nL ) det Z M

↔ |n1, n2, . . . , nLi

Y ’s define a L sites lattice and Z’s are boson occupancy [Berenstein, Correa, V´azquez 05]. The action of ∂Z from D1 when hitting the determinant and proper account of indices imply that integers in probe trace can be negative! • The action of D1 in the large N limit is closed on the set above and takes the form [Chen, Correa, Silva 07] D1 ∼ 2

L X l=1

a ˆ†l a ˆl −

L−1 X

(ˆ a†l a ˆl+1 + a ˆl a ˆ†l+1)

l=1

Where shift operators a ˆi , a ˆ†i lower and raise the occupation at each site with  √ M 1 + γ |ni − 1i if ni > 0 † √ γ= a ˆi |nii = |ni + 1i a ˆi|ni = γ |ni − 1i if ni ≤ 0 N All information of LLM dual geometry is contained in the a ˆi , a ˆ†i definition.

Consider the two-sites (impurities) Hamiltonian: †

†

†

†

H = λD1 = 2λ(a1a1 + a2a2 − a2a1 − a1a2) • Total occupation J is conserved (virtue of droplets axial symmetry), =⇒ Eigenstates con be chosen to have definite J ∞ X fm|J − m, mi |φ, Ji = m=−∞

Consider the two-sites (impurities) Hamiltonian: †

†

†

†

H = λD1 = 2λ(a1a1 + a2a2 − a2a1 − a1a2) • Total occupation J is conserved (virtue of droplets axial symmetry), =⇒ Eigenstates con be chosen to have definite J ∞ X solving a simple recursive eq for fm fm|J − m, mi |φ, Ji = m=−∞

The spectra for large ±J are

En(+J) = (1 + γ)8π 2n2 Jλ2 + O( J13 ) En(+J) = γ 8π 2n2 Jλ2 + O( J13 )

Consider the two-sites (impurities) Hamiltonian: †

†

†

†

H = λD1 = 2λ(a1a1 + a2a2 − a2a1 − a1a2) • Total occupation J is conserved (virtue of droplets axial symmetry), =⇒ Eigenstates con be chosen to have definite J ∞ X solving a simple recursive eq for fm fm|J − m, mi |φ, Ji = m=−∞

The spectra for large ±J are

En(+J) = (1 + γ)8π 2n2 Jλ2 + O( J13 ) En(+J) = γ 8π 2n2 Jλ2 + O( J13 )

Perfect agreement with dual string spectra zooming along ext/int edges s

(1 + γ)8π 2 n2 λ 2 2 λ ≈ J + 2 + (1 + γ)8π n 2 J2 J s γ 8π 2 n2 λ (−J) 2 2 λ Eppwave = −J + 2 + 2 1 + ≈ −J + 2 + γ 8π n 2 J2 J

(+J)

Eppwave = J + 2 + 2

1+

Consider the two-sites (impurities) Hamiltonian: †

†

†

†

H = λD1 = 2λ(a1a1 + a2a2 − a2a1 − a1a2) • Total occupation J is conserved (virtue of droplets axial symmetry), =⇒ Eigenstates con be chosen to have definite J ∞ X solving a simple recursive eq for fm fm|J − m, mi |φ, Ji = m=−∞

The spectra for large ±J are

En(+J) = (1 + γ)8π 2n2 Jλ2 + O( J13 ) En(+J) = γ 8π 2n2 Jλ2 + O( J13 )

Perfect agreement with dual string spectra zooming along ext/int edges s

(1 + γ)8π 2 n2 λ 2 2 λ ≈ J + 2 + (1 + γ)8π n 2 J2 J s γ 8π 2 n2 λ (−J) 2 2 λ Eppwave = −J + 2 + 2 1 + ≈ −J + 2 + γ 8π n 2 J2 J

(+J)

Eppwave = J + 2 + 2

1+

Large positive (negative) occupation is the field theory realization of short strings localized in the exterior (interior) edge.

BMN limit: Going to strong coupling

[Berenstein, Maldacena, Nastase 02]

Key idea: work with operators carrying a large quantum numbers, say J. This new parameter appears when performing computations and the outcome is that an effective coupling appears λ λef f = 2 J

BMN limit: Going to strong coupling

[Berenstein, Maldacena, Nastase 02]

Key idea: work with operators carrying a large quantum numbers, say J. This new parameter appears when performing computations and the outcome is that an effective coupling appears λ λef f = 2 J E.g.: O1 = tr(Z J Y 2) ↔ O2 = tr(Z nY Z J−nY ) mix to 1-loop order. The anomalous dimension looks λ ∆ − J − 2 = 2 # + O(λ2ef f ) J

BMN limit: Going to strong coupling

[Berenstein, Maldacena, Nastase 02]

Key idea: work with operators carrying a large quantum numbers, say J. This new parameter appears when performing computations and the outcome is that an effective coupling appears λ λef f = 2 J E.g.: O1 = tr(Z J Y 2) ↔ O2 = tr(Z nY Z J−nY ) mix to 1-loop order. The anomalous dimension looks λ ∆ − J − 2 = 2 # + O(λ2ef f ) J BMN limit: take λ and J (Z fields in trace) → ∞, keeping λef f fixed and small λ0  1 ⇒ perturbative expansion in λ0 valid λ  1 ⇒ computation valid in string regime

BMN limit: Going to strong coupling

[Berenstein, Maldacena, Nastase 02]

Key idea: work with operators carrying a large quantum numbers, say J. This new parameter appears when performing computations and the outcome is that an effective coupling appears λ λef f = 2 J E.g.: O1 = tr(Z J Y 2) ↔ O2 = tr(Z nY Z J−nY ) mix to 1-loop order. The anomalous dimension looks λ ∆ − J − 2 = 2 # + O(λ2ef f ) J BMN limit: take λ and J (Z fields in trace) → ∞, keeping λef f fixed and small λ0  1 ⇒ perturbative expansion in λ0 valid λ  1 ⇒ computation valid in string regime Dual viewpoint: nearly-BPS operators ↔ almost null geodesic. Suggest study the Penrose limit to the massless geodesic corresponding to BPS operator. 2 fixed and small, N → ∞, J → (BMN limit: gYM

√

N

Thermodynamic/semiclassical limit for D :

[Kruczenski 03] [V´azquez 06] [Chen, Correa, Silva 07]

Take L (number of lattice sites) to infinity `a la BMN:  λ →√ ∞ λ/L2  1 as L→ λ These allows perturbative results to be extrapolated to strong coupling.

Thermodynamic/semiclassical limit for D :

[Kruczenski 03] [V´azquez 06] [Chen, Correa, Silva 07]

Take L (number of lattice sites) to infinity `a la BMN:  λ →√ ∞ λ/L2  1 as L→ λ These allows perturbative results to be extrapolated to strong coupling. • A semiclassical σ -model action for D can be obtained by using a coherent state basis in a path integral representation of the evolution operator (taking λD1 as the Hamiltonian of the bosonic lattice).

Thermodynamic/semiclassical limit for D :

[Kruczenski 03] [V´azquez 06] [Chen, Correa, Silva 07]

Take L (number of lattice sites) to infinity `a la BMN:  λ →√ ∞ λ/L2  1 as L→ λ These allows perturbative results to be extrapolated to strong coupling. • A semiclassical σ -model action for D can be obtained by using a coherent state basis in a path integral representation of the evolution operator (taking λD1 as the Hamiltonian of the bosonic lattice). Defining coherent states for a ˆ as:

a ˆ|zi = z|zi

Thermodynamic/semiclassical limit for D :

[Kruczenski 03] [V´azquez 06] [Chen, Correa, Silva 07]

Take L (number of lattice sites) to infinity `a la BMN:  λ →√ ∞ λ/L2  1 as L→ λ These allows perturbative results to be extrapolated to strong coupling. • A semiclassical σ -model action for D can be obtained by using a coherent state basis in a path integral representation of the evolution operator (taking λD1 as the Hamiltonian of the bosonic lattice). Defining coherent states for a ˆ as:

a ˆ|zi = z|zi

The classical action for coherent states parameters z1(t), ...zL(t) is

Z S =

  ∂ dt ihz1(t)...zL(t)| |z1(t)...zL(t)i − hz1(t)...zL(t)|H|z1(t)...zL(t)i ∂t

Taking a continuum limit |z1(t)i ⊗ · · · ⊗ |zL(t)i

L→∞

−→

|z(t, σ)i 0 ≤ σ ≤ 1

Taking a continuum limit |z1(t)i ⊗ · · · ⊗ |zL(t)i One obtains

L→∞

−→

|z(t, σ)i 0 ≤ σ ≤ 1

[V´azquez 06] [Chen, Correa, Silva 07]

Z

Z

1





i λ i V z¯˙ − V¯ z˙ − λ0|z 0|2 λ0 ≡ 2 2 2 L 0 Here V (z, z¯) is given in terms of the coherent states norms as S=L

dt

dσ

V (z, z¯) = ∂z¯ log(hz|zi).

Taking a continuum limit |z1(t)i ⊗ · · · ⊗ |zL(t)i One obtains

L→∞

−→

|z(t, σ)i 0 ≤ σ ≤ 1

[V´azquez 06] [Chen, Correa, Silva 07]

Z

Z

1





i λ i V z¯˙ − V¯ z˙ − λ0|z 0|2 λ0 ≡ 2 2 2 L 0 Here V (z, z¯) is given in terms of the coherent states norms as S=L

dt

dσ

V (z, z¯) = ∂z¯ log(hz|zi). Each BPS operator will give a different coherent state norm functions.

Taking a continuum limit |z1(t)i ⊗ · · · ⊗ |zL(t)i One obtains

L→∞

−→

|z(t, σ)i 0 ≤ σ ≤ 1

[V´azquez 06] [Chen, Correa, Silva 07]

Z

Z

1





i λ i V z¯˙ − V¯ z˙ − λ0|z 0|2 λ0 ≡ 2 2 2 L 0 Here V (z, z¯) is given in terms of the coherent states norms as S=L

dt

dσ

V (z, z¯) = ∂z¯ log(hz|zi). Each BPS operator will give a different coherent state norm functions. In all cases λ hz1...zL|H|z1...zLi → L

Z 0

1

dσ|z 0|2

Taking a continuum limit |z1(t)i ⊗ · · · ⊗ |zL(t)i One obtains

L→∞

−→

|z(t, σ)i 0 ≤ σ ≤ 1

[V´azquez 06] [Chen, Correa, Silva 07]

Z

Z

1





i λ i V z¯˙ − V¯ z˙ − λ0|z 0|2 λ0 ≡ 2 2 2 L 0 Here V (z, z¯) is given in terms of the coherent states norms as S=L

dt

dσ

V (z, z¯) = ∂z¯ log(hz|zi). Each BPS operator will give a different coherent state norm functions. In all cases λ hz1...zL|H|z1...zLi → L

Z

1

dσ|z 0|2

0

For large L the semiclassical description is valid

Annular droplet case:

[Chen, Correa, Silva 07]

Fock space states for shift operators defined for positive and negative occupation a ˆ|zi = z|zi

Coherent state defined as:  √ a ˆ|ni =

|zi =

1 + γ |n − 1i √ γ |n − 1i

−1 X n=−∞

(γ)

−n/2

if n > 0 if n ≤ 0

z n|ni +

∞ X n=0

(1 + γ)

γ= −n/2

z n|ni

M N

Annular droplet case:

[Chen, Correa, Silva 07]

Fock space states for shift operators defined for positive and negative occupation a ˆ|zi = z|zi

Coherent state defined as:  √ a ˆ|ni =

|zi =

1 + γ |n − 1i √ γ |n − 1i

−1 X n=−∞

(γ)

−n/2

if n > 0 if n ≤ 0

z n|ni +

∞ X

M N

γ=

(1 + γ)

−n/2

z n|ni

n=0

The norm hz|zi is finite in an annular domain

√

γ < |z| <

√

1 + γ.

Annular droplet case:

[Chen, Correa, Silva 07]

Fock space states for shift operators defined for positive and negative occupation a ˆ|zi = z|zi

Coherent state defined as:  √ a ˆ|ni =

|zi =

1 + γ |n − 1i √ γ |n − 1i

−1 X

(γ)

−n/2

if n > 0 if n ≤ 0

z n|ni +

n=−∞

∞ X

(1 + γ)

−n/2

z n|ni

n=0

The norm hz|zi is finite in an annular domain V (z, z¯) = ∂z¯ log(hz|zi) =

M N

γ=

√

γ < |z| <

√

1 + γ.

z γ + 1 + γ − |z|2 z¯(γ − |z|2)

We recover the V function of the annular LLM droplet.

Semiclassical strings in LLM geometries

[Kruczenski, Ryzhov, Tseytlin 04]

We now show that the semiclassical description for D computed before matches the Polyakov action when • a “gauge” is chosen such that the angular momentum L in the transverse sphere is uniformly distributed along the string (realization of bosonic labelling)

Semiclassical strings in LLM geometries

[Kruczenski, Ryzhov, Tseytlin 04]

We now show that the semiclassical description for D computed before matches the Polyakov action when • a “gauge” is chosen such that the angular momentum L in the transverse sphere is uniformly distributed along the string (realization of bosonic labelling) • in a “non-relativistic” limit (first order in σ 0 derivatives)

Semiclassical strings in LLM geometries

[Kruczenski, Ryzhov, Tseytlin 04]

We now show that the semiclassical description for D computed before matches the Polyakov action when • a “gauge” is chosen such that the angular momentum L in the transverse sphere is uniformly distributed along the string (realization of bosonic labelling) • in a “non-relativistic” limit (first order in σ 0 derivatives) Natural formulation is to start with Polyakov action in momentum space LPol

1 B µν µ ν (G pµpν + Gµν ∂1X ∂1X ) + pµ∂1X µ = pµ∂0X + 2A A √ A = √−hh00 ν ν pµ = Gµν (A∂0X + B∂1X ) B = −hh01 µ

Semiclassical strings in LLM geometries

[Kruczenski, Ryzhov, Tseytlin 04]

We now show that the semiclassical description for D computed before matches the Polyakov action when • a “gauge” is chosen such that the angular momentum L in the transverse sphere is uniformly distributed along the string (realization of bosonic labelling) • in a “non-relativistic” limit (first order in σ 0 derivatives) Natural formulation is to start with Polyakov action in momentum space LPol

1 B µν µ ν (G pµpν + Gµν ∂1X ∂1X ) + pµ∂1X µ = pµ∂0X + 2A A √ A = √−hh00 ν ν pµ = Gµν (A∂0X + B∂1X ) B = −hh01 µ

This is the convenient formulation to fix some pµ to be uniformly distributed.

For a string restricted to a dark region in the LLM plane, calling η ↔ L  2 i i ds2 = −h−2 dt + V d¯ z + h−2dη 2 z − V¯ dz + h2dzd¯ 2 2

For a string restricted to a dark region in the LLM plane, calling η ↔ L  2 i i ds2 = −h−2 dt + V d¯ z + h−2dη 2 z − V¯ dz + h2dzd¯ 2 2 Instead of standard conformal gauge fixing we demand t=τ

L 1 pη = const. = √ ≡ √ λ λ0

For a string restricted to a dark region in the LLM plane, calling η ↔ L  2 i i ds2 = −h−2 dt + V d¯ z + h−2dη 2 z − V¯ dz + h2dzd¯ 2 2 Instead of standard conformal gauge fixing we demand L 1 pη = const. = √ ≡ √ λ λ0

t=τ

After imposing Virasoro constraints and solving eom for pz and pz¯ LPol = L(V (z), h(z), z, ˙ z 0) L is a complicated function.

For a string restricted to a dark region in the LLM plane, calling η ↔ L  2 i i ds2 = −h−2 dt + V d¯ z + h−2dη 2 z − V¯ dz + h2dzd¯ 2 2 Instead of standard conformal gauge fixing we demand t=τ

L 1 pη = const. = √ ≡ √ λ λ0

After imposing Virasoro constraints and solving eom for pz and pz¯ LPol = L(V (z), h(z), z, ˙ z 0) L is a complicated function. Assuming that the evolution of the string is almost rigid, i.e. imposing z˙ ∼ λ0  1 and expanding Z S=L

" 2

d σ

i i 0 0 2 02 V z¯˙ − V¯ z˙ − λ |z | + λ 2 2

00 2

0 4

|z | +

|z | h4

!

# 03

+ O(λ )

Summary We discussed the 1-loop planar computations of the anomalous dimensions of the gauge theory duals to strings probing LLM geometries.

Summary We discussed the 1-loop planar computations of the anomalous dimensions of the gauge theory duals to strings probing LLM geometries. BMN limits were taken to extrapolate to strong coupling (string regime)

Summary We discussed the 1-loop planar computations of the anomalous dimensions of the gauge theory duals to strings probing LLM geometries. BMN limits were taken to extrapolate to strong coupling (string regime) Topological and geometrical aspects of LLM backgrounds emerge from the gauge theory analysis: • The number of edges of LLM droplet is realized by the number of different ways the sites can be occupied (annulus+disk as well).

Summary We discussed the 1-loop planar computations of the anomalous dimensions of the gauge theory duals to strings probing LLM geometries. BMN limits were taken to extrapolate to strong coupling (string regime) Topological and geometrical aspects of LLM backgrounds emerge from the gauge theory analysis: • The number of edges of LLM droplet is realized by the number of different ways the sites can be occupied (annulus+disk as well). • We matched the near-BPS spectra.

Summary We discussed the 1-loop planar computations of the anomalous dimensions of the gauge theory duals to strings probing LLM geometries. BMN limits were taken to extrapolate to strong coupling (string regime) Topological and geometrical aspects of LLM backgrounds emerge from the gauge theory analysis: • The number of edges of LLM droplet is realized by the number of different ways the sites can be occupied (annulus+disk as well). • We matched the near-BPS spectra. • Non-connected constituents of the droplet are realized by complimentary semiclassical limits of the dilatation operator (annulus+disk).

Summary We discussed the 1-loop planar computations of the anomalous dimensions of the gauge theory duals to strings probing LLM geometries. BMN limits were taken to extrapolate to strong coupling (string regime) Topological and geometrical aspects of LLM backgrounds emerge from the gauge theory analysis: • The number of edges of LLM droplet is realized by the number of different ways the sites can be occupied (annulus+disk as well). • We matched the near-BPS spectra. • Non-connected constituents of the droplet are realized by complimentary semiclassical limits of the dilatation operator (annulus+disk). • We reproduced some of the functions of the LLM metric from a gauge theory computation.

Carrying on a similar analysis till 2-loops would allow to re-obtain the remaining h function of the LLM metric

Carrying on a similar analysis till 2-loops would allow to re-obtain the remaining h function of the LLM metric

For excitations in S 3, (dark regions) −2

2

ds = −h



i¯ i z − V dz dt + V d¯ 2 2

2

+ h2dzd¯ z + h−2dΩ23 z

dz 0 1 V = 0 2 ∂D 2πi |z − z| I dz 0 z¯0 − z¯ 4 h = 0 4 ∂D 2πi |z − z| I

D

Disk z V = 2 R − |z|2 4

h

R2 = (R2 − |z|2)2

The full metric is AdS5 × S 5

Half-plane i V = |z − z¯|

This is the maximally supersymmetric pp-wave

Annulus R12 z + V = 2 2 R2 − |z| z¯(R12 − |z|2) h4

R12 R22 + = (R22 − |z|2)2 (R12 − |z|2)2

Ellipse z(1 + 2)R − 2¯ z R V = (1 − 2)R2 + (z 2 + z¯2) − |z|2(1 + 2) h4 = ∂z V D: y 2 −

1+ ε 1− ε

x2 = R2



Single trace operators on top of e 2 trZ n1

tr(Y Z Y Z

n2

...Y Z

nL

)e

2

2  2 trZ

↔ |n1, n2, . . . , nLi

• The action of D1 is in this case of the form [Vazquez 06] D1 ∼ 2

L X l=1

where

ˆ †L ˆ L l l−

L−1 X

ˆ †L ˆ l+1 + L ˆ lL ˆ† ) (L l l+1

l=1

 1 ˆ a ˆ+√ a ˆ† L=√ 1 − 2 1 − 2

and a ˆ and a ˆ† are standard shift operators

ˆ Example 3: Coherent states for L

a ˆ|zi = z|zi

1  ˆ √ √ L= a ˆ+ a ˆ† 1 − 2 1 − 2 |zi =

∞ X

fn(z)|ni satisfying

n=0

1  zfn(z) = √ fn+1(z) + √ fn−1(z) 2 2 1− 1−

which allows to calculate ∞ X n=0

|fn(z)|2 =

1 1 − 2 + (z 2 + z¯2) − |z 2|(1 + 2)

Then, z(1 + 2) − 2¯ z V (z, z¯) = ∂z¯ log(hz|zi) = 1 − 2 + (z 2 + z¯2) − |z 2|(1 + 2) is precisely the V function of the elliptical LLM droplet

Coherent states for the standard shift operators: a ˆ|ni = |n − 1i a ˆ|0 i = 0 a ˆ†|ni = |n + 1i |zi =

∞ X

z n|ni

n=0

The norm hz|zi is finite in a disk |z| < 1 and V (z, z¯) = ∂z¯ log(hz|zi) =

z 1 − |z|2

is precisely the V function of the circular LLM droplet

a ˆ|zi = z|zi