Comments on the q-deformed AdS

(1)

Comments on the q-deformed AdS

₅

× S

⁵

superstring

Sergey Frolov

with Gleb Arutyunov and Riccardo Borsato: 1312.3542 and in progress

Hamilton Mathematics Institute and School of Mathematics, Trinity College Dublin and

Institut f ür Mathematik und Institut f ür Physik, Humboldt-Universit ät zu Berlin and

Steklov Mathematical Institute, Moscow

Supersymmetric Field Theories, Nordita, August 15, 2014

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Outline

1

Introduction

2

AdS

5

× S

⁵

q 3

Small κ limit

4

Large κ limit

5

Folded string

6

RR-fields

7

L.c. gauge and S-matrix

8

Summary

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Motivation

“Harmonic oscillator” of AdS/CFT: Maldacena ’97

IIB strings in AdS5× S⁵ ⇐⇒ N = 4 SU(N) SYM In planar limit one getsintegrablequantum string sigma model Energies of string states ≡ Scaling dimensions of SYM operators Integrable deformations

Orbifolds

γ-deformedAdS5× S⁵ (TsT-transformed)

Both can be described by thesameAdS5× S⁵ action but with twisted boundary conditions for the world-sheet fields

New type of deformation with real parameter Delduc, Magro, Vicedo ’13

Uses solutions of classical YBE Klimcik ’08

We show that the (bosonic part of the) world-sheet S-matrix coincides with thepsu_q(2|2) ⊕ psu_q(2|2)-invariant S-matrix

psu(2, 2|4) 7−→ psu_q(2, 2|4) ⊃ psu_q(2|2) ⊕ psu_q(2|2)Delduc, Magro, Vicedo ’14

Dual field theory is not Lorentz invariant.

New type of noncommutative field theory?

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Motivation

Orbifolds

(5)

Motivation

Orbifolds

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Motivation

Orbifolds

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Matrix realisation of su(2, 2|4)

M =

m θ η n

1 str M ≡ tr m − tr n = 0

2 M^†H + HM = 0 , H =

Σ 0 0 14

, Σ =

12 0 0 −12

3 m^†= −Σm Σ , n^†= −n , η^†= −Σ θ , m ∈ u(2, 2) , n ∈ u(4)

4 Bosonic subalgebra ofsu(2, 2|4):su(2, 2) ⊕ su(4) ⊕u(1)

5 psu(2, 2|4)is aquotient algebraofsu(2, 2|4)over thisu(1)

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Z

₄

-grading

M =M⁽⁰⁾+M⁽¹⁾+M⁽²⁾+M⁽³⁾

M^(0,2)= ¹₂

m ∓ Km^tK⁻¹ 0 0 n ∓ Kn^tK⁻¹

M^(1,3)= ¹₂

0 θ ∓iK η^tK⁻¹ η ±iK θ^tK⁻¹ 0

K = diag(−iσ2, −iσ2) M⁽⁰⁾andM⁽²⁾are even;M⁽¹⁾andM⁽³⁾are odd M⁽⁰⁾formso(4, 1) ⊕ so(5) ⊂ su(2, 2) ⊕ su(4).

M⁽²⁾form the tangent space of the coset SU(2, 2) × SU(4)/SO(4, 1) × SO(5) = AdS5× S⁵ [M₁^{(k )},M₂^(m)] =M₃^{(k +m)}moduloZ4

Notation:Piare projectors onM⁽ⁱ⁾:Pi(M) = M⁽ⁱ⁾

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Z

₄

-grading

M =M⁽⁰⁾+M⁽¹⁾+M⁽²⁾+M⁽³⁾

M^(0,2)= ¹₂

m ∓ Km^tK⁻¹ 0 0 n ∓ Kn^tK⁻¹

M^(1,3)= ¹₂

0 θ ∓iK η^tK⁻¹ η ±iK θ^tK⁻¹ 0

K = diag(−iσ2, −iσ2) M⁽⁰⁾andM⁽²⁾are even;M⁽¹⁾andM⁽³⁾are odd M⁽⁰⁾formso(4, 1) ⊕ so(5) ⊂ su(2, 2) ⊕ su(4).

M⁽²⁾form the tangent space of the coset SU(2, 2) × SU(4)/SO(4, 1) × SO(5) = AdS5× S⁵ [M₁^{(k )},M₂^(m)] =M₃^{(k +m)}moduloZ4

Notation:Piare projectors onM⁽ⁱ⁾:Pi(M) = M⁽ⁱ⁾

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Strings on AdS

₅

× S

⁵ Metsaev, Tseytlin ’98

Letg∈ SU(2, 2|4). Introduce the one-formA ∈ su(2, 2|4) A = −g⁻¹dg = A⁽⁰⁾+A⁽²⁾+A⁽¹⁾+A⁽³⁾ Action S =R dσdτL for superstrings onAdS5× S⁵

L = −g 2 h

γ^αβstr A⁽²⁾α A⁽²⁾_β + ^αβstr A⁽¹⁾α A⁽³⁾_β i g = _2πα^R²0 =

√ λ

2π, ^{τ σ}=1 and γ^αβ=h^αβ√

−h, detγ = −1 L = −g

4 γ^αβ− ^αβ strh

(P3+2P2− P1)(Aα).Aβ

i

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q-deformed AdS

₅

× S

⁵

: L and R-operator

Delduc, Magro, Vicedo ’13

L = −g

4(1 + η²) γ^αβ− ^αβ strh ˜d (Aα) 1

1 − ηRg◦ d(Aβ)i ,

q = e^−ν/g, ν =_1+η^2η2 Arutyunov, Borsato, SF ’13; Delduc, Magro, Vicedo ’14

P_iare projections onM⁽ⁱ⁾:P_i(M) = M⁽ⁱ⁾ d = P1+ 2

1 − η²P2−P3, d = P˜ 3+ 2

1 − η²P2−P1, str A d (B) = str ˜d (A) B Rg(M) = g⁻¹R(gMg⁻¹)g = adj_g−1◦ R ◦ adj_g(M)

Rsatisfies the modified classical Yang-Baxter equation

[R(M), R(N)]−R [R(M), N]+[M, R(N)] = [M, N] , str R(M) N = −str M R(N)

We choose

R(M)jk =







−i Mjk if j < k 0 if j = k i Mjk if j > k L is invariant underg→ gh,h∈ SO(4, 1) × SO(5) × U(1) L isκ-symmetric, and a Lax connection is known

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q-deformed AdS

₅

× S

⁵

: L and R-operator

L = −g

4(1 + η²) γ^αβ− ^αβ strh ˜d (Aα) 1

1 − η²P2−P3, d = P˜ 3+ 2

We choose

R(M)jk =







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q-deformed AdS

₅

× S

⁵

: L and R-operator

L = −g

4(1 + η²) γ^αβ− ^αβ strh ˜d (Aα) 1

1 − η²P2−P3, d = P˜ 3+ 2

We choose

R(M)jk =







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Explicit bosonic Lagrangian: metric parts

Arutyunov, Borsato, SF ’13

L = −g

2(1 + κ²)¹² γ^αβ− ^αβstrh A⁽²⁾α

1 1 − κRg◦ P2

(Aβ)i

, κ = 2η 1 − η² Choose a coset representativeg

Invert the operator 1 − κRg◦ P2

Bosonic Lagrangian is given by the sum of the AdS and sphere parts L = La+Ls=La^G+La^WZ+Ls^G+Ls^WZ

The metric pieces

La^G= −g

2(1 + κ²)¹²γ^αβ

−∂αt∂βt 1 + ρ²

1 − κ²ρ² + ∂αρ∂βρ (1 + ρ²) (1 − κ²ρ²) + ∂αζ∂βζρ²

1 + κ²ρ⁴sin²ζ+∂αψ1∂βψ1ρ²cos²ζ

1 + κ²ρ⁴sin²ζ + ∂αψ2∂βψ2ρ²sin²ζ

,

Ls^G= −g

2(1 + κ²)¹²γ^αβ

∂αφ∂βφ 1 − r²

1 + κ²r² + ∂αr ∂βr (1 − r²) (1 + κ²r²) + ∂αξ∂βξr²

1 + κ²r⁴sin²ξ+∂αφ1∂βφ1r²cos²ξ

1 + κ²r⁴sin²ξ + ∂αφ2∂βφ2r²sin²ξ ,

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Explicit bosonic Lagrangian: metric parts

L = −g

2(1 + κ²)¹² γ^αβ− ^αβstrh A⁽²⁾α

1 1 − κRg◦ P2

(Aβ)i

The metric pieces La^G= −g

2(1 + κ²)¹²γ^αβ

−∂αt∂βt 1 + ρ²

1 − κ²ρ² + ∂αρ∂βρ (1 + ρ²) (1 − κ²ρ²) + ∂αζ∂βζρ²

,

Ls^G= −g

2(1 + κ²)¹²γ^αβ

∂αφ∂βφ 1 − r²

1 + κ²r² + ∂αr ∂βr (1 − r²) (1 + κ²r²) + ∂αξ∂βξr²

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Explicit bosonic Lagrangian: metric parts

L = −g

2(1 + κ²)¹² γ^αβ− ^αβstrh A⁽²⁾α

1 1 − κRg◦ P2

(Aβ)i

The metric pieces La^G= −g

2(1 + κ²)¹²γ^αβ

−∂αt∂βt 1 + ρ²

1 − κ²ρ² + ∂αρ∂βρ (1 + ρ²) (1 − κ²ρ²) + ∂αζ∂βζρ²

,

Ls^G= −g

2(1 + κ²)¹²γ^αβ

∂αφ∂βφ 1 − r²

1 + κ²r² + ∂αr ∂βr (1 − r²) (1 + κ²r²) + ∂αξ∂βξr²

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Explicit bosonic Lagrangian: WZ parts

The Wess-Zumino parts La^WZ =g

2κ(1 + κ²)¹²^αβ ρ⁴sin 2ζ

1 + κ²ρ⁴sin²ζ∂αψ1∂βζ , Ls^WZ = −g

2κ(1 + κ²)¹²^αβ r⁴sin 2ξ

1 + κ²r⁴sin²ξ∂αφ1∂βξ . The deformed action is invariant under the shifts oft, ψk, φ, φk

The ranges ofρandr:0 ≤ ρ ≤ ¹

κ and0 ≤ r ≤ 1 The deformed AdS issingularat ρ =1/κ

In the undeformed case:−Z₀²+Z₁²+Z₂²+Z₃²+Z₄²− Z5²= −1 Z1+iZ2= ρcos ζ e^iψ¹, Z3+iZ4= ρsin ζ e^iψ², Z0+iZ5=p

1 + ρ²e^it

whileφ , φ1, φ2, ξ ,r are related toYA,Y_A²=1as

Y1+iY2=r cos ξ e^iφ¹, Y3+iY4=r sin ξ e^iφ², Y5+iY6=p

1 − r²e^iφ

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Explicit bosonic Lagrangian: WZ parts

2κ(1 + κ²)¹²^αβ ρ⁴sin 2ζ

2κ(1 + κ²)¹²^αβ r⁴sin 2ξ

1 + ρ²e^it

1 − r²e^iφ

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Explicit bosonic Lagrangian: WZ parts

2κ(1 + κ²)¹²^αβ ρ⁴sin 2ζ

2κ(1 + κ²)¹²^αβ r⁴sin 2ξ

1 + ρ²e^it

1 − r²e^iφ

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Small κ limit

SF, unpublished

TheS_q⁵ part reduces toS⁵andLs^WZvanishes. Letg(1 + κ²)¹² =1

ds_a²= −dt² 1 + ρ²

1 − κ²ρ² + d ρ² (1 + ρ²) (1 − κ²ρ²) + d ζ²ρ²

1 + κ²ρ⁴sin²ζ + d ψ₁²ρ²cos²ζ

1 + κ²ρ⁴sin²ζ+d ψ₂²ρ²sin²ζ , Ba= κ

2

ρ⁴sin 2ζ

1 + κ²ρ⁴sin²ζd ψ1∧ dζ

1 κ → 0, all the coordinates are finite. One getsAdS5withR = −20

2 κ → 0, ρ →1/ρ/κ, t → κt, ψ2→ κψ2; ρ >1 Babecomes a total derivative and the metric

ds²_a= − dt²

ρ²− 1+ d ρ²

ρ²− 1 + ρ² d ζ²

sin²ζ + ρ²cot²ζd ψ²₁+sin²ζ ρ² d ψ²₂ The curvatureR = −4 − 2 ^ρ²⁺¹

ρ²(ρ²−1)− 2^{cos 2ζ}_ρ2 <0

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Small κ limit

SF, unpublished

ds_a²= −dt² 1 + ρ²

1 − κ²ρ² + d ρ² (1 + ρ²) (1 − κ²ρ²) + d ζ²ρ²

1 + κ²ρ⁴sin²ζ+d ψ₂²ρ²sin²ζ , Ba= κ

2

ρ⁴sin 2ζ

1 κ → 0, all the coordinates are finite. One getsAdS5withR = −20

2 κ → 0, ρ →1/ρ/κ, t → κt, ψ2→ κψ2; ρ >1 Babecomes a total derivative and the metric

ds²_a= − dt²

ρ²− 1+ d ρ²

ρ²− 1 + ρ² d ζ²

sin²ζ + ρ²cot²ζd ψ²₁+sin²ζ ρ² d ψ²₂ The curvatureR = −4 − 2 ^ρ²⁺¹

ρ²(ρ²−1)− 2^{cos 2ζ}_ρ2 <0

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Small κ limit

SF, unpublished

ds_a²= −dt² 1 + ρ²

1 − κ²ρ² + d ρ² (1 + ρ²) (1 − κ²ρ²) + d ζ²ρ²

1 + κ²ρ⁴sin²ζ+d ψ2²ρ²sin²ζ , Ba= κ

2

ρ⁴sin 2ζ

3 κ → 0, ρ → ρ/√

κ, t →√

κt, ζ → ζ0+√ κζ, ψ1→√

κψ1/cos ζ0, ψ2→√

κψ2/sin ζ0. Baand the metric

ds²_a= ρ²(−dt²+d ψ2²) +ρ²(d ζ²+d ψ²₁) 1 + ρ⁴sin²ζ0

+d ρ² ρ² , Ba= ρ⁴sin ζ0

1 + ρ⁴sin²ζ0

d ψ1∧ dζ

This is the Maldacena-Russo background! Maldacena, Russo ’99

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Small κ limit

SF, unpublished

ds_a²= −dt² 1 + ρ²

1 − κ²ρ² + d ρ² (1 + ρ²) (1 − κ²ρ²) + d ζ²ρ²

1 + κ²ρ⁴sin²ζ+d ψ2²ρ²sin²ζ , Ba= κ

2

ρ⁴sin 2ζ

3 κ → 0, ρ → ρ/√

κ, t →√

κt, ζ → ζ0+√ κζ, ψ1→√

κψ1/cos ζ0, ψ2→√

κψ2/sin ζ0. Baand the metric

ds²_a= ρ²(−dt²+d ψ2²) +ρ²(d ζ²+d ψ²₁) 1 + ρ⁴sin²ζ0

+d ρ² ρ² , Ba= ρ⁴sin ζ0

1 + ρ⁴sin²ζ0

d ψ1∧ dζ

This is the Maldacena-Russo background! Maldacena, Russo ’99

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Small κ limit

SF, unpublished

The curvature

R = −4 + 40 1 + ρ⁴sin²ζ0

− 56

(1 + ρ⁴sin²ζ0)²

2 4 6 8

-20 -15 -10 -5

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Large κ limit

Hoare, Roiban, Tseytlin ’14

ds_a²=g(1 + κ²)¹²

−dt² 1 + ρ²

1 − κ²ρ² + d ρ² (1 + ρ²) (1 − κ²ρ²) + d ζ²ρ²

1 + κ²ρ⁴sin²ζ+d ψ₂²ρ²sin²ζ

ds_s²=g(1 + κ²)¹²d φ² 1 − r²

1 + κ²r² + dr² (1 − r²) (1 + κ²r²) + d ξ²r²

1 + κ²r⁴sin²ξ+ d φ²₁r²cos²ξ

1 + κ²r⁴sin²ξ+d φ²2r²sin²ξ

κ → ∞,g → κg,ρ →1/ρ,ψ2→ ψ2/κ,r → 1/r,φ2= φ2/κ The WZ term becomes a total derivative and the metric

1

gds²_a=dt² 1 + ρ²

− d ρ²

1 + ρ²+d ζ²ρ²

sin²ζ +d ψ₁²ρ²cot²ζ +d ψ₂² ρ² sin²ζ 1

gds²_s=d φ² r²− 1

+ dr²

r²− 1+d ξ²r²

sin²ξ +d φ²1r²cot²ξ + d φ²₂ r² sin²ξ T-duality along theψ2andφ2directions givesdS5×H⁵

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Large κ limit

Hoare, Roiban, Tseytlin ’14

ds_a²=g(1 + κ²)¹²

−dt² 1 + ρ²

1 − κ²ρ² + d ρ² (1 + ρ²) (1 − κ²ρ²) + d ζ²ρ²

ds_s²=g(1 + κ²)¹²d φ² 1 − r²

1 + κ²r² + dr² (1 − r²) (1 + κ²r²) + d ξ²r²

κ → ∞,g → κg,ρ →1/ρ,ψ2→ ψ2/κ,r → 1/r,φ2= φ2/κ The WZ term becomes a total derivative and the metric

1

gds²_a=dt² 1 + ρ²

− d ρ²

1 + ρ²+d ζ²ρ²

sin²ζ +d ψ₁²ρ²cot²ζ +d ψ₂² ρ² sin²ζ 1

gds²_s=d φ² r²− 1

+ dr²

r²− 1+d ξ²r²

sin²ξ +d φ²1r²cot²ξ + d φ²₂ r² sin²ξ T-duality along theψ2andφ2directions givesdS5×H⁵

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Large κ limit: T-duality along the time direction

SF, unpublished

ds_a²=g(1 + κ²)¹²

−dt² 1 + ρ²

1 − κ²ρ² + d ρ² (1 + ρ²) (1 − κ²ρ²) + d ζ²ρ²

ds_s²=g(1 + κ²)¹²d φ² 1 − r²

1 + κ²r² + dr² (1 − r²) (1 + κ²r²) + d ξ²r²

κ → ∞,g → κg,ρ → ρ/κ,t → t/κ,r → r /κ,φ → φ/κ The WZ term becomes 0, and the metric

1

gds²_a= − dt²

1 − ρ² + d ρ²

1 − ρ² + ρ² d ζ² +d ψ₁²cos²ζ +d ψ₂²sin²ζ 1

gds²s= d φ²

1 + r² + dr²

1 + r² +r² d ξ²+d φ²1cos²ξ +d φ²2sin²ξ T-duality along thetandφdirections givesdS5×H⁵

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Large κ limit: T-duality along the time direction

SF, unpublished

ds_a²=g(1 + κ²)¹²

−dt² 1 + ρ²

1 − κ²ρ² + d ρ² (1 + ρ²) (1 − κ²ρ²) + d ζ²ρ²

ds_s²=g(1 + κ²)¹²d φ² 1 − r²

1 + κ²r² + dr² (1 − r²) (1 + κ²r²) + d ξ²r²

κ → ∞,g → κg,ρ → ρ/κ,t → t/κ,r → r /κ,φ → φ/κ The WZ term becomes 0, and the metric

1

gds²_a= − dt²

1 − ρ² + d ρ²

1 − ρ² + ρ² d ζ² +d ψ₁²cos²ζ +d ψ₂²sin²ζ 1

gds²s= d φ²

1 + r² + dr²

1 + r² +r² d ξ²+d φ²1cos²ξ +d φ²2sin²ξ T-duality along thetandφdirections givesdS5×H⁵

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Mirror of AdS

₅

× S

⁵

≡ AdS

5

× S

⁵

κ=∞ Arutyunov, van Tongeren ’14

Metric and curvature (g = 1)

ds²_a= − dt²

1 − ρ² + d ρ²

1 − ρ²+ ρ² d ζ² +d ψ²₁cos²ζ +d ψ₂²sin²ζ , Ra=41 − 2ρ² 1 − ρ²

dss²= d φ²

1 + r² + dr²

1 + r²+r² d ξ²+d φ²1cos²ξ +d φ²2sin²ξ , Rs=−41 + 2r² 1 + r²

Dilaton Φ = Φ0−¹₂log(1 − ρ²)(1 + r²) The five-form F = 4e^−Φ(ωφ− ωt)

No Killing spinors =⇒ susy is not realised by superisometries In l.c. gauget = τ,pφ=1, it is the mirror of theAdS5× S⁵ strings Particular case of the general mirror duality Arutynov, de Leeuw, van Tongeren ’14

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Mirror of AdS

₅

× S

⁵

≡ AdS

5

× S

⁵

κ=∞ Arutyunov, van Tongeren ’14

Metric and curvature (g = 1)

ds²_a= − dt²

1 − ρ² + d ρ²

1 − ρ²+ ρ² d ζ² +d ψ²₁cos²ζ +d ψ₂²sin²ζ , Ra=41 − 2ρ² 1 − ρ²

dss²= d φ²

1 + r² + dr²

1 + r²+r² d ξ²+d φ²1cos²ξ +d φ²2sin²ξ , Rs=−41 + 2r² 1 + r²

Dilaton Φ = Φ0−¹₂log(1 − ρ²)(1 + r²) The five-form F = 4e^−Φ(ωφ− ωt)

No Killing spinors =⇒ susy is not realised by superisometries In l.c. gauget = τ,pφ=1, it is the mirror of theAdS5× S⁵ strings Particular case of the general mirror duality Arutynov, de Leeuw, van Tongeren ’14

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(S, J) folded string

SF, Roiban in progress

Folded string spinning in AdS5

qand rotating about a circle in S⁵

q. Ansatz is the same as for the undeformed case SF, Tseytlin ’02

t = κτ , ψ1= ωτ , ρ = ρ(σ) , φ = ντ , ζ = ψ2=r = ξ = φ1= φ2=0 The WZ term does not contribute. Virasoro constraint gives

(ρ⁰)²= (1 + ρ²) ω²κ²ρ⁴− (ω²− κ²− κ²ν²)ρ²+ κ²− ν² We wantρ⁰=0atρ = ρ0, andrhs > 0forρ < ρ0. Thus

κ > ν , ω²> κ²+ κ²ν² =⇒ ν²< ω² 1 + κ² ρ0satisfies

ω²κ²ρ⁴0− (ω²− κ²− κ²ν²)ρ²0+ κ²− ν²=0 ,

(ρ^∓₀)²=ω²− κ²− κ²ν²∓ q

(ω²− κ²− κ²ν²)²− 4ω²κ²(κ²− ν²) 2κ²ω²

Since(ρ⁰)²>0atρ =1/κboth roots must be less than1/κ² κ ≤ ωp

1 + κ²− κp ω²− ν² In what follows we denoteρ0≡ ρ⁻₀.

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(S, J) folded string

SF, Roiban in progress

Folded string spinning in AdS5

qand rotating about a circle in S⁵

q. Ansatz is the same as for the undeformed case SF, Tseytlin ’02

t = κτ , ψ1= ωτ , ρ = ρ(σ) , φ = ντ , ζ = ψ2=r = ξ = φ1= φ2=0 The WZ term does not contribute. Virasoro constraint gives

(ρ⁰)²= (1 + ρ²) ω²κ²ρ⁴− (ω²− κ²− κ²ν²)ρ²+ κ²− ν² We wantρ⁰=0atρ = ρ0, andrhs > 0forρ < ρ0. Thus

κ > ν , ω²> κ²+ κ²ν² =⇒ ν²< ω² 1 + κ² ρ0satisfies

ω²κ²ρ⁴0− (ω²− κ²− κ²ν²)ρ²0+ κ²− ν²=0 ,

(ρ^∓₀)²=ω²− κ²− κ²ν²∓ q

(ω²− κ²− κ²ν²)²− 4ω²κ²(κ²− ν²) 2κ²ω²

Since(ρ⁰)²>0atρ =1/κboth roots must be less than1/κ² κ ≤ ωp

1 + κ²− κp ω²− ν² In what follows we denoteρ0≡ ρ⁻₀.

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(S, J) folded string

The periodicity condition onρ(σ)implies 2π =

Z2π 0

d σ = 4 Zρ₀

0

d ρ q

(1 + ρ²) ω²κ²ρ⁴− (ω²− κ²− κ²ν²)ρ²+ κ²− ν² Angular momentumJ = J/g(1 + κ²)¹²

J = ν Z 2π

0

d σ = 2πν SpinS = S/g(1 + κ²)¹²

S = ω Z 2π

0

d σ ρ²=4ω Z ρ₀

0

d ρ ρ² q

(1 + ρ²) ω²κ²ρ⁴− (ω²− κ²− κ²ν²)ρ²+ κ²− ν² EnergyE = E/g(1 + κ²)¹²

E = κ Z2π

0

d σ 1 + ρ² 1 − κ²ρ² =4κ

Z ρ₀ 0

d ρp1 + ρ²

(1 − κ²ρ²)pω²κ²ρ⁴− (ω²− κ²− κ²ν²)ρ²+ κ²− ν²

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Large S expansion

corresponds toω → ∞, andρ0→ ρ⁺₀. Then κ → ωp

1 + κ²− κp

ω²− ν², ρ²0→ r

1 + 1 κ²

r 1 − ν²

ω² − 1< 1 2κ² No matter how fast the string rotates, itneverhits the singularity!

If ^ν²

ω² → ¹

1+κ² then ρ0→ 0 Let J = 0

E =w0

κ L+ 2 κ

logw0− 1 w0+1− 32

k0κ²

e^−L+32

2L

k0κ² −8 + 11κ²

k0κ⁴ − 16 w0k₀²κ³

e^−2L+ · · ·

L = κ k0

S + 2 k0

log 1 + k0

1 − k0

, w0= r

1 +√ κ

1 + κ², k0= r

1 −√ κ 1 + κ²

Compare with Gubser, Klebanov, Polyakov ’02

E = S + 2 log S + 8 log 2 − 2 +

∞

X

n=1

Pn(log S)e^{−n log S}

(35)

Large S expansion at κ = ∞

Take the limit κ → ∞ and rescale E → E/κ and S → S/κ² E = 2S + 4√

2 + 2 log 3 − 2√

2

− 32√

2e^−L+32

4S − 19√ 2

e^−2L+ · · · , L =√

2S + 4

Is there scattering theory atS = ∞? Can the terms∼ e^−nLbe obtained from TBA?

If ^J_S is constant

E = E0+E1e^−αS+· · · , α =√

L+ 2L

(L −1)S+2(L − 2)√

L(1 + L) (L −1)³S² +· · · ,

E0= LS + 4√ L

L− 1 − 4csch⁻¹√

L− 1

+ 8 (L − 2) (L −1)³S + · · · , E1= − 32√

L

(L −1)² −64(L − 2)(1 + 3L)

(L −1)⁴S + · · · , L=1 + r

1 + J² S²

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Large S expansion at κ = ∞

Take the limit κ → ∞ and rescale E → E/κ and S → S/κ² E = 2S + 4√

2 + 2 log 3 − 2√

2

− 32√

2e^−L+32

4S − 19√ 2

e^−2L+ · · · , L =√

2S + 4

Is there scattering theory atS = ∞? Can the terms∼ e^−nLbe obtained from TBA?

If ^J_S is constant

E = E0+E1e^−αS+· · · , α =√

L+ 2L

(L −1)S+2(L − 2)√

L(1 + L) (L −1)³S² +· · · ,

E0= LS + 4√ L

L− 1 − 4csch⁻¹√

L− 1

+ 8 (L − 2) (L −1)³S + · · · , E1= − 32√

L

(L −1)² −64(L − 2)(1 + 3L)

(L −1)⁴S + · · · , L=1 + r

1 + J² S²

(37)

Dilaton

Arutyunov, Borsato, SF in progress

Introduce the coordinates

x⁰=t , x¹= ψ2, x²= ψ1, x³=sin²ζ , x⁴= ρ², x⁵= φ3, x⁶= φ2, x⁷= φ1, x⁸=sin²ξ , x⁹=r².

The gravity fields can depend only onx³,x⁴,x⁸,x⁹.

The only nonvanishing components ofHµνρareH234andH789

Dilaton equation

4∇²Φ −4(∇µΦ)²+R − 1

12HµνρH^µνρ=0

Φ =Φ0+ Φa(x³,x⁴) + Φs(x⁸,x⁹) +Φas(x³,x⁴,x⁸,x⁹) 4∇²Φa− 4(∇µΦa)²+Ra−1

2H234H²³⁴= c(κ)

L² , L²=g(1 + κ²)¹²