Giant Gravitons and Gauge/ Gravity Duality

Giant Gravitons and gauge/ gravity duality

Yolanda Lozano (U. Oviedo)

Cape Town, March 2012

Tuesday 30 October 2012 - Motivation: Giant gravitons are useful states in which to test the AdS/CFT correspondence Use them to test the novel ⇒ AdS_4/CFT_3 duality of ABJM - How: Construct giant graviton solutions in the 7 3 AdS4 S /Zk and AdS4 CP dual backgrounds, work out × × the operator mapping, emergence of geometry.. - Results: 7 - M5-brane giant graviton in AdS4 S /Zk × - New SUSY NS5-brane in AdS CP3 4 × - Partial results on giant gravitons in AdS CP3 4 × (Based on arXiv:1107.5475 [hep-th], JHEP, with M. Herrero and M. Picos) (Also, work in progress with J. Murugan and A. Prinsloo, from U. Cape Town)

Tuesday 30 October 2012 Outline 1. Introduction 1.1. Gauge/gravity (holographic) correspondence 1.2. D-Branes 1.3. Giant gravitons and AdS/CFT

2. GST giant gravitons 3. Giant gravitons in ABJM 3.1. A few words about AdS_4/CFT_3 3.2. Giant graviton solutions 7 3.3. An M5-brane giant graviton in AdS4 S /Zk × 3.4. Giant gravitons in Type IIA? 4. Giant gravitons as fuzzy manifolds 5. Conclusions

Tuesday 30 October 2012 Outline 1. Introduction 1.1. Gauge/gravity (holographic) correspondence 1.2. D-Branes 1.3. Giant gravitons and AdS/CFT

2. GST giant gravitons 3. Giant gravitons in ABJM 3.1. A few words about AdS_4/CFT_3 3.2. Giant graviton solutions 3.3. An M5-brane giant graviton in 3.4. Giant gravitons in Type IIA? 4. Giant gravitons as fuzzy manifolds 5. Conclusions

Tuesday 30 October 2012 1.1. Gauge/gravity (holographic) correspondence

The gauge/gravity correspondence states that some gauge theories in ﬂat space at strong (weak) coupling are dual to weakly (strongly) coupled string theories with one extra dimension. Possibility to apply string theory techniques to strongly coupled systems (quark-gluon plasma, condensed matter systems). Possibility to learn something about quantum gravity by studying low dimensional systems with a holographic description (concrete realization of QG and spacetime as emergent phenomena).

Tuesday 30 October 2012 Original proposal by Maldacena:

Equivalence between Type IIB string theory on AdS S5 5 × and 4 dim =4 supersymmetric SU(N) Yang-Mills theory N

AdS5 : anti-de Sitter space in 5 dim (maximally symmetric solution of the Einstein eqs. with a negative cosmological constant)

Tuesday 30 October 2012 Original proposal by Maldacena:

Equivalence between Type IIB string theory on AdS S5 5 × and 4 dim =4 supersymmetric SU(N) Yang-Mills theory N

AdS5 : anti-de Sitter space in 5 dim (maximally symmetric solution of the Einstein eqs. with a negative cosmological constant) The symmetry group of 5 dim anti-de Sitter space matches precisely with the group of conformal symmetries of =4 SYM AdS/CFT N ⇒

Tuesday 30 October 2012 Original proposal by Maldacena:

Equivalence between Type IIB string theory on AdS S5 5 × and 4 dim =4 supersymmetric SU(N) Yang-Mills theory N

Tuesday 30 October 2012 Starting point: Study of N coincident D3-branes in Type IIB in the large N limit, based on two dual descriptions: - As solution of the classical eqs. of motion of ST/SUGRA - As a D-brane system

5 AdS5 S emerges as the near horizon geometry of the × 4 4 D3-brane solution. The radius is given by R =4πgsNls

5 AdS5 S emerges as the near horizon geometry of the × 4 4 D3-brane solution. The radius is given by R =4πgsNls The U(N) gauge theory living on the branes becomes 4 dim 2 =4SYM, with gYM = gs : N UV ﬁnite, conformally invariant Topological large N expansion with ‘t Hooft parameter: 2 λ = gYMN

4 4 Through the correspondence: R =4πλ ls

Tuesday 30 October 2012 4 4 λ = gsNR=4πλ ls implies

N , ﬁnite λ gs 0 Planar limit Classical →∞ ⇔ → ⇒ ↔ limit of IIB ST

Tuesday 30 October 2012 4 4 λ = gsNR=4πλ ls implies

N , ﬁnite λ gs 0 Planar limit Classical →∞ ⇔ → ⇒ ↔ limit of IIB ST

λ ls 0 Strong ‘t Hooft coupling limit →∞ ⇔ → ⇒ ↔ SUGRA limit

Tuesday 30 October 2012 4 4 λ = gsNR=4πλ ls implies

N , ﬁnite λ gs 0 Planar limit Classical →∞ ⇔ → ⇒ ↔ limit of IIB ST

λ ls 0 Strong ‘t Hooft coupling limit →∞ ⇔ → ⇒ ↔ SUGRA limit Strong-weak coupling duality ⇒

Tuesday 30 October 2012 4 4 λ = gsNR=4πλ ls implies

N , ﬁnite λ gs 0 Planar limit Classical →∞ ⇔ → ⇒ ↔ limit of IIB ST

λ ls 0 Strong ‘t Hooft coupling limit →∞ ⇔ → ⇒ ↔ SUGRA limit Strong-weak coupling duality ⇒

For each ﬁeld in the 5 dim bulk Operator in the dual CFT ↔ Hard.. Easier for certain operators due to their symmetries.

Tuesday 30 October 2012 1.2. D-branes The fundamental string (perturbative state of the spectrum)

occurs as a solitonic solution, electric source for B2 A class of p-branes, Dp-branes, are the electric sources for Cp+1. They are non-perturbative.

Tuesday 30 October 2012 1.2. D-branes The fundamental string (perturbative state of the spectrum)

occurs as a solitonic solution, electric source for B2 A class of p-branes, Dp-branes, are the electric sources for Cp+1. They are non-perturbative. They are also (p+1)-dimensional hypersurfaces on which fundamental strings can end ⇒ Dynamics determined at weak coupling by open string perturbation theory U(1) gauge theory (U(N) for a system of Dp-branes).⇒

D0D1 D2

Tuesday 30 October 2012 D-branes are related to other branes through the web of dualities that relate the different string theories in 10 dim:

IIA IIB

T M2 M5 M

M I

S F1 D2 D4 NS5 IIA T HET SO(32) HET E8xE8

Tuesday 30 October 2012 D-branes are related to other branes through the web of dualities that relate the different string theories in 10 dim:

IIA IIB

T M2 M5 M

M I

S F1 D2 D4 NS5 IIA T HET SO(32) HET E8xE8

7 N coincident M2-branes M-theory on AdS4 S dual → × to the 3 dim =8 gauge theory living on the M2-branes N 4 N coincident M5-branes M-theory on AdS7 S dual → × to the 6 dim (2,0) ﬁeld theory living on the M5-branes

Tuesday 30 October 2012 Recently, M2-branes on an orbifold M-theory on 7 → AdS4 S /Zk dual to the =6 Chern-Simons matter × N theory living on the M2: ABJM model (Aharony, Bergman, Jafferis, Maldacena’08)

New AdS/CFT pair in which to test the gauge/gravity correspondence 3 dimensions Applications in condensed matter →

Other examples with less SUSY, conﬁning, etc

Tuesday 30 October 2012 1.3. Giant gravitons and AdS/CFT Giant gravitons have proven very useful in the context of the AdS/CFT correspondence: - Matching D-brane conﬁgs. in string/M-theory with gauge invariant operators in the dual CFT - Explicit realization of the stringy exclusion principle - Emergent geometry (global and local geometries of the dual branes encoded in the gauge theory)

CFT AdS Family of chiral primary Graviton states with angular operators momentum in Sn Maximum weight Upper bound on the angular momentum

CFT AdS Family of chiral primary Graviton states with angular operators momentum in Sn Maximum weight Upper bound on the angular momentum Stringy exclusion principle (Maldacena, Strominger’98)

Tuesday 30 October 2012 Outline 1. Introduction 1.1. Gauge/gravity (holographic) correspondence 1.2. D-Branes 1.3. Giant gravitons and AdS/CFT 2. GST giant gravitons 3. Giant gravitons in ABJM 3.1. A few words about AdS_4/CFT_3 3.2. Giant graviton solutions 3.3. An M5-brane giant graviton in 3.4. Giant gravitons in Type IIA? 4. Giant gravitons as fuzzy manifolds 5. Conclusions

Tuesday 30 October 2012 2. GST giant gravitons

Gravitons expanding into a sphere in the Sn part (giant gravitons) with radius proportional to the angular momentum (Mc.Greevy, Susskind and Toumbas’00) Radius < radius of the n-sphere implies upper bound on the angular momentum, exactly as predicted by the stringy exclusion principle

Tuesday 30 October 2012 2. GST giant gravitons

n In AdSm S × 2 2 2 2 2 2 2 2 ds = dsAdS + L (dθ + cos θdφ +sin θdΩn 2) m − take θ = const , φ = φ(τ) ,r=0

2 2 2 2 2 2 2 2 ds = −dt + L cos θdφ + L sin θdΩn−2 ⇒

Tuesday 30 October 2012 2. GST giant gravitons

n In AdSm S × 2 2 2 2 2 2 2 2 ds = dsAdS + L (dθ + cos θdφ +sin θdΩn 2) m − take θ = const , φ = φ(τ) ,r=0

2 2 2 2 2 2 2 2 ds = −dt + L cos θdφ + L sin θdΩn−2 ⇒ Spherical (n-2)-brane with radius L sin θ orbiting the Sn in the φ direction

Tuesday 30 October 2012 The effective action of this brane is:

S(n 2) = Tn 2 detg + Tn 2 Cn 1 − − − | | − −

Tuesday 30 October 2012 The effective action of this brane is:

S(n 2) = Tn 2 detg + Tn 2 Cn 1 − − − | | − − and the Hamiltonian:

2 Pφ 2 N n 3 H = 1 + tan θ 1 sin − θ = H(θ) L − Pφ

Tuesday 30 October 2012 The effective action of this brane is:

S(n 2) = Tn 2 detg + Tn 2 Cn 1 − − − | | − − and the Hamiltonian:

2 Pφ 2 N n 3 H = 1 + tan θ 1 sin − θ = H(θ) L − Pφ The equilibrium radius of the spherical brane is then determined by H(θ)=0:

θ =0→ H = Pφ/L : point-like graviton n 3 sin − θ = P /N H = P /L : giant graviton φ → φ

Tuesday 30 October 2012 The effective action of this brane is:

S(n 2) = Tn 2 detg + Tn 2 Cn 1 − − − | | − − and the Hamiltonian:

2 Pφ 2 N n 3 H = 1 + tan θ 1 sin − θ = H(θ) L − Pφ The equilibrium radius of the spherical brane is then determined by H(θ)=0:

θ =0→ H = Pφ/L : point-like graviton n 3 sin − θ = P /N H = P /L : giant graviton φ → φ R = L sin θ ≤ L implies that P N Upper bound on φ ≤ ⇒ the angular momentum

Tuesday 30 October 2012 Also: - M = P /L , Q = P /L BPS φ φ ⇒ ˙ - φ =1/L , as the point-like graviton - Preserves same SUSYs than point-like graviton: (1 + Γ0φ) =0

SUSY algebra:

012 Qα,Qβ = P0 + Γ Z12 (membrane) { } Q ,Q = P + Γ0iP (grav. wave) { α β} 0 i (Bergshoeff, Kallosh, Ortin’93)

Tuesday 30 October 2012 Also: - M = P /L , Q = P /L BPS φ φ ⇒ ˙ - φ =1/L , as the point-like graviton - Preserves same SUSYs than point-like graviton: (1 + Γ0φ) =0

SUSY algebra:

012 Qα,Qβ = P0 + Γ Z12 (membrane) { } Q ,Q = P + Γ0iP (grav. wave) { α β} 0 i (Bergshoeff, Kallosh, Ortin’93)

0 0i P = Pi Qα,Qβ = P0 (1 + Γ ) SUSYs ⇒ { } ⇒ preserved by a gravitational wave propagating on φ

Tuesday 30 October 2012 Dual giant gravitons Gravitons can also expand into a sphere on the AdS part (dual giant gravitons) (Grisaru, Myers, Tafjord’00; Hashimoto, Hirano, Itzakhi’00) Radius proportional to the angular momentum, but AdS non-compact No upper bound on the angular momentum ⇒ Realization of the stringy exclusion principle? Giant gravitons dual to subdeterminant operators (Balasubramanian, Berkooz, Naqvi, Strassler’01) Dual giant gravitons dual to semiclassical coherent states (Hashimoto, Hirano, Itzakhi’00)

Tuesday 30 October 2012 Outline 1. Introduction 1.1. Gauge/gravity (holographic) correspondence 1.2. D-Branes 1.3. Giant gravitons and AdS/CFT

Tuesday 30 October 2012 3.1. A few words about AdS_4/CFT_3

AdS4/CF T3 relates the Type IIA superstring (M-theory) on 3 7 AdS4 CP ( AdS4 S /Zk ) to the =6 Chern-Simons × × N matter theory with gauge group U(N)k U(N) k known × − as ABJM

Tuesday 30 October 2012 3.1. A few words about AdS_4/CFT_3

AdS4/CF T3 relates the Type IIA superstring (M-theory) on 3 7 AdS4 CP ( AdS4 S /Zk ) to the =6 Chern-Simons × × N matter theory with gauge group U(N)k U(N) k known × − as ABJM

Type IIA (M-theory) is a good description when N 1/5 << k ( N 1/5 >> k )

Tuesday 30 October 2012 3.1. A few words about AdS_4/CFT_3

AdS4/CF T3 relates the Type IIA superstring (M-theory) on 3 7 AdS4 CP ( AdS4 S /Zk ) to the =6 Chern-Simons × × N matter theory with gauge group U(N)k U(N) k known × − as ABJM

Type IIA (M-theory) is a good description when N 1/5 << k ( N 1/5 >> k )

Like AdS5/CF T4 it is a strong weak coupling duality, with ‘t Hooft coupling λ = N/k : - The string background describes the ‘t Hooft limit of the theory: N,k with λ = N/k ﬁxed →∞ - IIA weakly curved when k<

Tuesday 30 October 2012 3.2. Giant gravitons solutions

D2-brane spherical dual giant graviton (Nishioka, Takayanagi’08) (Hamilton, Murugan, Prinsloo, Strydom’09)

D4-brane giant graviton expanding in the CP^3 (Herrero, Y.L., Picos’12; Giovannoni, Murugan, Prinsloo’12)

Tuesday 30 October 2012 3.2. Giant gravitons solutions

D2-brane spherical dual giant graviton (Nishioka, Takayanagi’08) (Hamilton, Murugan, Prinsloo, Strydom’09)

D4-brane giant graviton expanding in the CP^3 (Herrero, Y.L., Picos’12; Giovannoni, Murugan, Prinsloo’12)

7 Useful approach: Reduce from AdS4 S /Zk ×

Tuesday 30 October 2012 3.2. Giant gravitons solutions

D2-brane spherical dual giant graviton (Nishioka, Takayanagi’08) (Hamilton, Murugan, Prinsloo, Strydom’09)

D4-brane giant graviton expanding in the CP^3 (Herrero, Y.L., Picos’12; Giovannoni, Murugan, Prinsloo’12)

7 Useful approach: Reduce from AdS4 S /Zk × Giant gravitons, but also new supersymmetric states by reducing on the orbifold: - Toroidal D2-brane (Nishioka, Takayanagi’08) - Spherical D2-brane with D0-brane charge (Nishioka, Takayanagi’08) - NS5-brane with D0-brane charge (Herrero, Picos, Y.L.’12)

Tuesday 30 October 2012 7 3.3. An M5-brane giant graviton in AdS4 S /Zk × 7 1 3 S /Zk : S /Zk bundle over the CP 2 2 1 2 ds 7 = dτ + + ds 3 S /Zk k A CP

Tuesday 30 October 2012 7 3.3. An M5-brane giant graviton in AdS4 S /Zk × 7 1 3 S /Zk : S /Zk bundle over the CP 2 2 1 2 ds 7 = dτ + + ds 3 S /Zk k A CP 5 7 M5-brane wrapping an S S /Zk and propagating on τ ⊂

2 2 2 1 2 2 2 2 2 ds = dt + R dτ + sin µ (dχ + A)+sin µds 5 M5 − k2 k S 2 2 2 dsS5 =(dχ + A) + dsCP2

R6 C = sin6 µdχ dτ dVol(CP2) 6 k ∧ ∧

2 2 2 1 2 2 2 2 2 ds = dt + R dτ + sin µ (dχ + A)+sin µds 5 M5 − k2 k S 2 2 2 dsS5 =(dχ + A) + dsCP2

R6 C = sin6 µdχ dτ dVol(CP2) 6 k ∧ ∧

M5 wrapped on χ , isometric Use the action for M5W ⇒

Tuesday 30 October 2012 M2-branes ending on this brane must also be wrapped on the isometric direction Self-dual 2-form replaced by a vector ⇒ Closed form for the action ⇒

1 SDBI = T4 k det( + k− ) − M5 | G F | 1 k1 SCS = T4 ikC6 + 2 + ... M 2 k ∧ F ∧ F 5

Tuesday 30 October 2012 M2-branes ending on this brane must also be wrapped on the isometric direction Self-dual 2-form replaced by a vector ⇒ Closed form for the action ⇒

1 SDBI = T4 k det( + k− ) − M5 | G F | 1 k1 SCS = T4 ikC6 + 2 + ... M 2 k ∧ F ∧ F 5 kµ : Killing vector pointing on the isometric direction

2 µν = gµν k− kµkν : reduced metric G −

ikC6 : interior product

2 µ k− k1 = gµχ/gχχ dx : Momentum operator on χ

Tuesday 30 October 2012 Substituting for our background:

2 k 2 N 4 H = Pτ 1 + tan µ 1 sin µ = H(µ) R − Pτ Minimum energy solution: µ =0 Point-like graviton → 1/4 Pτ sin µ = Giant graviton. For this sol. Pτ N N → ≤ k For both E = P R τ

Tuesday 30 October 2012 In Type IIA: Reduce along τ Pτ charge becomes D0-brane ⇒ charge, and the M5 a NS5

Tuesday 30 October 2012 In Type IIA: Reduce along τ Pτ charge becomes D0-brane ⇒ charge, and the M5 a NS5 New SUSY state: NS5 brane wrapping a twisted 5-sphere ⇒ inside the CP3 , with D0 charge

2 2 2 2 2 2 2 ds = dt + L sin µ cos µ(dχ + A) + ds 2 NS5 − CP Also C1,C5

Tuesday 30 October 2012 In Type IIA: Reduce along τ Pτ charge becomes D0-brane ⇒ charge, and the M5 a NS5 New SUSY state: NS5 brane wrapping a twisted 5-sphere ⇒ inside the CP3 , with D0 charge

2 2 2 2 2 2 2 ds = dt + L sin µ cos µ(dχ + A) + ds 2 NS5 − CP Also C1,C5 Quite non-trivial state from the NS5-brane point of view

Tuesday 30 October 2012 In Type IIA: Reduce along τ Pτ charge becomes D0-brane ⇒ charge, and the M5 a NS5 New SUSY state: NS5 brane wrapping a twisted 5-sphere ⇒ inside the CP3 , with D0 charge

2 2 2 2 2 2 2 ds = dt + L sin µ cos µ(dχ + A) + ds 2 NS5 − CP Also C1,C5 Quite non-trivial state from the NS5-brane point of view Action for a wrapped NS5:

S = T i C dc CS 4 k 5 ∧ 0 ( c0 associated to D0-branes)

Tuesday 30 October 2012 2φ 2 5 2φ 2 2φ 2 2 k 2 SDBI = T4 d ξ e− k + e (ikC1) det + 2 2φ 2 1 − | G k + e (ikC1) F | = dc + P [C ] F1 0 1

c0 cyclic Conserved conjugate momentum ⇒ k N 2 H = M 1 + tan2 µ 1 sin4 µ L − M

Tuesday 30 October 2012 2φ 2 5 2φ 2 2φ 2 2 k 2 SDBI = T4 d ξ e− k + e (ikC1) det + 2 2φ 2 1 − | G k + e (ikC1) F | = dc + P [C ] F1 0 1

c0 cyclic Conserved conjugate momentum ⇒ k N 2 H = M 1 + tan2 µ 1 sin4 µ L − M Minimum energy solution:

µ =0 Point-like D0 → M 1/4 sin µ = Giant D0. For this sol. M N N → ≤ k For both E = M L

Tuesday 30 October 2012 2φ 2 5 2φ 2 2φ 2 2 k 2 SDBI = T4 d ξ e− k + e (ikC1) det + 2 2φ 2 1 − | G k + e (ikC1) F | = dc + P [C ] F1 0 1

c0 cyclic Conserved conjugate momentum ⇒ k N 2 H = M 1 + tan2 µ 1 sin4 µ L − M Minimum energy solution:

µ =0 Point-like D0 → M 1/4 sin µ = Giant D0. For this sol. M N N → ≤ k For both E = M New realization of the stringy L exclusion principle

Tuesday 30 October 2012 k In the maximal case sin µ =1 M = N E = N ⇒ ⇒ L Bound state of N dimonopoles, with energy ED0 = k/L ⇒ or k dibaryons, with energy ED4 = N/L

Realization of the duality of Young Tableaux with N rows and k columns ( instability realized in the string theory side in ↔ terms of a NS5 instanton that turns the k D4 into N D0) (ABJM)

Tuesday 30 October 2012 3.4. Giant gravitons in Type IIA?

- M5 with magnetic ﬂux to induce momentum on χ , through

T4 k1 2 2 2 S = F F ds 5 =(dχ + A) + ds 2 CS 2 k2 ∧ ∧ S CP In Type IIA: NS5 with D0 charge and momentum. Giant grav. only in the maximal case

Tuesday 30 October 2012 3.4. Giant gravitons in Type IIA?

- M5 with magnetic ﬂux to induce momentum on χ , through

T4 k1 2 2 2 S = F F ds 5 =(dχ + A) + ds 2 CS 2 k2 ∧ ∧ S CP In Type IIA: NS5 with D0 charge and momentum. Giant grav. only in the maximal case

1 1 - M5 wrapped on S /Zk and moving on the S : In Type IIA: D4-brane wrapped on the squashed CP2

2 2 2 2 2 2 2 ds = dt + L sin µ cos µ(dχ + A) + ds 2 D4 − CP and moving on χ

Tuesday 30 October 2012 Introduce an electric ﬂux (to undo the squashing):

Ei = L sin µ cos µAi C =0 5 P N 2 H = χ 1 + tan2 µ 1 sin4 µ L sin µ − Pχ N Ground state: sin µ =1 P = N and H = ⇒ χ L

Tuesday 30 October 2012 Introduce an electric ﬂux (to undo the squashing):

Ei = L sin µ cos µAi C =0 5 P N 2 H = χ 1 + tan2 µ 1 sin4 µ L sin µ − Pχ N Ground state: sin µ =1 P = N and H = ⇒ χ L Dibaryon: Limiting case of a D4-brane wrapping a → squashed CP2

Tuesday 30 October 2012 Introduce an electric ﬂux (to undo the squashing):

Ei = L sin µ cos µAi C =0 5 P N 2 H = χ 1 + tan2 µ 1 sin4 µ L sin µ − Pχ N Ground state: sin µ =1 P = N and H = ⇒ χ L Dibaryon: Limiting case of a D4-brane wrapping a → squashed CP2

No giant graviton away from the maximal case

Tuesday 30 October 2012 The giant graviton in this space is a more complicated solution (Giovannoni, Murugan, Prinsloo’12) :

Different ansatz: CP3 as a U(1) bundle over S2 S2 with × an extra coordinate that controls the sizes of the two S2 At maximal size the giant factorizes into two dibaryons:

Our ansatz: Select a CP2 submanifold.

Tuesday 30 October 2012 Outline 1. Introduction 1.1. Gauge/gravity (holographic) correspondence 1.2. D-Branes 1.3. Giant gravitons and AdS/CFT

Tuesday 30 October 2012 4. Giant gravitons as fuzzy manifolds

Giant graviton: p-brane with S p or CPp/2 topology with

angular momentum Macroscopical (R>>ls) ⇒

Tuesday 30 October 2012 4. Giant gravitons as fuzzy manifolds

Giant graviton: p-brane with S p or CPp/2 topology with

angular momentum Macroscopical (R>>ls) ⇒ Can we give a microscopical description, in terms of expanding gravitons?

Tuesday 30 October 2012 4. Giant gravitons as fuzzy manifolds

Giant graviton: p-brane with S p or CPp/2 topology with

angular momentum Macroscopical (R>>ls) ⇒ Can we give a microscopical description, in terms of expanding gravitons? Connect with known examples of expanded brane conﬁgs: Myers dielectric effect

Tuesday 30 October 2012 4. Giant gravitons as fuzzy manifolds

Giant graviton: p-brane with S p or CPp/2 topology with

angular momentum Macroscopical (R>>ls) ⇒ Can we give a microscopical description, in terms of expanding gravitons? Connect with known examples of expanded brane conﬁgs: Myers dielectric effect

Giant gravitons as dielectric gravitational waves → (Janssen, Y.L., Rodriguez-Gomez’02-05)

Tuesday 30 October 2012 Myers dielectric effect (an example):

n D0’s in a constant RR F4 :

dτSTr(i i C(3))= 1 dτSTr(XjXiXk)F (4) + ... X X − 3 0ijk

Tuesday 30 October 2012 Myers dielectric effect (an example):

n D0’s in a constant RR F4 :

Tuesday 30 October 2012 Myers dielectric effect (an example):

n D0’s in a constant RR F4 :

dτSTr(i i C(3))= 1 dτSTr(XjXiXk)F (4) + ... X X − 3 0ijk (4) F =2fijk Ground state: n D0’s expanded into a 0ijk → fuzzy S2 Xi = f J i, [J i,Jj]=2iijkJ k . . . 2 . .. .. 3 f . . (Xi)2 =( )2(n2 1)I = R2 I .. .. ⇒ i=1 2 − . .. C3 dipole moment Fuzzy D2-brane ⇒ 2 2 Regime of validity: 4πR /n << ls In the context of the gauge/gravity duality: λ << n2 ⇒ Finite ‘t Hooft coupling regime

Tuesday 30 October 2012 D0-branes are gravitons in M-theory, moving along the 11th direction: M0 M2 M5

wave D0 F1 D2 D4 NS5

Uplift Myers action for N D0-branes ⇒

Tuesday 30 October 2012 D0-branes are gravitons in M-theory, moving along the 11th direction: M0 M2 M5

wave D0 F1 D2 D4 NS5

Uplift Myers action for N D0-branes ⇒ Also, Matrix theory calculation: - Matrix string theory: (Non-Abelian) Type IIA strings with non-zero light-cone momentum - Sen-Seiberg limit + static gauge (Non-Abelian) massless → particles with spatial momentum (IIA gravitational waves)

Tuesday 30 October 2012 Dielectric couplings?: Matrix string theory in a weakly curved background Precise agreement with the linear expansion → of Myers action

The action for M-theory gravitons

1 1 i kj S = T0 dτ STr k− E00 + E0i(Q− I) E Ej0 detQ {− − k 2 i (3) 1 2 (6) k− k ∂X + i(i i )C + (i i ) i C + ... − i X X 2 X X k }

kµ : Killing vector pointing on the direction of propagation (isometric momentum eigenstate) ↔

1 (3) i i i k E = + k− (i C ) ,Q= δ + ik[X ,X ]E G k j j kj

Tuesday 30 October 2012 In the Abelian limit: Legendre transformation:

T0 1 µ ν S[γ]= dτ γ γ− ∂X ∂X g − 2 | | µν

Tuesday 30 October 2012 In the Abelian limit: Legendre transformation:

T0 1 µ ν S[γ]= dτ γ γ− ∂X ∂X g − 2 | | µν With this action:

Giant Fuzzy manifold

S2 Fuzzy S2

3 1 2 S S over Sfuzzy

S1 Fuzzy cylinder

5 1 2 S S over CPfuzzy

Tuesday 30 October 2012 7 The M5-brane in AdS4 S /Zk microscopically × p/2 p p - CP : coset manifold SU( 2 + 1)/U( 2 )

Tuesday 30 October 2012 7 The M5-brane in AdS4 S /Zk microscopically × - CPp/2 : coset manifold SU( p + 1)/U( p ) . Submanifold of 2 2 2 p +p R 4 determined by the constraints:

p2 p2 4 +p 4 +p p i 2 ijk j k 2 1 i (x ) =1, d x x = p −p x i=1 4 ( 2 + 1) j,k=1 p dimensional manifold →

Tuesday 30 October 2012 7 The M5-brane in AdS4 S /Zk microscopically × - CPp/2 : coset manifold SU( p + 1)/U( p ) . Submanifold of 2 2 2 p +p R 4 determined by the constraints:

p2 p2 4 +p 4 +p p i 2 ijk j k 2 1 i (x ) =1, d x x = p −p x i=1 4 ( 2 + 1) j,k=1 p dimensional manifold → p Fubini-Study metric of the CP 2 given by p2 4 +p 2 p i 2 ds p = p (dx ) CP 2 4( + 1) 2 i=1

Tuesday 30 October 2012 7 The M5-brane in AdS4 S /Zk microscopically × - CPp/2 : coset manifold SU( p + 1)/U( p ) . Submanifold of 2 2 2 p +p R 4 determined by the constraints:

p2 p2 4 +p 4 +p p i 2 ijk j k 2 1 i (x ) =1, d x x = p −p x i=1 4 ( 2 + 1) j,k=1 p dimensional manifold → p Fubini-Study metric of the CP 2 given by p2 4 +p 2 p i 2 ds p = p (dx ) CP 2 4( + 1) 2 i=1 p - Matrix level deﬁnition Fuzzy CP 2 : ↔ 1 p Xi = T i , T i : generators of SU( + 1) in the (m, 0) irrep √Cn 2

Tuesday 30 October 2012 Substituting in the action for M-theory gravitons:

2 k 2 2nN 4 H(µ)= Pτ 1 + tan µ 1 2 sin µ R − Pτ (m +3m)

Tuesday 30 October 2012 Substituting in the action for M-theory gravitons:

2 k 2 2nN 4 H(µ)= Pτ 1 + tan µ 1 2 sin µ R − Pτ (m +3m) Given that n = (m+1)(m+2) when m we reproduce 2 →∞ the macroscopical result:

2 k 2 N 4 H = Pτ 1 + tan µ 1 sin µ = H(µ) R − Pτ

Tuesday 30 October 2012 Outline 1. Introduction 1.1. Gauge/gravity (holographic) correspondence 1.2. D-Branes 1.3. Giant gravitons and AdS/CFT

2. GST giant gravitons 3. Giant gravitons in ABJM 3.1. A few words about AdS_4/CFT_3 3.2. (Dual) giant graviton solutions 3.3. An M5-brane giant graviton in 3.4. Giant gravitons in Type IIA? 4. Giant gravitons as fuzzy manifolds 5. Conclusions

Tuesday 30 October 2012 5. Conclusions

7 M5-brane giant graviton in AdS4 S /Zk : × New NS5-brane with non-trivial geometry, satisfying → the stringy exclusion principle and realizing on the gravity side the duality of Young tableaux

3 D4-brane giant graviton in AdS4 CP only in the maximal × case: Dibaryon: Limiting case of a D4-brane with momentum → wrapping a squashed CP2

Microscopical description: These conﬁgurations exist also at ﬁnite ‘t Hooft coupling → (useful if they are not BPS)

Tuesday 30 October 2012 Thanks!

Tuesday 30 October 2012 In progress: