A SOLITON MENAGERIE IN ADS Mukund Rangamani Durham University

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1 A SOLITON MENAGERIE IN ADS Mukund Rangamani Durham University Holography at finite density APC, Paris Nov 17, 011 Simon Gentle, Benjamin Withers, MR

2 Surprises in AdS Gravity in AdS shows some interesting surprises: Gravitational dynamics behaving like field theory dynamics (eg., black hole thermodynamics). Black holes with scalar hair Geons, etc.. One crucial element that underlies many of these phenomena is the gravitational box provided by AdS boundary conditions. We will explore some consequences of such behaviour for scalar solitons which we will view as non-linear condensates of scalar field in global AdS.

3 An invitation to solitons Our interest is going to be in solitonic solutions to a class of bulk gravity theories: static spherically symmetric configurations states of the dual CFT with fixed mass m and charge generically form a phase boundary in the micro-canonical ensemble Focus will be on understanding: when do these global solitons get large, ie., when does the planar limit exist? micro-canonical phase diagram role of external dials: CFT spectrum, deformations

4 OUTLINE Motivation Solitons in global AdS Bottom up theories A tale of two truncations Discussion

5 Solitons in global AdS Consider a AdS/CFT pair and focus on the bosonic operators: discrete spectrum of single particle states (CFT primary and descendants)! = +n + l Solitons: Bose condensed state with macroscopic occupation of a single energy level. not a linear superposition of states, since want occupation numbers large enough to cause gravitational backreaction. state of the dual CFT has fixed charge and mass and has a non-trivial vev for the dual operator analogs of charged neutron stars discussed earlier in AdS/CFT context Arsiwalla, de Boer, Papadodimas, Verlinde

6 Bulk set-up: Solitons in global AdS Einstein-Maxwell-scalar theory characterized by scalar potential, gauge coupling: S b = 1 16 G 4 Z d 4 x p g R 1 4 F (@ ) 1 Q( 1 ` )A V ( ) ` focus on solutions with no phase excitation for the scalar i..e, no superfluid configuration with non-trivial superfluid velocity. Q( ) V ( ) mass ` charge q 6+m ` m q p 1 sinh cosh p 7+cosh p 1 1 sinh p +cosh p 1

7 V ( ). s:gsanz when discussing the phenomenological theory we will specialise to the.1.1 Ansatz this for global solitons Throughout paper we will emplo s:gsanz scussing the phenomenological theory will specialise to thethe case m which =. determines the following sm IR,.1.1we Ansatz for global solitons Throughout this paper wesymmetric, will employ the follo Rt )AdS for spherically static s:gsanz Solitons in global t ) for spherically symmetric, static global so Ansatz for global solitons Throughout this paper we willremploy the following metric ansatz (pre co ds = R ) for spherically symmetric, static global solutions, t hout this paper we will employ the following metric ansatz (preserving SO(3) ds = g(r)e g(r)e (r) dt + ` Working with dimensionless r has a corefalloffs sphericallygravitational symmetric, staticansatz: global solutions, dr (r) for a where d is the line element ` d factors to (r). g(r)e dt + ` + r d where is the line element for a ds =some unit S. g(r) dr (r) AA== A(r)dt and and = (r). ansatz, ds = g(r)e dt + ` + r d (.4) A(r)dt = With (r).this With thist g(r) where d is the line element for a unit S. fields For the vector andthe scalsfa.1. Boundary conditions for rr! 1, where the admit the following! 1, where the fields admit the sec:bcscalar d is the line element for a unitas=. A(r)dt For the vector=and(r). scalar fieldthis we ansatz, take and With the boundary of global A +.. Since theasymptotics scalar field has A(r) mass=mµ ` r = conditions: regularity at core and AdS r)dt and Boundary = (r). With this ansatz, the boundary of global AdS is located at r! 1, where the fields admit the following asymptotic expansion, A(r) = µ 1 ( 1 = where the fields admit the following asymptotic expansion, we can not only impose the standard (r) = + + r r A(r) = µ +... conditions for the scalar, but we can also (r) = A(r) ==Aµ r Aasym A(r) (.5a) g(r)and = r alternat +1+ r conditions.1 For the standard (r) = c1 + (r) = (r) =O g(r).=.. r 1, + (r) = (.5b) operator rdual rtophiasym as O 1 and with g(r) = 1 r+ rr + m mhave h O i = 1 we as a result Note that 1 (r) g(r) r(.5c) + the 1 +theories +.. in. table for = each of listed 1. = We g(r) = r asymfalloffs. gasym r (r) = 0 + r dimension 1 operator is in supersymmetric most cases that t 1 = 0. We will require a (r) = (r) = (.5d) betaasym asymfalloffsin [4, for5]each of the theories listedconditi in ta the multi-trace boundary 8 of theories listed in will use coordinate h of the theories listed in 1.for Weeach willm usethe coordinate freedom in table t to0.set functional (We ).. will Of particular to = require ininterest mostfree ca = we can For asymfalloffs choice oftable scalar mass choose 1. 1We 1 origin. We will require in most cases that the0.solutions regularin(o atmost the We willare require cases that the turn solutions regular ) whichin we will on toare deform th 1 = 8 8 SCF T

8 Preliminaries: Stars in AdS Consider a slightly simpler problem, where we replace the matter in the bulk by a self-gravitating fluid (r), P(r), q(r) This gravitational system has a set of static solutions (stars) which are parameterized by their core density 0 Ρ r r m r r Hubeny, Liu, MR

9 Preliminaries: Stars in AdS Curiously, this system exhibits an interesting critical behaviour M a Ρ0 Page, Phillips Hubeny, Liu, MR Note that this also happens for asymptotically flat stars. Sorkin, Wald, Zhang Interestingly, the behaviour is dimension dependent. For d>d c ' 11 we have M( 0 ) become a monotone function of 0 Hammersley Intuitively, gravitational attraction becomes weaker... Vaganov

10 OUTLINE Motivation Solitons in global AdS Bottom up theories A tale of two truncations Discussion

11 Bulk Abelian Higgs model We ll first consider a simple theory in the bulk: charged scalar with mass and charge. Fix m = and study dynamics as a function of q. Q( ) V ( ) mass ` charge q p 6+m ` p m q p Solitons exist and can be parameterized by core value of scalar c : convince oneself by perturbative construction for small values of. Basu et al (B 3 LMU) Hartnoll, Herzog, Horowitz Bhattacharyya, Minwalla, Papadodimas CFT (boundary quantities) are determined as a function of core scalar. For small core scalar, they are monotone, but interesting things happen as we increase. c m( c ), ( c ), hoi( c ) c

12 Bulk Abelian Higgs model: I m( c ) 5 Planar limit 4 3 q =1.4 q =1.3 1 m( c )=m 1 + m q ( c ) cos(! log( c )+ ) q =1. q = c AdS 4 vacuum

13 Bulk Abelian Higgs model: I q q y vev 1.0 mass m vev and mass as function of bulk scalar charge for Δ =

14 Bulk Abelian Higgs model: I y vev q mass m 30 vev and mass as function of bulk scalar charge for Δ = 1

15 Solitons: summary 1 For fixed scalar m & bc, the theory has critical behaviour at some. Focussing on solutions which are connected perturbatively to global AdS we see that there is a maximal mass attained by solitons for q<q c. Solutions with q>q c appear to be smoothly connected for large masses to the planar zero temperature superfluids. m, (qµ) (qµ), (qµ) 0.0 q c Horowitz, Roberts qµ 8

16 Solitons: Questions 1 Below q c the fact that solutions have bounded mass suggests that the global solutions are not smoothly connected to the planar AdS solutions. Where are the planar superfluid solutions at zero temperature? singular solitons (which admit planar limti) new branches of solutions (not connected to global AdS) Both of these possibilities are a-priori equally likely; for 5 dimensional analogs Bhattacharyya, Minwalla, Papadodimas predict that the planar limit is characterized by singular global solitons. NB: large core scalar limit asymptotes to a singular solution. Story in 4d case turns out to be a lot richer.

17 Bubbles in solution space! m( c ) 8 q =1. 8 q =1.5 8 q = q =1.3 8 q = q = c A second branch of solution exists even for q < q c ; it controls the planar limit. Curious feature, closed loops in solution space!

18 m( c ) 4 Bubbles in solution space! c The bubbles seem to exist for q apple q? and disappear subsequently.

19 Microcanonical phase diagram: I m( c ) 8 m m c m( c ) 5 m m c The microcanonical phase diagram for sample charges q < q c & q > q c

20 OUTLINE Motivation Solitons in global AdS Bottom up theories Truncated tale of two truncations Discussion

21 The equal charged susy truncation 11 dimensional sugra admits a consistent truncation to 4d where we retain a single Maxwell field and a complex scalar (hyperscalar): p p Q( ) p V ( ) sinh +cosh p mass ` charge q 1 p 6+m ` p m q 1 Chong, Lu, Pope Donos, Gauntlett Quantization the theory with Δ = 1 boundary conditions for the scalar is supersymmetric, preserving 1/8-susy. Geometry of M-branes spinning equally in the transverse R 8 with hyperscalars turned on. Not only is this an interesting theory to consider from M brane perspective, but it also has curious implications for the non-conformal D1- brane theory. Withers, MR Kanitscheider, Skenderis; David, Mahato, Thakur, Wadia

22 Global susy solitons m( c ) m log( c ) Curiously, susy solutions in this theory seem to be monotonic in the core scalar. Asymptotic growth of charges is logarithmic wrt core scalar value. Strong contrast with the corresponding truncation in 5d which was studied earlier. Bhattacharyya, Minwalla, Papadodimas

23 Puzzle about planar limit ho 1 i T 0.8 F µ T µ T µ Donos, Gauntlett Planar hairy black holes exist in this theory, but they behave very differently than conventional examples of holographic superfluids! In particular, no zero T planar hairy black hole! So where does the large global soliton go to?

24 The large susy soliton The large susy soliton ends up as a very curious planar solution: one that is neutral. The solution has nontrivial scalar, Maxwell and metric profiles, but the asymptotic conserved charges all vanish. Possible because of the scalar boundary condition (the theory admits designer gravity solitons). ds = r µ dx µ dx dr 1+r, (r) =p sinh 1 1 r r r +1

25 More solitons (sans bubbles) m( c ) c There is much more to the story: new branches of solutions satisfying the same boundary conditions exist.

26 More solitons (sans bubbles) m( c ) c Study solution space using double trace deformation: interesting behaviours vis a vis new branches of solutions (though no change in phase structure).

27 Micro-canonical phase diagram 3.5 m( c ) m c

28 OUTLINE Motivation Solitons in global AdS Bottom up theories A tale of two truncations Discussion

29 Discussion Story of solitons in AdS with the micro-canonical phase diagram being controlled by four distinguished solutions: global AdS and associated perturbative solitons planar solitons (zero T superfluids) Two attractor solutions dominating large core scalar solitons The behaviour seen in bottom-up models can be reproduced in consistent truncations of M-theory. Towards a complete phase diagram for 1/8 BPS-solutions of M brane world-volume theory.

30 Open questions Global soliton in AdS can play an important role in understanding the degeneracy of BPS states. Open question: counting of susy black hole degeneracy (bh solutions preserve 1/16 susy). Equilibrium states of superfluids on compact spaces. rotating AdS black holes were shown in fluid/gravity correspondence to correspond to stationary rigidly rotating fluids. are there rigidly rotating superfluid configurations? Low dimesional (1+1) superfluids in large N limit?

31 r A c The End