SLIDE 1
A soliton menagerie in AdS
Simon Gentle
Durham University
3 April 2012
Based on [1112.3979] with M. Rangamani and B. Withers
SLIDE 2 Motivation
◮ Hairy black holes are commonplace in Anti-de Sitter space. ◮ Charged, asymptotically AdS black branes develop charged
scalar hair below a critical temperature.
Gubser; Hartnoll, Herzog & Horowitz ◮ AdS/CFT motivation: charged hairy black brane ↔ superfluid
◮ What happens at low temperature? Fernandez-Gracia & Fiol; Gubser & Nellore; Horowitz & Roberts
SLIDE 3
Main point
The planar limit of charged scalar solitons in global AdS4 coincides generically with the zero-temperature limit of charged hairy black branes.
SLIDE 4 Theory
◮ Einstein-Maxwell-scalar theory with Λ < 0:
S =
ℓ2 − 1 4F 2 − (∂φ)2 − q2 ℓ2 φ2A2 − m2
φφ2
φℓ2 = −2 and set ℓ = 1 from now on.
SLIDE 5 Planar limit
◮ Global Schwarzschild-AdS4 (one-parameter family):
ds2 = −r2
r2 − m r3
dr2 r2 1 + 1
r2 − m r3
+ r2dΩ2
2 ◮ Consider the scaling limit
r → λr, t → t/λ, λ → ∞, λ2dΩ2
2 → d
x2
2 ◮ Leaves us with vacuum planar AdS4 unless we scale
m → λ3m too.
◮ When does a solution in global AdS have an interesting planar
limit?
- 1. Branch of solutions with unbounded asymptotic coefficient.
- 2. Other coefficients must grow fast enough to survive.
SLIDE 6
Solitons vs. Branes
Global solitons:
◮ Regular core, horizon-free and asymptotic to global AdS4. ◮ Non-topological ◮ Examples in this type of theory in 5D: Dias, Figueras, Minwalla, Mitra, Monteiro & Santos
Charged hairy black branes:
◮ Regular horizon and asymptotic to planar AdS4.
SLIDE 7
Construction of solitons
◮ Ansatz:
ds2 = −g(r)e−χ(r)dt2 + dr2 g(r) + r2dΩ2
2
A = At(r)dt, φ = φ(r)
◮ Asymptotic expansions:
g = r2 + 1 + φ2
1
2 − m r + . . . , χ = χ∞ + . . . At = µ − ρ r + . . . , φ = φ1 r + φ2 r2 + . . .
◮ Integrate numerically with remaining boundary conditions
using a shooting method.
◮ One-parameter family at a given q.
SLIDE 8
Initial soliton results
◮ Critical qc ◮ Physical intuition: force balance
⇒ Planar limit exists
◮ Similar plots for charged boson stars in flat space: Kleihaus, Kunz, L¨ ammerzahl & List
SLIDE 9 Coincidence of limits
◮ Plot scaling-invariant quantities. Here is an example at
q2 = 1.3 > q2
- c. The dashed lines indicate these quantities for
charged hairy black branes at low temperature.
2 4 6 8 0.00 0.05 0.10 0.15 0.20 0.25
qµ
m (qµ)3 , ρ (qµ)2 , φ2 (qµ)2 ◮ Regularity?
SLIDE 10 New soliton branches
◮ Low temperature charged hairy black branes exist for q < qc.
⇒ Find new global soliton branches:
◮ disconnected from AdS vacuum ◮ planar limit exists ∀ q ◮ closed bubbles in space of solutions
◮ Coincidence of limits is generic.
SLIDE 11 New soliton branches
1 2 3 4 2 4 6 8 1 2 3 4 50 100 150 200 250
m φc
◮ Contours of the function q(m, φc): colour represents the value
- f q. Inset: behaviour over a larger range of m.
SLIDE 12
Summary
◮ Charged hairy black brane at zero T ↔ planar soliton ◮ Coincidence of limits is generic. ◮ Useful technique for finding new connections between
solutions.
◮ Starting point for understanding superfluid phases at low
temperatures.
SLIDE 13 Generic coincidence of limits
◮ Scaling-invariant quantities. The right panel zooms in to the
interesting bits.
0.00 0.05 0.10 0.15 0.20 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.160 0.165 0.170 0.175 0.180 0.095 0.100 0.105 0.110 0.115 0.120
m µ3 φ2 µ2 φ2 µ2
– – – Low temperature charged hairy black brane
- - - - Large-φc global soliton
