Sources and V evs for Lifshitz Holography Jelle Hartong Niels - - PowerPoint PPT Presentation

Sources and V evs for Lifshitz Holography

Jelle Hartong Niels Bohr Institute Gauge/Gravity Duality 2013 Munich Tuesday, July 30

In collaboration with Morten Christensen, Niels Obers, Blaise Rollier, to appear

Outline of the Talk

• Motivation
• The Model and Lifshitz Solutions
• AlLif Space-Times and Beyond
• Frame Fields and Boundary Geometry
• W

ard Identities

• Counting Sources and V

evs

• Summary

Motivation

• Lifshitz holography: new type of holographic duality

with non-AdS asymptotics and Lorentz violating dual field theories ( )

• Interesting for: i). black holes as they form near-horizon

geometries of extremal black branes, ii). AdS/CMT, iii). extending holography beyond AdS/CFT.

• Boundary stress-energy tensor for Lifshitz holography
• Boundary geometry
• Irrelevant deformations of AlLif space-times
• Identify the sources and vevs and holographic

reconstruction

[t] = 2 , [~ x] = 1

The Model

• T

runcation of type IIB on

• Scherk-Schwarz reduce to 4D using standard KK ansatz

for metric and dilaton and

• [Donos, Gauntlett, 2010],[Cassani, Faedo, 2011],

[Chemissany, J.H., 2011]

• Counterterms and reduction: [Papadimitriou, 2011],

[Chemissany, Geissbuehler, J.H., Rollier, 2012]

S = Z d5x p −ˆ g ✓ ˆ R + 12 − 1 2(∂ ˆ φ)2 − 1 2e2 ˆ

φ(∂ ˆ

χ)2 ◆ ˆ χ = ku + χ S5

S = Z d4x√−g ✓ R − 3 2(∂Φ)2 − 1 4e3ΦF 2 − 1 2(∂φ)2 − 1 2e2φ(Dχ)2 − V ◆

Dχ = dχ − kA V = k2 2 e−3Φ+2φ − 12e−Φ F = dA

Lifshitz Solutions

• A z=2 Lifshitz background:
• In 5D:
• Size of the circle:
• V

alidity of the approximation:

• [Costa, Taylor, 2010], [McGreevy, Balasubramanian, 2011]

ds2 = l2eΦ(0) ✓dr2 r2 − e−2Φ(0)l2 dt2 r4 + 1 r2 (dx2 + dy2) ◆

eΦ = eΦ(0) = lkgs 2

dˆ s2 = l2 ✓dr2 r2 + 1 r2 (2dudt + dx2 + dy2) ◆ + l2k2g2

s

4 du2

2πLphys = Z 2πL du p ˆ guu = 2πLlkgs 2 , u ∼ u + 2πL

l ls 1 , gs ⌧ 1 , Lphys ls 1

Lphys l = Lkgs 2 1 (decompactification: N = 4 SYM) , ⇠ 1 (DLCQ with nonzero theta angle) , ⌧ 1 (below KK mass scale: 3D LCS)

AlLif Solutions and Beyond

• Asymptotically locally Lifshitz (AlLif) spaces [Ross, 2011]:
• and are asymptotically constant with .
• Boundary gauge field and axion:
• For HSO there exists an ADM slicing with
• Irrelevant deformations: and is not HSO.

Leading log deformations, i.e. in radial gauge.

• Cannot lift the constraint:

φ(0) 6= cst τ(0)a Φ φ r−4 log r τ(0)a

ds2 = eΦ dr2 r2 − etet + δijeiej

et

a = r−2τ(0)a + . . . , where τ(0)a is HSO

ei

a = r−1ei (0)a + . . .

Aa − e−3Φ/2et

a = A(0)a + . . .

χ = χ(0) + . . .

e2Φ(0) = −1 4e3Φ(0) ⇣ ✏abc

(0) ⌧(0)a@b⌧(0)c

⌘2 + k2 4 e2φ(0)

[t] = 2 , [~ x] = 1

Φ(0) = log kgs 2

Boundary Geometry I

• Define and such that:
• There is no Lorentzian boundary metric! Cannot raise/

lower indices.

• Local bulk Lorentz transformations act on the boundary

frame fields as Galilean boosts and rotations:

• Spin connection coefficients determined by the vielbein

postulate:

τ(0)aτ a

(0) = −1

ei

(0)aτ a (0) = 0

τ(0)aea

(0)i = 0

ei

(0)aea (0)j = δi j

τ a

(0)

ea

(0)i

δτ(0)a = 0 δei

(0)a = ξb (0)

⇣ ωi

(0)bτ(0)a + ω(0)b i je j (0)a

⌘ δτ a

(0) = ξb (0)ωi (0)bea (0)i

δea

(0)i = −ξb (0)ω(0)b j iea (0)j

D(0)aei

(0)b = ∂aei (0)b − Γc (0)abei (0)c + ωi (0)aτ(0)b + ω(0)a i je j (0)b = 0

Boundary Geometry II

• The vielbein postulate constraints such that:
• W

e define by for AlLif space-times but we keep this also for the Lifshitz UV completion so that:

• When the boundary geometry is

Newton-Cartan for which we have .

• For applications of Newton-Cartan to CMT see e.g. [Son,

2013].

Γc

(0)ab

⇣ δd

b + τ(0)bτ d (0)

⌘ ⇣ δe

c + τ(0)cτ e (0)

⌘ r(0)aΠ(0)de = 0 ∂aτ(0)b − ∂bτ(0)a = 0

Π(0)ab = δijei

(0)ae j (0)b

Πab

(0) = δijea (0)ieb (0)j

Γc

(0)ab = −1

2τ c

(0)(∂aτ(0)b + ∂bτ(0)a) + 1

2Πcd

(0)

• ∂aΠ(0)bd + ∂bΠ(0)ad − ∂dΠ(0)ab
• −1

2e3Φ(0)/2Πcd

(0)

• F(0)daτ(0)b + F(0)dbτ(0)a
• F(0)ab = ∂aA(0)b − ∂bA(0)a

ˆ Γc

(0)ab = Γc (0)ab

Γc

(0)ab

r(0)aτ(0)b = r(0)aΠbc

(0) = 0

W ard Identities I

• On-shell variation of the action:
• Local symmetries:
• diffeomorphisms
• anisotropic W

eyl:

• gauge:
• local tangent space
• Only at leading order:

δS ⇠ Z d3xe(0) ⇣ St

(0)aδτ a (0) + Si (0)aδea (0)i + T t (0)δA(0)t + T i (0)δA(0)i

+hOΦiδΦ(0) + hOφiδφ(0) + hOχiδχ(0) A(0) δr r ◆

δA(0)a = ∂aΛ(0) , δχ = kΛ(0)

δτ a

(0) = 2w(0)τ a (0) , δea (0)i = w(0)ea (0)i

δA(0)t = 2w(0)A(0)t , δA(0)i = w(0)A(0)i

δr = w(0)r δτ a

(0) = −ω(0)τ a (0) , δea (0)i = ω(0)ea (0)i

δA(0)t = −ω(0)A(0)t , δA(0)i = ω(0)A(0)i

δΦ(0) = −2ω(0)

W ard Identities II

• The local tangent space rotation and gauge invariant

boundary stress-energy tensor:

• W

ard identities for assuming

• Similar to HIM boundary stress tensor [Hollands,

Ishibashi, Marolf, 2005]

• Conserved currents [Ross, Saremi, 2009]

T b

(0)a = −τ b (0)

✓ St

(0)a + T t (0)

1 k ∂aχ(0) ◆ + eb

(0)i

✓ Si

(0)a + T i (0)

1 k ∂aχ(0) ◆

T b

(0)a

B(0)a = A(0)a − 1 k ∂aχ(0) r(0)bT b

(0)a = T c (0)b

⇣ τ(0)cr(0)aτ b

(0) + ei (0)cr(0)aeb (0)i

⌘ T t

(0)∂aB(0)t T i (0)∂aB(0)i hOΦi∂aΦ(0) hOφi∂aφ(0)

∂aτ(0)b − ∂bτ(0)a = 0

A(0) = −2τ(0)bτ a

(0)T b (0)a + ei (0)bea (0)iT b (0)a + 2T t (0)B(0)t + T i (0)B(0)i

r(0)b ⇣ Ka

(0)AT b (0)a

⌘ = 0

LK(0)Aτ a

(0) = LK(0)AΠab (0) = LK(0)Abdry scalar = 0

Counting Sources & V evs

• The anomaly is given by a 3D z=2 Horava-Lifshitz action

coupled to matter fields.

• #sources=15(components)-8(local

symmetries)-1(c0nstraint)=6

• #vevs=15(components)-9(W

ard identities including the

• ne for the leading order symmetry)=6
• In the Fefferman-Graham expansion there appears

another function which does not appear in the W ard identities.

• W

e thus have 6+6+1 free functions controlling the expansion.

Summary

• There exists a Lifshitz model that is a well-defined low

energy limit of IIB string theory.

• Found all irrelevant deformations of AlLif space-times.
• Frame fields useful for boundary geometry (Newton-

Cartan) and coordinate independent definition of boundary stress-energy tensor.

• There are 6+6 sources & vevs and one additional free

function that control the Fefferman-Graham expansion.