String theory effects on 5D black strings Alejandra Castro - - PowerPoint PPT Presentation

String theory effects on 5D black strings

Alejandra Castro University of Michigan

Great Lakes String Conference 2008 University of Wisconsin, Madison

Work in collaboration with J. Davis, P. Kraus and F. Larsen

hep-th/0702072, hep-th/0703087, 0705.1847[hep-th], 0801.1863 [hep-th]

Motivation

Black hole entropy Small black holes

• Classical description:

Singular solutions in SUGRA with vanishing area at horizon.

• Quantum description:

Two-charge system (momentum + winding) with non-zero entropy Higher order corrections should provide a string/Planck scale horizon and reproduce entropy according to Wald’s formula Thermodynamics Statistical mechanics General relativity Bekenstein-Hawking area law for entropy Corrections from higher dimension operators String theory Degeneracy of D-branes bound states Weakly coupled regime (large charge limit)

Maldacena, Strominger & Witten

Outline

• Motivation
• Challenges and progress of higher derivative corrections
• Theoretical framework: 5D conformal supergravity
• Black strings: detailed analysis
• Small black holes
• Conclusion

Why five dimensions?

HD corrections: Large versus small black holes Anomaly inflow: Take advantage of AdS3 /CFT2 First determine all corrections up to a given order (very hard task, but doable!) For large black hole at leading order the solution is regular. Expansion is systematic and higher orders are small For small black holes the leading order solution is singular. Size of the horizon is of the same order as corrections. All terms in expansion contribute. (very hard task!) Gauge and Gravitational anomalies: Entropy formula controlled by left & right moving central charges

• f 1+1 CFT.

cIJK AI FJ FK c2I AI Tr(R2) KEY: Central charge governed by the coefficients of Chern-Simons terms in supergravity.

Kraus & Larsen

The theory and the victim…

Laboratory: Five dimensional black strings Theoretical framework: 5D Supergravity

• N=2 supergravity

coupled to vector multiplets

• Can be obtained as arising from a reduction of M-theory on CY3

. Four-form flux describing M-branes wrap cycles in CY3 leading to sources in 5D. M2-branes source of electric charge -- qI . M5-branes source of magnetic charge -- pI.

• New feature: off-shell formalism
• Auxiliary fields are introduced to make

susy transformations independent of action.

• Extended one dimensional objects carrying magnetic charge.
• Near horizon geometry: AdS3

x S2.

• M5-branes wrapping four-cycles. Scales

at the horizon are set by cIJK pIpJpK.

• Small black hole:

cIJK pIpJpK=0.

Precision Measurements

Off-shell conformal supergravity

δψμ = µ Dμ + 1 2vabγμab − 1 3γμγ · v ¶ ² = 0 δΩI = µ −1 4γ · F I − 1 2γa∂aM I − 1 3M Iγ · v ¶ ² = 0 δχ = µ D − 2γcγabDavbc − 2γa²abcdevbcvde + 4 3(γ · v)2 ¶ ² = 0

• Supersymmetry

transformations close off-shell

• Inclusion of a scalar (D) and two-form (vab

) auxiliary field

• This formalisms allows unambiguously to find supersymmetric

completion of four derivative mixed gravitational Chern-Simons term

L0 = −1 2D − 3 4R + v2 + N µ1 2D − 1 4R + 3v2 ¶ + 2NIvabF I

ab

+NIJ µ1 4F I

abF Jab + 1

2∂aM I∂aM J ¶ + 1 24ecIJKAI

aF J bcF K de²abcde

eL1 = c2I 24 · 16²abcdeAIaRbcfgRde

fg + . . .

Bergshoeff et al Fujita et al Hanaki et al

Attractors & maximal SUSY

Strategy

• Near-horizon

enhancement

• f

supersymmetry: AdS3 x S2

• Entropy depends only on near-horizon

data, given uniquely by the charges carried by the BH

• . Extremization

principles: Minimizing central charge and/or entropy function.

Ferrara, Kallosh & Strominger Kraus & Larsen

• A. Sen

I = 1 4π2 Z d5x√gL + SCS + Sbndy

Starting from 2d Conformal Anomaly:

T i

i = − c

12R(2)

c = −6`3

A`2 SL

∂c ∂φi = 0

φi = {`A, `S, M I, . . .}

with

Extremize: c-extremization

AdS3 AdS2

Example: Black string

• Thermodynamics
• f a black string excited to level q0

c = 1 2 (cL + cR)

cR = 6p3 + 1 2c2 · p

cL = 6p3 + c2 · p

S = 2π r 1 6|q0|(cIJKpIpJpK + c2 · p) S = 2π r 1 6cLL0

Cardy’s Cardy’s Formula Formula 5D Attractor 5D Attractor +

`S = 1 2`A , M I = pI `A

`3

A = p3 + 1

12c2 · p

c = 6p3 + 3 4c2 · p

SUSY: AdS3

x S2 c-extremization Central charge

ds2 = `2

Ads2 AdS + `2 SdΩ2 2

Agreement with microscopic analysis and results obtained by looking at a subsector

• f

possible 4D higher derivative corrections.

We can do better…

Full black string geometry

ds2 = e2U(r) ¡dt2 − dx2

5

¢ − e−4U(r) ¡dr2 + r2dΩ2

2

¢

F I = −pI 2 ²2 M Ie−2U = HI = M I

∞ + pI

2r e−6U = 1 6cIJKHIHJHK + c2I 24 ¡∇HI∇U + 2HI∇2U¢

• Solving gaugino

variation relates FI with the scalars MI. Imposing the BI for this solutions, determines the harmonic functions

• Special geometry constraint (eom

for D), governs the profile U(r)

• Most of the geometry is captured by solving the gravitino

variation

• Magnetic charges are topological and determined by the Bianchi Identity

Small but significant

–1 –0.9 –0.8 –0.7 –0.6 –0.5 –0.4 –0.3 2 4 6 8 10 12 14 16

• Small black string:

CY3 =T2 x K3 where c2 (K3)=24, p M5-branes wraps cycle of K3. Dual to p heterotic string with 8 bosons + 8 fermions in right movers and 24 bosons in left movers cL = 24p cR = 12p

• Plot displays numerical solution and

asymptotic solutions of the metric function for a small black string.

• Asymptotic flat boundary conditions

match the near horizon geometry.

cR = 6p3 + 1 2c2 · p

cL = 6p3 + c2 · p

Conclusions

• Exploiting symmetries

and anomalies

• f 5D black strings, we can obtain and

explain agreement between macroscopic and microscopic entropy. Off-shell supergravity Chern-Simons terms Attractor mechanism Geometry + Fields Anomalies Near horizon

• Other scenarios:

Static and rotating black holes

• n asymptotically flat space and

Taub-NUT spaces. Comparison with claims from OSV conjecture and 4D-5D lift. All the ingredients combined allow us to study in detail the full geometry and compute the relevant thermodynamic quantities (central charge in 5D, entropy in 4D).

Degeneracy formula for dyons