Insightful D-branes Albion Lawrence Brandeis University 1
Outline I. Introduction II. A singularity with a gauge theory dual III. Gauge theory vs. spacetime coordinate transformations IV. Gauge theory dynamics and hyperbolic black holes V. Conclusions Based on work with G. Horowitz and E. Silverstein arxiv:0904.3922 2
I. Introduction Large black hole: curvature remains weak well inside the horizon. 1. Infalling observer (B) remains semiclassical until it reaches the singularity. A 2. External observer (A) sees black hole evaporate via long-wavelength, thermal, B Hawking radiation. Infalling observer is “cooked” near the horizon and re-emitted as Hawking radiation. “Black hole complementarity”: ‘t Hooft, Susskind Unitarity of BH evaporation implies that these two pictures are equivalent (dual). If so, what is the map? 3
Black holes in AdS/CFT AdS 5 black hole ∼ 4d gauge theory at temperature T = T ( M ) %' ! (' 8 ? Gauge theory time ∼ Schwarzschild time !"#$%&#%'% (Time experienced by observer at fixed distance from BH). Bulk: infalling objects approach horizon as t → ∞ Gauge theory: excitations spread and thermalize Gauge theory description of semiclassical physics behind the horizon? Gauge theory description of singularity? 4
II. A singularity with a gauge theory dual Green, Lawrence, McGreevy, Morrison, and Silverstein HLS Poincare coordinates: r 2 + ℓ 2 dr 2 t 2 + d � ds 2 � − d ˜ x 2 � ( ℓ is AdS radius) 5 = p p 3 ℓ 2 r 2 constant r p p Patch of R 3 , 1 : ds 2 4 = − dt 2 p + t 2 p d σ 2 H 3 singularity constant ˜ t Orbifold: Σ = H 3 / Γ t p → 0 − becomes singular t p → ∞ constant t p constant t p patch covered by t p , σ Final bulk metric: r 2 + ℓ 2 dr 2 � � ds 2 − dt 2 p + t 2 p d σ 2 p p 5 = patch covered by ℓ 2 Σ r 2 p cosmological coordinates Dual gauge theory lives on ds 2 4 = − dt 2 p + t 2 p d σ 2 Σ “collapsing cone” metric: 5
1. Similar in spirit to other time-dependent QFTs Das et. al. Awad et. al. Craps, Sethi, et. al. Martinec et. al. 2. Distinct from example of unstable QFTs Horowitz and Hertog Craps, Hertog, and Turok • D 3 branes at constant r p are solutions of e.o.m. Bernamonti and Craps • Stretched W bosons have mass m W constant in time. • Momentum m KK along Σ grows with t p → 0 − . • Dimensionless ratio m W /m KK → 0. Singularity associated with IR of QFT 3. Unclear if QFT is well defined as t p → 0 − 6
Static coordinate system � t 2 p r 2 p − ℓ 2 � r = − r p t p t = − ℓ 2 ln ℓ 2 r 2 ℓ p 5 = − f ( r ) dt 2 + dr 2 ds 2 f ( r ) + r 2 d σ 2 Σ f ( r ) = r 2 t → ∞ ℓ 2 − 1 constant t “M=0 topological black hole”: Emparan constant r 5d version of BTZ black hole constant t p r → ∞ • Negatively curved horizon at r = ℓ . • Temperature T ∼ 1 / ℓ . Shaded patch covered by t p , r p • Horizon area ∼ ℓ 3 ; entropy ∼ N 2 . Dual: gauge theory on Σ × R at T = 1 /R Σ 7
Conformal transformation: t p → 0 − t → ∞ t p = − ℓ e − t/ ℓ conformal cyl = e 2 t/ ℓ ds 2 ds 2 cone → ds 2 transformation cone Σ Σ Seems to map QFT variables describing Scwharzschild observers to QFT variables describing infalling observers • How does the map act on bulk probes? • Can this be generalized to other BHs? 8
III. Gauge theory vs. spacetime coordinate transformations A. N = 4 SYM on Σ × R , T = 1 /R Σ Consider D3-brane wrapping Σ and moving in r, t String theory: D3-brane probe dynamics described by DBI action �� � det ∂ α X µ ∂ β X ν G µ ν ( X ) − A (4) 1 d τ d 3 σ � S DBI = g s ( α ′ ) 2 RR = t τ ↓ = r ( t ) r σ i : Σ → Σ one − to − one � r 3 � � ˆ f ( r ) − r 4 − ℓ 4 r 2 ˙ V � S static = − f ( r ) − dt g s ( α ′ ) 2 ℓ f ( r ) = r 2 ℓ 2 − 1 9
Gauge theory: • t = gauge theory time. • Take adjoint scalar out on Coulomb branch, φ = α ′ r . • Integrate out W-bosons charged under U (1) × U ( N − 1). • S static is resulting e ff ective action for φ 10
� r 3 � � ˆ f ( r ) − r 4 − ℓ 4 r 2 ˙ V � S static = − f ( r ) − dt g s ( α ′ ) 2 ℓ f ( r ) = r 2 ℓ 2 − 1 r 2 < f ( r ) 2 : ”scalar speed limit”. 1. ˙ Silverstein and Tong t → ∞ r → ℓ as t → ∞ constant t 2. Let r = r 0 ( t ) + δ r ( t ): • r 0 ( t ) solves classical e.o.m. • Expand S static in δ r Expansion in δ r breaks down as f → 0. Semiclassical physics in (t, r) breaks down at horizon 11
B. N = 4 SYM on collapsing cone Consider D3-brane wrapping Σ and moving in r p , t p String theory: �� � det ∂ α X µ ∂ β X ν G µ ν ( X ) − A (4) 1 d τ d 3 σ � S DBI = g s ( α ′ ) 2 RR = t p τ ↓ = r p ( t p ) r p σ i : Σ → Σ one − to − one � � � r 4 p t 3 r 2 ˆ r p 2 ℓ 2 − ℓ 2 ˙ t 3 p r 3 V � S cosmo = − dt p p p p − p g s ( α ′ ) 2 r 2 ℓ 4 Gauge theory: • t p = gauge theory time. • Take adjoint scalar out on Coulomb branch, φ = α ′ r p . • Integrate out W-bosons charged under U (1) × U ( N − 1). 12
� � � r 4 p t 3 r 2 ˆ r p 2 ℓ 2 − ℓ 2 ˙ V t 3 p r 3 � S cosmo = − dt p p p p − p g s ( α ′ ) 2 r 2 ℓ 4 1. Horizon at r p t p = ℓ 2 , singularity as t p → 0 − . Horizon reached in finite time t p → 0 − 2. Let r p = r p, 0 ( t p ) + δ r p ( t p ): • r p, 0 ( t p ) solves classical e.o.m. constant t p • Expand S static in δ r p Patch covered Expansion in δ r p by t p , r p • regular at horizon • breaks down at singularity 13
C. Transformation of QM variables of probes 1. Conformal transformation of QFT t p = − ℓ e − ˜ t ℓ maps collapsing cone to Σ × R cyl = e 2˜ t/ ℓ ds 2 ds 2 cone → ds 2 cone φ = r/ α ′ dimension-1 field: r p = e ˜ t/ ℓ ˜ r ˜ r � = t, r t, ˜ Conformal transformation: S cosmo �→ S static � � � ℓ 2 − ℓ 2 ( ˙ ˆ r/ ℓ ) 2 r 4 r +˜ ˜ S cosmo = ˜ V r 2 ˜ − ˜ r 3 � d ˜ S = − t ˜ g s ( α ′ ) 2 r 2 ˜ ℓ 14
This is not surprising: t p → 0 − (a) Coordinate transformations t p = − ℓ e − ˜ t/ ℓ re ˜ t/ ℓ r p = ˜ constant t p do not change equal-time slices in bulk Patch covered (b) Bulk metric: by t p , r p t 2 + ˜ r + ℓ 2 r ) d ˜ r d ˜ Σ + 2 ℓ r 2 ds 2 r 2 d σ 2 r 2 r ) = ˜ 5 = − f (˜ td ˜ r 2 d ˜ f (˜ ℓ 2 − 1 ˜ ˜ �� � det ∂ α X µ ∂ β X ν G µ ν ( X ) − A (4) 1 d τ d 3 σ � S DBI = g s ( α ′ ) 2 RR ˜ = ↓ t τ r (˜ ˜ = ˜ t ) r σ i : Σ → Σ � � � ℓ 2 − ℓ 2 ( ˙ ˆ r/ ℓ ) 2 r 4 r +˜ ˜ r 2 ˜ V ˜ − ˜ d ˜ r 3 � S = − t ˜ g s ( α ′ ) 2 r 2 ˜ ℓ 15
2. Bulk coordinate transformations in dual QFT r = − r p t p ℓ map ”cosmological” to ”static” coordinates � t 2 p r 2 p − ℓ 2 � t = − ℓ 2 ln ℓ 2 r 2 p In QFT: • t, t p are QFT times Map ( r p , t p ) → ( r, t ) includes a field- dependent time reparametrization • r, r p are quantum fields constant r p Generic to any nontrivial change of bulk equal-time slicings constant t Seems exotic in QFT constant r constant t p (but remember gauge transformations for quantum inflaton fluctuations) 16
Solution: add gauge invariance Let t p = t p ( τ ) , r p = r p ( τ ). � � � p ˙ t p t 3 ˙ p r 4 r 2 ℓ 2 − ℓ 2 ˙ t 2 r 2 ˆ V t 3 p r 3 � S cosmo → S DBI = − d τ p p p p − p g s ( α ′ ) 2 ℓ 3 r 2 ℓ Invariant under reparametrizations of τ Reduces to S cosmo if we gauge fix t p = τ r = − r p t p is now a simple field redefinition ℓ � r 2 p t 2 p − ℓ 4 � t = ℓ 2 ln ℓ 2 r 2 p Fix t = τ : � r 3 � � ˆ f ( r ) − r 4 − ℓ 4 r 2 ˙ V � S DBI → S static = − f ( r ) − dt g s ( α ′ ) 2 ℓ (Up to boundary term/RR gauge transformation) 17
Conformal transformation to Σ × R t 2 + ˜ r + ℓ 2 r ) d ˜ r d ˜ Σ + 2 ℓ ds 2 r 2 d σ 2 r 2 5 = − f (˜ td ˜ r 2 d ˜ ˜ ˜ ↓ ˜ r → ∞ r 2 t 2 + d σ 2 r 2 + ℓ 2 d ˜ 5 ∼ ˜ � − d ˜ � ds 2 ℓ 2 Σ r 2 ˜ Field theory dual seems to live on Σ × R Nonsingular field theory on nonsingular space: t p → 0 − mapped to ˜ t → ∞ . ˜ t, ˜ r extend behind horizon. constant t ˜ S [˜ r ] well behaved at horizon Better variables to see behind horizon constant ˜ t Which action arises as QFT effective action? 18
Transformation of quantum observables Conjugate momenta: rf ( r ) p t + r 4 − ℓ 4 ℓ ˆ = p ˜ p r − V N r ℓ 4 rf ( r ) ˆ V N = p t + p ˜ t ℓ Hamiltonians: H t = − p t ; H ˜ t = − p ˜ t Equal up to a constant r → ∞ : ˜ r → ∞ Equal- t slices asymptotically identical to equal-˜ t slices 19
D. Transformation of full QFT Puzzles: 1. Yang-Mills action invariant under conformal transformation 2. S cosmo �→ S static under conformal transformation 3. S cosmo , S static e ff ective actions for SYM? 4. Is S static or ˜ S e ff ective action for SYM on Σ × R ? 20
Gauge theory: • t = gauge theory time. • Take adjoint scalar out on Coulomb branch, φ = α ′ r . • Integrate out W-bosons charged under U (1) × U ( N − 1). • S static is resulting e ff ective action for φ Hidden step: must fix gauge before integrating out W bosons. Functional form of e ff ective action depends on gauge choice. 21
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