Asymptotic Symmetries and Holography

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1 Asymptotic Symmetries and Holography Rashmish K. Mishra Based on: Asymptotic Symmetries, Holography and Topological Hair (RKM and R. Sundrum, )

2 Unification of diverse topics IR structure of QFTs, inclusive quantities, dressed states Known and unknown holography: AdS/CFT and flat space Large gauge transformations and diffeomorphisms Black Hole information paradox Soft theorems in QFT: Weinberg soft photon and soft graviton theorem Gravitation and EM memory. Symmetries of General Relativity Generalizations of the traditional AS approach Lectures on the Infrared Structure of Gravity and Gauge Theory (Strominger, )

3 Unification of diverse topics IR structure of QFTs, inclusive quantities, dressed states Known and unknown holography: AdS/CFT and flat space Large gauge transformations and diffeomorphisms Black Hole information paradox Soft theorems in QFT: Weinberg soft photon and soft graviton theorem Gravitation and EM memory. Symmetries of General Relativity Generalizations of the traditional AS approach Lectures on the Infrared Structure of Gravity and Gauge Theory (Strominger, )

4 In this talk Asymptotic Symmetries of AdS4 space are derived by coupling the holographically dual CFT3 to Chern-Simons theory in a probe limit. Such symmetries are seen in a set of CFT3 correlators smeared with external CS lines. An AdS4 analog of 4D Minkowski super-rotation symmetry is probed by 3D Einstein gravity, giving a CFT2 structure via AdS3/CFT2 and AdS3 foliations of AdS4.

5 Outline What are Asymptotic symmetries Observational aspects Connection to holography 4D Anti de-sitter space, AdS/CFT AS in AdS: Why not and why. Ingredients of the construction Non-standard correlators Chern-Simons on manifolds with boundary Large level limit Putting it all together: an example no-go of AdS4 Emergent Chern-Simons AdS vs Mink: Lessons Conclusion and Outlook

6 What are Asymptotic Symmetries? Gauge transformations and diffeomorphism that preserve the asymptotic structure of spacetime. ASG = Allowed gauge transformations Trivial gauge transformations 1 2 Both these have associated conserved charges Lorentz Boost Is a coordinate transformation Changes energy of a system, i.e. a state with a given energy changes into another distinct state.

7 What are Asymptotic Symmetries? Story is similar in Gauge theory and gravity. Historically first noticed in GR in 1962 by Bondi, van der Burg, Metzner and Sachs (given the name BMS group) What are the symmetries of GR? or Is there a limit in which Special Relativity comes from General Relativity Formulation: Look at asymptotically flat spacetime, (leads to fall-off condition on metric components, boundary conditions.) Look for coordinate transformations that preserve these fall-off/ boundary conditions. Remove the coordinate transformations that do nothing to boundary data (i.e. act trivially).

8 What are Asymptotic Symmetries? Massless particles: go to null-infinity. Metric near future null infinity: Penrose diagram of Minkowski space where refers to sub-leading terms. Asymptotic symmetries change the sub-leading terms, but keep the leading part same.

9 Flat space AS Super-translations: Super-rotations: Penrose diagram of Minkowski space In particular: Super-rotations Suggestive of a 2D CFT? Virasoro Algebra! (without central extension) (Barnich & Troessaert, 2010) 4D scattering amplitudes and asymptotic symmetries from 2D CFT (Chueng et al, ) 2D Stress Tensor for 4D Gravity (Kapec et al, )

10 Infinite number of charges in QED? QED is an extremely well tested theory. If there are new charges, and infinite number of them, where are they? Have we seen them? Missed them? IR divergences in QED Two ways for IR finiteness: (Usually) consider inclusive cross-section. Certain dressed states as asymptotic states give IR finite answers. (Faddeev and Kulish, 1970) Role of asymptotic charges clarified (Kapec, Perry, Raclariu, Strominger, )

11 What is the observable? Memory Gravitational memory: distance between two observers changes after a GW passes. (Zeldovich and Polnarev, 74) Electromagnetic memory: a superconducting grid on a large sphere. (Susskind, 2015) Detectable by LIGO: Detecting gravitational-wave memory with LIGO: implications of GW (2016, )

12 Connection to Holography Asymptotic Symmetries as a precursor to AdS/CFT: Famously, symmetries of AdS3 were found to be Virasoro (Brown and Henneaux, 1986) with central extension. Separating 4D Minkowski space into Rindler and Milne regions, using hyperbolic foliations (Euclidean AdS3 in Milne and ds3 in Rindler) to describe them, one arrives at a 2D CFT on the celestial sphere. Soft factors agree to the form of bulk-boundary propagator: both in massless and massive case. EAdS3 = H3 ds3 4D scattering amplitudes and asymptotic symmetries from 2D CFT (Chueng et al, ) A holographic reduction of Minkowski space-time (Jan de Boer et al, hep-th/ )

13 Asymptotic Symmetries of AdS space Generalizations? Lessons for Flat space? de-sitter space?

14 4 dimensional Anti-de Sitter space Global coordinates: Topology: Boundary: Dual 3D CFT lives at the conformal boundary: Isometries of Symmetries of

15 AdS/CFT gauge/diffeomorphism invariant observables in AdS are boundary CFT correlators. Witten diagrams External lines: Internal lines: Vertices: Bulk-Boundary propagators Bulk-Bulk propagators AdS interactions

16 AS of AdS space 3D AdS space: Infinite dimensional symmetries of a 2D CFT (Brown and Henneaux, 86) 4D AdS space: finite? Traditional perspective: Boundary of AdS in 4 or higher dimensions have finite dimensional isometries. Modern perspective: There is no soft limit because AdS is a gapped spectrum How does one make progress?

17 Construction 1 Consider CFT3 on AdS3 with reflecting boundary conditions on Dual to AdS4/2 (Bousso and Randall, 01) 2 Gauge the CFT3 with 3D Chern-Simons (with level ) 3 Take the large level limit ( ) We will consider the gauge invariant CFT3 correlators in this limit.

18 Construction 1 Consider CFT3 on AdS3 with reflecting boundary conditions on Dual to AdS4/2 (Bousso and Randall, 01) AdS3/CFT2 2 Gauge the CFT3 with 3D Chern-Simons (with level ) 3 Take the large level limit ( ) An analog of soft limit in 4D Minkowski space We will consider the gauge invariant CFT3 correlators in this limit.

19 1 Construction on with Dirichlet b.c. at, projecting Fixed τ r fixed r AdS 3 r = 0 ρ = 0 : AdS 3 boundary Decompose fields into mass eigenstates (KK decomposition)

20 1 Product Space r Fixed τ fixed r AdS 3 ξ = π/2 fixed ξ AdS 3 r = 0 ρ = 0 ξ = 0 ξ ξ = π/2 ρ = 0 : AdS 3 boundary AdS4 fields can be KK decomposed to go to 3d mass basis. For simplicity consider massless YM field, which is not sensitive to at tree level. Mass of (AdS units) AdS3/CFT2: KK modes dual to

21 2 Chern-Simons theory on a manifold with boundary 2D WZW model : generator of gauge group, : CS level Switch to: EOM:, work in (bulk) gauge. Boundary conditions: Boundary conditions too stringent. Lead to trivial solution. Need to relax boundary conditions.

22 2 Chern-Simons theory on a manifold with boundary 2D WZW model Add at the boundary: EOM modified at boundary: At boundary:, while is unconstrained at boundary. Even though we have fixed gauge,, we must include the EOM. Non-Abelian field strength:. On boundary, and using, AdS/CFT:

23 2 Chern-Simons theory on a manifold with boundary Chirally conserved current: Fourier decomposition: Kac-Moody algebra: Non-trivial, but predictive and universal evolution: Gravity 3D gravity can be formulated as a CS theory. Get algebra, generalizing algebra. Central charge:

24 3 Witten diagrams in a large CS level limit Typical Witten diagram O 2 O 1 j t j CS propagators: Interaction vertex: Choose a useful normalization: To leading order in large level, all internal CS lines are removed from Witten diagrams (with CFT operators).

25 3 Witten diagrams in a large CS level limit Abelian CS Leading Witten diagram with external CFT lines j O j Leading Witten diagram without external CFT lines j j Restrict to a subset of correlators with matter ( particles) for a finite limit. Clarifies a confusion from flat space. We are in large (not vanishing) central charge limit.

26 Putting it all together Holography up and down. 2D 3D 4D Preons Quarks & gluons Tower of Hadrons Two ways to probe bulk physics. Smeared by dξ cos((2l 2 +1)ξ) l1 l2 ξ ξ l3 l4 Standard Correlator Non-standard Correlator These are the only gauge invariant correlators, once the CS is switched on.

27 Putting it all together z CS y l1 4D photon X l2 l3 l4 (CFT) charges are already manifest in this expression! There is a connection to the traditionally seen large gauge transformations and memory in Asymptotic symmetries analysis.

28 Putting it all together Away from the boundary, Bulk-boundary propagator solves the sourceless equation from the quadratic part of the bulk action, approaching a delta function at boundary. Approaching boundary: A 3D pure gauge lifts to a 4D pure gauge trivially: Shadow effect:

29 Obstruction in AdS4 A necessary condition for large diffeomorphisms to correspond to asymptotic symmetries is that they preserve its boundary structure, in particular those reducing to conformal isometries on the boundary geometry. at at

30 Obstruction in AdS4 Fixed τ r AdS3 foliations of AdS4 clarify the two different structure of the asymptotic boundary. ρ

31 Two line summary A set of CFT3 correlators (smeared with external Chern Simons bulk-boundary propagator) are considered, which are dual to CFT2 correlators from AdS3/CFT2. The infinite AS charges are made manifest by CFT2 correlator, which are seen to correspond to non-standard AdS4/2 correlators.

32 Emergent CS Poincare Patch: Clarifies some important things: Emergence of CS and meaning of soft limit Large but finite CS level Mirror symmetry and Abelian CS: A with a U(1) gauging by CS is dual to another, with modified boundary conditions. (Witten hep-th/ ) Generalizations possible for non-abelian case.

33 AdS4 vs Mink4 Manifest Unitarity Massive particles Non-perturbative effects Charges as a useful form of hair

34 Outlook Soft limit in other variables Connection to Mirror symmetry Similar steps in de-sitter space Connection to traditional AS approach

35 EXTRA SLIDES

36 Practicalities of Memory Bad news: SNR too small for detection. DC effect, experimental sensitivity low at small frequencies. Numerical Relativity not developed fully to simulate the waveforms. Good news: Accumulated effect. Data taking has started. Detecting gravitational-wave memory with LIGO: implications of GW (2016, ) The gravitational-wave memory effect (2010, )