Eric Perlmutter, DAMTP, Cambridge

Transcription

1 Eric Perlmutter, DAMTP, Cambridge Based on work with: P. Kraus; T. Prochazka, J. Raeymaekers ; E. Hijano, P. Kraus; M. Gaberdiel, K. Jin TAMU Workshop, Holography and its applications, April 10, 2013

2 1. Why study higher spin gravity in 2013? 2. What are the conjectured holographic dualities involving higher spins? 3. What do we know about 3d higher spin black holes? In gravity and CFT? 4. Can we probe the AdS 3 /CFT 2 conjecture beyond symmetries and spectra? Many of the following results will not depend on the details of the dual CFT in consideration But some will. Namely, the matching of certain 4-point functions. This is different from ordinary AdS/CFT: we are not making a strongweak interpolation, nor using non-renormalization theorems. In the spirit of strong form of AdS/CFT!

3 1. Symmetry is special and interesting 2. String theory is expected to possess enhanced global symmetry at high energies 3. AdS/CFT Free limits of large N gauge theories Classical string theory in strongly curved AdS In the limit, CFTs possess higher spin symmetry! Direct comparison 4. Naturalness: an all-orders derivative expansion e.g. cubic couplings of 4d higher spins to gravity: Consistency requires infinite tower of spins nonlocal theory (Note: requires nonzero Λ!)

4 Theory has vector-like spectrum: c ~ N Conditions for CFTs to have classical gravity duals: 1. Large c 2. Parametric separation of scales: O(1) fields of low dimension 3. These fields are generalized free fields : correlators obey large N factorization `Large N means classical spacetime There may, but need not, be another scale ~ λ: field theory vs. string theory Vasiliev theories have one scale ~ N They capture, morally, the λ =0 endpoint of AdS/CFT dualities When we match bulk and boundary, we match at same coupling. This is powerful test of holography!

5 Main ingredients: master fields [Prokushkin, Vasiliev] Spacetime 1-form: higher spin sector Fields are functions of spacetime + internal variables At linearized level, Spacetime 0-form: matter sector 3d Vasiliev gravity = hs[λ] + hs[λ] Chern-Simons theory w/ gauge-invariant matter coupling Infinite tower of massless integer spin fields, s = 2,3, Spacetime 0-form: enforces higher spin symmetry # scalars (and/or fermions): masses, couplings fixed by symmetry (a la string theory) In 3d, tower of spins can be truncated to s<n: then hs[n] SL(N,R) Generators: Spin index, s=2,3, Mode index, m <s Identity: Low spin examples:

6 SL(2) SL(N) hs[λ] Virasoro (W 2 ) W N W [λ] Generalized Brown-Henneaux boundary conditions give extended conformal algebras [Campoleoni, Fredenhagen, Pfenninger, Theisen; Gaberdiel, Hartman; Henneaux, Rey]

7 In both: One can add SUSY, non-abelian internal symmetries Little is known beyond linearized order around given solution In 3d: One can truncate spins to finite tower Higher spin fields are non-propagating (a la pure gravity) There is continuum of inequivalent AdS vacua (parameterized by λ) Exact higher spin symmetry does not imply that dual CFT is free, unlike 4d In 4d: The theory contains a scalar interaction ambiguity: huge set of consistent higher spin theories (with family of CFT duals?)

8 W N minimal models: [Gaberdiel, Gopakumar] where Currents of spin-s=2, 3,, N, and tower of scalar primaries Consider two very different large c limits at fixed λ: 1. t Hooft limit: Dual to 3d Vasiliev gravity with 0 λ < 1 Theory recently proven to have classical W [λ] symmetry in this limit 2. Semiclassical limit : Dual to Vasiliev gravity with λ =-N: the sl(n) theory + matter c > N-1 implies non-unitarity, e.g. Δ<0 Lots of evidence: symmetries, spectra, partition function/one-loop determinants, correlation functions, black holes [Gaberdiel, Gopakumar, Hartman, Raju; Papadodimas, Raju; Chang, Yin]

9 Higher spin symmetry transformations extend diffeomorphisms Singularities, event horizons, causal structure: Not gauge-invariant (Gauge transformation) Turn on chemical potential (α) for spin-3 charge (W) A smooth higher spin black hole obeys: Central element of group HS[λ] H = Euclidean thermal holonomy A = bulk Chern-Simons gauge field 1. There exists a gauge with manifestly smooth black hole metric (+higher spin fields) 2. Geometric smoothness in `black hole gauge = holonomy condition Holonomy condition fixes charges; ensemble is thermodynamic

10 Linearizing around flat connections, matter equation is: C = spacetime 0-form containing scalar + derivatives Star operation = Associative multiplication underlying hs[lambda] Physical scalar field is identity piece of C: [Pope, Romans, Shen] Gauge symmetry: Locally, all is pure gauge: A=0 gauge Physical gauge Solutions without differential equations!

11 Scalar bulk-boundary propagator: Note two features: 1. Invariant under thermal shifts: 2. Mixed correlator non-singular Boundaries causally disconnected CFT L CFT R

12 1. Easy to show thermal periodicity: 2. Explicitly: e.g. 1 st order correction to one-sided correlator: 3. Explicitly: e.g. 1 st order correction to mixed correlator: Argued to imply usual black hole causal structure

13 In CFT on torus, where W = holomorphic spin-3 current, and. So, O(α) result fixed by 3-pt function, integrated over the torus O(α) is universal: on the plane, conformal symmetry fixes where f(λ) = spin-3 eigenvalue of O. Result: bulk result is reproduced in CFT via the contour integral

14 The preceding calculations were fixed by symmetry and the assumption of the existence of certain scalar primaries. Can we compute something which specifically probes the W N minimal models? 4-point functions

15 Minimal model reps labeled by pair of SU(N) highest weights: In semiclassical limit, 1. Perturbative excitations: ( Single trace ) ( Multi-trace ) [Gaberdiel, Gopakumar; EP, Prochazka, Raeymaekers] 2. Classical backgrounds ( conical defects ): `Conical defect = Smooth, asymptotically AdS solution of sl(n,c) Chern-Simons theory with nonzero higher spin charges [Castro, Gopakumar, Gutperle, Raeymaekers] Higher spin charges Smoothness fixes {Q} to contain precisely the information in an SU(N) Young diagram, viz. that of Λ -

16 A precision check: Match bulk and CFT 4-pt functions in large c limit How? Compute scalar 2-pt function in defect background: (Defect background) (Single particle excitation = bulk scalar) Result is function of Young diagram data [Hijano, Kraus, EP] One easily computes scalar 2-pt function (from the bulk!) e.g. Λ - =q-fold sym rep: q

17 Compute using Coulomb gas of N-1 free bosons Result: matches bulk for arbitrary D representation, in semiclassical limit e.g. Λ - = For general N, k: q In the semiclassical limit, Simplicity: only one conformal block contributes

18 Backreaction and Witten diagrams in 3d Vasiliev theory = + Σ s s Can we form a black hole? The CFT has a global hs[λ] symmetry. Can we see this in the bulk, beyond linearized order? More observables 3d higher spins from string theory 4d higher spin black holes : what do those words mean? F-theorem from 4d higher spins = Entanglement entropy across S 1