Entanglement Entropy in Three-Dimensional Higher Spin Theories Juan Jottar Based on: arXiv:1302.0816,1306.4347 with Jan de Boer + to appear with G. Compère and W. Song Juan Jottar (U. of Amsterdam) Gauge/Gravity Duality 2013 Munich July 29th, 2013 1
Outline Motivation: higher spin theories in AdS 3 1 Entanglement entropy proposal 2 Testing the proposal 3 Outlook 4 Juan Jottar (U. of Amsterdam) Gauge/Gravity Duality 2013 Munich July 29th, 2013 2
Motivation: higher spin theories in AdS 3 Outline Motivation: higher spin theories in AdS 3 1 Entanglement entropy proposal 2 Testing the proposal 3 Outlook 4 Juan Jottar (U. of Amsterdam) Gauge/Gravity Duality 2013 Munich July 29th, 2013 3
Motivation: higher spin theories in AdS 3 Higher spin theories Broad definition: interacting theories of gravity coupled to a finite (or infinite) number of massless fields of spin s > 2 . Motivation from holography: explore AdS/CFT in a regime where the bulk theory is not just classical (super-)gravity, and the dual theory is not necessarily strongly-coupled: ◮ Critical O ( N ) vector models in 3 d in the large- N limit dual to higher spin theories of Fradkin-Vasiliev type in AdS 4 . (Klebanov, Polyakov 2002; Giombi, Yin 2010-12; Maldacena, Zhiboedov 2011-12) ◮ Two-dimensional minimal model CFTs with extended ( W -)symmetries in the large- N limit dual to higher spin theories in AdS 3 . (Gaberdiel, Gopakumar 2010) Motivation from GR: curvature, causal structure, etc are not invariant under the higher spin gauge symmetries ⇒ challenge traditional geometric notions. Juan Jottar (U. of Amsterdam) Gauge/Gravity Duality 2013 Munich July 29th, 2013 4
Motivation: higher spin theories in AdS 3 AdS 3 gravity as a Chern-Simons theory Standard AdS 3 gravity can be written as an SL ( 2 , R ) × SL ( 2 , R ) Chern-Simons theory. (Achucarro, Townsend 1986; Witten 1988) Take 3 d gravity with a negative cosmological constant Λ = − 1 /ℓ 2 . Combine dreibein e a and (dual) spin connection ω a = ( 1 / 2 !) ǫ abc ω bc into SL ( 2 , R ) connections A = A a J a = ω + e A a J a = ω − e A = ¯ ¯ ℓ , ℓ c where the J a satisfy the so ( 2 , 1 ) ≃ sl ( 2 , R ) algebra [ J a , J b ] = ǫ ab J c . Defining CS ( A ) = A ∧ dA + 2 3 A ∧ A ∧ A one finds ℓ I CS ≡ k � k = � � CS ( A ) − CS ( ¯ Tr A ) 4 G 3 4 π M 1 �� � R + 2 � � � ω a ∧ e a d 3 x � = | g | − 16 π G 3 ℓ 2 M ∂ M Juan Jottar (U. of Amsterdam) Gauge/Gravity Duality 2013 Munich July 29th, 2013 5
Motivation: higher spin theories in AdS 3 Higher spin theories in AdS 3 The higher spin theories in 3 d are constructed by generalizing the gauge algebra. The pure higher spin sector of the 3 d Vasiliev theory is hs [ λ ] ⊕ hs [ λ ] Chern-Simons theory. The dual CFT has an infinite tower of conserved currents and W ∞ [ λ ] symmetry. (Henneaux, Rey 2010; Gaberdiel, Hartman 2011) For λ = N ∈ Z it is possible to truncate the tower of higher spin fields to s ≤ N . The bulk theory reduces to sl ( N , R ) ⊕ sl ( N , R ) Chern-Simons. Different sl ( 2 ) embeddings are possible. The asymptotic symmetry algebra is of W N type. (Campoleoni, Fredenhagen, Pfenninger, Theisen 2010) Juan Jottar (U. of Amsterdam) Gauge/Gravity Duality 2013 Munich July 29th, 2013 6
Motivation: higher spin theories in AdS 3 Extending the AdS 3 /CFT 2 dictionary The BTZ black hole entropy (via Bekenstein-Hawking) and holographic entanglement entropy (via Ryu-Takayanagi) match universal CFT results: Cardy entropy formula � � c c ¯ S thermal = 2 π 6 L 0 + 2 π L 0 6 (Single interval) Entanglement entropy at finite temperature T = β − 1 � β � π ∆ x �� S A = c 3 log π a sinh β where c is the central charge and a the UV cutoff. Question: How do we compute these entropies in higher spin theories? Juan Jottar (U. of Amsterdam) Gauge/Gravity Duality 2013 Munich July 29th, 2013 7
Entanglement entropy proposal Outline Motivation: higher spin theories in AdS 3 1 Entanglement entropy proposal 2 Testing the proposal 3 Outlook 4 Juan Jottar (U. of Amsterdam) Gauge/Gravity Duality 2013 Munich July 29th, 2013 8
Entanglement entropy proposal Entanglement entropy Consider a quantum system in a pure (or mixed) state, with density operator ρ = | Ψ �� Ψ | (or ρ = e − β H ). Partition the Hilbert space as H = H A ⊗ H B ( B = A c ), the reduced density matrix for subsystem A is defined as ρ A = Tr B ρ . The entanglement entropy S A associated with A is then given by the Von Neumann entropy of ρ A : S A = − Tr A ρ A log ρ A . Universal results in 2 d CFTs. (Holzhey, Larsen, Wilczek 1994; Calabrese, Cardy, 2004) If A is a single interval of length ∆ x in an 1 d system: ◮ System on the ∞ line at zero temperature: S A ≃ ( c / 3 ) log (∆ x / a ) � L ◮ Periodic system of total length L : S A ≃ ( c / 3 ) log � π a sin ( π ∆ x / L ) ◮ System on the ∞ line in a thermal (mixed) state: � � β S A ≃ ( c / 3 ) log π a sinh ( π ∆ x /β ) Juan Jottar (U. of Amsterdam) Gauge/Gravity Duality 2013 Munich July 29th, 2013 9
Entanglement entropy proposal Holographic entanglement entropy Generically, entanglement entropy is notoriously difficult to compute in field theory, even for free theories. However, in theories with a gravity dual it can be computed using the Ryu-Takayanagi prescription: t = const. y A x S A = Area ( γ A ) r 4 G N bulk Juan Jottar (U. of Amsterdam) Gauge/Gravity Duality 2013 Munich July 29th, 2013 10
Entanglement entropy proposal Entanglement entropy for higher spin theories What replaces the R-T prescription in a 3 d higher spin theory?. Write something that: ◮ Involves the natural objects in the h.s. theory: Wilson lines. ◮ Is invariant under the diagonal gauge group (rotations of the local frame), and under gauge transformations that do not change the state in the dual theory. ◮ Encodes the geodesic length in the pure gravity case. One is lead to consider �� P �� Q � � �� ¯ W ( P , Q ) ≡ Tr R P exp P exp A A Q P Juan Jottar (U. of Amsterdam) Gauge/Gravity Duality 2013 Munich July 29th, 2013 11
Entanglement entropy proposal Entanglement entropy for higher spin theories For AAdS 3 solutions of pure gravity, placing P and Q on the boundary, and at equal times, we find W 2 d ( P , Q ) | ρ P = ρ Q = ρ 0 − ρ 0 →∞ exp d ( P , Q ) − − − → where d ( P , Q ) is the geodesic distance. Since ent ent in AdS 3 is related to the geodesic distance via the Ryu-Takayanagi prescription, we propose �� P �� Q ��� � � ¯ � S ent = k cs log Tr R P exp P exp A A � � Q P ρ P = ρ Q = ρ 0 →∞ This is a well-defined object in the higher spin theory as well. In the absence of field-theoretical results to compare against, we will perform some plausibility tests. Juan Jottar (U. of Amsterdam) Gauge/Gravity Duality 2013 Munich July 29th, 2013 12
Testing the proposal Outline Motivation: higher spin theories in AdS 3 1 Entanglement entropy proposal 2 Testing the proposal 3 Outlook 4 Juan Jottar (U. of Amsterdam) Gauge/Gravity Duality 2013 Munich July 29th, 2013 13
Testing the proposal Test 1: Recover AdS 3 results For solutions of pure gravity we obtain � ∆ x � S PAdS 3 = c Poincaré-patch: 3 log a � ℓ � ∆ ϕ �� S AdS 3 = c global: 3 log a sin 2 � β + β − � � � �� S BTZ = c π ∆ x π ∆ x black hole: 6 log π 2 a 2 sinh sinh β + β − in agreement with CFT results and holographic calculations. (Ryu,Takayanagi, 2006; Hubeny, Rangamani, Takayanagi, 2007) The higher spin theory with diagonally-embedded sl ( 2 ) contains a truncation to Einstein gravity plus Abelian Chern-Simons fields. Our prescription also gives the right result for black holes carrying U ( 1 ) charges, as recently confirmed by an independent CFT calculation. (Caputa, Mandal, Sinha 2013) Juan Jottar (U. of Amsterdam) Gauge/Gravity Duality 2013 Munich July 29th, 2013 14
Testing the proposal Test 2: Thermal entropy The entanglement entropy should approach the thermal entropy in the limit ∆ x ≫ β (i.e. as subsystem A approaches the full system). E.g. � β S A = c � π ∆ x �� π c 3 log π a sinh − − − − → 3 β ∆ x = s thermal ∆ x β ∆ x ≫ β Since the causal structure is not invariant under the higher spin gauge symmetries, we cannot calculate the entropy with the usual methods. Proceed indirectly by demanding existence of a well-defined partition function. E.g. N = 3: � e 2 π i ( τ ˆ L + α ˆ W ) � = e S + 2 π i ( τ L + α W ) Z ( τ, α ) = Tr with the integrability condition ∂α = ∂ W ∂ L τ = i ∂ S α = i ∂ S and ∂ L , ∂τ 2 π 2 π ∂ W Juan Jottar (U. of Amsterdam) Gauge/Gravity Duality 2013 Munich July 29th, 2013 15
Testing the proposal Gutperle and Kraus showed that integrability follows from demanding smoothness (connection has trivial holonomy around contractible cycle of the Euclidean torus) and obtained: � � c 1 − 3 S = 4 π 6 L 4 C with C = C ( W / L 3 / 2 ) . Using the canonical formalism for charges in GR it was found instead (Perez, Tempo, Troncoso 2013) � − 1 � � c � 1 − 3 1 − 3 S can = 4 π 6 L 2 C 4 C which agrees with a perturbative result obtained using the metric formulation of the higher spin theory and Wald’s entropy formula. (Campoleoni et. al. 2012) Juan Jottar (U. of Amsterdam) Gauge/Gravity Duality 2013 Munich July 29th, 2013 16
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