The action for higher spin black holes Max Ba nados (Santiago) - PowerPoint PPT Presentation

The action for higher spin black holes Max Ba˜ nados (Santiago) with R. Canto (Santiago) and S. Theisen (Golm) arXiv:1204.5105 [hep-th]. Published in JHEP.

Fields and their interactions Fields with spins lower or equal than two ( s ≤ 2) interact happily with each other and with themselves, s = 1 s = 3 s = 0 s = 1 s = 2 2 2 φ Ψ Ψ µ A µ g µν φ � � � � � Ψ � � � � A µ � � � Ψ µ � � g µν � but the situation changes dramatically for fields with s > 2:

Free higher spin field theories (Fronsdal equations) Free equations of motion for higher spin fields can be built in a very symmetric and algorithmic way, spin equation gauge symmetry � A σ − ∂ σ ∂ ν A ν = 0 s = 1 δ A σ = ∂ σ ǫ � h σν − ∂ σ ∂ ρ h ρν − ∂ ν ∂ ρ h σρ s = 2 δ h σν = ∂ σ ǫ ν + ∂ ν ǫ σ + ∂ σ ∂ ν h = 0 � h σνσ − ∂ σ ∂ ρ h ρνσ ..... = 0 s = 3 δ h σνρ = ∂ σ ǫ νρ + ∂ ρ ǫ σν + ∂ ν ǫ ρσ . . . . . . . . . but adding interactions and/or self-interactions is a difficult problem.

Interactions – preserving gauge invariance. ◮ Self interactions for vector ( s = 1) and tensor ( s = 2) fields are not restricted (by gauge invariance): � � F σν F σν + f ( F σν F σν ) � = I s =1 � √ g ( R + f ( R ) + · · · ) I s =2 = (Of course, at s = 2, √ gR is already an interacting theory.) ◮ Self-interactions of s > 2 higher spin fields described by symmetric tensors g µνρ , g µνρσ , ... are severely restricted. The only known interacting action (Vasiliev) involves the whole tower of fields with all s .

In three dimensions life is easier The magic is provided by the Chern-Simons action, I [ A µ ] = k � � AdA + 2 � A = A µ dx µ ∈ G . 3 A 3 , 4 π ◮ This action has a cubic A 3 interaction. ◮ Gauge and diffeomorphism (trivial) invariant µν ξ ν ≈ 0 δ λ A a µ = D µ λ a , δ ξ A a µ = F a ◮ Possess non-trivial solutions on topologically non-trivial manifolds. And applications to knot theory ◮ Provides a gauge field theory formulation of three-dimensional gravity

SL (2 , ℜ ) × SL (2 , ℜ ) and three-dimensional gravity Consider two SL (2 , ℜ ) Chern-Simons fields, � a µ � ¯ ¯ � � b ν a µ b ν ¯ A µ = , A µ = c µ − a µ ¯ c µ − ¯ a µ then, the following equality follows (Ach´ ucarro-Townsend 1986) 1 � d 3 x √ g ( R + Λ) . I [ A ] − I [¯ A ] = 16 π G The dictionary between A , ¯ A and metric variables is e µ = A µ − ¯ g µν = Tr( e µ e ν ) where A µ ( e − 1 w e + e − 1 ∂ e ) µ Γ µ w µ = A µ + ¯ = where A µ λρ λρ ℓ = k 4 G

SL ( N , ℜ ) × SL ( N , ℜ ) and higher spin fields Let A σ , ¯ A σ be two SL ( N , ℜ ) Chern-Simons fields and e µ = A µ − ¯ A µ . Define now N − 1 metrics (Cayley-Hamilton theorem) = Tr( e µ e ν ) g µν g µνρ = Tr( e ( µ e ν e ρ ) ) g µνρσ = Tr( e ( µ e ν e ρ e σ ) ) . . . = Tr( e ( σ 1 e σ 2 · · · e σ N ) ) g σ 1 σ 2 ...σ N 1. These fields satisfy Fronsdal equations, when linearized, on the AdS background. Thus Chern-Simons theory provides interactions for higher spin gauge fields, preserving gauge invariance. 2. Asymptotic symmetries are W N algebras (Henneaux et al, Campoleoni et al (2010))

Black holes Not a lot is known yet about these theories.... But black holes have been found. For N = 3, g µν and g µνρ have the structure: f 2 ( r ) dt 2 + dr 2 g σν dx σ dx ν f 2 ( r ) + r 2 d φ 2 , = dr 2 � � f 3 ( r ) dt 2 + g σνρ dx σ dx ν dx ρ χ ( r ) f 3 ( r ) + z 2 3 ( r ) d φ 2 = d φ where f 2 ( r ) and f 3 ( r ) vanishes at the same point. See, for example, Gutperle-Kraus (2011) and Castro el at (2012).

Our plan 1. Topological characterization of solutions. 2. Regularity conditions ( → Hawking temperature) 3. The Euclidean on-shell action (‘free energy’) for black holes. We shall not discuss the emergence of W N algebras. See the extensive recent –and not too recent– literature for details: SL( N , ℜ ) Chern-Simons → 2d affine algebras | reduced → W N algebras

3d Euclidean black holes live on a solid torus dr 2 ds 2 = +( r 2 − M ) dt 2 + r 2 − M + r 2 d φ 2 Example: In the Euclidean geometry, the time coordinate is compact. horizon ρ=0 ρ = r − M 1 / 2 0 < ρ < ∞ 0 ≤ t < β, contractible loop t φ ρ 0 ≤ φ < 2 π, non-contractible loop The three dimensional spacetime topology can be seen as a torus × ℜ + = disc × S 1 .

Interesting (not zero), regular, solutions Interesting solutions A µ = { A t , A r , A φ } ∈ SL ( N , ℜ ) must satisfy: 1. The Chern-Simons equations of motion F µν = 0 2. Must have a non-trivial holonomy along φ : A φ d φ � = 1 � Pe If this holonomy was trivial, the solution can be set to zero by a gauge transformation. 3. Must have a trivial holonomy along t . A t dt = 1 . � Pe If this holonomy is not trivial, the field will be singular, because the time cycle is contractible. Solutions are then characterized by conditions on A t and A φ . Note, that A t and A φ are coupled through F t φ = 0.

Building the general solution in radial gauge A r = 0 ◮ F µν = 0 in the gauge A r = 0 imply A t ( t , φ ) , A φ ( t , φ ) , ∂ t A φ − ∂ φ A t + [ A t , A φ ] = 0 . ◮ Furthermore, for black holes, we consider static and spherically symmetric fields. That is, we take A t , A φ to be constant matrices. The equations reduce to: [ A t , A φ ] = 0 In summary, our game will be to find constant SL ( N , ℜ ) matrices A φ , A t that commute, and satisfy the holonomy conditions. A φ d φ = e 2 π A φ � = 1 , A t dt = e A t = 1 � � Pe Pe

A φ and charges Let A φ be a general SL ( N , ℜ ) matrix,   a 11 a 12 ... a 1 N a 21 a 22 .. a 2 N   A φ =  , a ij ∈ ℜ  . . .  ... . . .   . . .  a N 1 ... ... a NN with � 2 π A φ � = 1 Tr( A φ ) = 0 , Pe 0 The coefficients a ij are not really relevant, but only the N − 1 Casimirs or charges , Q 2 = Tr( A 2 Q 3 = Tr( A 3 ... Q N = Tr( A N φ ) , φ ) , φ ) , All physical quantities will depend on these charges. See below.

A t and chemical potentials For a given A φ we seek A t such that [ A t , A φ ] = 0, for all charges. One concludes that A t must be a function of A φ , A t = f ( A φ ) . Furthermore, the most general function is (Cayley-Hamilton theorem) σ 2 A φ + σ 3 A 2 φ + · · · + σ N A N − 1 A t = − Trace φ Besides the N − 1 charges, A t brings in N − 1 new parameters σ 2 , σ 3 , ...σ N . Solutions are characterized by pairs { σ 1 , Q 1 } , { σ 2 , Q 2 } ... which turn out to be canonically conjugated. Finally, the trivial holonomy condition (regularity) A t = 1 , � Pe imply exactly N − 1 equations that fix the chemical potentials σ n as functions of the charges Q n , or the other way around.

Example N = 3 A good parametrization for A φ is:   0 1 0 � φ − 1 �  , A 2 3Tr( A 2 A φ = 0 0 1 A t = σ 2 A φ + σ 3 φ )  Q 3 Q 2 0 ( Q 2 = Tr( A 2 φ ), Q 3 = Tr( A 3 φ ).) A t dt = 1 becomes: � The condition Pe 2 + 72 σ 3 (1 + σ 2 ) 2 Q 2 − 16 σ 3 3 Q 3 2 + 54 σ 2 0 = 3 (1 + σ 2 ) Q 3 Q 2 3 + 27 (1 + σ 2 ) 3 Q 3 , +27 σ 3 3 Q 2 4 Q 2 (1 + σ 2 ) 2 + 6 Q 3 (1 + σ 2 ) σ 3 + 8 8 π 2 3 Q 2 2 σ 2 = 3 . These 2 equations express Q 2 , Q 3 as functions of σ 2 , σ 3 . Then, the black hole has two independent parameters (spin 2 and spin 3 charge).

Integrability and free energy From these relations Gutperle and Kraus discover a “coincidence”: = − 16 σ 3 Q 2 ∂ Q 2 2 + 9 (1 + σ 2 ) Q 3 = ∂ Q 3 8 σ 2 3 Q 2 − 3 (1 + σ 2 ) 2 ∂σ 3 ∂σ 2 This equality implies that there must exists a function W ( σ 2 , σ 3 ) such that, Q 2 = ∂ W ( σ 2 , σ 3 ) Q 3 = ∂ W ( σ 2 , σ 3 ) , , ∂σ 2 ∂σ 3 What does W mean?

The partition function Gutperle and Kraus conjectured the identification, e − W ( σ 2 ,σ 3 ) = Tr H e − ( σ 2 Q 2 + σ 3 Q 3 ) , W ( σ 2 , σ 3 ) is the free energy and Q 2 = ∂ W ( σ 2 , σ 3 ) Q 3 = ∂ W ( σ 2 , σ 3 ) , , ∂σ 2 ∂σ 3 is not a “coincidence” but derived from the partition function. ◮ We will see that this idea is indeed correct. In the semiclassical limit, we shall prove that the on-shell action �� � � AdA + 2 k 3 A 3 � ≡ W ( σ 2 , σ 3 ) � 4 π � solution is in fact the solution to the integrability condition.

Plugging a solution into its action seems easy... ◮ Often the solution in globally defined coordinates is not known. Black holes, for example, need two patches. For free theories, this is easily solved: � � 1 � dx √ g − g σν ∂ Φ ∂ Φ ∂ x ν − m 2 Φ 2 I [Φ] = , ∂ x σ 2 1 � dx √ g Φ � d Σ ν Φ ∂ ν Φ , � Φ − m 2 Φ � � = − 2 r →∞ � d Σ ν Φ ∂ ν Φ = 0 − r →∞ ◮ The bulk part is zero for all solutions. We only need the field in one patch (at infinity) to find I [Φ]on-shell But for an interacting theory, life is not as easy.