Dynamical fields in de Sitter from CFT entanglement Michał P. Heller Perimeter Institute for Theoretical Physics, Canada based on 1509.00113 with J. de Boer, R. Myers and Yasha Neiman
Introduction
Differential entropy and hole-ography Balasubramanian, de Boer, Chowdhury, Czech & Heller 1310.4204 Consider a CFT2 in the vacuum (say on S 1 of length of an interval of an angular opening 2 L ) and entanglement entropy S( ) = c 3 log(2l µ sin ) Consider now an infinite family of intervals given by ( ) and compute E = Z 2 0 d S 0 ( ( )) This novel UV-finite quantity is called differential entropy. In a holographic CFT2 it measures areas of closed curves on a constant-t slice of AdS3 1/11
Kinematic space Czech, Lamprou, McCandlish & Sully 1505.05515 One can introduce a partial order on the set of intervals for which we calculate EE A A A B B B Timelike Spacelike Lightlike This (partially) motivates introducing the light-cone coordinates v = + and considering space with the volume form u =! = @ u @ v Sdu^dv and SSA turns out to guarantee @ u @ v S 0. For the vacuum we obtain c! = 12 sin 2 du ^ dv ( ) The minimal metric compatible with this volume form and the partial order is ds 2 = c 12 sin 2 ( ) du dv = c 12 sin 2 ( ) 2/11 d 2 + d 2 de Sitter2!
Question behind this work Is there more to this Lorentzian structure than merely ordering intervals on a constant time slice of a CFT2? 3/11
Dynamics in de Sitter
Entanglement first law Consider small perturbations of some reference density matrix = 0 + The change in the entropy is equal to the change in the modular Hamiltonian S = tr ( log ) S 0 = hh mod i In general, H mod = log is unknown, but for reduced density matrices for spherical entangling surfaces in the CFTD vacuum it turns out to be fixed by conformality Z ~x 2< 2 H mod = c 0 2 d (D 1) x 2 ~x 2 2 T Casini, Huerta & Myers 1102.0440 As a result, the change in the entanglement entropy for small perturbations of 0i is S = 2 Z ~x 2< 2 d (D 1) x 2 ~x 2 2 ht i 4/11
Entanglement first law yields dynamics in de Sitter In two dimensions (on a plane) we simply have S(, ) = 2 Z + dx 2 ( x) 2 2 ht i(x) at fixed ht i(x) S(, ) + x It is straightforward to check that 1 p g @ a p gg ab @ b S 2 L 2 S =0 with g ab dx a dx b = L2 being the inflationary patch of de Sitter2! 2 d 2 + d 2 5/11
Specific example: universal excited state Any CFT has T µ operator, hence T ( 0, 0 ) 0i is an universal excitation in CFT2 s cf. Nozaki, Numasawa & Takayanagi 1401.0539 ht i = h0 T ( 0 +, 0 ) T (,x) T ( 0, 0 ) 0i h0 T ( 0 +, 0 ) T ( 0, 0 ) 0i 4 2 4 6 2 2 + 4 ( 2 + 2 ) 4 Entanglement first law: S(, ) = 2 Z + dx 2 ( x) 2 2 ht i(x) ds2 analogue of the bulk-boundary smearing function by Kabat et al. 6/11
Generalization to higher spins in CFT2 The exploratory idea of Hijano & Kraus 1406.1804 is the following: H mod = Z z2 z 1 (z z2 )(z z 1 ) dz T (z) + antiholomorphic z 2 z 1 H mod,3 = 3 Z z2 z 1 dz so in the presence of a higher spin current (here spin-3 current ) one can try 2 (z z2 )(z z 1 ) W (z) + antiholomorphic z 2 z 1 W Any quantity behaving as fields in de Sitter with Z + dx 2 ( x) 2 2 f(x) leads to a dynamical m 2 L 2 = 2 perturbation in entanglement entropy Hijano & Kraus construction m 2 L 2 = 2 m 2 L 2 = 6 7/11
Generalization to higher dimensions The first law for any CFT in D spacetime dimensions S = 2 Z ~x 2< 2 d (D 1) x 2 ~x 2 2 ht i implies propagation in D-dimensional de Sitter 1 p g @ a p gg ab @ b S D L 2 S =0 with g ab dx a dx b = L2 being the inflationary patch of de SitterD. 2 ( d 2 + d~x 2 ) Note that free field theories have 1 many higher spin currents. see, e.g., Maldacena & Zhiboedov 1112.1016 By extrapolating 1406.1804 we might expect then 8/11 1 many fields in de SitterD
Relation to Einstein s equations in AdS In 2013 several groups were able to derive linearized Einstein s equations from the Ryu-Takayanagi proposal and the first law of entanglement entropy. In 1304.7100 Takayanagi et al. showed that linearized Einstein s equations in AdS3 give Linear combination gives our wave equation on ds2: S A (,l,t)=0. apple @ 2 l/2 + @2 2 (l/2) 2 In 1308.3792 Bhattacharya and Takayanagi obtained from Einstein s equations in AdS4 We see here that this equation in any D has nothing to do with a 2-derivative geometry and follows purely from the underlying conformal symmetry. 9/11
Summary and open problems
Summary ongoing work with Rob Myers and Guifre Vidal Entanglement in excited states organizes itself in a Lorentzian way: r a r a ds D S m 2 S =0 with m 2 L 2 ds D = D, e.g. for in T ( 0, 0) 0i ECFT2 and = L2. ds 2 ds 2 2 ( d 2 + d 2 ) This statement applies to any CFT in any D provided the first law holds! The statement concerns constant time slices in a CFT. For theories with higher spin charges: one dynamical field in dsd for each charge. 10/11
Some open problems Is recent explosion of emergent geometries an artifact of considering CFTs? Does auxiliary de Sitter = kinematic space? Higher dimensional generalization of differential entropy and hole-ography? Other fields in de Sitter, in particular gravitons? Nonlinear eoms? New holography? Local operators in de Sitter (generalization of Kabat et al.)? see Xiao Xiao 1402.7080 Link with MERA / cmera? 11/11