Dynamical fields in de Sitter from CFT entanglement
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1 Dynamical fields in de Sitter from CFT entanglement Michał P. Heller Perimeter Institute for Theoretical Physics, Canada based on with J. de Boer, R. Myers and Yasha Neiman
2 Introduction
3 Differential entropy and hole-ography Balasubramanian, de Boer, Chowdhury, Czech & Heller 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
4 Kinematic space Czech, Lamprou, McCandlish & Sully 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 =! v Sdu^dv and SSA turns out to 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!
5 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
6 Dynamics in de Sitter
7 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 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
8 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 a p gg 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
9 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 ht i = h0 T ( 0 +, 0 ) T (,x) T ( 0, 0 ) 0i h0 T ( 0 +, 0 ) T ( 0, 0 ) 0i ( ) 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
10 Generalization to higher spins in CFT2 The exploratory idea of Hijano & Kraus 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
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 a p gg 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 By extrapolating we might expect then 8/11 1 many fields in de SitterD
12 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 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. 2 l/2 2 (l/2) 2 In 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
13 Summary and open problems
14 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
15 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 Link with MERA / cmera? 11/11
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