Holographic Renyi Entropy

Transcription

1 Holographic Reyi Etropy Dmitri V. Fursaev Duba Uiversity & Bogoliubov Laboratory JINR th Marcel Grossma Meetig, Stockholm

2 black hole etropy etropic origi of gravity etaglemet & quatum gravity holography ad AdS gravity

3 Motivatio: To study etaglemet Reyi etropy (ERE) as a more geeral otio, to use ERE to get more isights ad resolve some ope problems

4 Pla of the talk:. Itroductio. Reyi etropy: - ew results (4D CFT, weak couplig); - possible holographic represetatio of Reyi etropy; - strog couplig results 3. Semiclassical derivatio: - more argumets to a proof of holographic etaglemet etropy - etaglemet etropy ad etropic origi of gravity

5 Quatum etaglemet quatum mechaics: states of subsystems may ot be described idepedetly = states are etagled importace: studyig correlatios of differet systems (especially at strog coupligs), critical pheomea ad etc

6 reduced desity matrix ( A, a B, b) ( A B) ( A, a B, a), ( a b) ( A, a A, b), a A A a Tr, Tr,

7 Etropy as a measure of etaglemet Tr reduced desity matrix S l Tr ( ) etaglemet Reyi etropy I geeral, 0, ad Next we cosider iteger values,3,4,...

8 Basic properties I. S 0, ( S 0, ( ) ( ) if ad oly if is pure state) Differet limits: S S S ( ) Tr, l etaglemet etropy II. S "Symmetry" i a pure state S ( ) ( )

9 etaglemet has to do with quatum gravity: possible source of the etropy of a black hole (states iside ad outside the horizo); d=4 supersymmetric BH s are equivalet to, 3, qubit systems etaglemet etropy allows a holographic iterpretatio for CFT s with AdS duals

10 Holographic Formula for Etaglemet Etropy (=) Ryu ad Takayaagi, hep-th/060300, AdSd (bulk space) B miimal (least area) surface i the bulk 4d space-time maifold (asymptotic boudary of AdS) B separatig surface etropy of etaglemet S A ( ) 4G d is measured i terms of the area of B ( d ) G is the gravity couplig i AdS

11 Holographic formula eables oe to compute etaglemet etropy i strogly correlated systems with the help of geometrical methods (the Plateau problem); Ryu-Takayaagi formula passes several o-trivial tests: - i D ad 4D CFT s (at weak couplig); -for differet quatum states; H -for differet shapes ad topologies of the separatig surface i boudary CFT Is it possible to fid a holographic descriptio of etaglemet Reyi etropy?

12 Etaglemet Reyi Etropy i CFT s at weak couplig D.F. JHEP 05 (0) 080, e-prit: arxiv:0.70 [hep-th]

13 if st step: represetatio i terms of a partitio fuctio H / T H / T ( ) H / T ( ) where e / Tr H / T Tr e, H / T (, ) Tr (Tr e ) e is a thermal desity matrix S l Z( T / ) l Z( T ), Z( T ) Tr e i geeral a aalog of this relatio: S Z T ( T) l Z(, T ) l Z( T ) a "partitio fuctio", "iverse temperature" Z( T ) Z(, T )

14 d step: relatio of a partitio fuctio to a effective actio o a curved space W(, T) l Z(, T) effective actio { } { ' } /T these itervals are idetified 3 Tr { } { } 0 oe glues (=3) copies a curved space with coical sigularity at the separatig poit (surface)

15 W L k 3 d step: use results of spectral geometry k l det Lk, k Laplace operators of differet spi fields o d d p p W A l( / )... for dimesio d p0 k pd tlk p k k, p where k, p k, p k p0 is a UV cutoff; A p d d M is a physical scale (mass, iverse syze etc) eve, A A, A : Tr e t A...; a example: a scalar Laplacia L0 : A0 O( ), A R ( ), 4 M B There are o-trivial cotributios from coical sigularities located at the 'separatig' surface B

16 computatios S d s d p p sd l( / )..., d p p S s l( / )..., d ( ) ( ) d p p ( ) s d p d p Reyi etropies s lim( ) A ( ), s ( ) p p p A () A ( ) p p s s 0, s s 0 ( ) ( ) 0 0 k k (if boudaries are abset)

17 4D N=4 super SU(N) Yag-Mills theory at weak coup. 6 scalar multiplets, 4 multiplets of Weyl spiors, multiplet of gluo fields ( ) ( ) ( ) S s s 4l( / )... s A( B) area of the separatig surface B ( ) d( N) 4 is a UV cutoff

18 ( ) 4 Coformal ivariace s d( N )( a( ) F c( ) F b( ) F ) a c b F a B d x R B ( ), F d x C c i j i j B b Tr( i )Tr( i ) Tr( i i ), B F d x k k k k, R B B B C ( ) scalar curvature of, a pair of uit orthogoal ormals to, i M B, ( k ) B Weyl tesor of at extrisic curvatures of i F F F g x e g x ( x) a, b, c ivariat with respect to the Weyl trasformatios '( ) ( ) the problem is to determie a( ), c( ), b( )

19 ( ) 4 4 relatio to the trace aomaly i Etaglemet etropy (=) s lim s cf af bf c lim c( ), a lim a( ), 4 4 a c T ae ci 4 4 c a b D 4 E4 RR 4R R R 6 I4 C C 6 C R ( g R g R g R g R Cardy's cojecture: "charge" a decreases mootoically alog RG flows R ) ( g g g g ) 6

20 Computatio of coefficiet fuctios s d( N )( a( ) F c( ) F b( ) F ) ( ) 4 a c b cotributios to heat kerel coefficiets from coical sigularities A a F c F b F () i 4( ) i( ) a i( ) c i( ) b a a a a 3 3 c( ) (6 c0( ) 4 c/ ( ) c( )) ( 3 3) 3 3 ( ) (6 0( ) 4 / ( ) ( )) ( 7 5) lim b( ), ( 'holographic' argumets by S.N. Solodukhi, arxiv:080.37)

21 Toward a holographic descriptio of Etaglemet Reyi Etropy i CFT s

22 Holographic Reyi Etropy (a suggestio) ( ) ( ) ( ( ) ( ) ( ) )... ( ) S f (5) A B a Fa c Fc b Fb 4GN A B ( ) volume of B; F F F B,, are some local (bulk) ivariat fuctioals set o ; a c b f ( ), a( ), c( ), b( ) ( ) (5) N some coefficiet fuctios; to reproduce Ryu-Takayaagi formula for etaglemet etropy S 4G A( B) f (), a() c() b() 0

23 F, F, F a c b the couplig) are fixed by coformal ivariace (should ot deped o F F, F F, F F, a a c c b b the strategy is to fix F, F, F a c b i the limit of weak coupligs f ( ), a( ), c( ), b( ) iformatio is required may deped o the couplig, extra

24 Asymptotics at AdS If we put: 3 K L M N Fc d y RKLMNl m l m B 3 MN Fb d y KMN K B it follows that: Fb, c Fb, c l l... z o other ivariats which yield Weyl ivariat structures appear: RR, l KL the 3d fuctioal, are costat (gravity eqs.) R B, is ot idepedet (Gauss-Codazzi eq.)

25 Asymptotics at AdS (cotiued) F F a Fal... 4l z z caot be defied as a local fuctioal (similar to ) oe (but ot sigle) optio: F AB ( ) ll 3 l other optios are possible; F, F a b c a to match weak couplig calculatios oe should choose a 4 96 c 4 3 b( ) b( )? 4 ( ) a( ) ( )( 5), ( ) c( ) ( )( 3),,

26 coefficiet fuctios at strog couplig? holographic type computatio by Hug, Myers, Smolki, Yale [hep-th] for spherical etaglig surface i Mikowsky spacetyme (sice reduced desity is themal CFT dual to some 5D black hole i the AdS bulk) ( ) S N astrog ( ) # A #l( A) #- some umerical coefficiets, A 4 a ( ) ( x ( x )), x 8 4 because F F 0, F 4, it appears that a strog strog ( ) c b a should correspod to a( ) R to compare with our result (weak couplig): ( ) 3 S N A 3 (5 7 )l( A ) 8 48

27 Derivatio of Holographic Etaglemet Etropy D.F. JHEP 0609 (006) 08, e-prit: hep-th/ critics by: M. Headrick, Phys. Rev. D8 (00) 600, arxiv: [hep-th]

28 the idea of the proof Z CFT Z AdS Z M Dg e W[ g] AdS ( ) [ ], M : M M partitio fuctio for Reyi etropy of ordrer l Z ( M ) W ( ) AdS actio at a statioary poit, the holographic etropy is S AdS ( ) l Z AdS ( M ) l Z AdS ( M ) takig a aive limit ( d [ ] [ ] ( ) [ ], 6 G M / B d 4Gd d saddle poit approximatio requires ) oe has (assumig bulk spaces have coical sigularities) W ( ) W () W M I M R gd x A B S AdS ( ) A[ B] S 4G CFT AB [ ] to be a miimal hypersurface!

29 critics of this derivatio (M.Headrick):. The derivatio does ot reproduce the etaglemet Reyi etropy;. Bulk maifolds with coical sigularities caot be statioary cofiguratios of the AdS partitio fuctio Studyig the holographic Reyi etropy allows oe to remove these problems

30 Avoidig st critical argumet: WM [ ] is a effective (ot classical) actio, W[ M ] I[ M ] cotributio of coical sigularities i the effective actio at a "o-classical" form (experiece with oe-loop computatios) WM [ ] 6 G S(, B) A( B), d / B d R gd x S B SAdS ( ) W ( ) W () S(, B); S S B AdS M ( ) coicides with ERE if (, ) is a holographic ERE Thus, ERE is reproduced by this way of argumets ( ) (, ), has

31 Avoidig d critical argumet: the resolutio is i specific boudary coditios i AdS partitio fuctio which are obeyed by bulk maifolds with coical sigularities the maifolds with coical sigularities should be put i a separate set of geometries i the AdS partitio fuctio sice their actio is o-classical, W[ M ] I[ M ] ; the saddle poit approximatio o this set picks up a sigular geometry, o matter source is eeded

32 If derivatio of holographic etaglemet etropy i the AdS/CFT cotext is successful Why caot we ask about etaglemet i quatum gravity itself (by assumig it has a uderlyig microscopic structure) ad use a semiclassic approximatio? details: D.V. Fursaev, Phys. Rev. D77 (008) 400, e-prit: arxiv:07. [hep-th]

33 Suggestio (DF, 06,07): EE i quatum gravity betwee degrees of freedom separated by a surface B is B is a least area miimal hypersurface i a costat-time slice coditios: SB ( ) AB ( ) 4G static space-times etaglemet etropy i quatum gravity is a dyamical quatity, variatios of the area satisfy the equatios suggested i the framework of etropic gravity hypothesis (E.Verlide,arXiv: [hep-th])

34 thak you for attetio