Conformal anomaly, entanglement entropy and boundaries Sergey N. - PDF document

Conformal anomaly, entanglement entropy and boundaries Sergey N. Solodukhin University of Tours Talk at Oxford Holography Seminar February 9, 2016

Plan of the talk: 1. Brief review: local Weyl anomaly, entangle- ment entropy 2. Integral Weyl anomaly in presence of bound- aries a) d=4 b) d=6 3. Integral Weyl anomaly in odd dimensions 4. Entanglement entropy and boundaries 5. Some open questions 1

Based on 1. S.S., “Boundary terms of conformal anomaly,” Phys. Lett. B 752 , 131 (2016) [arXiv:1510.04566 [hep-th]]. 2. Dima Fursaev and S.S. “Anomalies, entropy and boundaries,” arXiv:1601.06418 [hep-th]. 3. work in progress with Amin Astaneh, Clement Berthiere 2

Other recent relevant works: 1. D. Fursaev, “Conformal anomalies of CFT’s with boundaries,” arXiv:1510.01427 [hep-th] 2. C. P. Herzog, K. W. Huang and K. Jensen, “Universal Entanglement and Boundary Geom- etry in Conformal Field Theory,” arXiv:1510.00021 [hep-th]. 3

Earlier relevant works: 1. J. S. Dowker and J. P. Schofield, “Confor- mal Transformations and the Effective Action in the Presence of Boundaries,” J. Math. Phys. 31 , 808 (1990). 2. J. Melmed, “Conformal Invariance and the Regularized One Loop Effective Action,” J. Phys. A 21 , L1131 (1988). 3. I. G. Moss, “Boundary Terms in the Heat Kernel Expansion,” Class. Quant. Grav. 6 , 759 (1989). 4. T. P. Branson, P. B. Gilkey and D. V. Vas- silevich, “The Asymptotics of the Laplacian on a manifold with boundary. 2,” Boll. Union. Mat. Ital. 11B , 39 (1997) 4

Let me first remind you briefly the standard story 5

Local Weyl anomaly c g µν � T µν � = 24 πR , d = 2 a b g µν � T µν � = − 1920 π 2 Tr W 2 , d = 4 5760 π 2 E 4 + Tr W 2 = R αβµν R αβµν − 2 R µν R µν + 1 3 R 2 E 4 = R αβµν R αβµν − 4 R µν R µν + R 2 . (For scalar field a = b = 1) g µν � T µν � = 0 , d = 2 n + 1 6

Entanglement entropy and Weyl anomaly Σ is compact 2d entangling surface S d =4 = A (Σ) 4 πǫ 2 + s 0 ln ǫ a b � k 2 ] Σ [ W abab − Tr ˆ s 0 = 180 χ [Σ] − 240 π χ [Σ] is Euler number of Σ W abab is projection of Weyl tensor on subspace orthogonal to Σ, n a , a = 1 , 2 is a pair of normal vectors 1 k a µν = k a d − 2 γ µν k a , a = 1 , 2 is trace-free ˆ µν − extrinsic curvature of Σ 7

EE in d dimensions In d dimensions compact entangling surface Σ is ( d − 2)-dimensional Logarithmic term s 0 in entanglement entropy is given by integral over Σ of a polynomial in- variant constructed from Weyl tensor W µανβ , even number of covariant derivatives of Weyl k a tensor, extrinsic curvature ˆ µν and projections on normal vectors n a µ . If d is odd no such invariant exists so that s 0 = 0 if d = 2 n + 1 8

In this talk: What changes if manifold has boundaries? 9

Conformal boundary conditions General (mixed) boundary condition is a com- bination of Robin and Dirichlet b.c. ( ∇ n + S )Π + ϕ | ∂ M = 0 , Π − ϕ | ∂ M = 0 , Π + +Π − = 1 10

Conformal scalar field in d di- mensions Dirichlet b.c. (Π + = 0) φ | ∂ M = 0 Conformal Robin b.c. (Π − = 0) ( ∇ n + ( d − 2) 2( d − 1) K ) φ | ∂ M = 0 Remark: in d = 4 exists one more (complex) Robin b.c. S = 1 3 K ± i K µν = K µν − 1 � K 2 , 10Tr ˆ ˆ 3 γ µν K 10 for which (classical and quantum) theory is conformal 11

Dirac field in d = 4 dimensions Π − ψ | ∂ M = 0 , ( ∇ n + K/ 2)Π + ψ | ∂ M = 0 Π ± = 1 2 (1 ± iγ ∗ n µ γ µ ), γ ∗ is chirality gamma function 12

Integral Weyl anomaly Variation of effective action under constant rescaling of metric � A ≡ ∂ σ W [ e 2 σ g µν ] = � T µ � µ M d For free fields integral Weyl anomaly reduces to computation of heat kernel coefficient A d . 13

General structure √ g � T µν � g µν = a χ ( M d ) + b k √ γI k ( W ) � � M d M d √ γJ k ( W, ˆ √ γ K n ( ˆ � � + b ′ K ) + c n K ) , k ∂ M d ∂ M d χ [ M d ] is Euler number of manifold M d , I k ( W ) are conformal invariants constructed from the Weyl tensor, K n ( ˆ K ) are polynomial of degree ( d − 1) of the trace-free extrinsic curvature, 1 K µν = K µν − d − 2 γK is trace free extrinsic cur- K µν → e σ ˆ ˆ vature of boundary; K µν if g µν → e σ g µν . 14

Q : Does it mean that there are new conformal charges b ′ n , c n ? A : we suggest that in appropriate normaliza- tion b ′ n = b n and the corresponding boundary term J k ( W, ˆ K ) is in fact the Hawking-Gibbons type term for the bulk action I k ( W ) c n are indeed new boundary conformal charges 15

Gibbons-Hawking type terms Re-writing functional of curvature in a form linear in Riemann tensor � � U αβµν R αβµν − U αβµν V αβµν + F ( V ) � I bulk = M d In order to cancel normal derivatives of the metric variation on the boundary one should add a boundary term, � U αβµν P (0) I boundary = − αβµν ∂ M d P (0) αβµν = n α n ν K βµ − n β n ν K αµ − n α n µ K βν + n β n µ K αν n µ is normal vector and K µν is extrinsic curva- ture of ∂ M d Barvinsky-SS (95) 16

For a bulk invariant expressed in terms of Weyl tensor only, � U αβµν W αβµν − U αβµν V αβµν + F ( V ) � � I [ W ] = M d � U αβµν P αβµν − ∂ M d 1 P αβµν = P (0) d − 2( g αµ P (0) βν − g αν P (0) βµ − g βµ P (0) αβµν − αν P (0) + g βν P (0) αµ ) + ( d − 1)( d − 2)( g αµ g βν − g αν g βν ) P (0) = n µ n α K αβ + n µ n α K αν − K µν − n µ n ν K µν P (0) = − 2 K P αβµν has same symmetries as the Weyl tensor. In particular, P α µαν = 0. P αβµν can be expressed in terms of ˆ K µν 17

Examples � � Tr ( W n ) − n Tr ( PW n − 1 ) 1 . M d ∂ M d � � Tr ( W ∇ 2 W ) − 2 Tr ( P ∇ 2 W ) 2 . M d ∂ M d 18

Integral Weyl anomaly in d = 4: anomaly of type A First of all, bulk integral of E 4 is supplemented by some boundary terms to form a topological invariant, the Euler number, 1 � χ [ M 4 ] = E 4 32 π 2 M 4 − 1 ( K µν R nµnν − K µν R µν − KR nn + 1 � 2 KR 4 π 2 ∂ M 4 − 1 3 K 3 + K Tr K 2 + 2 3Tr K 3 ) R µnνn = R µανβ n α n β and R nn = R µν n µ n ν Dowker-Schofield (90) Herzog-Huang-Jensen (2015) 19

Integral Weyl anomaly in d = 4: anomaly of type B Gibbons-Hawking type boundary term: � � Tr W 2 − 2 Tr ( WP ) M 4 ∂ M 4 Due to properties of Weyl tensor: Tr ( WP ) = Tr ( WP (0) ) = 4 W µναβ n µ n β ˆ K να 20

Integral Weyl anomaly in d = 4 � T � = − a � 180 χ [ M 4 ] M 4 �� � b � Tr W 2 − 8 W µναβ n µ n β ˆ + K να 1920 π 2 M 4 ∂ M 4 c � K 3 Tr ˆ + 280 π 2 ∂ M 4 For B-anomaly balance between bulk and bound- ary terms agrees with calculation for free fields of spin s=0,1/2, 1 Fursaev (2015) also Herzog-Huang-Jensen (2015) 21

Values of boundary charge c : (Malmed (88), Dowker-Schofield (95), Fursaev (2015)) c = 1 for s = 0 (Dirichlet b.c.) c = 7 / 9 for s = 0 (Robin b.c.) c = 5 for s = 1 / 2 (mixed b.c.) c = 8 for s = 1 (absolute or relative b.c) 22

Local Weyl anomaly in d = 6 � T � = A = aE 6 + b 1 I 1 + b 2 I 2 + b 3 I 3 + TD where E 6 is the Euler density in d = 6 and we defined I 1 = Tr 1 ( W 3 ) = W αµνβ W µσρν W αβ σ ρ µν I 2 = Tr 2 ( W 3 ) = W σρ αβ W W µν σρ αβ I 3 = Tr ( W ∇ 2 W ) + Tr 2 ( WXW ) ν − 6 = X [ µ [ α δ ν ] µν β ] , X µ ν = 4 R µ 5 Rδ µ X ν αβ 23

Integral Weyl anomaly in d = 6 � � T � = a ′ χ [ M 6 ] M 6 �� � � Tr 1 W 3 − 3 Tr 1 ( PW 2 ) + b 1 M 6 ∂ M 6 �� � � Tr 2 W 3 − 3 Tr 2 ( PW 2 ) + b 2 M 6 ∂ M 6 � � Tr ( W ∇ 2 W ) − 2 Tr ( P ∇ 2 W ) + b 3 [ M 6 ∂ M 6 � � + Tr 2 ( WXW ) − Tr 2 ( WQW )] M 6 ∂ M 6 � K 3 + c 2 Tr ˆ � K 2 Tr ˆ K 5 � c 1 Tr ˆ + ∂ M 6 two new boundary charges c 1 and c 2 there may exist additional invariant with deriva- tives of extrinsic curvature 24

Integral Weyl anomaly in odd di- mensions Euler number of M d vanishes if d is odd Euler number of boundary ∂ M d may appear in integral anomaly d = 3 : � T � = c 1 c 2 � � K 2 Tr ˆ 96 χ [ ∂ M 3 ] + 256 π M 3 ∂ M 3 ( c 1 , c 2 ): ( − 1 , 1) for scalar filed (Dirichlet b.c.) (1 , 1) for scalar field (conformal Robin b.c) (0 , 2) for Dirac field (mixed b.c.) Remark: similar anomaly for defects Jensen- O’Bannon (2015) 25

Integral Weyl anomaly in d = 5 � � T � = c 1 χ [ ∂ M 5 ] M 5 � [ c 2 Tr W 2 + c 3 W αnβn W α β n n + c 4 W nαβµ W αβµ + n ∂ M 5 + c 5 W αµβν ˆ K µν + c 6 W α β K σ K αβ ˆ n n ˆ K ασ ˆ β K 2 ) 2 + c 8 Tr ˆ K 4 + c 9 Tr ( ˆ + c 7 (Tr ˆ K D ˆ K )] D is conformal operator acting on trace free symmetric tensor in 4 dimensions values of c k for conformal scalar field: work in progress with Clement Berthiere 26

Entanglement entropy: d = 3 (recent work with Fursaev) Renyi entropy S ( n ) ≃ c ( n ) L/ǫ − ln( ǫ ) s ( n ) s ( n ) = ηnA 3 (1) − A 3 ( n ) n − 1 A 3 ( n ) is heat kernel coefficient on replica man- ifold M n 27