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 µν c g Tµν = R, d = 2 24π µν a b 2 g Tµν = E4+ Tr W , d = 4 −5760π2 1920π2 2 αβµν µν 1 2 Tr W = R R 2RµνR + R αβµν − 3 αβµν µν 2 E = R R 4RµνR + R . 4 αβµν − (For scalar field a = b = 1) µν g Tµν = 0 , d = 2n + 1 6 Entanglement entropy and Weyl anomaly Σ is compact 2d entangling surface A(Σ) S =4 = + s0 ln ǫ d 4πǫ2 a b 2 s0 = χ[Σ] [Wabab Tr ˆk ] 180 − 240π ZΣ − χ[Σ] is Euler number of Σ Wabab is projection of Weyl tensor on subspace orthogonal to Σ, na, a = 1, 2 is a pair of normal vectors a a 1 a ˆk = k γµνk , a = 1, 2 is trace-free µν µν − d 2 extrinsic curvature− of Σ 7 EE in d dimensions In d dimensions compact entangling surface Σ is (d 2)-dimensional − Logarithmic term s0 in entanglement entropy is given by integral over Σ of a polynomial in- variant constructed from Weyl tensor Wµανβ, even number of covariant derivatives of Weyl ˆa tensor, extrinsic curvature kµν and projections a on normal vectors nµ. If d is odd no such invariant exists so that s0 = 0 if d = 2n + 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 ∂ = 0 , Π ϕ ∂ = 0 , Π++Π = 1 ∇ | M − | M − 10 Conformal scalar field in d di- mensions Dirichlet b.c. (Π+ = 0) φ ∂ = 0 | M Conformal Robin b.c. (Π = 0) − (d 2) ( n + − K)φ ∂ = 0 ∇ 2(d 1) | M − Remark: in d = 4 exists one more (complex) Robin b.c. 1 i 2 1 S = K 10Tr Kˆ , Kˆµν = Kµν γµνK 3 ± 10q − 3 for which (classical and quantum) theory is conformal 11 Dirac field in d = 4 dimensions Π ψ ∂ = 0 , ( n + K/2)Π+ψ ∂ = 0 − | M ∇ | M 1 µ Π = (1 iγ n γµ), γ is chirality gamma ± 2 ± ∗ ∗ function 12 Integral Weyl anomaly Variation of effective action under constant rescaling of metric 2σ µ ∂σW [e gµν]= T A≡ µ ZMd D E For free fields integral Weyl anomaly reduces to computation of heat kernel coefficient Ad. 13 General structure µν √g Tµν g = a χ( )+ b √γI (W ) Md k k ZMd ZMd ˆ ˆ +bk′ √γJk(W, K)+ cn √γ n(K) , ∂ ∂ K Z Md Z Md χ[ ] is Euler number of manifold , I (W ) Md Md k are conformal invariants constructed from the Weyl tensor, n(Kˆ) are polynomial of degree K (d 1) of the trace-free extrinsic curvature, − 1 Kµν = Kµν d 2γK is trace free extrinsic cur- − − σ vature of boundary; Kˆµν e Kˆµν if gµν σ → → e gµν. 14 Q: Does it mean that there are new conformal charges bn′ , cn? A: we suggest that in appropriate normaliza- tion bn′ = bn and the corresponding boundary term Jk(W, Kˆ) is in fact the Hawking-Gibbons type term for the bulk action Ik(W ) cn are indeed new boundary conformal charges 15 Gibbons-Hawking type terms Re-writing functional of curvature in a form linear in Riemann tensor I = UαβµνR UαβµνV + F (V ) bulk αβµν − αβµν ZMd In order to cancel normal derivatives of the metric variation on the boundary one should add a boundary term, αβµν (0) Iboundary = U P − ∂ αβµν Z Md (0) P = nαnνK n nνKαµ nαnµK +n nµKαν αβµν βµ− β − βν β µ n is normal vector and Kµν is extrinsic curva- ture of ∂ Md Barvinsky-SS (95) 16 For a bulk invariant expressed in terms of Weyl tensor only, I[W ]= UαβµνW UαβµνV + F (V ) αβµν − αβµν ZMd αβµν U Pαβµν − ∂ Z Md (0) 1 (0) (0) (0) P = P (gαµP gανP g Pαν αβµν αβµν−d 2 βν − βµ − βµ − (0) (0) P +g Pαµ )+ (gαµg gανg ) βν (d 1)(d 2) βν − βν − − (0) α α Pµν = nµn K + nµn Kαν Kµν nµnνK αβ − − P (0) = 2K − Pαβµν has same symmetries as the Weyl tensor. α In particular, P µαν = 0. Pαβµν can be expressed in terms of Kˆµν 17 Examples n n 1 1. Tr (W ) nTr (PW − ) − ∂ ZMd Z Md 2. Tr (W 2W ) 2 Tr (P 2W ) ∇ − ∂ ∇ ZMd Z Md 18 Integral Weyl anomaly in d = 4: anomaly of type A First of all, bulk integral of E4 is supplemented by some boundary terms to form a topological invariant, the Euler number, 1 χ[ 4]= E4 M 32π2 ZM4 1 µν µν 1 (K Rnµnν K Rµν KRnn + KR −4π2 ∂ − − 2 Z M4 1 2 K3 + KTr K2 + Tr K3) −3 3 α β µ ν Rµnνn = Rµανβn n and Rnn = Rµνn n 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 ) − ∂ ZM4 Z M4 Due to properties of Weyl tensor: (0) µναβ Tr (WP ) = Tr (WP ) = 4W nµnβKˆνα 20 Integral Weyl anomaly in d =4 a T = χ[ ] −180 M4 ZM4 b 2 µναβ + Tr W 8 W nµnβKˆνα 1920π2 − ∂ ! ZM4 Z M4 c + Tr Kˆ 3 280π2 ∂ Z M4 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 = = aE + b I + b I + b I + TD A 6 1 1 2 2 3 3 where E6 is the Euler density in d =6 and we defined 3 µσρν αβ I1 = Tr 1(W )= WαµνβW Wσ ρ 3 µν σρ αβ I2 = Tr 2(W )= Wαβ Wµν Wσρ I = Tr (W 2W )+ Tr (WXW ) 3 ∇ 2 [ ] 6 X µν = X µδν , Xµ = 4Rµ Rδµ αβ [α β] ν ν − 5 ν 23 Integral Weyl anomaly in d =6 T = a χ[ ] ′ M6 ZM6 3 2 +b1 Tr 1W 3 Tr 1(PW ) − ∂ ! ZM6 Z M6 3 2 +b2 Tr 2W 3 Tr 2(PW ) − ∂ ! ZM6 Z M6 2 2 +b3[ Tr (W W ) 2 Tr (P W ) ∇ − ∂ ∇ ZM6 Z M6 + Tr 2(WXW ) Tr 2(WQW )] − ∂ ZM6 Z M6 2 3 5 + c1Tr Kˆ Tr Kˆ + c2Tr Kˆ ∂ Z M6 two new boundary charges c1 and c2 there may exist additional invariant with deriva- tives of extrinsic curvature 24 Integral Weyl anomaly in odd di- mensions Euler number of vanishes if d is odd Md Euler number of boundary ∂ d may appear in integral anomaly M d = 3 : c1 c2 2 T = χ[∂ 3]+ Tr Kˆ 96 M 256π ∂ ZM3 Z M3 (c1,c2): ( 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 χ[∂ ] 1 M5 ZM5 2 α β αβµ + [c2Tr W +c3WαnβnW n n+c4WnαβµWn ∂ Z M5 αµβν ˆ ˆ α β ˆ ˆ σ +c5W KαβKµν + c6W n nKασK β +c (Tr Kˆ 2)2 + c Tr Kˆ 4 + c Tr (Kˆ Kˆ)] 7 8 9 D is conformal operator acting on trace free D symmetric tensor in 4 dimensions values of ck 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) ≃ − nA (1) A (n) s(n) = η 3 − 3 n 1 − A3(n) is heat kernel coefficient on replica man- ifold n M 27 Consider = R2 L, L is an interval with 2 M × end points P1 and P2 Entangling surface Σ = L, replica space n = M n L C × A (n)= A ( n) A (L) , 3 2 C × 1 1 2 A2( n)= (1 n ) C 12n − is the heat kernel coefficient on two-dimensional cone, and 1 A1(L)= tr χ, χ =Π+ Π 4 − − XPk 28 for scalar field c n + 1 c s(n) = 1 , s(n=1) = 1 48 n 24 XP XP for Dirac field s(n) = 0 , s(n=1) = 0 INTERESTING PREDICTION: dependence on angle between entangling sur- face Σ and boundary ∂M cos α = (n,t), nµ normal vector to ∂ , M3 tµ tangent vector to Σ.