v = ( x − x 0 ) 2 + ( y − y 0 ) 2 ( x − x 0 ) 2 + ( y + y 0 ) 2 Boundary T race Anomalies and Boundary Conformal Field Theory Christopher Herzog (YITP, Stony Brook University) K.-W. Huang and CPH arXiv:1707.06224 � August 9, 2017 Huang, Jensen, and CPH to appear
Outline ✤ Why bCFT? � ✤ Review of trace anomalies and especially boundary contributions. � ✤ New results: relations between boundary trace anomalies and two and three point functions of the displacement operator. � ✤ Demonstration that the b 2 charge can depend on marginal couplings.
What do D-branes, AdS/CFT, topological insulators, and entanglement entropy for field theories have in common? B A
What do D-branes, AdS/CFT, topological insulators, and entanglement entropy for field theories have in common? B The importance of boundaries. A
D-branes ✤ Boundary conditions for open strings � ✤ Non-perturbative insight into string theories led to the second superstring revolution and the realization that the different string theories were part of one larger structure Polchinski ’89-‘95 01
AdS/CFT ✤ Conformal boundary of anti-de Sitter space is where the conformal field theory “lives”. � ✤ Myriad developments — quantum gravity, strongly coupled field theories, black holes, … — perhaps best summarized by the fact that Maldacena’s original paper now has over 15,000 citations. 01
Topological Insulators ✤ The insulating bulk material has conducting surface states that are protected by symmetry � ✤ Predicted in the late 80s for time reversal symmetry (Pankratov, Pakhomov, Volkov) , the effects were first observed in real materials in 2007 (Konig, Wiedmann, et al.) and have now been studied and seen in many other shapes and forms as well. 01
Entanglement Entropy ✤ In field theory, entanglement is B often measured with respect to spatial regions, leading to the importance of the “entangling surface” � A ✤ Led to new proofs of the “a” and “c” theorems for renormalization group flow of field theories in 2 and 4 dimensions. Also given a new perspective to black hole physics. 01
Would all of these developments have been “obvious” if we just understood quantum field theory in the presence of a boundary a little better to begin with?
Boundary Conformal Field Theory ✤ Surprisingly unexplored. McAvity and Osborn ’93 and ’95 papers on two point functions have fewer than 100 citations each. � ✤ Flat space: A planar boundary breaks the SO(d,2) symmetry to SO(d-1,2). � ✤ Curved space: Require that the boundary and boundary terms in the action preserved Weyl invariance.
Stress Tensor T race Today’s talk: To understand new terms in the trace anomaly � associated with the presence of a boundary. A quick review: J µ = x ν T ν µ ; r µ J µ = 0 classically, Weyl invariance implies r µ T µ ν = 0 ) T µ µ = 0 given quantum mechanically, there are anomalies…
T race Anomaly in 2d (no boundary) c h T µ µ i = 24 π R Ricci scalar curvature ✤ c is also the coefficient of the T µ n two-point function. � ✤ Zamolodchikov c -theorem (’86), � c UV > c IR ✤ Huerta-Casini (’04) entanglement entropy proof of c - theorem —towards a map of 2d QFTs
T race Anomaly in 4d (no boundary) 1 16 π 2 ( cW 2 � aE.D. + d ⇤ R ) h T µ µ i = scheme dependent, � Weyl curvature ignore Euler density ✤ c is also the coefficient of the T µ n two-point function. � ✤ Cardy (’88) a -conjecture and later Komargodski- Schwimmer proof (’11) � a UV > a IR ✤ Casini-Teste-Torroba (’17) entanglement entropy proof of a -theorem —towards a map of 4d QFTs
T race Anomaly in 6d (no boundary) µ i ⇠ a E.D. + c 1 W 3 + c 2 W 3 + c 3 W ⇤ W h T µ hints from supersymmetry of a 6d a -theorem
T race Anomaly with a Codimension One Boundary K AB extrinsic curvature hat on K removes trace c 2D h T µ µ i = 24 π ( R + 2 K δ ( x n )) Jensen-O’Bannon (’15) b -theorem a UV > a IR µ i = 1 4 π ( � aR + b tr ˆ 3D K 2 ) δ ( x n ) h T µ 1 16 π 2 ( cW 2 � aE.D. + ( � b 1 tr ˆ K 3 + b 2 K AB W nAnB ) δ ( x n )) h T µ 4D µ i = Solodukhin-Fursaev (’16) conjecture µ i ⇠ δ ( x ⊥ )( b 1 W 2 + b 2 K 4 + . . . ) h T µ b 2 = 8 c 5D µ i ⇠ a E.D. + c 1 W 3 + c 2 W 3 + c 3 W ⇤ W + δ ( x ⊥ )( b 1 KW 2 + . . . ) h T µ 6D
Today’s Talk Can we say anything more about no aR 3d yes b tr ˆ K 2 Related to displacement operator � yes b 1 tr ˆ K 3 two and three point functions 4d b 2 K AB W AnBn yes
Displacement Operator definition: � operator sourced by small � δ I δ x n ≡ D n � � � � � changes in the embedding D n diffeomorphism Ward identity: ∂ µ T µn = D n δ ( x n ) T nn ∂ µ T µA = 0 tangential components � still conserved pill box argument implies T nn ( ~ x, x n ) | x n =0 = D n ( ~ x )
x ) D n (0) i = c nn h D n ( ~ | ~ x | 2 d Results c nnn h D n ( ~ x ) D n ( ~ x 0 ) D n (0) i = x | d | ~ x 0 | d | ~ x 0 | d | ~ x � ~ Argument analogous to Osborn � and Petkou’s result for c in 4d. � aR ??? (Fails for a in 3d because R is � 3d topological.) b = π 2 b tr ˆ K 2 8 c nn b 1 = 2 π 3 plan: � b 1 tr ˆ K 3 35 c nnn •where do these results 4d come from? � b 2 = 2 π 4 •what about ? b 2 K AB W AnBn b 2 = 8 c 15 c nn
in 3d b tr ˆ K 2 The term in the trace anomaly can be produced from � an effective anomaly action in limit e goes to zero where � µ is a UV regulator: µ ✏ I ( b ) = b Z tr ˆ K 2 4 ⇡ ✏ @ M For small deviations from planarity K AB ≈ ∂ A ∂ B x n ⇒ scale dependence of displacement 2-pt function = x ) D n (0) i = b 4 ⇡ ⇤ 2 � ( ~ µ @ µ h D n ( ~ x ) x ) D n (0) i = c nn The short distance behavior of h D n ( ~ x | 6 | ~ x ) D n (0) i = c nn 512 ⇤ 3 (log µ 2 ~ x 2 ) 2 can be regulated by writing instead h D n ( ~ (Freedman, Johnson, Latorre ’92)
in 4d b 1 tr ˆ K 3 Analogous to b in 3d. Now we need to � match scale dependent contributions of µ ✏ b 1 Z I ( b 1 ) = K 3 tr ˆ 16 ⇡ 2 ✏ @ M and c nnn h D n ( ~ x ) D n ( ~ x 0 ) D n (0) i = | ~ x | d | ~ x 0 | d | ~ x � ~ x 0 | d scale dependence of 3-pt function � relies on recent work by � Bzowski, Skenderis, and McFadden ‘13
in 4d b 2 K AB W nAnB Could extract from h D n i W 6 =0 We’ll do something more roundabout. δ 2 I ( b 2 ) µ ✏ b 2 Z I ( b 2 ) = We will cancel KW δ g µ ν δ g λρ 16 ⇡ 2 ✏ @ M δ 2 I ( c ) c µ ✏ Z against a boundary term in I ( c ) = W 2 δ g µ ν δ g λρ 16 ⇡ 2 ✏ M An order of limits issue spoils the naive relation between b 2 and c .
Aside on Stress Tensor 2-point function with boundary ( x − x 0 ) 2 Depends on a function a ( v ) of a cross ratio v = ( x − x 0 ) 2 + 4 x n x 0 n : boundary limit v → 1 : coincident limit v → 0 1 ✓ 4 α v α )ˆ 3 I µ ν , ρσ + ( v 2 ∂ 2 h T µ ν ( x ) T ρσ ( x 0 ) i = I µ ν , ρσ ( x � x 0 ) 8 ◆ � 1 6( v ∂ v α )(ˆ β µ ν , ρσ + 7ˆ I µ ν , ρσ ) same tensor structure that appears in absence of a boundary ˆ ˆ precise form of and not so important β µ ν , ρσ I µ ν , ρσ essential point is that they can’t cancel the b 2 boundary term
b 2 K AB W nAnB in 4d (part 2) c µ ✏ Z I ( c ) = W 2 is valid in the coincident limit. 16 ⇡ 2 ✏ M To reproduce the scale dependence of the 2-pt function in � the boundary limit, , we should use this effective action � v → 1 with a different value of c , in order to match to a (1). δ 2 I ( b 2 ) δ 2 I ( c ) Canceling against δ g µ ν δ g λρ δ g µ ν δ g λρ then leads to a relation between a (1), i.e. c nn , and b 2 .
Checks for Free Fields Using heat kernel methods, a number of these charges were � computed for free fields in the late 80s and early 90s � (Melmed, Moss, Dowker, Schofield) and later revisited in the � last couple of years (Solodukhin, Fursaev, Jensen, Huang, CPH) . b s =0 = 1 2 = 1 b s = 1 64 (D or R) , 32 , = 2 = 2 = 2 = 16 s = 1 b s =0 35(D) , b s =0 7 , b s =1 45(R) , b 2 35 , 1 1 1 1 b 2 = 8 c The displacement operator correlation functions � yield the same results!
Why is for free fields? b 2 = 8 c : boundary limit v → 1 α ( v ) ∼ 1 + v 2 d For free theories : coincident limit v → 0 effect of image � theory without � points on other side � boundary of the boundary = ⇒ 2 α (0) = α (1) by the old Osborn-Petkou (’93) argument α (0) ∼ c
What about interactions? Wilson-Fisher fixed point for f 4 scalar field theory, starting in 4d McAvity and Osborn (’93, ’95) showed, both in the e expansion � and in a large N expansion that 2 α (0) 6 = α (1) Downside: We need to be in exactly 4d to connect � to b 2 , and in exactly 4d f 4 scalar field theory is free We need some more examples…
Mixed Dimensional QED Gorbar, Gusynin, Miransky ‘01 � S = − 1 Z Z d 3 x ( i ¯ d 4 x F µ ν F µ ν + ψ / D ψ ) S.-J. Rey ‘07 � 4 ∂ M M Kaplan, Lee, Son, Stephanov ‘09 � S. Teber ‘12 where D µ = r µ � igA µ boundary conditions: F nA = g ¯ ψγ A ψ •relation to graphene � • b -function for g vanishes perturbatively � •behavior under electric-magnetic duality (Son ’17) � •relation to large N f QED 3 (Kotikov-Teber ’13)
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