Boundary T race Anomalies and Boundary Conformal Field Theory - - PowerPoint PPT Presentation
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
August 9, 2017
Christopher Herzog (YITP, Stony Brook University)
v = (x − x0)2 + (y − y0)2 (x − x0)2 + (y + y0)2
K.-W. Huang and CPH arXiv:1707.06224 Huang, Jensen, and CPH to appear
✤ Why bCFT? ✤ Review of trace anomalies and especially boundary
✤ New results: relations between boundary trace anomalies
✤ Demonstration that the b2 charge can depend on marginal
A B
A B
01
✤ 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
✤ 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
✤ 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
01
✤ In field theory, entanglement is
spatial regions, leading to the importance of the “entangling surface”
✤ 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.
✤ Surprisingly unexplored. McAvity and Osborn ’93
✤ Flat space: A planar boundary breaks the SO(d,2)
✤ Curved space: Require that the boundary and
classically, Weyl invariance implies ) T µ
µ = 0
given rµT µν = 0 Jµ = xνT νµ; rµJµ = 0 quantum mechanically, there are anomalies… Today’s talk: To understand new terms in the trace anomaly associated with the presence of a boundary. A quick review:
✤ c is also the coefficient of the Tµn two-point function. ✤ Zamolodchikov c-theorem (’86), ✤ Huerta-Casini (’04) entanglement entropy proof of c-
hT µ
µi =
c 24π R cUV > cIR —towards a map of 2d QFTs Ricci scalar curvature
✤ c is also the coefficient of the Tµn two-point function. ✤ Cardy (’88) a-conjecture and later Komargodski-
✤ Casini-Teste-Torroba (’17) entanglement entropy proof
—towards a map of 4d QFTs hT µ
µi =
1 16π2 (cW 2 aE.D. + d⇤R) Euler density scheme dependent, ignore aUV > aIR Weyl curvature
hT µ
µi ⇠ a E.D. + c1W 3 + c2W 3 + c3W⇤W
hints from supersymmetry of a 6d a-theorem
2D 4D 6D 3D 5D hT µ
µi ⇠ δ(x⊥)(b1W 2 + b2K4 + . . .)
hT µ
µi ⇠ a E.D. + c1W 3 + c2W 3 + c3W⇤W + δ(x⊥)(b1KW 2 + . . .)
Jensen-O’Bannon (’15) b-theorem Solodukhin-Fursaev (’16) conjecture
hT µ
µi =
c 24π (R + 2Kδ(xn)) hT µ
µi = 1
4π (aR + b tr ˆ K2)δ(xn) hT µ
µi =
1 16π2 (cW 2 aE.D. + (b1 tr ˆ K3 + b2KABWnAnB)δ(xn)) b2 = 8c aUV > aIR KAB extrinsic curvature hat on K removes trace
Can we say anything more about 3d 4d aR b tr ˆ K2 b1 tr ˆ K3 b2KABWAnBn no yes yes yes Related to displacement operator two and three point functions
definition: operator sourced by small changes in the embedding δI δxn ≡ Dn diffeomorphism Ward identity: ∂µT µn = Dnδ(xn) ∂µT µA = 0 tangential components still conserved
pill box argument implies
3d 4d aR b tr ˆ K2 b1 tr ˆ K3 b2KABWAnBn ??? b = π2 8 cnn hDn(~ x)Dn(0)i = cnn |~ x|2d hDn(~ x)Dn(~ x0)Dn(0)i = cnnn |~ x|d|~ x0|d|~ x ~ x0|d b1 = 2π3 35 cnnn b2 = 2π4 15 cnn Argument analogous to Osborn and Petkou’s result for c in 4d. (Fails for a in 3d because R is topological.) plan:
come from?
b2 = 8c
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: I(b) = b 4⇡ µ✏ ✏ Z
@M
tr ˆ K2 For small deviations from planarity KAB ≈ ∂A∂Bxn = ⇒ scale dependence of displacement 2-pt function µ@µhDn(~ x)Dn(0)i = b 4⇡ ⇤2(~ x) hDn(~ x)Dn(0)i = cnn |~ x|6 hDn(~ x)Dn(0)i = cnn 512⇤3(log µ2~ x2)2 The short distance behavior of can be regulated by writing instead
(Freedman, Johnson, Latorre ’92)
I(b1) = b1 16⇡2 µ✏ ✏ Z
@M
tr ˆ K3 hDn(~ x)Dn(~ x0)Dn(0)i = cnnn |~ x|d|~ x0|d|~ x ~ x0|d Analogous to b in 3d. Now we need to match scale dependent contributions of and scale dependence of 3-pt function relies on recent work by Bzowski, Skenderis, and McFadden ‘13
Could extract from hDniW 6=0 We’ll do something more roundabout. I(c) = c 16⇡2 µ✏ ✏ Z
M
W 2 We will cancel δ2I(b2) δgµνδgλρ δ2I(c) δgµνδgλρ against a boundary term in I(b2) = b2 16⇡2 µ✏ ✏ Z
@M
KW An order of limits issue spoils the naive relation between b2 and c.
Depends on a function a(v) of a cross ratio v = (x − x0)2 (x − x0)2 + 4xnx0n hTµν(x)Tρσ(x0)i = 1 (x x0)8 ✓4α 3 Iµν,ρσ + (v2∂2
vα)ˆ
Iµν,ρσ 1 6(v∂vα)(ˆ βµν,ρσ + 7ˆ Iµν,ρσ) ◆ same tensor structure that appears in absence of a boundary ˆ Iµν,ρσ ˆ βµν,ρσ precise form of and not so important essential point is that they can’t cancel the b2 boundary term v → 1 v → 0 : boundary limit : coincident limit
I(c) = c 16⇡2 µ✏ ✏ Z
M
W 2 is valid in the coincident limit. To reproduce the scale dependence of the 2-pt function in the boundary limit, , we should use this effective action with a different value of c, in order to match to a(1). v → 1 δ2I(b2) δgµνδgλρ δ2I(c) δgµνδgλρ Canceling against then leads to a relation between a(1), i.e. cnn, and b2.
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). bs=0 = 1 64 (D or R) , bs= 1
2 = 1
32 , b2 = 8c bs=0
1
= 2 35(D) , bs=0
1
= 2 45(R) , b
s= 1
2
1
= 2 7 , bs=1
1
= 16 35 , The displacement operator correlation functions yield the same results!
For free theories theory without boundary effect of image points on other side
= ⇒ 2α(0) = α(1)
v → 1 v → 0 : boundary limit : coincident limit by the old Osborn-Petkou (’93) argument α(0) ∼ c α(v) ∼ 1 + v2d
Wilson-Fisher fixed point for f4 scalar field theory, starting in 4d McAvity and Osborn (’93, ’95) showed, both in the e expansion and in a large N expansion that 2α(0) 6= α(1) Downside: We need to be in exactly 4d to connect to b2, and in exactly 4d f4 scalar field theory is free We need some more examples…
S = −1 4 Z
M
d4x F µνFµν + Z
∂M
d3x(i ¯ ψ / Dψ) FnA = g ¯ ψγAψ Dµ = rµ igAµ where boundary conditions:
Gorbar, Gusynin, Miransky ‘01 S.-J. Rey ‘07 Kaplan, Lee, Son, Stephanov ‘09
S = −1 4 Z
M
d4x F µνFµν + Z
∂M
d3x(i ¯ ψ / Dψ) FnA = g ¯ ψγAψ Dµ = rµ igAµ where boundary conditions:
Gorbar, Gusynin, Miransky ‘01 S.-J. Rey ‘07 Kaplan, Lee, Son, Stephanov ‘09
The usual Ward identity for QED relates Zψ = Zg The superficial degree of divergence of the photon self energy is one (compared with two in four dimensional QED). The gauge invariant prefactor pµpν − δµνp2 Πµν(p)
cuts down the degree of divergence to -1. In other words, Zγ is finite. coupling is not perturbatively renormalized. = ⇒
hAA(x)AB(x0)i = AB ✓ 1 (x x0)2 + 1 (~ x ~ x0)2 + (y + y0)2 ◆ In standard Feynman gauge, for Maxwell theory with a boundary The coupling to the boundary produces a small change (Neumann boundary condition for tangential components.) hAA(x)AB(x0)i = AB ✓ 1 (x x0)2 + 1 O(g2) (~ x ~ x0)2 + (y + y0)2 ◆ = ⇒ α(1) = α(0)(2 − O(g2)) For the stress tensor two point function
a(1) and hence b2 depended on the exactly marginal coupling. Unlike the situation for the bulk charges a and c in 4d. Wess-Zumino consistency forces a to be constant along marginal directions. No such argument for c. However, SUSY fixes c to be a constant, and it’s unknown how to construct 4d CFTs with marginal directions but without SUSY.
✤ Related boundary central charges in three and four
✤ Argued that in 4d, the relation b2 = 8c is special to free
✤ Discussed mixed dimensional QED as an example
b tr ˆ K2 b1 tr ˆ K3 b2KABWAnBn
✤ Verify our proposal for b2 by computing it directly, for mixed QED in
a curved space-time.
✤ Higher codimension defects. (Billo, Goncalves et al. ’16) ✤ Find bounds on these boundary central charges. (Hofman-Maldacena ‘08) ✤ Computation of these central charges in AdS/CFT, for example in
Janus solutions. (Takayanagi ’11, Miao et al., Astaneh et al. ’17)
✤ Models with only boundary interactions, like mixed QED. ✤ Boundary bootstrap. (Liendo et al. ‘12)
✤ Constrain QFT by constraining CFTs ✤ Provide a more local view of QFT by figuring out how
bubuzuke
A B ` SE = ↵Area(@A) d−2 + . . . + 4a(−1)d/2 ln ` + . . . UV cutoff
(Solodukhin 2008; Casini-Huerta-Myers 2011)
ball entanglement entropy and a ρA = trB ρ SE = − tr(ρA log ρA) where and H = HA × HB critically, assumes factorization of Hilbert space: a more local way to thinking about physics
result fails for a Maxwell field in 4d — Hilbert space doesn’t factorize across the boundary of A (Trivedi, Wall, Donnelly, Huang, Dowker,…) failure of perturbative corrections to SE for free scalar (Herzog, Nishioka) Resolution: Free scalar requires an extra boundary term. SE = ↵Area(@A) d−2 + . . . + 4a(−1)d/2 ln ` + . . .
✤ small mass ✤ small temperature ✤ anomalous dimensions of twist operators