40 years of the Weyl anomaly M. J. Duff Physics Department Imperial College London CFT Beyond Two Dimensions Texas A&M March 2012 1 / 50
Abstract Classically, Weyl invariance S ( g , φ ) = S ( g ′ , φ ′ ) under φ ′ = Ω( x ) α φ g ′ µν ( x ) = Ω( x ) 2 g µν ( x ) implies g µν T µν = 0 But in the quantum theory g µν < T µν > � = 0 Over the period 1973-2012 this Weyl anomaly has found a variety of applications in quantum gravity, black hole physics, inflationary cosmology, string theory and statistical mechanics. 2 / 50
Recall flat space ancestry For spaces admitting conformal Killing vectors ξ i µ ( x ) µ = 2 ∇ µ ξ i ν + ∇ ν ξ i D g µν ∇ ρ ξ i ρ there is a classically conserved current J i ν = ξ i µ T µν For example SO ( D , 2 ) in flat Minkowski space But anomaly resides in the divergence of the dilatation current ∇ ν < J i ν > = 1 ρ g µν < T µν > � = 0 D ∇ ρ ξ i Coleman and Jackiw 1970 3 / 50
Timeline 1973 Discovery of the Weyl anomaly using dimensional regularization Capper and Duff 4 / 50
Timeline 1975 Supermultiplet of anomalies Ferrara and Zumino 5 / 50
Timeline 1976 Non-local effective lagrangian for trace anomalies Deser, Duff and Isham Zeta functions, heat kernels and anomalies Christensen Dowker Hawking 6 / 50
The heat kernel The one-loop effective action is given by S A = ln [ det ∆] − 1 / 2 where ∆ is a conformally invariant d-dimensional operator. Its kernel F ( x , y , ρ ) obeys the heat equation ∂ ∂ρ F ( x , y , ρ ) + ∆ F ( x , y , ρ ) = 0 with the initial conditions F ( x , y , 0 ) = δ ( x , y ) 7 / 50
The heat kernel One can express F as � e − ρ ∆ φ n ( x ) φ n ( y ) F ( x , y , ρ ) = n � e − ρλ n φ n ( x ) φ n ( y ) = n where φ n are the eigenfunctions of ∆ with eigenvalues λ n : ∆ φ n = λ n φ n normalized according to d d x √ g ( x ) φ n ( x ) φ m ( x ) = δ mn � 8 / 50
b 4 coefficients The action may thus be written as d ρ d d x ρ − 1 √ g ( x ) A ( x , ρ ) � S A = where A ( x , ρ ) = F ( x , x , ρ ) . A ( x , ρ ) obeys an asymptotic expansion, valid for small ρ , B n ( x ) ρ n − d � A ( x , ρ ) ∼ 2 n where d d x √ gb n ( x ) � B n = (1) 9 / 50
Zeta functions The Schwinger-DeWitt coefficients b n are scalar polynomials, which are of order n in derivatives of the metric. In d = 4, for example, when ∆ is the conformally invariant Laplacian acting on scalars: ∆ = − � + 1 6 R 1 2880 π 2 [ R µνρσ R µνρσ − R µν R µν + 30 � R ] b 4 = Furthermore, B 4 = n 0 + ζ ( 0 ) where n 0 is the number of zero modes and ζ ( s ) = Σ n λ − s n is defined only over the non-zero eigenvalues of ∆ . 10 / 50
Timeline 1977 CFTs and the a and c coefficients Duff Trace anomalies and the Hawking effect Christensen and Fulling 11 / 50
CFTs Weyl anomalies appear in their most pristine form when CFTs are coupled to an external gravitational field. In this case 1 A = g µν � T µν � = ( 4 π ) 2 ( cF − aG ) where F is the square of the Weyl tensor: F = C µνρσ C µνρσ = R µνρσ R µνρσ − 2 R µν R µν + 1 3 R 2 , G is proportional to the Euler density: G = R µνρσ R µνρσ − 4 R µν R µν + R 2 , Note no R 2 term. We ignore � R terms whose coefficient can be adjusted to any value by adding the finite counterterm d 4 x √ gR 2 . � 12 / 50
Central charges c and a In the CFT a and c are the central charges given in terms of the field content by ¯ a ≡ 720 a = 2 N 0 + 11 N 1 / 2 + 124 N 1 ¯ c ≡ 720 c = 6 N 0 + 18 N 1 / 2 + 72 N 1 where N s are the number of fields of spin s . In the notation of Duff 1977 ( 4 π ) 2 b ′ = − a ( 4 π ) 2 b = c 13 / 50
Euler number When F − G vanishes, anomaly reduces to 1 32 π 2 R ∗ µνρσ R ∗ µνρσ A = A where 360 A = ¯ c − ¯ a = 4 N 0 + 7 N 1 / 2 − 52 N 1 so that in Euclidean signature d 4 x √ gg µν T µν = A χ ( M 4 ) � where χ ( M 4 ) is the Euler number of spacetime. 14 / 50
Timeline 1978 Conformal (and axial) anomalies for arbitrary spin Christensen and Duff 15 / 50
Arbitrary spin Calculate b 4 for arbitrary ( n , m ) reps of Lorentz group, then physical anomaly given by A = A ( n , m ) + A ( n − 1 , m − 1 ) − 2 A ( n − 1 / 2 , m − 1 / 2 ) so in total A total = 4 N 0 + 7 N 1 / 2 − 52 N 1 − 233 N 3 / 2 + 848 N 2 where N s are the number of fields of spin s . The b 4 coefficient for chiral reps (1/2,0) (1,0) etc also involve R*R unless we add (0,1/2) (0,1) etc 16 / 50
1980 Anomaly-driven inflation Starobinsky Vilenkin p -forms and inequivalent anomalies Duff and van Nieuwenhuizen Grisaru et al Siegel The path-integral approach to anomalies Fujikawa Bastianelli and van Nieuwenhuizin 17 / 50
Timeline 1981 Critical dimensions for bosonic and super strings Polyakov 18 / 50
Bosonic string In the first quantized theory of the bosonic string, one starts with a Euclidean functional integral � D γ DX e − Γ = Vol ( Diff ) e − S [ γ, X ] where 1 � d 2 ξ √ γγ ij ∂ i X µ ∂ j X ν η µν S [ γ, X ] = 4 πα ′ As shown by Polyakov, the Weyl anomaly in the worldsheet stress tensor is given by 1 γ ij < T ij > = 24 π ( D − 26 ) R ( γ ) D is the contribution of the scalars while the − 26 arises from the diffeomorphism ghosts that must be introduced into the functional integral. 19 / 50
Fermionic string In the case of the fermionic string, the result is 1 γ ij < T ij > = 16 π ( D − 10 ) R ( γ ) Thus the critical dimensions D = 26 and D = 10 correspond to the preservation of the two dimensional Weyl invariance γ ij → Ω 2 ( ξ ) γ ij . 20 / 50
Timeline 1983 Conformal anomaly and W-Z consistency (no R 2 ) Bonora et al Anomaly in conformal supergravity Fradkin and Tseytlin 21 / 50
Timeline 1984 Local version of effective action Riegert 22 / 50
Local action Conformal operators √ g ∆ d = � g ′ ∆ ′ d ∆ 2 = � ∆ 4 ≡ � 2 + 2 R µν ∇ µ ∇ ν + 1 3 ( ∇ µ R ) ∇ µ − 2 3 R � Riegert Subsequent work by Antoniadis, Mazur and Mottola Local action d 4 x √ gF φ − b ′ S anom = b � � d 4 x √ g [ φ ∆ 4 φ − ( G − 2 3 � R ) φ ] 2 2 23 / 50
Timeline 1985 Spacetime Einstein equations from vanishing worldsheet anomalies Callan et al 24 / 50
Timeline 1986 The c -theorem Zamolodchikov 25 / 50
Timeline 1988 c -theorem and/or a -theorem in four dimensions? Cardy Osborn Capelli et al Shore Shapere Antoniadis et al 26 / 50
Timeline 1993 Geometric classification of conformal anomalies in arbitrary dimensions Deser and Schwimmer 27 / 50
Timeline 1998 The holographic Weyl anomaly Henningson and Skenderis Graham and Witten Imbimbo et al Einstein manifolds and the a and c coefficients Gubser 28 / 50
Holography A distinguished coordinate system, boundary at ρ = 0 G MN dx M dx N = L d + 12 ρ − 2 d ρ d ρ + ρ − 1 g µν dx µ dx ν 4 The effective action may be written d ρ d d x ρ − 1 √ g ( x ) B ( x , ρ ) � S B = where the specific form of B ( x , ρ ) depends on initial action. b n ( x ) ρ n − d � B ( x , ρ ) ∼ 2 n Formal similarity with Schwinger-DeWitt coefficients, indeed A ∼ b 4 same for N=4 Yang-Mills but not in general. 29 / 50
Timeline 2000 Anomaly-driven inflation revived Hawking et al a and c and corrections to Newton’s law Duff and Liu Anomalies and entropy bounds Nojiri et al 30 / 50
Corrections to Newton’s law In his 1972 PhD thesis under Abdus Salam, the author calculated one-loop CFT corrections to Newton’s law (Schwarzschild solution) � � V ( r ) = G 4 M 1 + α G 4 , r 2 r where G 4 is the four-dimensional Newton’s constant, � = c = 1 and α is a purely numerical coefficient, soon recognized as the c coefficient in the Weyl anomaly α = 8 3 π c 31 / 50
N=4 Yang-Mills A particularly important example of a CFT is provided by N = 4 super Yang-Mills with gauge group U ( N ) , for which ( N 1 , N 1 / 2 , N 0 ) = ( N 2 , 4 N 2 , 6 N 2 ) Then a = c = N 2 4 and hence N 2 c 2 R µν R µν − 2 R µν R µν − 1 � 3 R 2 � � 3 R 2 � A = = ( 4 π ) 2 32 π 2 The contribution of a single N = 4 U ( N ) Yang-Mills CFT is 1 + 2 N 2 G 4 V ( r ) = G 4 M � � . 3 π r 2 r 32 / 50
Randall-Sundrum Now fast-forward to 1999 when Randall and Sundrum proposed that our four-dimensional world is a 3-brane embedded in an infinite five-dimensional universe. They showed that there is an r − 3 correction coming from the massive Kaluza-Klein modes 1 + 2 L 52 V ( r ) = G 4 M � � . 3 r 2 r where L 5 is the radius of AdS 5 . Superficially, our 4D quantum correction seems unrelated to their 5D classical one. But through the miracle of AdS/CFT N 2 = π L 3 G 4 = 2 G 5 5 2 G 5 L 5 the two are in fact equivalent. Duff and Liu 33 / 50
Timeline 2001 a and c and the graviton mass Dilkes et al Weyl cohomology revisited Mazur and Mottola 34 / 50
Timeline 2005 Anomalies as an infra-red diagnostic; IR free or interacting? Intriligator 35 / 50
Timeline 2006 Macroscopic effects of the quantum trace anomaly Mottola et al 36 / 50
Timeline 2007 Anomalies and the hierarchy problem Meissner 37 / 50
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