Conformal Anomalies and Gravitational Waves Hermann Nicolai MPI f - PowerPoint PPT Presentation

bla Conformal Anomalies and Gravitational Waves Hermann Nicolai MPI f¨ ur Gravitationsphysik, Potsdam (Albert Einstein Institut) Work based on: K. Meissner and H.N.: arXiv:1607.07312 H. Godazgar, K. Meissner and H.N.: in progress

Executive Summary Is the cancellation of conformal anomalies required • Quantum mechanically: to ensure quantum consis- tency of perturbative quantum gravity? ... in analogy with cancellation of gauge anomalies for Standard Model (where they are required to maintain renormalizability), and/or • already at classical level: corrections from induced anomalous non-local action to Einstein Field Equa- tions may potentially overwhelm smallness of Planck scale ℓ PL ⇒ huge corrections to any solution? If so, cancellation requirement could lead to very strong restrictions on admissible theories!

Conformal Symmetry Conformal symmetry comes in two versions: 1. Global conformal symmetry = extension of Poincar´ e group by dilatations D and conformal boosts K µ 2. Local dilatations = Weyl transformations g µν ( x ) → e 2 σ ( x ) g µν ( x ) Important consequence: flat space limit of Weyl and diffeomorphism invariant theories exhibits full (global) conformal symmetry (via conformal Killing vectors) → important restrictions on effective actions Γ = Γ[ g ] .

Conformal Anomaly ≡ Trace Anomaly Conformal anomaly ( ≡ trace anomaly) [Deser,Duff,Isham(1976)] µ ( x ) = a E 2 ( x ) ≡ aR ( x ) T µ ( D = 2) µ ( x ) = A ( x ) ≡ a E 4 ( x ) + c C µνρσ C µνρσ ( x ) ( D = 4) T µ where E 4 ( x ) ≡ Euler number density E 4 ≡ R µνρσ R µνρσ − 4 R µν R µν + R 2 C µνρσ C µνρσ = R µνρσ R µνρσ − 2 R µν R µν + 1 3 R 2 Coefficients c s and a s for fields of spin s (with s ≤ 2 ) were computed already long ago. [Duff(1977);Christenses,Duff(1978);Fradkin,Tseytlin(1982); Tseytlin(2013); see also: Eguchi,Gilkey,Hanson, Phys.Rep.66(1980)213]

Anomalous Effective Action Anomaly can be obtained by varying anomalous effec- tive action Γ anom = Γ anom [ g ] 2 g µν ( x ) δ Γ anom [ g ] A ( x ) = − � δg µν ( x ) − g ( x ) but this effective action is necessarily non-local . Simplest example: string theory in non-critical dimen- sion has a trace anomaly T µµ ∝ R ⇒ leads to anoma- lous effective action = Liouville theory. [Polyakov(1981)] d 2 x √− gR � − 1 � Γ D =2 anom ∝ g R • new propagating degree of freedom (longitudinal mode of world sheet metric = Liouville field) ⇒ changes physics in dramatic ways!

Analog for gravity in D = 4 : non-local actions that give anomaly exactly are known, for instance [Riegert(1984)] � E 4 − 2 � � E 4 − 2 � � � � d 4 xd 4 y ( x ) G P ( x, y ) Γ anom [ g ] = − g ( x ) − g ( y ) 3 � g R 3 � g R ( y ) with △ P G P ( x ) = δ (4) ( x ) , and the 4th order operator � R µν − 1 � △ P ≡ � g � g + 2 ∇ µ 3 g µν R ∇ ν However, no closed form actions are known that have the correct conformal properties (as would be obtained from Feynman diagrams), despite many efforts. [Deser,Schwimmer(1993);Erdmenger,Osborn(1998);Deser(2000);Barvinsky et al.(1998); Mazur,Mottola(2001);...] In lowest order d 4 x √− g E 4 � − 1 � Γ D =4 anom ∝ g R + · · · where · · · stands for infinitely many (non-local) terms.

While 2 g µν ( x ) δ Γ anom [ g ] A ( x ) = − � δg µν ( x ) − g ( x ) is local, contribution to Einstein equations R µν − 1 2 δ Γ anom [ g ] � � ℓ − 2 2 g µν R = − δg µν ( x ) + · · · PL � − g ( x ) in general remains non-local for non-scalar modes. non-localities from � − 1 Claim: in Γ anom [ g ] can ‘over- g whelm’ smallness of Planck scale and produce observ- able deviations for Einstein’s equations! Typical correction is (symmetrized traceless part of) G ret ⋆ E 4 G ret ⋆ R � � � � ∝ ∇ µ ∇ ν + · · · with retarded propagator G ret in space-time background given by metric g µν solving classical Einstein equations.

For order of magnitude estimate, evaluate this integral for a (conformally flat) cosmological background ds 2 = a ( η ) 2 ( − dη 2 + d x 2 ) by integrating from end of radiation era ( = t rad ) back to t 0 = n ∗ ℓ PL , with a ( η ) = η/ (2 t rad ) and η = 2 √ tt rad and with retarded Green’s function [Waylen(1978)] 4 π | x − y | · δ ( η − η ′ − | x − y | ) 1 G ret ( η, x ; η ′ , y ) = a ( η ) a ( η ′ ) Resulting correction on r.h.s. of Einstein’s equations ∼ 10 − 5 t − 1 T anom rad ( n ∗ ℓ PL ) − 3 00 ‘beats’ factor ∼ ( t rad ℓ PL ) − 2 on l.h.s. even for n ∗ ∼ 10 8 ! Similar results from evaluating contribution of Riegert action → could be a generic phenomenon, and thus af- fect any solution of Einstein equations. [Godazgar,Meissner,HN]

Cancelling conformal anomalies massless massive c s a s c s ¯ ¯ a s 3 2 ( 3 − 1 2 ( 179 3 − 1 0 ( 0 ∗ ) 2 ) 2 ) 2 ( ∅ ) 2 ( ∅ ) 1 9 − 11 9 − 11 2 2 4 2 4 39 − 63 1 18 − 31 2 2 3 − 411 589 289 − 201 2 2 4 2 1605 − 1205 2 783 − 571 2 2 • ¯ c s and ¯ a s include lower helicities: ¯ c 1 = c 1 + c 0 , etc. • Gravitinos and supergravity needed for cancellation • No cancellation possible for N ≤ 4 supergravities

NB: gravitino contribution may evade positivity prop- erties because there does not exist a gauge invariant traceless energy momentum tensor for s = 3 2 . [A.Schwimmer] c 2 + 5 c 3 2 + 10 c 1 + 11 c 1 2 + 10 c 0 = 0 ( N = 5) c 2 + 6 c 3 2 + 16 c 1 + 26 c 1 2 + 30 c 0 = 0 ( N = 6) c 2 + 8 c 3 2 + 28 c 1 + 56 c 1 2 + 70 c 0 = 0 ( N = 8) Old result: combined contribution � s ( c s + a s ) vanishes for all N ≥ 3 theories with appropriate choice of field representations for spin zero fields [Townsend,HN(1981)] . Thus: conformal anomalies for � s a s and � s c s cancel only for N ≥ 5 supergravities! [Meissner,HN] ... as they do for ‘composite’ U(5), U(6) and SU(8) R-symmetry anomalies. [Marcus(1985)] Implications for finiteness of N ≥ 5 supergravities? [Cf. Carrasco,Kallosh,Roiban,Tseytlin(2013);Bern,Davies,Dennen(2014)]

Idem for D=11 SUGRA compactified AdS 4 × S 7 SO (8) representations [ n +2 0 0 0] , [ n 0 2 0] , [ n − 2 2 0 0] , 0 [ n − 2 0 0 2] , [ n − 2 0 0 0] [ n +1 0 1 0] , n − 1 1 1 0] , 1 2 [ n − 2 1 0 1] , [ n − 2 0 0 1] 1 [ n 1 0 0] , [ n − 1 0 1 1] , [ n − 2 1 0 0] 3 [ n 0 0 1] , [ n − 1 0 1 0] 2 2 [ n 0 0 0] ‘Floor-by-floor’ cancellation [Cf.Gibbons,HN(1985)] : for all n c 2 f 2 ( n ) + ¯ ¯ c 3 2 f 3 2 ( n ) + ¯ c 1 f 1 ( n ) + ¯ c 1 2 f 1 2 ( n ) + ¯ c 0 f 0 ( n ) = 0 where f s ( n ) ≡ � (dimensions of SO(8) spin- s irreps) at Kaluza-Klein level n (no anomalies for odd D ).

Conceptual Issues Why worry about conformal anomalies in theories that are not even classically conformally invariant? HOWEVER: recall axial anomaly and anomalous con- servation of axial current ψγ 5 ψ + α 8 πF µν ˜ µ = 2 im ¯ ∂ µ J A F µν → anomaly is crucial even in presence of explicitly broken axial symmetry ( m � = 0 ) . Idem for gauge anomalies in Standard Model: these must cancel even when quarks and leptons acquire masses via spontaneous symmetry breaking. Is there a hidden conformal structure behind N ≥ 5 supergravities (and M Theory)? But cannot be con- formal supergravity in any conventional sense...

Outlook V. Mukhanov: “You cannot figure out the fundamental theory by simply looking at the sky!” But maybe there is a way...