Conformal Symmetry in Einstein-Cartan Gravity Lucat Stefano & Tomislav Prokopec ArXiv: 1606.02677 & 1709.00330 1
Weyl symmetry and motivation • Renormalisable theories posses an UV fixed point of RG flow, where the theory becomes conformally invariant. � β i = µ ∂λ i � = 0 � ∂ µ � � λ ∗ i • Generalisation of scale (conformal) transformation: " # r ( µ ξ ν ) = r λ ξ λ g µ ν → Ω 2 ( x ) g µ ν g µ ν D • Conformal symmetry, • Weyl symmetry, in background dependent arbitrary manifolds 1
The conformal (trace) anomaly • If the theory is Weyl invariant, we have, δ S δ g µ ν ω g µ ν = T µ ) h T µ µ i = 0 µ = 0 = • However, a background field analysis for arbitrary metric reveals this is not realised in a renormalised field theory: ∂ L µ i = C 1 E 4 + C 2 W αβγδ W αβγδ + C 3 ⇤ R + X h T µ β i ∂λ i i 1
The link between torsion and Weyl symmetry • Why should torsion be linked to Weyl symmetry? • The torsion trace is naturally linked to scale transformations. • Transforming torsion and vierbein leaves the Cartan connection invariant. 1
Geometrical properties Proper time R λ σ µ ν → R λ reparametriz- σ µ ν • Riemann curvature and ation geodesics trajectories γ µ ! e − θ ( x ) r ˙ γ µ r ˙ γ ˙ γ ˙ are frame invariant. d τ g . i . = Φ × d τ R µ ν − 1 2 g µ ν R = κ T µ ν • Trajectories of free falling bodies invariant up to a reparametrization of time. T µ ν → e − ( D − 2) θ ( x ) T µ ν • Absence of dimension-full α α Φ 2 ( x ) → e ( D − 2) θ ( x ) parameters requires dynamical κ ≡ Φ 2 ( x ) Planck Mass. 1
Scale symmetry and dilatation current • Scale invariant theory possess a Π µ = − D − 2 φ∂ µ φ Noether charge, the dilatation 2 current • If scale invariance is exact on the ∂ µ Π µ = 0 state of the field, the scale current is conserved and energy tensor is traceless T µ µ = 0 • If the theory is scale invariant, the equation of motion imply ⇒ Π µ = T µ ν x ν if g µ ν = η µ ν = T µ µ = − ∂ µ Π µ 1
Interactions in scalar theory Π µ = D − 2 • For scalars the dilatation current is: ⇒ ∂ µ Π µ = T µ φ∂ µ φ = µ 2 • Idea: couple dilatation current to ◆ 2 ✓ D − 2 L int = T µ Π µ + T µ T µ φ 2 torsion trace (and complete theory 2 by requiring symmetry). • Extension of gravitational field δ S 1 δ S 2 Π µ = T µ ν = √− g √− g sources. Equation of motion imply δ T µ δ g µ ν the fundamental equation: r µ Π µ + T µ µ = 0 1
Weyl symmetry in the quantum theory(formally) π = δ S • Phase space quantisation is [ � , ⇡ ] = i ~ � ( D � 1) ( ~ x 0 ) x − ~ δ ˙ manifestly Weyl invariant: φ • This means that the Weyl Z D � D ⇡ exp ( iS [ � , ⇡ ]) = symmetry Ward identities are preserved: Z ⇣ √− gg 00 � ( D � 1) ( ~ ⌘ 1 x 0 ) = D � det exp ( iS [ � ]) x − ~ 2 hr µ ˆ Π µ i + h ˆ µ i = 0 T µ • Source dilatation current by generating Energy momentum trace • Identity “broken” by terms which ( D � 4) λ h ˆ φ 4 i vanish upon regularisation, e.g. 1
Can we then show this in a renormalised field theory? • Callan et al. showed that it is possible, in any generic renormalisable field theory, to construct a energy momentum tensor whose trace satisfies, ∂ L All dimension X Θ µ Θ µ µ = T µ µ + r µ Π µ Λ i full couplings µ = ∂ Λ i in the theory i 1 r µ r ν � g µ ν r 2 � φ 2 � Θ µ ν = T µ ν � D � 1 • This work shows that our Ward identity is in fact satisfied in the full quantum theory, at least in the flat space limit. 1
So what about local anomaly? √− g � ¯ Z R ν µ + ¯ R 2 − 4 ¯ R µ ν ¯ R µ νλσ ¯ d D x R λσ µ ν � • Local anomaly action: S e ff = lim D − 4 D → 4 � ¯ R 2 � 4 ¯ R ν µ + ¯ h ˆ R µ ν ¯ R αβγδ ¯ R γδαβ � µ i = C T µ 6 = 0 Local anomaly • Including torsion trace this is µ + r µ Π µ = 0 T µ compensated, and does not violate the fundamental Ward identity. R 2 � 4 ¯ R ν µ + ¯ R γδαβ = r µ V µ • This is because the Gauss Bonnet ¯ R µ ν ¯ R αβγδ ¯ ) Π µ ! Π µ + V µ integral is a boundary term, and = gets absorbed in the divergence of the dilatation current. 1
(Some) physical discussion • Breaking of the Ward identity for 5 = r µ h ¯ ψγ 5 γ µ ψ i 6 = 0 r µ J µ chiral transformations: • Means that the number of d Z x J 0 d ~ 5 ( ~ x ) = N F � N ¯ F = d t ( N F � N ¯ F ) 6 = 0 ) fermions is not conserved Σ anymore. x ) } i = 1 Z Z Z p h ( ie 2 i ! t a † p a † p � ie − 2 i ! t a ~ x Π 0 ( ~ d ~ x ) = d ~ x h { ⇡ ( ~ x ) , � ( ~ d ~ p ) i p a − ~ ~ − ~ 2 Σ Σ • Measures somehow the mixing of the state. If anomaly gets generated this is not conserved anymore. What exactly this means we still do not know… 1
Summary • We constructed a theory of gravity and torsion which is locally Weyl invariant. • Formal arguments led us to propose that the trace anomaly is actually just a manifestation of sourcing the dilatations current, but does not actually break the local symmetry, just the global (scale symmetry) part. • This solves the local anomaly, and predicts that only explicit violations of the Weyl symmetry result in violations of its Ward identities. • We think this might indicate that the torsion has a role to play in UV completion of gravity, if such a theory can be described by a curved spaces CFT. 1
Thanks for attention Questions? 1
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