Conformal Symmetry in Einstein-Cartan Gravity Lucat Stefano & - - PowerPoint PPT Presentation

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Conformal Symmetry in Einstein-Cartan Gravity

Lucat Stefano & Tomislav Prokopec ArXiv: 1606.02677 & 1709.00330

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Weyl symmetry and motivation

• Generalisation of scale (conformal) transformation:

gµν → Ω2(x)gµν

• Renormalisable theories posses an UV fixed point
• f RG flow, where the theory becomes conformally

invariant.

" r(µξν) = rλξλ D gµν #

• Weyl symmetry, in

arbitrary manifolds

• Conformal symmetry,

background dependent

βi = µ∂λi ∂µ

• λ∗

i

= 0

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The conformal (trace) anomaly

• If the theory is Weyl invariant, we have,

δS δgµν ωgµν = T µ

µ = 0 =

) hT µ

µ i = 0

• However, a background field analysis for arbitrary metric reveals

this is not realised in a renormalised field theory:

hT µ

µ i = C1E4 + C2WαβγδW αβγδ + C3⇤R +

X

i

βi ∂L ∂λi

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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.

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Geometrical properties

• Riemann curvature and

geodesics trajectories are frame invariant. Rλ

σµν → Rλ σµν Proper time reparametriz- ation

r ˙

γ ˙

γµ ! e−θ(x)r ˙

γ ˙

γµ

• Trajectories of free falling bodies

invariant up to a reparametrization

• f time.
• Absence of dimension-full

parameters requires dynamical Planck Mass.

Rµν − 1 2gµνR = κTµν

Tµν → e−(D−2)θ(x)Tµν

κ ≡ α Φ2(x) → e(D−2)θ(x) α Φ2(x)

dτg.i. = Φ × dτ

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Scale symmetry and dilatation current

• Scale invariant theory possess a

Noether charge, the dilatation current

• If scale invariance is exact on the

state of the field, the scale current is conserved and energy tensor is traceless

Πµ = −D − 2 2 φ∂µφ ∂µΠµ = 0

• If the theory is scale invariant, the

equation of motion imply

= ⇒ Πµ = T µ

ν xν if gµν = ηµν

T µ

µ = 0

T µ

µ = −∂µΠµ

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Interactions in scalar theory

• For scalars the dilatation current is:
• Idea: couple dilatation current to

torsion trace (and complete theory by requiring symmetry).

Πµ = 1 √−g δS δTµ

Tµν = 2 √−g δS δgµν

• Extension of gravitational field
• sources. Equation of motion imply

the fundamental equation:

rµΠµ + T µ

µ = 0

Πµ = D − 2 2 φ∂µφ = ⇒ ∂µΠµ = T µ

µ

Lint = TµΠµ + ✓D − 2 2 ◆2 TµT µφ2

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Weyl symmetry in the quantum theory(formally)

• Phase space quantisation is

manifestly Weyl invariant:

[, ⇡] = i~(D1)(~ x − ~ x0)

π = δS δ ˙ φ

• This means that the Weyl

symmetry Ward identities are preserved:

hrµ ˆ Πµi + h ˆ T µ

µi = 0

• Source dilatation current by

generating Energy momentum trace

• Identity “broken” by terms which

vanish upon regularisation, e.g.

(D 4)λhˆ φ4i

Z DD⇡ exp (iS[, ⇡]) = = Z D det

1 2

⇣√−gg00(D1) (~ x − ~ x0) ⌘ exp (iS[])

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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,

Θµ

µ =

X

i

Λi ∂L ∂Λi

Θµν = Tµν 1 D 1

• rµrν gµνr2

φ2

All dimension full couplings in the theory

• This work shows that our Ward identity is in fact satisfied in the full

quantum theory, at least in the flat space limit.

Θµ

µ = T µ µ + rµΠµ

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So what about local anomaly?

• Local anomaly action:

Local anomaly

• Including torsion trace this is

compensated, and does not violate the fundamental Ward identity.

T µ

µ + rµΠµ = 0

• This is because the Gauss Bonnet

integral is a boundary term, and gets absorbed in the divergence of the dilatation current.

h ˆ T µ

µ i = C

¯ R2 4 ¯ Rµν ¯ Rνµ + ¯ Rαβγδ ¯ Rγδαβ 6= 0 ¯ R2 4 ¯ Rµν ¯ Rνµ + ¯ Rαβγδ ¯ Rγδαβ = rµVµ = ) Πµ ! Πµ + Vµ

Seff = lim

D→4

Z dDx √−g D − 4 ¯ R2 − 4 ¯ Rµν ¯ Rνµ + ¯ Rµνλσ ¯ Rλσµν

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(Some) physical discussion

• Breaking of the Ward identity for

chiral transformations:

rµJµ

5 = rµh ¯

ψγ5γµψi 6= 0

Z

Σ

d~ x J0

5(~

x) = NF N ¯

F =

) d dt (NF N ¯

F ) 6= 0

• Means that the number of

fermions is not conserved anymore.

• 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…

Z

Σ

d~ x Π0(~ x) = Z

Σ

d~ x h{⇡(~ x), (~ x)}i = 1 2 Z d~ ph(ie2i!ta†

~ pa† −~ p ie−2i!ta~ pa−~ p)i

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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.

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Thanks for attention

Questions?