Scale vs Conformal invariance from holographic approach

Yu Nakayama 　（ IPMU & Caltech ）

Scale invariance= Conformal invariance?

Scale = Conformal?

•QFTs and RG-groups are classified by scale invariant IR fixed point 　（ Wilson’s philosophy ）

• Conformal invariance gave a (complete?) classification of 2D critical phenomena

• But scale invariance does not imply conformal invariance???

Scale invariance

Conformal invariance

Scale = conformal?

•　 Scale invariance doe not imply confomal invariance ！

•　 A fundamental (unsolved) problem in QFT

•　 AdS/CFT

•　 To show them mathemtatically in lattice models is notoriously difficult 　（ cf Smirnov ）

In equations…

• Scale invariance

Trace of energy-momentum (EM) tensor is a divergence of a so-called Virial current

• Conformal invariance

• EM tensor can be improved to be traceless

Summary of what is known in field theory

• Proved in (1+1) d (Zamolodchikov Polchinski)

• In d+1 with d>3, a counterexample exists (pointed out by us)

• In d = 2,3, no proof or counter example

In today’s talk

• I’ll summarize what is known in field theories with recent developments.

• I’ll argue for the equivalence between scale and conformal from holography viewpoint

Part 1. From field theory

Free massless scalar field • Naïve Noether EM tensor is

• Trace is non-zero (in d ≠ 2)

but it is divergence of the Virial current by using EOM it is scale invariant• Furthermore it is conformal because the Vi

rial current is trivial• Indeed, improved EM tensor is

QCD with massless fermions• Quantum EM tensor in perturbatinon theory

• Banks-Zaks fixed point at two-loop

• It is conformal• In principle, beta function can be non-zero at sca

le invariant fixed point, but no non-trivial candidate for Virial current in perturbation theory

• But non-perturbatively, is it possible to have only scale invariance (without conformal)? No-one knows…

Maxwell theory in d > 4

• Scale invariance does NOT imply conformal invariance in d>4 dimension.

• 5d free Maxwell theory is an example (Nakayama et al, Jackiw and Pi)

– note ： assumption (4) in ZP is violated

• It is an isolated example because one cannot introduce non-trivial interaction

Maxwell theory in d > 4• EM tensor and Virial current

• EOM is used here• Virial current 　　　　 is not a de

rivative so one cannot improve EM tensor to be traceless

• Dilatation current is not gauge invariant, but the charge is gauge invariant

Zamolodchikov-Polchinski theorem (1988): A scale invariant field theory is conformal invariant in (1+1) d when

1.　 It is unitary2.　 It is Poincare invariant (causal) 3. It has a discrete spectrum(4). Scale invariant current exists

(1+1) d proof

According to Zamolodchikov, we define

At RG fixed point, 　　　 , which means

C-theorem!

a-theorem and ε- conjecture• conformal anomaly a in 4 dimension is monotoni

cally decreasing along RG-flow• Komargodski and Schwimmer gave the physical

proof in the flow between CFTs • However, their proof does not apply when the fix

ed points are scale invariant but not conformal invariant

• Technically, it is problematic when they argue that dilaton (compensator) decouples from the IR sector. We cannot circumvent it without assuming “scale = conformal”

• Looking forward to the complete proof in future

Part 2. Holgraphic proof

Hologrpahic claim

Scale invariant field configurationAutomatically invariant under the isometry of conformal transformation (AdS space)

Can be shown from Einstein eq + Null energy condition

Start from geometry

d+1 metric with d dim Poincare ＋ scale invariance automatically selects AdSd+1 space

Can matter break conformal?

Non-trivial matter configuration may break AdS isometry

Example 1: non-trivial vector field

Example 2: non-trivial d-1 form field

But such a non-trivial configuration violates Null Energy Condition

Null energy condition:

(Ex)

Basically, Null Energy Condition demands m2 and λare positive (= stability) and it shows 　 a = 0

More generically, strict null energy condition is sufficient to show

scale = conformal from holography

Null energy condition:

strict null energy condition claims the equality holdsif and only if the field configuration is trivial

• The trigial field configuration means that fields are invariant under the isometry group, which means that when the metric is AdS, the matter must be AdS isometric

On the assumptions• Poincare invariance 　

– Explicitly assumed in metric

• Discreteness of the spectrum– Number of fields in gravity are numerable

• Unitarity– Deeply related to null energy condition.

E.g. null energy condition gives a sufficient condition on the area non-decreasing theorem of black holes.

On the assumptions: strict NEC• In black hole holography

– NEC is a sufficient condition to prove area non-decreasing theorem for black hole horizon

– Black hole entropy is monotonically increasing• What does strict null energy condition

mean?– Nothing non-trivial happens when the black

hole entropy stays the same• No information encoded in “zero-energy

state”• Holographic c-theorem is derived from the

null energy condition

Summary• Scale ＝　 Conformal invariance ？

• Holography suggests the equivalence (but what happens in d>4?)

• Relation to c-theorem ？• Chiral scale vs conformal invariance

• Direct proof ? Counterexample ?

Holographic c-theorem• In AdS CFT radial direction = scale of RG-group

• A’(r) determines central charge of CFT • By using Einstein equation, A’ is given by

• Here we used null energy condition• In 1+1 dimension the last term is

　　 so strict null energy condition gives the complete understanding of field theory theorem