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 Virial 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 scale 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 derivative 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 　It is unitary 　It is Poincare invariant (causal) 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 monotonically decreasing along RG-flow Komargodski and Schwimmer gave the physical proof in the flow between CFTs However, their proof does not apply when the fixed 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 configuration Automatically 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 (Ex) Basically, Null Energy Condition demands m2 and λ are 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 holds if 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 Discreteness of the spectrum 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