Scale vs Conformal invariance from holographic approach Yu

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 2 D 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 • 　Ad. S/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. • 5 d 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 1. 　It is unitary 2. 　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 C-theorem! At RG fixed point, 　　　 , which means

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 (Ad. S space) Can be shown from Einstein eq + Null energy condition

Start from geometry d+1 metric with d dim Poincare ＋ scale invariance automatically selects Ad. Sd+1 space

Can matter break conformal? Non-trivial matter configuration may break Ad. S 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 m 2 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 Ad. S, the matter must be Ad. S 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 nondecreasing 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 Ad. S 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