Scale vs Conformal invariance from holographic approach

1. Scale vs Conformal invariance from holographic approach Yu Nakayama　（IPMU & Caltech）

2. Scale invariance= Conformal invariance?

3. 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???

4. Scale invariance

5. Conformal invariance

6. 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）

7. 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

8. 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

9. 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

10. Part 1. From field theory

11. 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

12. 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…

13. 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

14. 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

15. 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

16. (1+1) d proof According to Zamolodchikov, we define At RG fixed point,, which means   C-theorem!

17. 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

18. Part 2. Holgraphic proof

19. 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

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

21. 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

22. But such a non-trivial configuration violates Null Energy Condition Null energy condition: (Ex) Basically, Null Energy Condition demands m2 and λ are positive (= stability) and it showsa = 0

23. 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

24. 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.

25. 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

26. 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 ?

27. 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