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
(Ex)
Basically, Null Energy Condition demands m2 and λ
are positive (= stability) and it shows　a = 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 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.

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