1. Holographic judgement on Scale vs Conformal invariance
Yu Nakayama　（IPMU & Caltech）
Field theory part: based on appendix of
arxiv/
Holographic part: many earlier papers of mine
Japanese article will appear in JSPS periodical

2. Q: Is N=4 SYM really conformal invariant?
It may be more non-trivial than you think…

3. A: Of course it is. Because it is dual to AdS5XS5
BEST ANSWER
A: Of course it is. Because it is dual to AdS5XS5

4. Part 1. Field theory analysis
Scale vs Conformal Proof in 4d Counterexample Which is correct?

5. The response to Weyl transform in QFT Require Weyl invariance:
Scale vs Conformal 101
The response to Weyl transform in QFT
Require Weyl invariance:
Instead require only constant Weyl inv:
May be improved to be traceless when
: virial current
Obviously conformal is stronger than scale inv

6. Simple counterexample: Maxwell theory
Consider U(1) free Maxwell theory in d >4.
EM tensor and Virial current
Virial current 　　　　 is not a derivative
 cannot improve EM tensor to be traceless
Does not satisfy conformal Ward identity
Dilatation current is not gauge invariant,
but charge is gauge invariant

7. A scale invariant field theory is conformal invariant in (1+1) d when
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

8. (1+1) d old proof According to Zamolodchikov, we define
At RG fixed point,　　　 , which means 
 C-theorem!

9. Theorem: (Reeh-Schlierder) Proof is highly non-trivial (Try it!)
Remark on the last line
In CFT, trivially true by state-operator correspondence. Is it true in general QFTs?
Unitarity tells
Theorem: (Reeh-Schlierder)
Proof is highly non-trivial (Try it!)
Causality is essential.
By the way, in chiral version of the “theorem” by H & S, they abused the theorem. They can never prove the above statement within their assumptions… Counterexamples do exist!

10. Modern compensator approach (Komargodski)
Start with the UV CFT perturbed by relevant deformations.
Add compensator (= dilaton) to preserve the conformal invariance
Compute the IR effective action
Conformal anomaly matching 
Kinetic term must be positive
The rate of the change is governed by
Scale  Conformal

11. d=4 (perturbative) old “Proof”
In early 90’s, the perturbative proof was done by Osborn though he did not emphasize it
Wess-Zumino consistency condition for RG-flow in curved background
One of the constraint (a-theorem?)
Assume a is constant at fixed point, scale  conformal?
Caveat: positivity of G is not shown (unlike d=2)
Perturbatively, it is positive though…

12. Compensator approach (Komargodski-Scwhimmer, Luty-Polchinski-Rattazi)
We start with the UV CFT perturbed by relevant deformations + dilaton compensation
Compute the IR effective action
Conformal anomaly matching 
Must be positive (due to causality/unitarity)
The rate of the change is governed by
When T is small: Scale  Conformal?

13. Non-trivial exists at three loops
“Counterexample”
Grinstein et al studied beta functions of gauge/fermions/bosons at three loops
Non-trivial exists at three loops
If , then it is scale invariant but non-conformal invariant!
The RG trajectory is cyclic
Perturbative. Clearly in contradiction.
What’s wrong? Which is correct??
Can we write by using EOM?

14. Resolution of the debate 1
How do we know ?
CS-eq does not tell total derivative part
Beta functions are ambiguous (“gauge choice”: cf scheme choice): gauge invariant B function
What we really have to show
Grinstein et al later computed
Lo and behold:
So their theory is actually conformal

15. Resolution of the debate 2
The “proof” part (LPR) was also too quick
They didn’t introduce “partial virial current” contribution explicitly.
Extra contribution from or ?
We can imagine their computation was in B-gauge (or unitary gauge in holography)
After careful reformulation of the problem, we can show
Still, perturbative proof, but contradiction gone.

16. Not enough to compute beta functions for conformal invariance
Lessons
Not enough to compute beta functions for conformal invariance
Can we compute B function directly?
With manifest SUSY preserving regularization, B = beta to all order in perturbation theory (Nakayama)
Scale = conformal in perturbation theory, but non-perturbative regime remains open

17. Part 2. Holographic verdict
Holographic c-theorem revisited Holographic realization of the debate and resolutions Scale  Conformal

18. Holographic c-theorem revisited
According to holographic c-theorem:
Null energy-condition leads to strong c-theorem
Suppose the matter is given by NSLM
Strict null energy-condition demands the positivity of the metric (unitarity needs positivity of the metric)
Is scale  conformal??

19. More precise version 1 Implement the operator identity:
Achieved by gauge transformation
In Poincare patch
We may have the non-zero beta functions (B gauge)
But this can be gauge equivalent to the vector condensation (virial gauge)
In both cases, the field configuration is scale inv but non-conformal (holographic cyclic RG). Is it possible?

20. More precise version 2 Reconsider holographic RG-flow
Null energy-condition leads to strong c-theorem
Matter is given by gauged NLSM
Assume positive metric from strict NEC
Scale inv but non-conformal config forbidden

21. Holographic verdict Rather trivial modification  gauging
Scale  Conformal under the same assumption
What was the confusion/debates then?
Suppose we start with manifestly conformal background
Gauge transform: apparently non-conformal but scale invariant (and it looks cyclic?)
Just a gauge artifact!

22. General theorem Scale inv  Conformal inv in Holography when
Gravity with full diff (cf Horava gravity)
Matter: must satisfy strict NEC
When saturate the NEC, matter must be trivial configuration
Sufficient condition for strong c-theorem
Sufficient to protect unitarity in NLSM
In any space-time dimension
Scale inv  Conformal inv in Holography

23. Lessons and outlooks
A lot of confusions in field theory, but crystal-clear in holography
Debate was gauge artifact (new ambiguity)
It seems converging to the point where holography predicts. I’m happy!
Beyond perturbation in 4d?
In other dimensions: 3d? 6d?
With defects and boundaries?