Holographic judgement on Scale vs Conformal invariance Yu Nakayama　（IPMU & Caltech） Field theory part: based on appendix of arxiv/1208.4726 Holographic part: many earlier papers of mine Japanese article will appear in JSPS periodical

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

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

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

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

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

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

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

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!

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

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…

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?

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?

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

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.

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

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

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??

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?

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

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!

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

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?