Holographic judgement on Scale vs Conformal invariance

Yu Nakayama　（ IPMU & Caltech）Field theory part: based on appendix of

arxiv/1208.4726Holographic part: many earlier papers of mineJapanese 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

Part 1. Field theory analysis

Scale vs ConformalProof in 4d CounterexampleWhich is correct?

Scale vs Conformal 101The response to Weyl transform in QFT

Require Weyl invariance:

Instead require only constant Weyl inv:

May be improved to be traceless when

Obviously conformal is stronger than scale inv

: virial current

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

Zamolodchikov-Polchinski theorem (1988): A scale invariant field theory is conformal invariant in (1+1) d when

1.　 It is unitary2.　 It is Poincare invariant (causal) 3. 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!

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”

• 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…

In early 90’s, the perturbative proof was done by Osborn though he did not emphasize it

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?

“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” contri

bution 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.

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 1Implement 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 2Reconsider 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

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?