Scale vs Conformal invariance
from holographic approach
Yu Nakayama （IPMU & Caltech）
Scale invariance
= Conformal invariance?
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???
Scale invariance
Conformal invariance
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）
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
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
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
Part 1. From field theory
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
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…
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
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
Zamolodchikov-Polchinski
theorem (1988):
A scale invariant field theory is
conformal invariant in (1+1) d
when
1. It is unitary
2. It is Poincare invariant (causal)
3. It has a discrete spectrum
(4). Scale invariant current exists
(1+1) d proof
According to Zamolodchikov, we define
 C-theorem!
At RG fixed point,

, which means
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
Part 2. Holgraphic proof
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
Start from geometry
d+1 metric with d dim Poincare ＋
scale invariance automatically selects
AdSd+1 space
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
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 shows a = 0
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
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 nondecreasing theorem of black holes.
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
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 ?
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