Transcript ppt file
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