Constraining theories with higher
spin symmetry
Juan Maldacena
Institute for Advanced Study
Based on http://arxiv.org/abs/1112.1016 & to appear
by J. M. and A. Zhiboedov & to appear.
• Elementary particles can have spin.
• Even massless particles can have spin.
• Interactions of massless particles with spin are very
highly constrained.
Spin 1 = Yang Mills
Spin 2 = Gravity
Spin s>2 (higher spin) = No interacting theory in asymptotically flat space
• Coleman Mandula theorem : The flat space S-matrix
cannot have any extra spacetime symmetries beyond
the (super)poincare group. Needs an S-matrix. Higher
spin gauge symmetries  become extra global
symmetries of the S-matrix.
• Yes go: Vasiliev: Constructed interacting theories with
massless higher spin fields in AdS4 .
Witten Sundborg Sezgin Sundell
Polyakov – Klebanov Giombi Yin
…
• AdS4  dual to CFT3
• Massless fields with spin s ≥ 1  conserved
currents of spin s on the boundary.
• Conjectured CFT3 dual: N free fields in the
singlet sector
• This corresponds to the massless spins fields
in the bulk.
• 1/N = ħ = coupling of the bulk gravity theory.
The bulk theory is not free.
• What are the CFT’s with higher spin symmetry
(with higher spin currents) ?
• We will answer this question here:
• They are simply free field theories
• This is the analog of the Coleman Mandula
theorem for CFT’s, which do not have an Smatrix.
• We will also constrain theories where the
higher spin symmetry is “slightly broken”.
Why do we care ?
• This is an interesting phase of gravity, or spacetime.
• Any boundary CFT that has a weak coupling limit has a
higher spin conserved currents at zero coupling.
• In examples, such as N=4 SYM, this is smoothly connected
to the phase where the higher spin fields are massive.
Presumably by some sort of Higgs mechanism.
• In weakly coupled string theory, at high energies, we expect
to have higher spin ``almost massless’’ fields. So it is
interesting to understand the implications of this
spontaneously broken symmetry.
• We will not address these more interesting questions here.
We will just address the more restricted question posed in
the previous slide.
• Vasiliev theory + boundary conditions that break
the higher spin symmetry  Dual to the large N
Wilson Fischer fixed point…
Polyakov – Klebanov
Giombi Yin
• Two approaches to CFT’s :
- Write Lagrangian and solve it in perturbation
theory
- Bootstrap: Use the symmetries to constrain the
answer. Works nicely when we have a lot of
symmetry.
• We can also view this as constraining the
asymptotic form of the no boundary
wavefunction of the universe in AdS.
Assumptions
• We have a CFT obeying all the usual
assumptions: Locality, OPE, existence of the
stress tensor with a finite two point function,
etc.
• The theory is unitary
• We have a conserved current of spin, s>2.
• We are in d=3
• (We have only one conserved current of spin
2.)
Conclusions
• There is an infinite number of higher spin
currents, with even spin, appearing in the OPE of
two stress tensors.
• All correlators of these currents have two
possible forms:
• 1) Those of N free bosons in the singlet sector
• 2) Those of N free fermions in the singlet sector
Outline
• Unitarity bounds, higher spin currents.
• Simple argument for small dimension
operators
• Outline of the full argument
• Then: cases with slightly broken higher spin
symmetry.
Unitarity bounds
• Scalar operator: Δ ≥ ½ (in d=3)
Bounds for operators with spin
• Operator with spin s
. (Symmetric traceless
indices)
• Bound: Twist = Δ -s ≥ 1 .
• If Twist =1 , then the current is conserved
• We consider minus components only:
Spin s-1 , Twist =0
Removing operators in the twist gap
• Scalars with 1 > Δ ≥ ½
• Assume we have a current of spin four.
• The charge acting on the operator can only
give (same twist  only scalars )
• Charge conservation on the four point
function implies (in Fourier space)
Of course we
also have:
• This implies that the momenta are equal in pairs
 the four point function factorizes into a
product of two point functions.
• We can now look at the OPE as 1  2 , and we
see that the stress tensor can appear only if Δ=½ .
• So we have a free field !
• Intuition: Transformation = momentum
dependent translation  momenta need to be
equal in pairs. Same reason we get the Coleman
Mandula theorem !
• Observations:
• We need to constrain both the correlators and
the action of the higher spin symmetry. Of
course three point functions determine the
action of the symmetry.
• We used twist conservation and unitarity to
constrain the action of the generator.
• Then we used this to constrain the correlators.
Twist one
• Now we have:
• Sum over S’’ has finite range
• Some c’s are non-zero , e.g.
Structure of three point functions
• Three point functions of three conserved currents
are constrained to only three possible structures:
- Bosons
Giombi, Prakash, Yin
- Fermions
Costa, Penedones, Poland, Rychkov
- Odd (involves the epsilon symbol).
- We have more than one because we have spin
- The theory is not necessarily a superposition of free bosons
and free fermions (think of s=2 !)
Brute Force method
• Acting with the higher spin charge, and writing the
most general action of this higher spin charge we get a
linear combination of the rough form
Coefficients in
Transformation law
• The three point functions are constrained to three
possible forms by conformal symmetry  lead to a
large number of equations that typically fix many of
the relative coefficients of various terms.
• The equations separate into three sets, one for the
bosons part, one for the fermion part and one for the
odd part.
• In this way one constrains the transformation
laws.
• Then one constrains the four point function.
• Same as in a theory with N bosons or
fermions. One can also show that N is an
integer.
Quantization of Ñ, or the coupling in
Vasiliev’s theory
• We can show that the single remaining
parameter, call it Ñ, is an integer.
• It is simpler for the free fermion theory
• It has a twist two scalar operator
• Consider the two point function of
• If Ñ is not an integer some of these are
negative.
• So Ñ=N
Conclusions
• Thus, we have proven the conclusion of our
statement. Proved the Klebanov-Polyakov
conjecture (without ever saying what the Vasiliev
theory is !).
• Generalizations:
- More than one conserved spin two current 
expect the product of free theories (we did the case
of two)
- Higher dimension.
Almost conserved higher spin currents
• There are interesting theories where the
conserved currents are conserved up to 1/N
corrections.
• Vasiliev’s theory with bounday conditions that
break the higher spin symmetry
• N fields coupled to an O(N) chern simons
gauge field at level k.
• ‘t Hooft-like coupling
Giombi, Minwalla, Prakash,
Trivedi, Wadia, Yin
Aharony, Gur-Ari, Yacoby
Giombi, Minwalla, Prakash,
Trivedi, Wadia, Yin
Aharony, Gur-Ari, Yacoby
Fermions + Chern Simons
(6.20), (6.14). All t hat remains is t he int egral of t he right hand side of (7.1). In order for
t his t o vanish, we need t hat
2
• Spectrum of ``singlea trace’’
operators as in the(7.3)
= − a .
5
free case.
So t his relat ive coefficient is fixed in t his simple way, for all λ, t o leading order in 1/ N .
This
a somewhat t rivial
result since it
also follows from demanding
hattot he
• isViolation
of current
conservation:
(2pt fnstset
1 ) special
2
1
conformal generat or K − annihilat es t he right hand side of (7.1). We have spelled it out in
order t o illust rat e t he use of t he broken symmet ry.
Breaks parity
As a less t rivial example, consider t he insert ion of t he same broken charge conservat ion
ident ity in t he t hree point funct ion of t he st ress t ensor. We will do t his t o leading order
• Insert this into correlation functions
in λ. We get
i
Si
a1
j −n − − j 2 (x 1 )j 2 (x 2 )j 2 (x 3 ) ∼ √
N
d3 x [∂ ˜j 0 j 2 −
2˜
j 0 ∂j 2 ](x)j 2 (x 1 )j 2 (x 2 )j 2 (x 3 ) .
5
(7.4)
Now let ’s t ake t he large N limit in t his equat ion. In t he left hand side we can subst it ut e
t he act ion of t he charges on each of t he operat ors. This gives
• Conclusion: All three point functions are
• Two parameter family of solutions
• We do not know the relation to the
microscopic parameters N, k.
• As
we can rescale the operator
and we get the large N limit of the Wilson
Fischer fixed point.
• The operator
becomes the operator
which has dimension two (as opposed to the
free field value of one). It also becomes parity
even.
Discussion
• In principle, it could be extended to higher point functions…
Future
• It is interesting to consider theories which have other ``single trace”
operators (twist 3) that can appear in the right hand side of the
divergence of the currents. (e.g. Chern Simons plus adjoint fields).
• These are Vasiliev theories + matter.
• What are the constraints on “matter’’ theory added to a system
with higher spin symmetry?. Conjecture : String theory-like.
• Of course, this will be an alternative way of doing usual
perturbation theory. The advantage is that one deals only with
gauge invariant quantities.
Conclusions
• Proved the analog of Coleman Mandula for
CFT’s. Higher spin symmetry  Free theories.
• Used it to constrain Vasiliev-like theories
• A similar method constrains theories with a
higher spin symmetry violated at order 1/N.
A final conjecture
• Assume that we have a theory in flat space
with a weakly coupled S-matrix.
• The the theory contains massive higher spin
fields , s > 2 .
• The tree level S-matrix does is well behaved at
high energies.
• Then it should be a kind of string theory.