Juan Maldacena Institute for Advanced Study Based on httparxivorgabs 11121016 amp to appear by J M and A Zhiboedov amp to appear Elementary particles can have spin ID: 192344
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Slide1
Constraining theories with higher spin symmetry
Juan MaldacenaInstitute for Advanced Study
Based on
http://arxiv.org/abs/
1112.1016
& to appear
by J. M. and A.
Zhiboedov
& to appear. Slide2
Elementary particles can have spin. Even massless particles can have spin.
Interactions of massless particles with spin are very highly constrained.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 .
Spin 1 = Yang Mills
Spin 2 = Gravity
Spin s>2 (higher spin) = No interacting theory in asymptotically flat spaceSlide3
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.
Witten
Sundborg
Sezgin
Sundell
Polyakov
–
Klebanov
Giombi
Yin …Slide4
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 S-matrix. We will also constrain theories where the higher spin symmetry is “slightly broken”.Slide5
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. Slide6
Vasiliev theory + boundary conditions that break the higher spin symmetry
Dual to the large N Wilson Fischer fixed point… 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.
Polyakov
–
Klebanov
Giombi
YinSlide7
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 unitaryWe have a conserved current of spin, s>2. We are in d=3(We have only one conserved current of spin 2.)Slide8
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 sectorSlide9
Outline
Unitarity bounds, higher spin currents.Simple argument for small dimension operatorsOutline of the full argumentThen: cases with slightly broken higher spin symmetry. Slide10
Unitarity bounds
Scalar operator: Δ ≥ ½ (in d=3) Slide11
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 =0Slide12
Removing operators in the twist gap
Scalars with 1 > Δ ≥ ½Assume 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
a
lso have:Slide13
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 !Slide14
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
. Slide15
Twist one
Now we have:
Sum over S’’ has finite rangeSome c’s are non-zero , e.g.Slide16
Structure of three point functions
Three point functions of three conserved currents are constrained to only three possible structures: - Bosons - Fermions - Odd (involves the epsilon symbol).
We have more than one because we have spinThe theory is not necessarily a superposition of free bosons and free fermions (think of s=2 !)
Giombi
,
Prakash
, Yin
Costa,
Penedones
, Poland,
RychkovSlide17
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 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.
Coefficients in
Transformation lawSlide18
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. Slide19
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
=NSlide20
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. ConclusionsSlide21
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 symmetryN 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,
YacobySlide22
Fermions + Chern
SimonsSpectrum of ``single trace’’ operators as in the free case. Violation of current conservation: (2pt fns
set to 1 ) Insert this into correlation functions
Breaks parity
Giombi
,
Minwalla
,
Prakash
,
Trivedi
,
Wadia
, Yin
Aharony
,
Gur
-Ari,
YacobySlide23
Conclusion: All three point functions are
Two parameter family of solutions We do not know the relation to the microscopic parameters N, k. Slide24
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. Slide25
Discussion
In principle, it could be extended to higher point functions… 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.
FutureSlide26
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. Slide27
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.