constraining theories with higher spin symmetry
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
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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.
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• 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 space
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• 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 …
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• 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”.
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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.
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• 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 Yin
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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.)
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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
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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.
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Unitarity bounds
• Scalar operator: Δ ≥ ½ (in d=3)
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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
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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 wealso have:
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• 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 !
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• 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.
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Twist one
• Now we have:
• Sum over S’’ has finite range• Some c’s are non-zero , e.g.
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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 spin- The theory is not necessarily a superposition of free bosons and
free fermions (think of s=2 !)
Giombi, Prakash, YinCosta, Penedones, Poland, Rychkov
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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 law
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• 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.
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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
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• 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.
Conclusions
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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, YinAharony, Gur-Ari, Yacoby
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Fermions + Chern Simons• Spectrum 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, YinAharony, Gur-Ari, Yacoby
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• Conclusion: All three point functions are
• Two parameter family of solutions
• We do not know the relation to the microscopic parameters N, k.
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• 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.
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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.
Future
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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.
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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.