(In)Formal Higher Spin Theories Best Higher Spin Conference, UMONS - - PowerPoint PPT Presentation

(In)Formal Higher Spin Theories

Best Higher Spin Conference, UMONS Based on a paper with Alexei Sharapov Zhenya Skvortsov

LMU, Munich and Lebedev Institute, Moscow

April, 27, 2017

E.Skvortsov

Intro Comments

Higher-spin theories contain ∞-many of massless fields, whose AdS/CFT duals are higher-rank conserved tensors. The latter implies that the dual theory is free. Therefore, intuitively, knowledge of some free conformal field theory should be necessary and sufficient in order to build the dual higher-spin theory. But how? Historically, the first idea was to set up some deformation problem in the bulk (Noether procedure) and find interaction terms together with the deformation that respects them Another idea is reconstruction: write down the AdS vertices such that the CFT correlators computed via Witten diagrams coincide with the correlators of the required free dual CFT

E.Skvortsov

Main Messages

There is a deformation procedure that is orthogonal to the usual weak-field expansion: unfolding. It treats higher-spin symmetries in the exact way. However, locality is not under control. The initial data is some higher-spin algebra. We show that the first vertex is determined by a Hochschild cocycle of the higher-spin algebra. There are no higher-order obstructions and other vertices go for a ride The cocycle can be generated by embedding everything into a bigger algebra, which mathematically provides a resolution. The first vertex gives description of fluctuations over higher-spin background given by any flat connection; Equations for such fluctuations can be easily completed to full nonlinear equations (e.g., Vasiliev equations)

E.Skvortsov

Unfolding

Unfolded equations are ’de Rham=wedge product of fields’: dW A = QA(W ) , QA(W ) =

• k

QA

B1...BkW B1 ∧ ... ∧ W Bk

Formal consistency d2 = 0, d = dxm∂m leads to 0 ≡ ddW A = dQA(W ) = dW B ∧ − → ∂ QA(W ) ∂W B and we find Jacobi-like quadratic contraints QB ∧ − → ∂ QA(W ) ∂W B ≡ 0 Example: flat connection of some Lie algebra dω = 1

2[ω, ω]

E.Skvortsov

Q-manifold

Unfolding is equivalent to Q-manifold: super-manifold with an

• dd Q vector-field that is nilpotent, Q2 = 0.

coordinates : W A vector-field : Q = QA(W ) − → ∂ ∂W A . It is important to remember that formal consistency Q2 = 0 leads to formal gauge invariance of the equations; any PDE’s can be written in the unfolded form; it is unclear which unfolded equations correspond to well-defined PDE’s (very easy to give examples of unfolded equations that are meaningless once d is taken seriously); ....

E.Skvortsov

Initial Data

HS algebras can be defined in many ways: directly in AdS by trying to build an algebra on Killing tensors ∇mξm(s−1) = 0 or HS connection (Fradkin-Vasiliev); symmetries of free conformally-invariant equations, for example, φ = 0 (Eastwood); symmetries generated by higher-rank conserved tensors φ∂...∂φ that exist in free theories; U(so(d, 2))/I where I some ideal, e.g. the one that is annihilated by Singleton, φ = 0; quantization of the algebra of functions on the co-adjoint

• rbit that corresponds to the free field as a representation
• f so(d, 2);

In any case it is ∞-dim; associative; so(d, 2) is a subalgebra; there is a trace; there is an automorphism π(f ) that flips the sign of AdS-transvections

E.Skvortsov

Initial Data

The minimalistic data is ∞-dimensional associative algebra hs; equipped with an automorphism π (can be trivial); Physics/reality can add few more details: free conformal field origin of the algebra; π(f ) = f (L, −P), where P are AdS-transvections;

• scillator realization, for example the 4d bosonic algebra

is the (even subalgebra) of Weyl algebra A2, [qi, pj] = 1, i, j = 1, 2, i.e. even functions f (q, p) or differential

• perators in two variables;

Disclaimer: locality is not under control, but something will still survive (what?) — we call it formal higher-spin theories. All Vasiliev equations lead to formal theories.

E.Skvortsov

Deformation Problem

Natural object is a connection ω of hs Just the knowledge of δω = dξ − [ω, ξ]⋆ gives (some) cubic interactions (Fradkin, Vasiliev) The only unfolded equations that we can write down are dω = ω ⋆ ω If the HS algebra admits a deformation (for example, deformed

• scillators viewed from the vantage point of undeformed ones),

then we can add dω = ω ⋆ ω + Ψ(ω, ω) The consistency tells exactly that Ψ is a two-cocycle responsible for a deformation of hs. Full theory: Prokushkin-Vasiliev theory (later)

E.Skvortsov

Deformation Problem

The study of free HS fields suggests that one should start with dω = ω ⋆ ω dC = ω ⋆ C − C ⋆ π(ω) On AdS (any flat HS background) the equation describes pure gauge Fronsdal fields plus degrees of freedom hidden in C (Bargmann-Wigner equations, Weyl tensors, etc.) Now we can have other deformations dω = ω ⋆ ω + V(ω, ω, C) + O(C 2) dC = ω ⋆ C − C ⋆ π(ω) + O(C 2)

E.Skvortsov

Weak-field vs. Unfolding

AdS is a particular exact solution with ω = Ω ∈ so(d, 2) and C = 0, i.e. dΩ = Ω ⋆ Ω. ω = Ω + ω(1) + ω(2) + ... , C = 0 + C (1) + C (2) + ... , Free equations see a part of the first vertex

Dω(1) = V(Ω, Ω, C (1)) ,

• DC (1) = 0

Each unfolded vertex contributes to ∞-many orders in the weak-field expansion:

Dω(2) − V(Ω, Ω, C (2)) = ω(1) ⋆ ω(1) + V(Ω, ω(1), C (1)) + V(Ω, Ω, C (1), C (1)) ,

• DC (2) = ω(1) ⋆ C (1) − C (1) ⋆ π(ω(1)) + V(Ω, C (1), C (1)) .

Looks like expanding Riemann tensor near g (0), which generates ∞-many terms via (g (0) + g (1) + ...)−1

E.Skvortsov

Hochschild

We need to find at least the first vertex in dω = ω ⋆ ω + V(ω, ω, C) + O(C 2) dC = ω ⋆ C − C ⋆ π(ω) + O(C 2) Matrix extensions (global symmetry on the CFT side) allows

• ne to reduce wild Chevalley-Eilenberg problem to the

Hochschild one. The answer is V(ω, ω, C) = Φ(ω, ω) ⋆ π(C) Φ(a, b) is a Hochschild two-cocycle of the HS algebra: −a ⋆ Φ(b, c) + Φ(a ⋆ b, c) − Φ(a, b ⋆ c) + Φ(a, b) ⋆ π(c) = 0

E.Skvortsov

Hochschild Complex

Hochschild differential acts on functions of n-arguments. In

• ur case it takes values in the twisted-adjoint representation
• f the higher-spin algebra.

Z : δHZ = a ⋆ Z − Z ⋆ π(a) δHZ = 0 would give (anti)-center of the HS algebra. Possible redefinitions ω → ω + ωC are coboundaries: f (a) : δHf = a ⋆ f (b) − f (a ⋆ b) + f ⋆ π(c) The vertex is the two-cocycle: −a ⋆ Φ(b, c) + Φ(a ⋆ b, c) − Φ(a, b ⋆ c) + Φ(a, b) ⋆ π(c) = 0 The two-cocycle valued in the adjoint of the algebra is responsible for a deformation of ⋆-product, e.g. hs(λ): −a ⋆ Ψ(b, c) + Ψ(a ⋆ b, c) − Ψ(a, b ⋆ c) + Ψ(a, b) ⋆ c = 0

E.Skvortsov

Higher-Spins on Background of Their Own

The first vertex gives for free equations that describe free HS fields over any HS-flat background: dΩ = Ω ⋆ Ω , dω = Ω ⋆ ω + ω ⋆ Ω + Φ(Ω, Ω) ⋆ π(C) , dC = Ω ⋆ C − C ⋆ π(Ω) , The Hochschild term is missing in 3d, but flat connections give back holes, for example. Fronsdal fields transform under global HS symmetries as δω = ξ ⋆ ω − ω ⋆ ξ + Φ(ξ, Ω) ⋆ π(C) − Φ(Ω, ξ) ⋆ π(C) , δC = ξ ⋆ C + C ⋆ π(ξ) ,

E.Skvortsov

Formal HS Theories aka Vasiliev Equations

dω = ω ⋆ ω + Φ(ω, ω) ⋆ π(C) + O(C 2) dC = ω ⋆ C − C ⋆ π(ω) + O(C 2) Given a HS algebra, the first vertex is given by a Hochschild cocycle; It can be shown at least in some of the cases that there are no obstructions at higher orders: the first vertex should be completed by higher-order terms; f ∗ g = f · g + {f , g} + ... Still, how to construct higher-order terms? Resolution: bigger complex where the initial complex is a cohomology itself. Now the cohomology can be computed in different ways and some of them can be simpler.

E.Skvortsov

Vasiliev Resolution

Having the Hochschild cocycle in front of your eyes, it is easy to see that it can be generated as Φ(f , g) = f (y) ♥ δ−1g(y) ♥ δ−1Z

• z=0

where Z = κ = exp i[z · y]dz ∧ dz and f (y)♥V (y, z) = f (y) exp i[← − ∂ y · (− → ∂ y − − → ∂ z)]V (y, z) Homotopy operator of the de Rham complex δ−1 δ−1fα1...αk(z) = 1 tk−1dt zνfνα2...αk(zt) There is a deeper reason: bi-complex with differential= Hochschild + de Rham in z-space

E.Skvortsov

Vasiliev Resolution

Having the resolution in front of your eyes Φ(f , g) = f (y) ♥ δ−1g(y) ♥ δ−1κ

• z=0

it is easy to see that the equations come from δS = C ⋆ κ dz ∧ dz δW = DS DW = 0 where D = d + [ω, •] replaces the arguments of Φ. Here d + W + S is the connection in x-z-space Moral: Vasiliev equations provide the field-theoretical realization of the resolution of the Hochschild complex

E.Skvortsov

Vasiliev Resolution

• 1. δ-cohomology, z-independent; we are back to Hochschild;
• 2. Hochschild cohomology in y − z; then δ Nontrivial!

Z − Φ = (δ + D)(S + W ) δS = C ⋆ κ dz ∧ dz ≡ Z δW = DS DW = 0 where D = d + [ω, •] replaces the arguments of Φ. Here d + W + S is the connection in x-z-space Moral: Vasiliev equations provide the field-theoretical realization of the resolution of the Hochschild complex

E.Skvortsov

Prokushkin-Vasiliev Theory

Weyl algebra [yα, yβ] = 2iǫαβ admits no deformations. Adding

• ne Klein operator one does find a deformation

[yα, yβ] = 2iǫαβ {yα, k} = 0 k2 = 1 The full deformed version is known as deformed oscillators: [qα, qβ] = 2iǫαβ(1 + νK) {qα, K} = 0 K 2 = 1 Ignoring the matrix extension by Clifford algebra (which is always possible) we observe that the Prokushkin-Vasiliev theory accounts for a deformation induced by ν. A related statement proved in the Prokushkin-Vasiliev paper is that the theory can be reduced to dA = A ∗ A dC = A ∗ C − C ∗ A where ∗ is the product in the deformed oscillators algebra.

E.Skvortsov

Three-dimensions are Empty?

The starting point for the 3d theory should be dA± = A± ∗ A± dC = A+ ∗ C − C ∗ A− where A± are left/right connections of hs(λ). we do no expect any deformations linear in C because there are no Weyl tensors in 3d; we do not expect any deformations quadratic in C because all stress-tensors built out of the scalar field are formally exact in AdS. The scalar is hidden in C together with all its derivatives (Prokushkin-Vasiliev, Lucena-Gomez et al).

E.Skvortsov

Three-dimensions are Empty?

The starting point for the 3d theory should be dA± = A± ∗ A± + V(h, h, C, C) dC = A+ ∗ C − C ∗ A− where A± are left/right connections of hs(λ). we do no expect any deformations linear in C because there are no Weyl tensors in 3d; we do not expect any deformations quadratic in C because all stress-tensors built out of the scalar field are formally exact in AdS. The scalar is hidden in C together with all its derivatives (Prokushkin-Vasiliev, Lucena-Gomez et al). It seems that beyond the flat-connection level the non-triviality

• f 3d higher-spin theories is due to carefully imposing locality!

The only result right now is CS+current interactions

(Lucena-Gomez et dudes).

E.Skvortsov

Type-A in Any Dimension

All Fronsdal fields in any d makes use of the same cocycle, which is thanks to a specific realization of the HS algebra: [Y A

α , Y B β ] = 2iηABǫαβ

Splitting Y A

α = y a α, yα, it is important that the twist map π is

yα → −yα. Therefore, one can think of the d-dim Vasiliev equations as the 3d-ones with dependence on y a

α attached via

∞-dim matrix factors. It is important that the deformation respects the definition of the HS algebra as sp(2)-singlets modulo sp(2)-gauge transformations. Therefore, the deformation is due to the same A1-cocycle as in 3d and 4d. All Vasiliev-like equations are built upon one and the same A1-cocycle Φ.

E.Skvortsov

Recipe

In order to construct a formal higher-spin theory

1 find some realization of the relevant higher-spin algebra

(oscillators?);

2 find the Hochschild cocycle (it is easy to see whether it

exists or not). This gives description of fluctuations over HS-flat backgrounds;

3 guess some resolution (it relates cycles to cocycles); 4 now one describe fluctuations as

D2 = 0 DG = Z

5 moving Z to DD = Z identifies background with

fluctuations and gives the full nonlinear system

E.Skvortsov

Some Other Results

Feigin, Felder and Shoikhet constructed cocycles of the Weyl algebras An (the case of A1 is used in Vasiliev equations); in particular, there is a four-cocycle of A2, Φ(ω, ω, ω, ω), which may be of interest for the 4d HS theory; we found resolution for all of these cocycles: doubling trick works for all An; in general, the extension of Kontsevich formality theorem can be used to construct the relevant cocycles — explicit formulas via Shoikhet-Kontsevich graphs; general philosophy: nontrivial HS objects correspond to some cohomologies of HS algebras (these are easy to identify). Nonlinear completions of the deformations can be obtained via resolutions; uniqueness of the deformation in 4d (phase vs. CS-matter)

E.Skvortsov

Conclusions

Formal higher-spin theories (aka Vasiliev equations): based on resolution of the Hochschild complex of higher-spin algebras; Related problems: on-shell action, various invariants, e.g. non-linear completion of traces that are reproduce free CFT correlators; More generally: formal AdS/CFT by conjecturing some invariants to match the CFT correlators (no integrals over space-time can be taken right now); Would be nice to constrain everything by locality or at least understand the relation to the Noether procedure; It would interesting to understand the relation to formal

• structures. Higher-spin related systems lead to explicit

formulas for cocycles that are hard to obtain in general.

E.Skvortsov

Thank You!

E.Skvortsov

Simplest Hochschild Cocycle

All the Vasiliev equations take advantage of one and same cocycle of Weyl algebra A1, [yα, yβ] = 2iǫαβ. Star-product (f ⋆ g)(y) = exp i

• ǫαβ ∂1

α ∂2 β

• f (y + y1)g(y + y2)
• yi=0 .

Then the cocycle can be written as Φ(f , g) =

• ei[(p0·p1)(1−2u1)+(p0·p2)(1−2u2)+(p1·p2)(1+2u1−2u2)]

× (p1 · p2) f (y1)g(y2)

• yi=0

where the integral is over 2d simplex 0 ≤ u1 ≤ u2 ≤ 1; p0 = iy, pi = ∂i. It was found implicitly by Vasiliev, while cocycles for general An were computed by Feigin, Felder and Shoikhet using the formality theorems.

E.Skvortsov

Resolution of Cocycles

In the general case of An we have a functional of 2n arguments: Φ(f1, ..., f2n) = det |p1, ..., p2n|

• ei[
• 0≤i<j≤2n(1+2ui−2uj)(pi·pj)]

The doubling trick yα → yα, zα works and the system of equations that yield Φ is D2 = 0 DG = Z

• DB = 0

where Z = (B ⋆ κ) dz1 ∧ · · · ∧ dz2n, D = d + W + S and G is a set of forms in x − z of total degree 2n − 1. The case of A1 is special in that G ∼ W , S and we can write D2 = Z

• DB = 0

E.Skvortsov