Generalization of Vasiliev Higher Spin Theorywith an Action Principle

Ergin Sezgin

Texas A & M University

w/ Nicolas Boulanger and Per Sundell, arXiv:1505.04957

Higher Spin Workshop, Penn State, August 29, 2015

Motivations for this work

• How unique is Vasiliev’s theory?

Extensions beyond Chan-Paton factors, supersymmetry and parity breakinginteraction ambiguity with enlarged the HS algebra?

• Higher spin invariants of relevance to the computation of tree-level n-pointamplitudes have been constructed. However, considered as part of aneffective action, their coefficients are arbitrary.

Can an extension of Vasiliev theory remove this arbitrariness?

• Can prospects for construction of an action be improved?

We will see that there is a remarkable extension with desired properties, leadingto non-commutative Frobenius-Chern-Simons gauge theory.

Bosonic Vasiliev theory in a nutshell

• Formulated in terms of differential forms on M4 ×Z4 with coordinates(xµ, zα, zα) and exterior derivative d = dx + dz + dz where Zα = (zα, zα)are Grassmann even and non-commutative SL(2, C) spinors.

• In the case of the minimal bosonic models, consisting of integer spins, thehigher-spin representation content can be encoded using a second set ofGrassmann even non-commutative SL(2, C) spinors Y α = (yα, yα).

• Introduce one-form W (x, Z;Y ) and zero-form Φ(x, Z;Y ).

• In the normal ordering of the Weyl algebra

[Yα, Yβ ]? = −[Zα, Zβ ]? = 2iCαβ , [Yα, Zβ ]? = 0

with respect to Yα ± Zα, symbol composition reads

P (Z, Y ) ? Q(Z, Y ) =

∫U,V

P (Z + U, Y + U) Q(Z − V, Y + V ) e−iU·V

• Twisted central two-form

J = − i4

(dzαdzα κ+ dzαdzα κ

),

defined in terms of the inner Klein operator

κ = eiyαzα , κ ? κ = 1 .

• By its definition

dJ = 0 , J ? W = π(W ) ? J , J ? Φ = π(Φ) ? J ,

where the star product outer automorphisms

π(y, y, z, z) = (−y, y,−z, z) , d π = π d

• Real integer spin fields

ππ(W,Φ, J) = (W,Φ, J) , (W †,Φ†, J†) = (−W,π(Φ),−J)

• Vasiliev equations

dW +W ?W + Φ ? (J − J) = 0 ,

dΦ +W ? Φ− Φ ? π(W ) = 0 .

• Cartan integrable, i.e. consistent with d2 = 0, hence invariant under

δW = dε+ [W, ε]? , δΦ = −ε ? Φ + Φ ? π(ε) ,

which define adjoint and twisted-adjoint representations, respectively.

• Spin s > 1 gauge fields arise in W |Z=0 while massless scalar field andmassless spin s > 1 Weyl tensors are packed into in Φ|Z=y=0.

• The adjoint two-form Φ ? (J − J), which can be replaced by a moregeneral interaction ambiguity (provided parity symmetry is yielded),deforms the symplectic structure on Z4.

Questioning the rational behind J being non-dynamical is the keyconsideration for the generalization that we have found.

Extension of Vasiliev theory

• In addition to M4 ×Z4, introduce an auxiliary dimension, such that thetotal manifold on which the fields live is

M9 = X5 ×Z4

where ∂Z4 = ∅ and ∂X4 contains M4. [Boulanger and Sundell, 2011].

(In the spirit of Kazinski, Lyakhovich and Sharapov, 2005).

• All fields valued in associative higher spin algebra A extended by inner andouter Klein operators with trace TrA.

• The algebra A star commutes to the algebra Ω(Z4) of forms on Z4:

A ?O(Z4) = O(Z4) ?A .

• Comparison with Vasiliev’s model requires taking

πyπy(Φ,W ) = πzπz(Φ,W ) = (Φ,W ) ,

which can indeed be imposed perturbatively and for the exact solutions ofimportance.

Formalism capable of incorporating more general noncommutativebase manifolds.

Bifundamental master fields

• Introduce a zero-form B, a pair of one-forms (A, A) and a two-form B onM9 such that

Φ ∈ B|∂M9 , W ∈ 12(A+A)|∂M9 ,

Φ ? (J − J) ∈ (B ? B)|∂M9 .

• Let B transform in a bifundamental representation of A (more below)

δB = −ε ? B +B ? ε ,

• Let B transform in the opposite bifundamental representation

δB = −ε ? B + B ? ε .

Drastic reduction of the number of HS invariants.

• Choose boundary conditions on Z4 such that

B = f(B, J, J) ,

B plays the role of a dynamical counter part of J and J .

Bifundamental master field equations

Postulate the following Cartan integrable system:

dA+A ? A−B ? B = 0 ,

dA+ A ? A− B ? B = 0 ,

dB +A ? B −B ? A = 0 ,

dB + A ? B − B ? A = 0 .

Gauge transformations:

δA = dε+ [A, ε]? + η ? B +B ? η ,

δA = dε+ [A, ε]? + η ? B + B ? η ,

δB = dη +A ? η + η ? A− ε ? B +B ? ε ,

δB = dη + A ? η + η ? A− ε ? B + B ? ε .

Frobenius algebra

Finite-dimensional, unital, associative algebra A defined over a field k, andequipped with a non-degenerate bilinear form σ : A×A→ k that satisfies theequation σ(a · b, c) = σ(a, b · c). For example, any matrix algebra with thebilinear form σ(a, b) = tr (a · b) is a Frobenius algebra.

• Employ an eight-dimensional associative algebra F with elements(eij , h eij) where eij are 2× 2 matrices with nonvanishing entry 1 only atthe ith row and jth column, and h is an outer Kleinian element satisfying

[h, e11] = 0 = [h, e22] , h, e12 = 0 = h, e12

• Frobenius algebra trace operation is defined as

TrF∑i,j

eijMij(h) = M11(0) +M22(0)

Assembling the master fields as

X =∑i,j

Xijeij =

(A B

B A

)the field equations read

FX := dX + hXh ? X = 0 ,

or equivalently(hd)(hX) + (hX) ? (hX) = 0 .

Lagrange multipliers and higher forms

• Inside the bulk of M9, introduce Lagrange multiplier master fields

P =∑i,j

P ijeij =

(V U

U V

)

• Duality extended field content

deg(B,A, A, B) ∈ (2n, 1 + 2n, 1 + 2n, 2 + 2n)n=0,1,2,3 ,

deg(U , V, V , B) = (8− 2n, 7− 2n, 7− 2n, 6− 2n)n=0,1,2,3 .

Superconnection and Frobenius-Chern-Simons action

Assemble X and P into superconnection

Z = hX + P

Use the trace operation TrF⊗A (more on A and TrA below) to construct theaction

SFCS =

∫M9

TrF⊗A(

12Z ? qZ + 1

3Z ? Z ? Z

), q := hd .

Globally defined generalized Hamiltonian action

SH = SFCS −1

4

∫∂M9

TrF⊗A [hπh(Z) ? Z]

=

∫M9

TrF⊗A(P ? FX + 1

3P ? P ? P

),

where q := hd and πh(h) = −h, and assuming structure group is associated toX and and that P |∂M9 = 0 .

The component form of the action:

SH =

∫M9

TrA[V ?

(F +B ? B

)+ U ? DB + 1

3V ?3 − V ? U ? U

+V ?(F + B ? B

)+ U ? DB + 1

3V ?3 − V ? U ? U

],

• The terms cubic in Lagrange multipliers require the duality extended fieldcontent.

• Using the boundary condition P |∂M9 = 0, the field equation reduce to

dA+A ? A+B ? B = 0 etc but note that these are duality extended.

Base manifold

• The topology of Z4 may be chosen in a variety ways as to achieve classesof symbols with well-defined star products and finite integrals.

• We chooseZ4 = S2 × S2

obtained from R4 by adding suitable points at infinity to R4.

• For X5, one simple alternative is the semi-infinite cylinder

X5 = X4 × [0,∞) , ∂X4 = ∅ .

though multiple boundaries may of course ultimately be much moreinteresting.

The key point is that the topology of M9 is now part of the modulispace of the theory, and the Poisson structures on it is deformed by the

vacuum expectation values of B ? B and B ? B.

The higher spin algebra

A = Π+ ?⊕

m,m=0,1

Pm,m(y, y, ky, ky) ? (κy)m ? (κy)m ?Π+ ,

Π± =1

2(1± ky ? ky)

• Pm,m consist of arbitrary polynomials in oscillators (yα, yα) and outerKlein operators (ky, ky). (Will comment later on suitable non-polynomialextension).

• The Π+ projection removes odd polynomials in (y, y) leading to a modelconsisting (perturbatively) of Lorentz tensorial higher spin fields.

• Inner Klein operators for the Weyl algebra

κy = 2πδ2(y) κy = 2πδ2(y)

are required to generate the vacuum expectation values for B

• Composition rule

P1 ? P2 =

∫R4

d2ξd2ξd2ηd2η

(2π)4ei(η

αξα+ηαξα)×

× P1(y + ξ, y + ξ; ky, ky)P2(y + η, y + η; ky, ky)

realizes the Weyl algebra of the oscillator algebra

[yα, yβ ]? = 2iεαβ , [yα, yβ ]? = 2iεαβ ,

ky, yα? = 0 = ky, yα? , ky ? ky = ky ? ky = 1 .

in terms of Weyl ordered symbols.

Trace operations and variational principle

• The trace operation TrA is defined as

TrAP (y, y; ky, ky) = P00(0, 0; 0, 0)

using symbols defined in the Weyl order.

• Recall the Frobenius algebra trace operation

TrF∑i,j

eijMij(h) = M11(0) +M22(0)

• General variation of the action:

δS =

∫M9

TrF⊗A(δX ? hRPh+ δP ? RX + d(δX ? P )

)

Assuming that X is free to fluctuate at ∂M9, the variational principle thusimplies that

RX := dX + hXh ? X + P ? P = 0

RP := dP +X ? P + hPh ? X = 0

P |∂M9 = 0

The bulk action is invariant under εX transformation, while under εP

transformation we get:

δεP SH =

∫M9

TrF⊗A d(εP ? RX

)This gives the condition

εP |∂M9 = 0

Reduction of the system

In order to more fully exhibit the topological degrees of freedom contained inthe field B, we consider the following Ansatz for the duality extended systemon X4 ×Z4:

V = V = 0 = U = U , A = A = W ,

B =N∑r=0

∑m,m=0,1

Wr;m,m(B) ? (J)?m ? (J)?m ? J(r)X ,

where J and J are the closed and central forms on Z4 defined above, J(r)X is a

basis for the even-degree de Rham cohomology on X4, which are central andclosed elements, and Wr;m,m are star functions. We assume that

J(0)X = 1 , W0;0,0 = 0 .

Substituting the above Ansatz into the field equations, we obtain

FW = −N∑r=0

∑m,m=0,1

B ?Wr;m,m(B) ? (J)?m ? (J)?m ? J(r)X ,

DWB = 0 ,

where FW = dW +W ?W and DWB = dB +W ? B −B ?W .

Vasiliev’s recent proposal

Let us compare the above result, once written using normal order, with therecent proposal of Vasiliev in which the field equations, adapted to ournotation, take the form

FW = V(B) ? J − V(B) ? J + U(B) ? J ? J + gJ ? J + L ,

DWB = 0 ,

where the star functions are polynomials vanishing at B = 0, g is a constantand

L = L[2] + L[4] ,

belong to the de Rham cohomology on X4.

Adapting the reduced FCS system

Upon taking the only nonvanishing star functions to be Wr;0,0 (r > 1),W0;1,0(B), W0;0,1(B) and W0;1,1(B), and assuming that

W0;1,0(ν) = 0 ,

one can redefineB → ν +B ,

after which the reduced FCS equations take the form

FW = −(ν+B)?[W0;1,0(ν +B) ? J +W0;0,1(ν +B) ? J +W0;1,1(ν +B) ? J ? J

]−

N∑r=1

(ν +B) ?Wr;0,0(ν +B) ? J(r)X ,

DWB = 0 .

Comparison: Equations of motion

Thus, we identify the curvature deformations of Vasiliev’s recent duality systemas a subset of the deformations above, with

V(B) = −(ν +B) ?W0;1,0(ν +B) , V(B) = (ν +B) ?W0;0,1(ν +B) ,

U(B) = −(ν +B) ?W0;1,1(ν +B)− g , g = −νW0;1,1(0) ,

L = −ν∑r

Wr;0,0(ν)J(r)X .

Compared with Vasiliev’s recent system, the above system also contains anextra non-central deformation given by

−N∑r=1

(ν +B) ?Wr;0,0(ν +B) ? J(r)X + ν

∑r

Wr;0,0(ν)J(r)X ,

the consequences of which remain to be investigated.

Comparison: Central terms versus boundary deformations

• The integral of L[2] evaluated on the Didenko–Vasiliev black hole solutionhas been interpreted as a “black hole charge”, and the integral of L[4] hasbeen proposed as a candidate for the generating functional of correlatorsin the AdS4/CFT3 higher spin holography [Vasiliev’15].

• These interpretations, however, make use of non-standard gauge fixingprocedures in which the globally defined gauge parameters associated withthe de Rham cohomology elements are used to remove locally definedgauge fields, which is only possible if the latter belong to trivial bundles.

• Moreover, the direct connection between the charges of L and thepartition function remains unclear in the absence of a path integral.

Instead, one may accommodate similar charges in an on-shell action derivedfrom a path integral of AKSZ type, and without having to employ the gaugefixing procedure mentioned above, by adding Chern classes to the generalizedHamiltomian action:

Stot = SH +∑p,n

∫X2p×Z4

TrA(xnF?n + xnF

?n) ,

where X2p ⊆ X4 (possibly augmented by Chern-Simons forms at boundaries).

Zero-form charges versus new HS invariants

• In the standard Vasiliev theory, a special classes of invariants areconstructed as

I2n =

∫Z4

TrA(ky ? ky ? B?(2n) ? J ? J)|p0 ,

referred to as zero-form charges, as they are evaluated at a single point.

• These functionals have been shown [Colombo, Sundell, Didenko,Skvortsov] to be building blocks for higher spin amplitudes.

• They can be employed as deformations of a previously constructed[Boulanger and Sundell] Hamiltonian action for the original Vasilievsystem. However, each building block comes with an arbitrary coefficient.

• In the FCS model, we have instead only two invariants that yield zero-formcharges in Vasiliev branch. These are:∫

X0×Z4

TrA(B ? B)?n for n = 1, 2

The consequences for amplitude computation under investigation.

Comments on perturbative analysis

• Define W = (A+ A)/2 and K = (A− A)/2 and expand perturb aboutthe vacuum solution

B(0) = J − J , W (0) = L−1 ? dL , K(0) = 0 , B(0) = 0

we have shown that the theory does not give rise to any additional degreesof freedom under certain natural assumptions.

• Classical moduli arise in the solution for B, accounting for an interactionambiguity in Vasiliev’s theory.

Comments on nonpoynomial extension of A and one-body solutions

• The polynomial Pm,m we have in the HS algebra A can be extended byinclusion of suitable nonpolynomials such that the trace operation stillmakes sense.

• This extension is associated with the tensor product of singleton andanti-singleton representation and gives rise to Coulomb/Schwarzschild-likecharged objects (AdS stationary solutions). Iazeolla and Sundell’13]

• These degrees of freedom arise also in Vasiliev’s theory even though theyare usually not considered to be part of the perturbative spectrum.

Details and consequences are currently under investigation.

Open Problems

• Construction of the boundary deformations and techniques for theirregularization.

• Quantization of the FCS model using TFT/AKSZ methods.

• Computation of amplitudes and making contact with CS-matter systems.

• Making contact with topological open strings and string partons arising intensionless limit in AdS.