Secrets of Higher-Spin Theories

M.A.Vasiliev

Lebedev Institute, Moscow

arXiv:1804.06520

Gelfond, MV arXiv:1805.11941 and 1904.?????

Didenko, Gelfond, Koribut, MV arXiv:1807.00001 and 1904.?????

Quantum Gravity in Paris

IHP, Paris, April 17, 2019

1-

Plan

I Introduction

II Nonlinear HS theories and spin-locality

III Coxeter HS theories

Coxeter groups and Cherednik algebras

Coxeter HS theories

IV Conclusions

2

Part I

Introduction

3

Quantum Gravity Challenge

QG effects should matter at ultra-high energies of Planck scale

mP ∼ 1019GeV

A distinguished theoretical possibility is to conjecture that the regime

of ultra high (transPlanckian) energies exhibits some high symmetries

that are spontaneously broken at low energies

Idea: to understand what kind of higher symmetries can be

introduced in relativistic theory and to see consequences

HS gauge theory:

theory of higher symmetries consistent with unitary QFT

Must be beautiful

and can affect fundamental concepts of gravity and quantum mechanics

4

Fronsdal Fields

Fronsdal fields 1978

All m = 0 HS fields are gauge fields

ϕn1...ns is a rank s symmetric tensor obeying ϕkkmmn5...ns = 0

Gauge transformation:

δϕn1...ns = ∂(n1εn2...ns)

, εmmn3...ns−1 = 0

In 60-70th it was argued (Weinberg, Coleman-Mandula) that

HS symmetries cannot be realized in a nontrivial local field theory in

Minkowski space

Green light: AdS background with Λ 6= 0 Fradkin, MV, 1987

In agreement with no-go statements the limit Λ→ 0 is singular

5

Historical Comments

In eighties-nineties people were claiming that

• HS theories with massless fields carrying spins s > 2 do not exist

• Peculiar background: why AdS?

• Why infinite-dimensional symmetries?

• etc

Majority did not care

The situation has changed with the AdS/CFT

Sundborg; Witten; Konstein, MV, Zaikin; Sezgin, Sundell

6

AdS/CFT Correspondence

That HS gauge theory is formulated in AdS4 fits naturally AdS/CFT

Klebanov and Polyakov conjecture (2002):

AdS4 HS theory is dual to 3d vectorial conformal σ-models of ϕi, ψiα.

Checked Giombi and Yin (2009) and extended to Chern-Simons BCFT

Aharony, Gur-Ari, Yacoby and Giombi, Minwalla, Prakash, Trivedi, Wadia, Yin (2011)

SB =k

4πSCS+

1

2

∫d3xDnφiD

nφi , SF =k

4πSCS+

∫d3xψiγ

nDnψi , i = 1, . . . N

3d bosonization as a consequence of duality

ϕ =π

2λB , ϕ =

π

2(1− λF ) (η = exp iϕ) λ :=

N

k

Unexpected possibility of lab tests of Quantum Gravity

AdS3/CFT2 HS correspondence Gaberdiel and Gopakumar (2010)

7

HS Gauge Theory Versus String Theory

HS theories: Λ 6= 0, m = 0, symmetric fields s = 0,1,2, . . .∞

First Regge trajectory

String Theory: Λ = 0, m 6= 0 except for a few zero modes

Infinite set of Regge trajectories

What is a HS symmetry of a string-like extension of HS theory?

MV 2012, 2018, Gaberdiel and Gopakumar 2014-2018

String Theory as spontaneously broken HS theory?! (s > 2,m > 0)

AdS/CFT Boundary interpretation

HS: φi(x) Vector fields

ST: Aij(x) Matrices

Tensor models ?!: φijk...(x) Tensors

8

Properties of HS Algebras

Global symmetry of symmetric vacuum of HS theory Fradkin, MV 1986

Let Ts be a homogeneous polynomial of degree 2(s− 1) in yα , ∂∂yα

[Ts1 , Ts2] = Ts1+s2−2 + Ts1+s2−4 + . . .+ T|s1−s2|+2

Once spin s > 2 appears, the HS algebra contains an infinite tower of

higher spins: [Ts, Ts] gives rise to T2s−2 as well as T2 of o(3,2) ∼ sp(4).

Since HS symmetries do not commute with space-time symmetries

[Tn , THS] = THS , [Tnm , THS] = THS =⇒ δHSϕnm ∼ ϕHS

Riemann geometry is not appropriate in HS theory

9

Non-Locality of HS Gauge Theory

HS interactions contain higher derivatives:

A.Bengtsson, I.Bengtsson, Brink (1983); Berends, Burgers, van Dam (1984)

infinite towers of spins imply infinite towers of derivatives

How (non)local is HS gauge theory?

The mildest possible nonlocality: each vertex with fields of definite spins

is local. I do not yet claim that this happens in all orders but all vertices

we have found so far up to the quintic order are spin-local: local in the

spinor space.

The worst: complete mess?!

Deja vu with seventies-eighties

10

Most Urgent Problems

Appropriate scheme leading to (minimally nonlocal) choice of variables.

Didenko, Gelfond, Korybut, MV 2016 - 2019

String-like and Tensor-like HS theory!?

MV 1804.06520; Degtev, MV 1905...

11

Unfolded dynamics

First-order form of differential equations

qi(t) = ϕi(q(t)) initial values: qi(t0)

Unfolded dynamics: multidimensional covariant generalization

∂

∂t→ d , qi(t)→WΩ(x) = dxn1 ∧ . . . ∧ dxnpWn1...np(x)

dWΩ(x) = GΩ(W(x)) , d = dxn∂n MV 1988

GΩ(W ) : function of “supercoordinates” WΦ

GΩ(W ) =∞∑n=1

fΩΦ1...ΦnW

Φ1 ∧ . . . ∧WΦn

d > 1: Nontrivial compatibility conditions

GΦ(W ) ∧∂GΩ(W )

∂WΦ≡ 0

Holographic duality between theories in different dimensions:

universal unfolded system admits different space-time interpretations.

12

Cohomological Interpretation

Group-theoretically, the problem is reformulated in terms of Chevalley-

Eilenberg (Hockshield) cohomology of a HS algebra h with coefficients

in certain h-modules C. MV 2007

Difficult problem: find an h-module C such that it

(i) contains necessary field representations like graviton and massless

fields

(ii) allows a deformation in all orders, i.e. full nonlinear theory.

What is String-like HS Theory?

13

HS Algebra and Modules

Free field analysis: realization of the HS algebra as Weyl algebra

[yα , yβ]∗ = 2iεαβ , [yα , yβ]∗ = 2iεαβ Fradkin, MV 1987

AdS4 algebra sp(4) ∼ o(3,2)

Naive way to extend the spectrum of fields

yα → ynα

does not lead to physically acceptable HS theories

Let hs1 be a HS algebra with the single set of oscillators

The Fock hs1-module F1 describes free boundary conformal fields

D|0〉 = h1|0〉

The lowest weight representations of the naively extended algebras hsp

built from p copies of oscillators have too high weights

14

Framed Oscillator Algebras

The problem is resolved in the framed oscillator algebras

replacing usual oscillator algebra with the single unit element I

[ynα , ymβ ]∗ = 2iδnmεαβI ,

by

[ynα , ymβ ]∗ = 2iδnmεαβIn

”Units” In are assigned to each specie of the oscillators forming a set of

commutative central idempotents (tensor product of the Weyl algebras)

IiIj = IjIi , IiIi = Ii

This allows us to consider Fock modules Fi obeying

IjFi = δijFi

equivalent to those of the single-oscillator case:

massless HS fields are in the spectrum

Idea from boundary OPE algebra in twistor variables: Gelfond, MV 2015

15

Part II

Nonlinear Higher-Spin Equations

and

Spin-Locality

16

Central On-Shell Theorem

Infinite set of integer spins

ω(y, y | x) , C(y, y | x) f(y, y) =∑∞n,m=0

1n!m!fα1...αn,α1...αmy

α1 . . . yαnyα1 . . . yαm

The full unfolded system for free bosonic fields is 1989

? R1(y, y | x) =i

4

(ηH

αβ ∂2

∂yα∂yβC(0, y | x) + ηHαβ ∂2

∂yα∂yβC(y,0 | x)

)?? D0C(y, y | x) = 0

Vacuum: sp(4) ∼ o(3,2)

Rαβ := dωαβ + ωαγωβγ − Hαβ = 0 , R

αβ:= d + ωαγh

γβ

+ ωβ δ

hαδ = 0

Hαβ := hαα ∧ hβα , Hαβ

:= hαα ∧ hαβ

R1(y, y | x) = Dad0 ω(y, y | x) Dad

0 = DL − hαβ(yα

∂

∂yβ+

∂

∂yαyβ

)

D0 = DL + hαβ(yαyβ +

∂2

∂yα∂yβ

)DL = dx −

(ωαβyα

∂

∂yβ+ ωαβyα

∂

∂yβ

)?? implies that higher-order terms in y and y describe higher-derivative

descendants of the primary HS fields

17

Fields of the AdS4 Nonlinear Equations

Infinite set of spins s = 0,1/2,1,3/2,2 . . . Fermions need doubling of fields

Doubled Weyl algebra connection: ωii(y, y | x) , i = 0,1

Twisted adjoint module: Ci1−i(y, y | x),

ωii(y, y | x) = ωii(y, y | x) , Ci1−i(y, y | x) = C1−i i(y, y | x)

A(y, y | x) = i∞∑

n,m=0

1

n!m!yα1 . . . yαnyβ1

. . . yβmAα1...αn,

β1...βm(x)

Nonlinear HS equations demand doubling of spinors and Klein operator

ω(Y |x) −→W (Z;Y ;K|x) , C(Y |x) −→ B(Z;Y ;K|x)

Some of the nonlinear HS equations determine the dependence on ZA

in terms of “initial data” ω(Y ;K|x) and C(Y ;K|x)

S(Z;Y ;K|x) = dZASA(Z;Y ;K|x) is a connection along ZA (A = (α, α))

Klein operators K = (k, k): k generates chirality automorphisms

kf(A) = f(A)k , A = (aα , aα) : A = (−aα , aα) , k2 = 118

Nonlinear HS Equations

Inner Klein operators:

κ = exp izαyα , κ = exp izαyα , κ ∗ f = f ∗ κ , κ ∗ κ = 1

dW +W ∗W = 0dB +W ∗B −B ∗W = 0dS +W ∗ S + S ∗W = 0 1992S ∗B−B ∗ S = 0

S ∗ S = i(dZAdZA + ηdzαdzαB ∗ k ∗ κ+ ηdzαdzαB ∗ k ∗ κ)

Dynamical content is located in the x-independent twistor sector

η = exp iϕ is a free phase parameter suggesting 3d bosonization.

The non-zero curvature has the form of Z2-Cherednik algebra

This form comes out from the consistency of nonlinear HS equations

19

Higher-Spin Star Product

(f ∗ g)(Z, Y ) =∫dSdT exp iSAT

A f(Z + S, Y + S)g(Z − T, Y + T )

[YA, YB]∗ = −[ZA, ZB]∗ = 2iCAB , Z − Y : Z + Y normal ordering

Via relation between space-time derivatives and Y, Z-derivatives

non-locality of the star product induces space-time non-locality

The form of HS star-product has remarkable properties

crucially important for the analysis of HS theory

20

Perturbative Analysis

Vacuum solution

B0 = 0 , S0 = dZAZA , W0 =1

2wAB(x)YAYB

dW0 +W0 ∗W0 = 0

wAB(x): describes AdS4.

First-order fluctuations

B1 = C(Y ) , S = S0 + S1 , W = W0(Y ) +W1(Y ) +W0(Y )C(Y )

[S0 , f ]∗ = −2idZf , dZ = dZA∂

∂ZA

21

Reconstruction of ZA Variables

Perturbatively, order n equations have the form

dZUn(Z;Y |dZ) = V [U<n](Z;Y |dZ) dZV [U<n](Z;Y |dZ) = 0 .

These can be solved as

Un(Z;Y |dZ) = d∗ZV [U<n](Z;Y |dZ) + h(Y) + dZε(Z; Y|dZ)

conventional homotopy operator ∂ = θA ∂∂ZA

and resolution d∗Z

d∗ZV (Z;Y |dZ) = ZA∂

dZA

∫ 1

0

dt

tV (tZ;Y |tdZ)

lead to nonlocalities beyond the free field level Prokushkin, Vasiliev 1999;

Boulanger, Kessel, Skvortsov, Taronna 2015; Vasiliev 2017

Alternative forms of d∗Z that differ by dZ-closed forms can also be used.

This is most important in the context of locality!

Space-time equations on ω(Y |x) and C(Y |x) are in dZ-cohomology

22

Interpretation

The indirect solution of the problem allowed to reduce the complicated

Hochschild cohomology problem in space-time to the simple De Rham

cohomology in the auxiliary Z-space.

The resolution freedom encodes

• All possible gauge choices in dz-exact forms

• All possible choices of field variables in dz−cohomology

Proves existence of field equations in all orders and provides general

solution: Any (non)local consistent unfolded HS equations are contained

in the nonlinear HS equations

Analysis of nonlinear HS equations is free of any additional assumptions

like, e.g., AdS/CFT ...

How to single out the proper (e.g., minimally nonlocal) frames?

23

Modified homotopy

An obvious freedom in the definition of homotopy operator is to replace

ZA → ZA + CA

∂ → ∂C =(ZA + CA

) ∂

∂θA,

∂

∂ZA(CB) = 0 .

The shift operator C commutes with ∂∂ZA

Local vertices were reproduced in the zero-form sector: include quartic

vertices Didenko, Gelfond, Korybut, MV 2018

24

Spin-Locality and Ultra-Locality

Spin-Locality:

any vertex with a fixed set of spins contains at most a finite num-

ber of contractions of spinorial indices between factors of zero-forms

C(Y )C(Y ) . . .

Ultra-Locality:

factors of C(Y ) are also y or y-independent if the contractions over

holomorphic and antiholomorphic variables are absent.

Ultra-Locality implies that further star-products will not induce nonlo-

calities in the higher orders.

Structure lemma states that, in the one-form sector proper shifted

homotopy implies ultra-locality as a consequence of spin-locality.

Gelfond, MV 2018

25

Results

Paradoxical results of the previous treatments on the meaningless con-

tribution of the stress tensor to field equations in

E ∼ α(η)T

with non-positive definite α(η) (η is the free phase parameter of the HS

theory) have been resolved Gelfond, MV 2017

E ∼ ηηT

HS AdS/CFT was checked for ∀η Didenko, MV 2017, Sezgin, Skvortsov 2017

Evaluation of all ω2C2 vertices.

Gelfond, MV 2019, Didenko, Gelfond, Korybut, MV 2019

All of them turn out to be ultra-local and contain a class of quintic local

vertices!

Pfaffian locality theorem suggests that HS theory is spin-local in all

orders.26

Misleading Statements in the Literature

Phase dependence of the current HS interactions following from non-

linear HS equations

Divergencies in the perturbative analysis of nonlinear HS equations

Impossibility of extracting local results

Risky statements on the nonlinear HS equations:

Terrible non-locality of HS theory

27

Part III

Towards Higher-Spin String-Like

andTensor-Like Theories

28

Coxeter Groups and Cherednik Algebras

A rank-p Coxeter group C is generated by reflections with respect to root

vectors va in a p-dimensional Euclidean vector space V . An elementary

reflection associated with va

Rvaxi = xi − 2via

(va, x)

(va, va), R2

va = I

Cherednik deformation of the semidirect product of the oscillator

algebra with the group algebra of C is

[qnα , qmβ ] = −iεαβ

2δnm +∑v∈R

ν(v)vnvm

(v, v)kv

, kvqnα = Rv

nmq

mα kv

It respects Jacobi identities (α = 1,2, n = 1; . . . , p)

Coupling constants ν(v) are invariants of C being constant on the con-

jugacy classes of root vectors under the action of C.

General framed Cherednik algebra

[qnα , qmβ ] = −iεαβ

2δnmIn +∑v∈R

ν(v)vnvm

(v, v)kv

, kv := kv∏Ii1(v) . . . Iik(v)

29

Bp–Coxeter System

Important case of the Coxeter root system is Bp with the roots

R1 = ±en 1 ≤ n ≤ p , R2 = ±en ± em 1 ≤ n < m ≤ p .

Apart from permutations Bp contains reflections of basis axes vn± = en.

R1 and R2 form two conjugacy classes of Bp.

The Coxeter group of 3d HS theory is A1 ∼ B1.

B2 underlies the string-like HS models.

Bp is conjectured to underly tensor-like HS models.

Lorentz symmetry of HS theories follows from the fundamental property

of Cherednik algebras that for any Coxeter root system the generators

tαβ :=i

4

p∑n=1

qnα , qnβ

obey the sp(2) commutation relations properly rotating all indices α

[tαβ , qnγ ] = εβγq

nα + εαγq

nβ

30

Coxeter HS Equations

Unfolded equations for 1804.06520 C-HS theories remain the same except

iS ∗ S = dZAndZAn +∑i

∑v∈Ri

Fi∗(B)dZαnv

ndZαmvm

(v, v)∗ κv

κv are generators of C acting trivially on all elements except for dZαn

κv ∗ dZnα = RvnmdZ

mα ∗ κv

Fi∗(B) is any star-product function of the zero-form B on the conjugacy

classes Ri of C. In the important case of the Coxeter group Bp

iS∗S = dZAndZAn+

∑v∈R1

F1∗(B)dZαnv

ndZαmvm

(v, v)∗κv+

∑v∈R2

F2∗(B)dZαnv

ndZαmvm

(v, v)∗κv

with arbitrary F1∗(B) and F2∗(B) responsible for the

HS and stringy/tensorial features, respectively

F2∗(B) 6= 0 for p ≥ 2.

The framed construction leads to a proper massless spectrum.

Jacobi for Cherednik imply consistency of field equations

31

Extensions

W , S and B can be valued in any associative algebra A: A∞ structure.

Multi-particle extensions are associated with the semi-simple Coxeter

groups. The simplest option with C = BNp is the product of N of Bp

systems

BNp := Bp ×Bp × . . .︸ ︷︷ ︸N

.

The limit N →∞ along with the graded symmetrization of the product

factors expressing the spin-statistics is related to the (graded symmet-

ric) multi-particle algebra M(h(C)) of the HS algebra h(C)

M(h(C)) = U(h(C)) : Hopf algebra.

Multi-particle algebra M(h(C)) is conjectured to underly the full fledged

string and tensor bulk models

32

Klein Operators and Single-Trace operators

Enlargement of the field spectra of the rank-p > 1 Coxeter HS models:

C(Y nα ; kv) depend on p copies of oscillators Y nα and Klein operators kv

Qualitative agreement with enlargement of the boundary operators in

tensorial boundary models.

Klein operators of Coxeter reflections permute master field arguments

At p = 2 the star product of two master fields C(Y1, Y2|x)k12 gives

(C(Y1, Y2|x)k12) ∗ (C(Y1, Y2|x)k12) = C(Y1, Y2|x) ∗ C(Y2, Y1|x) .

p = 2 system: strings of fields with repeatedly permuted arguments

Cnstring := C(Y1, Y2|x) ∗ C(Y2, Y1|x) ∗ C(Y1, Y2|x) . . .︸ ︷︷ ︸n

.

are analogous of the single-trace operators in AdS/CFT .

C(Y1, Y2|x) and C(Y1, Y2|x) ∗ C(Y2, Y1|x): single-trace-like

C(Y1, Y2|x) ∗ C(Y1, Y2|x) : double-trace-like.

For p > 2 fields carry p arguments permuted by Sp generated by kij

33

Conclusion

The shifted homotopy scheme is proposed leading to minimally non-

local HS vertices derived from the nonlinear equations. A class of ultra

local new vertices is found in the one-form sector containing vertices

up to quintic order. Didenko, Gelfond, Korybut, MV to appear soon

Coxeter HS theories extend minimal HS theories to String-like B2 models

and tensor-like Bp models of any rank p

The spectrum of the B2 HS model is analogous to that of

String Theory with the infinite set of Regge trajectories.

N = 4 SYM is argued to be a natural dual of the B2–HS model

Bp models are particularly interesting in the context of recent discovery

of fixed points of tensor models Benedetti, Gurau, Harribey 2019

Main problem on the agenda:

spontaneous breaking of HS symmetries in the Coxeter HS models

34