Osp(1|2M)-invariant higher-spin systems

Dmitri Sorokin

INFN, Padova Section

Based on arXiv:1401.1645, 1408.6675 + work in progress

with Ioannis Florakis (CERN) and Mirian Tsulaia (Canberra)

Penn State Uni, August 27-29, 2015

Preliminaries

The largest finite-dimensional symmetry of a HS system in D=4 is

extended conformal symmetry Sp(8)

it unifies in infinite multiplets fields of s=0,1,2. …, and s=1/2, 3/2,…,

• integer and half-integer spin fields can be further unified into an

Osp(1|8) supermultiplet

Free HS theory is a simple theory of a “hyper” scalar and spinor field

in a space-time extended with 6 extra (tensorial coordinates) “hyperspace”

in which Sp(8) acts as the conformal group

(Fronsdal ’85; Bandos, Lukierski, D.S. ’98-99; Vasiliev ’01; Didenko ,Vasiliev ‘03;

Plyushchay, Tsulaia, D.S. ‘03; Bandos, Bekaert et. al. ’05; Gelfond, Vasiliev ‘05-’15;

Fedoruk, Ivanov ‘07,…)

Description of the Sp(8)-invariant HS system with CFT methods

2

Sp(8) symmetry of 4d HS theory (Fronsdal, 1985)

Sp(8) acts on infinite spectrum of 4d HS single-particle states (s=0, 1/2, 1, 3/2, 2,…)

this is a consequence of Flato-Fronsdal Theorem, 1978:

Tensor product of two 3d singleton moduli comprises all the massless spin-s fields in 4d

Singletons are 3d massless scalar and spinor fields which enjoy 3d conformal symmetry

Can Sp(8) play a role similar to Poincaré or conformal symmetry acting geometrically on a hyperspace containing 4d space-time?

Whether a 4d HS theory can be formulated as a field theory on this hyperspace?

Fronsdal ’85:

minimal dimension of the Sp(8) hyperspace (containing 4d space-time) is 10

)R,4()3,2(

)R,8()4,2()3,2()3,1(

SpSO

SpSOSOSO

)4()3,2( SpSO

)8()4()4( S S SpSpSp

`Geometrically’ means: mn

n

mmnm

n

mm

conf xxkxkbxxlax 2 2

3

Particles and fields in Sp(8) hyperspace

• Bandos & Lukierski ‘98: Twistor-like (super) particle on a tensorial space

(their motivation was not related to HS theory, but to supersymmetry)

Most general N=1 susy in flat 4d:

1,2,3,4, ,

0],[ , },{

mm

nlmmn

mn

m

m ZPZPQQ

scoordinatematrix 44 -

s)coordinate extra (6 s),coordinate (4d

41

21

mn

mn

m

m

nmmn

mn

m

m

yxXX

yyZxP10d space

coordinates:

Superparticle action: ),(

iXdS - commuting twistor-like variable

possesses hidden (generalized superconformal) symmetry )8()8|1( SpOSp

Quantization (Bandos, Lukierski & D.S. ‘99)

0),(

XiX

- describes in 4d free fields of any spin s=0, 1/2, 1, 3/2, 2, … 4

(Curtright ‘88, “Are there superstrings in 11D?”)

),,( ; ,)( ,

ZiXP supertwistor

relations

Field theory in flat Sp(8) hyperspace

Field equations in flat hyperspace (Vasiliev ’01):

)( ...)()(),( ...1

1 XCXfXbXCk

k

0),( ),( ),(2

4

XCi

XXedXC

iForier transform Free unfolded

equations

)( and )( XfXb are independent scalar and spinor hyperfields satisfying the equations:

0)()(

0)( )(

XfXf

Xb

4d content of )( and )( XfXb are Higher-Spin curvatures (Vasiliev ’01, Bandos et. al.‘05):

Integer spins: ...))(()()(),(21

, pqmn

qnmppqmn

mn

mn

mnm yyxRyxFxyxb

½ integer spins: ...)()()(),(21 mn

nmmn

mnm yxxyxf

)(41

21 mn

mn

m

m yxXX

.... ;0 ,0 ;0 ,0 ;0 ,],[][

2 pqmn

mp

qpmnmn

m

mnl RRFF Eoms and Bianchi: 5

0),(

,0),(

22

lpmn

mnlp

mn

m yy

yxb

yx

yxb

Section condition in generalized

geometry of M-theory (Berman et. al.)

Sp(8) transformations in hyperspace

(symmetries of the field equations)

mn

n

mmnm

n

mm

conf xxkxkbxxlax 2 2

Conformal transformations:

Sp(8) transformations:

XkXXgaXSp )(

)8( 2

Sp(8) generators:

XXXK

XXL

XP

,2 ,

generators of GL(4)

Hyperspace is a coset space: )4(

)8(

GLK

SpP

KKLPPLLLL

LKPKKPP

],[ ,],[ ,],[

,],[ ,0],[ ,0],[

fXkgfXkgfXf

bXkgbXb

)()(

,)(

21

21

conformal weights of the fields

6

Hyperspace extension of AdS(4) (Bandos, Lukierski, Preitschopf, D.S. ’99; Vasiliev ‘01)

10d group manifold Sp(4,R) ~ SO(2,3)

)3,1(

)4(4

SO

SpAdS KP

GLK

SpSp

)4(

)8()4( - different Sp(8) coset realization

Like Minkowski and AdS(4) spaces, which are conformally flat,

the flat hyperspace and Sp(4) are (locally) related to each other

by a “generalized conformal” transformation

Sp(M) group manifolds are GL-flat (Plyushchay, D.S. & Tsulaia ‘03)

Algebra of covariant derivatives on Sp(4):

, ,)(],[ )()(21

CCCCr

r - is Sp(4) (or AdS4) radius

dXGGXXXG

XGG

r

1- 1-

41 )( ;)( ,

Sp(4) Cartan form

GL-flatness is important for the relation between the field equations in flat and Sp(4) hyperspace

7

HS field equations in Sp(4) (Didenko and Vasiliev ‘03; Plyushchay, D.S. & Tsulaia ‘03)

0)()(

0)( )(

XfXf

Xb

Flat hyperspace equations:

Sp(4) field equations (Plyushchay, D.S. & Tsulaia ‘03) :

Generalized conformal relations between flat and Sp(4) hyperfields (Florakis, D.S. & Tsulaia ‘14)

)( )()(41

FCFCFFr

Fermi:

BCCCCBCCCBBrr

)()( )()(32

1)()()(8

12 Bose:

),( det)( ),( det)(

41

XGXfGGXFXbGXBr

8

“Energy-momentum” tensors of hyperfields

and “HS” conserved currents (Vasiliev ’02)

9

)()()()(

)()()()(

XfXfXfXfT

XbXbXbXbT

f

b

Conservation law: 0 TTTT

Integration measure in hyperspace

Sp(4):

),( tr )(tr

41

11)4(

XG

GGdXEEEEEEEEEEE

r

Sp

Flat 𝑀10: dXdXdXdXdXdXdXdXdXdX ),( tr )(tr )10(

3 7

“HS” currents:

n

k

knkk bbknaTknkn

0

......

1

... )(1121),()1(

Closed 4-form:

4

4321

44332211

4321

)4()4()4( ,0 ,M

JQdJTdXdXdXdXJ

)(det)(det )(det )( )( )(

)(det)( )( )(

)(det )( )(

4

1

214

1

324

1

31321

2

1

21

1

2121

2

1

2121

XXXXXXcXbXbXb

XXXXcXfXf

XXcXbXb

bbb

ff

bb

Sp(8) invariant correlation functions

In flat hyperspace (Vasiliev ‘01, Vasiliev & Zaikin ’03) (also Didenko & Skovrtsov ‘12)

In Sp(4) hyperspace (Florakis, D.S. & Tsulaia‘14)

flat)Sp(

flat)Sp(

flat)Sp(

XbXbXbXGXGXGXBXBXB

XfXfXGXGXGXGXFXF

XbXbXGXGXBXB

)( )( )()(det )(det )(det)( )( )(

)( )( )( )( )(det )(det)( )(

)( )( )(det )(det)( )(

3213214321

212

1

214

21

2121421

)(

41

XXGr

conformal weight

10

Four-point functions

11

Bosonic:

Fermionic:

jiij XXX

Supersymmetry in hyperspace

12

motion of equationsinvariant -)81( - 0),(],[

22},{ ,

fieldsauxiliary )()(),(

|OSpXDD

ZPiDDiD

XfXbX

mn

mn

m

m

Higher spin fields form an infinite dimensionsal D=4 supermultiplet

Osp(1|8) – invariant correlation functions of scalar superfields

jii

jii

jiij

kkk

XXZ

ZZZcXXX

ZcXX

22

3123123332211

1222211

321321 |)det(||)det(||)det(|),(),(),(

|)det(|),(),(

0, ±1/2, ±1, ±3/2, ±2, ±5/2, …

On supergroup manifold Osp(1|4): 0),()8

( ][ XCr

i

Conclusion Free theory of the infinite number of massless HS fields in 4d flat and AdS4

space has generalized conformal Sp(8) symmetry and can be compactly formulated in 10d hyperspace with the use of one scalar and one spinor field.

May Sp(8) symmetry have something to do with the trivialization of the partition function of the infinite system of s=0,1,2, …(Beccaria & Tseytlin ‘14)

Higher dimensional extension to Sp(2M) invariant hyperspaces is straightforward (Bandos, Lukierski, D.S. ‘99, Vasiliev ’01,.., …)

known physically relevant cases are (Bandos, Bekaert, de Azcarraga, D.S., Tsulaia‘05)

M=2 (d=3), M=4 (d=4), M=8 (d=6), M=16 (d=10)

describe conformal HS fields in corresponding space-times.

Supersymmetric generalizations are available

(Bandos et. al, Vasiliev et. al., Ivanov et. al., P. West, Florakis et. al…)

Is there any relation to Doubled Field Theory?

Main problem: Whether one can construct an interacting field theory in hyperspace which would describe HS interactions in conventional space-time (at least with vertices composed of HS curvatures and their derivatives)

Gelfond & Vasiliev ‘15: HS current interactions break Sp(8) down to SU(2,2)

◦ How can one reveal this breaking using conformal analysis of correlation functions? 13