Can Chern-Simons or Rarita-Schwinger be a Volkov-Akulov Goldstone?

Dmitri Sorokin

INFN Padova Section and Uni PD

arXiv:1806.05945 with Sukriti Bansal

ESI Programme “Higher spins and holography”, Vienna, 20 March 2019

Motivation

• How to break higher-spin symmetries?

• Are they broken spontaneously? What is the mechanism?

• Can one construct consistent non-linear models of higher-spin Goldstone fields associated with spontaneous breaking of HS symmetries?

• To start with a simplified set-up based on simple but yet non-trivial finite HS algebras (similar to SUSY) and see….

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History • In 1975 Hietarinta constructed (super)algebras that generalize to higher-

spins conventional susy algebras and proposed to use their non-linear realizations for the description of HS Goldstone fields ala Volkov-Akulov

• Independently, in 1977 Baaklini considered the case of spin-3/2 superalgebra of this type in D=4. 89’-10’ Shima, Tanii, Tsuda, Pilot, Rajpoot…

• In 1984 Aragon & Deser constructed, in D=3, “Hypergravity” a system of (non-dynamical) gravity and fermionic spin-(2n+3)/2 gauge field invariant under local spin-(2n+1)/2 supersymmetry transformations

• Recently Hypergravity was revisited by Zinoviev ’14, Bunster et.al ’14, Fuentealba et.al ’15, Henneaux et.al ’15, …

Our motivation: study effects of spontaneous symmetry breaking and properties of non-linear Lagrangians for higher-spin goldstone fields

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Simplest cases to be considered: spin-1 and spin-3/2 Goldstones in D=2+1

Hietarinta algebras

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{ , } 2

[ , ] 0

( 0,1,..., 1)

- spinor indices

c

c

c

Q Q P

Q P

a D

Poincaré SUSY algebra

1 1 1 1

1

... ... ... , ... ,

...

{ , }

[ , ] 0

n m n m

n

a a b b a a b b c

c

a a

c

Q Q f P

Q P

Lorentz-invariant structure constants

Bosonic Hietarinta algebras 1 1 1 1

1

... ... ... , ... ,

...

[ , ]

[ , ] 0

n m n m

n

a a b b a a b b c

c

a a

c

S S f P

S P

Difference from the conventional HS algebras:

1 2 1 2 1 2 1 22 4 | | 2[ , ] ...s s s s s s s sT T T T T

(Anti)commutators of the HS generators in the Hietarinta algebras do not produce other yet higher spin generators, i.e. the Hietarinta algebras are finite dimensional.

Hietarinta algebras can be obtained by contractions of HS algebras

Volkov-Akulov Goldstino models ‘72

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(Describe effects of spontaneous susy breaking in terms of non-linear effective Lagrangians for Goldstone fields on which susy is realized non-linearly)

{ , } 2

[ , ] 0

( 0,1,..., 1)

c

c

c

Q Q P

Q P

a D

Poincaré SUSY algebra

( )Q x Spin ½ Goldstino

VA Construction :

i) Find a susy-invariant Cartan one-form:

SUSY transformations:

Volkov-Akulov Goldstino models

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Susy invariant one-form in space-time:

2 2( ) ( )a a a b a a b a

b b bE dx if d dx if dx E

iii) Construct susy invariant action:

21 2

1

2

...! det D

D

f a a a D a

a a bDS E E E f d x E

ii) Replace the Grassmann coordinate with the field:

3 2 2

1/2 ( ( ) )a abc

a a b cS d x i f f In D=3

susy breaking scale

2 2, a a b

bx if if susy variations:

Non-linear transformation

D=3 Chern-Simons Goldstones

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2 2

( )

( ) ,

a

a

dbc a dbc

a a b c d a b c

S A x

A x c f c A A x f c A

constant vector parameter

Invariant one-form: 2 2( )a a abc d a abc d a

b c d b d c dE dx f A dA dx f A A dx E

Invariant action:

2

2 3 3

2 (det 1)

fa abc abc dfg

d a b c a d b f c gS f d x E d x A A A A A A

Chern-Simons

Does the action have a gauge symmetry? ( ) ...a aA x

NO vector field acquires a propagating degree of freedom

Propagating CS Goldstone is Galileon

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Stueckelberg trick - make the action gauge invariant by replacing

1/2 1/2 ( ), , a a a a a aA A A f x A f )

Make this substitution in the action and consider a so-called decoupling limit f → ∞

Quartic Galileon Lagrangian (appears in modified gravities):

Galilean symmetry: a

ac c x

Consequence: Galileon Lagrangians are quadratic in time derivatives

Hamiltonian analysis

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2

3

2

fabc abc dfg

a b c a d b f c gS d x A A A A A A

The action is of the first order in time derivative 0 aA

Conjugate momenta are constrained: 2 2

0 2

( , ) 1,2, ( ) 0

( , )

i ij i ij

j b i c i j

i

b i c

p A f F A A i A O f

p f F A A

On the constraint surface Hamiltonian is

2 2 2

0 06 det , 6( ) detGalileon

i j i jH f A A H

Hamiltonian is not bounded from below. Fluctuations around certain zero-energy configurations may have negative energies leading to instabilities

secondary constraint

The 4 constraints are of the second class and reduce the number of dofs to 1

3d Rarita-Schwinger goldstino model

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Spin-3/2 Superalgebra

Special cases:

b=-4/5, c=8/5

b=4, c=-2

b=c=0

The case of our interest

3d Rarita-Schwinger goldstino model

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Spin-3/2 Superalgebra , 2 , , 0 , =1,2a b abc a

c bQ Q C P Q P

Action

2

4

3

2

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( )

+ ( )

fabc abc dfg afc dbg

a b c a b c d f g

if abc dfg abg dfc pqr

c p g a q b d r f

S d x i

Rarita-Schwinger

Gauge symmetry:

Susy invariant one-form

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Reduction of the non-linear 3/2-goldstino action to the quadratic RS action

Non-linear realization of the rigid spin-3/2 susy:

spin-3/2 goldstino is a pure gauge:

- closes on translation

linearized local «susy»

Summary and things to do • The simplest examples of spontaneous breaking of Hietarinta symmetries and

corresponding Goldstone models turned out to be peculiar non-linear generalizations of the Chern-Simons and Rarita-Schwinger Lagrangians

• The CS Goldstone propagates a scalar mode which is a Galileon field that appears in modified theories of gravity

It would be of interest to consider coupling of the CS Goldstone to a 3d bi-gravity theory which is invariant under local symmetry associated with the algebra

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2 spin-2 gauge fields

spin-connection

“Duality” relation with the 3d Maxwell algebra:

3d “Maxwell-Chern-Simons” gravity P. Salgado, R.J. Szabo and O. Valdivia ’14

L. Avilés, E. Frodden, J. Gomis, D. Hidalgo and J. Zanelli ‘18

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Higher spin generalization s=n+1

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Can this action be obtained by a contraction of the conventional CS HS theory based on SL(n,R) x SL(2,R)?

Conclusion and outlook • In the spin-3/2 case the spontaneous breaking of the rigid spin-3/2 susy retains

local symmetry of the Rarita-Schwinger Lagrangian. The non-linear spin-3/2 goldstino Lagrangian reduces to the quadratic RS Lagrangian upon a non-linear field redefinition. This causes the 3d RS goldstino be non-propagating field

• Problems to study:

To couple the RS goldstino to conventional 3d (super)gravity and Hypergravity (with spin-2 and spin-5/2 gauge fields) and to study properties of these models

To consider the RS goldstino model in D=4 associated with the algebra

Generalization to higher-spin Goldstone fields

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