Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France

Super-BMS algebras and the asymptotic
structure of supergravity at spatial infinity

Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras

Marc Henneaux
Université Libre de Bruxelles & Collège de France

Flat-spacetime holography workshop (OIST, Okinawa)
18 March 2021

Conclusions andcomments

1 / 25

Introduction
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

2 / 25

Introduction
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France

There are various motivations for studying the asymptotic
structure of gravitational theories at spatial infinity.

Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

2 / 25

Introduction
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction

There are various motivations for studying the asymptotic
structure of gravitational theories at spatial infinity.
1

The physical states in the quantum theory are naturally defined on
spacelike (Cauchy) hypersurfaces, which asymptote spatial infinity.

Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

2 / 25

Introduction
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France

There are various motivations for studying the asymptotic
structure of gravitational theories at spatial infinity.
1

Introduction
2
Results
Representationsof the Lorentz
algebraBMS4
supertranslations

The physical states in the quantum theory are naturally defined on
spacelike (Cauchy) hypersurfaces, which asymptote spatial infinity.
The existence of a smooth null infinity, given reasonable initial
data, is a non-trivial dynamical question. Obstructions to thesmoothness of null infinity have indeed been identified (Friedrich,
Valiente-Kroon). Does the BMS symmetry exhibited at null infinity
exist independently of the existence of a smooth null infinity ?

Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

2 / 25

Introduction
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France

There are various motivations for studying the asymptotic
structure of gravitational theories at spatial infinity.
1

Introduction
2
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation

The physical states in the quantum theory are naturally defined on
spacelike (Cauchy) hypersurfaces, which asymptote spatial infinity.
The existence of a smooth null infinity, given reasonable initial
data, is a non-trivial dynamical question. Obstructions to thesmoothness of null infinity have indeed been identified (Friedrich,
Valiente-Kroon). Does the BMS symmetry exhibited at null infinity
exist independently of the existence of a smooth null infinity ?

The BMS symmetry has been successfully shown to be a genuine
symmetry of Einstein theory of gravity at spatial infinity.

BMS4
superalgebras
Conclusions andcomments

2 / 25

Introduction
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France

There are various motivations for studying the asymptotic
structure of gravitational theories at spatial infinity.
1

Introduction
2
Results
Representationsof the Lorentz
algebraBMS4
supertranslations

The physical states in the quantum theory are naturally defined on
spacelike (Cauchy) hypersurfaces, which asymptote spatial infinity.
The existence of a smooth null infinity, given reasonable initial
data, is a non-trivial dynamical question. Obstructions to thesmoothness of null infinity have indeed been identified (Friedrich,
Valiente-Kroon). Does the BMS symmetry exhibited at null infinity
exist independently of the existence of a smooth null infinity ?

Infinitedimensional
spinorrepresentation

The BMS symmetry has been successfully shown to be a genuine
symmetry of Einstein theory of gravity at spatial infinity.

BMS4
superalgebras

We describe here the extension to supergravity.

Conclusions andcomments

2 / 25

Introduction
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

3 / 25

Introduction
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France

Results based on joint work with O. Fuentealba, S. Majumdar, J.
Matulich and T. Neogi (published and to appear)

Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

3 / 25

Introduction
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France

Results based on joint work with O. Fuentealba, S. Majumdar, J.
Matulich and T. Neogi (published and to appear)
which finds its roots in the “N = 0” work done with C. Troessart.

Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

3 / 25

Introduction
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results

Results based on joint work with O. Fuentealba, S. Majumdar, J.
Matulich and T. Neogi (published and to appear)
which finds its roots in the “N = 0” work done with C. Troessart.
Key features of the program initiated and developed with C.
Troessaert are :Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

3 / 25

Introduction
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraResults based on joint work with O. Fuentealba, S. Majumdar, J.
Matulich and T. Neogi (published and to appear)
which finds its roots in the “N = 0” work done with C. Troessart.
Key features of the program initiated and developed with C.
Troessaert are :The analysis is carried out in phase space, using the Hamiltonian
formulation of gravity.

BMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

3 / 25

Introduction
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation

Results based on joint work with O. Fuentealba, S. Majumdar, J.
Matulich and T. Neogi (published and to appear)
which finds its roots in the “N = 0” work done with C. Troessart.
Key features of the program initiated and developed with C.
Troessaert are :The analysis is carried out in phase space, using the Hamiltonian
formulation of gravity.
We insist throughout that the action be finite (on-shell andoff-shell) for all allowed phase space configurations. In particular,
there is a well-defined (finite) symplectic structure.

BMS4
superalgebras
Conclusions andcomments

3 / 25

Introduction
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras

Results based on joint work with O. Fuentealba, S. Majumdar, J.
Matulich and T. Neogi (published and to appear)
which finds its roots in the “N = 0” work done with C. Troessart.
Key features of the program initiated and developed with C.
Troessaert are :The analysis is carried out in phase space, using the Hamiltonian
formulation of gravity.
We insist throughout that the action be finite (on-shell andoff-shell) for all allowed phase space configurations. In particular,
there is a well-defined (finite) symplectic structure.
Symmetries are phase space transformations that leave the action
invariant and for which there is therefore a well-defined moment
map.

Conclusions andcomments

3 / 25

Introduction
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

4 / 25

Introduction
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France

It is well known that the (anti-)commutator of two local
supersymmetry transformations is a diffeomorphism
parametrized by ξµ = i²̄1 γµ ²2 .

Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

4 / 25

Introduction
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation

It is well known that the (anti-)commutator of two local
supersymmetry transformations is a diffeomorphism
parametrized by ξµ = i²̄1 γµ ²2 .
If the parameters of supersymmetry transformations go to O(1)functions of the angles at infinity (“improper gaugetransformations”),µ ¶1,²i → ²̃i (θ, ϕ) + Or
the resulting diffeomorphism is in general not a BMS4
transformations.

BMS4
superalgebras
Conclusions andcomments

4 / 25

Introduction
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

It is well known that the (anti-)commutator of two local
supersymmetry transformations is a diffeomorphism
parametrized by ξµ = i²̄1 γµ ²2 .
If the parameters of supersymmetry transformations go to O(1)functions of the angles at infinity (“improper gaugetransformations”),µ ¶1,²i → ²̃i (θ, ϕ) + Or
the resulting diffeomorphism is in general not a BMS4
transformations.
Indeed, i²̄1 γµ ²2 goes at infinity to an arbitrary “angle-dependent”
translation, which involves four functions of the angles instead ofone.

4 / 25

Introduction
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

It is well known that the (anti-)commutator of two local
supersymmetry transformations is a diffeomorphism
parametrized by ξµ = i²̄1 γµ ²2 .
If the parameters of supersymmetry transformations go to O(1)functions of the angles at infinity (“improper gaugetransformations”),µ ¶1,²i → ²̃i (θ, ϕ) + Or
the resulting diffeomorphism is in general not a BMS4
transformations.
Indeed, i²̄1 γµ ²2 goes at infinity to an arbitrary “angle-dependent”
translation, which involves four functions of the angles instead ofone.
This means that the supersymmetry transformations should be
restricted, and perhaps redefined, asymptotically.
4 / 25

Finite-dimensional supersymmetric extension
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

5 / 25

Finite-dimensional supersymmetric extension
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity

There are at least two ways to achieve this task.

Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

5 / 25

Finite-dimensional supersymmetric extension
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France

There are at least two ways to achieve this task.
The first approach takes as asymptotic conditions for thegravitino fieldµ ¶1ψk = O 2r

Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

5 / 25

Finite-dimensional supersymmetric extension
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France

There are at least two ways to achieve this task.
The first approach takes as asymptotic conditions for thegravitino fieldµ ¶1ψk = O 2r

Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation

as suggested by the fact that the superchargeZI
d2 Si ²T γij ψjQ ∼ d3 x²T S +2S∞should converge.

BMS4
superalgebras
Conclusions andcomments

5 / 25

Finite-dimensional supersymmetric extension
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France

There are at least two ways to achieve this task.
The first approach takes as asymptotic conditions for thegravitino fieldµ ¶1ψk = O 2r

Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

as suggested by the fact that the superchargeZI
d2 Si ²T γij ψjQ ∼ d3 x²T S +2S∞should converge.
This implies that the supersymmetry parameter ² should go to a
constant at infinityµ ¶1² → ²0 + O.
r
5 / 25

Finite-dimensional supersymmetric extension
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

6 / 25

Finite-dimensional supersymmetric extension
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results

The supercharge reduces then toZIQ ∼ d3 x²T S + ²T0

2S∞

d2 Si γij ψjand depends on four constants - the four components of ²0 .

Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

6 / 25

Finite-dimensional supersymmetric extension
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction

The supercharge reduces then toZIQ ∼ d3 x²T S + ²T0

2S∞

d2 Si γij ψjResults

and depends on four constants - the four components of ²0 .

Representationsof the Lorentz
algebraThe resulting super-BMS4 algebra has only a finite number of
fermionic generators. These close on ordinary translations.

BMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

6 / 25

Finite-dimensional supersymmetric extension
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction

The supercharge reduces then toZIQ ∼ d3 x²T S + ²T0

2S∞

d2 Si γij ψjResults

and depends on four constants - the four components of ²0 .

Representationsof the Lorentz
algebraThe resulting super-BMS4 algebra has only a finite number of
fermionic generators. These close on ordinary translations.

BMS4
supertranslations
Infinitedimensional
spinorrepresentation

This is the algebra considered some time ago by Awada, Gibbons
and Shaw, who showed that it could indeed be realized as asymmetry.

BMS4
superalgebras
Conclusions andcomments

6 / 25

Finite-dimensional supersymmetric extension
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction

The supercharge reduces then toZIQ ∼ d3 x²T S + ²T0

2S∞

d2 Si γij ψjResults

and depends on four constants - the four components of ²0 .

Representationsof the Lorentz
algebraThe resulting super-BMS4 algebra has only a finite number of
fermionic generators. These close on ordinary translations.

BMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras

This is the algebra considered some time ago by Awada, Gibbons
and Shaw, who showed that it could indeed be realized as asymmetry.
The work of Awada et al was carried out at null infinity.

Conclusions andcomments

6 / 25

Finite-dimensional supersymmetric extension
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction

The supercharge reduces then toZIQ ∼ d3 x²T S + ²T0

2S∞

d2 Si γij ψjResults

and depends on four constants - the four components of ²0 .

Representationsof the Lorentz
algebraThe resulting super-BMS4 algebra has only a finite number of
fermionic generators. These close on ordinary translations.

BMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

This is the algebra considered some time ago by Awada, Gibbons
and Shaw, who showed that it could indeed be realized as asymmetry.
The work of Awada et al was carried out at null infinity.
An equivalent analysis can be performed at spatial infinity.

6 / 25

Infinite-dimensional supersymmetric extension
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

7 / 25

Infinite-dimensional supersymmetric extension
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France

The second approach has an infinite-dimensional fermionicextension, parametrized by even spinor functions on the
two-sphere.

Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

7 / 25

Infinite-dimensional supersymmetric extension
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results

The second approach has an infinite-dimensional fermionicextension, parametrized by even spinor functions on the
two-sphere.
This second approaches involves also non-trivial asymptotic
redefinitions of the supersymmetry transformations at infinity,

Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

7 / 25

Infinite-dimensional supersymmetric extension
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraThe second approach has an infinite-dimensional fermionicextension, parametrized by even spinor functions on the
two-sphere.
This second approaches involves also non-trivial asymptotic
redefinitions of the supersymmetry transformations at infinity,
which are necessary to preserve the asymptotic conditions at
spatial infinity on the supergravity fields,

BMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

7 / 25

Infinite-dimensional supersymmetric extension
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation

The second approach has an infinite-dimensional fermionicextension, parametrized by even spinor functions on the
two-sphere.
This second approaches involves also non-trivial asymptotic
redefinitions of the supersymmetry transformations at infinity,
which are necessary to preserve the asymptotic conditions at
spatial infinity on the supergravity fields,
and to make the supersymmetry transformations with
non-vanishing supersymmetry parameters at infinity
well-defined canonical transformations.

BMS4
superalgebras
Conclusions andcomments

7 / 25

Infinite-dimensional supersymmetric extension
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras

The second approach has an infinite-dimensional fermionicextension, parametrized by even spinor functions on the
two-sphere.
This second approaches involves also non-trivial asymptotic
redefinitions of the supersymmetry transformations at infinity,
which are necessary to preserve the asymptotic conditions at
spatial infinity on the supergravity fields,
and to make the supersymmetry transformations with
non-vanishing supersymmetry parameters at infinity
well-defined canonical transformations.
There is a well-defined moment map, and well-definedsupersymmetry charges.

Conclusions andcomments

7 / 25

Infinite-dimensional supersymmetric extension
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

8 / 25

Infinite-dimensional supersymmetric extension
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France

The algebra of the asymptotic symmetries is the BMS4 algebra in
the bosonic sector.

Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

8 / 25

Infinite-dimensional supersymmetric extension
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction

The algebra of the asymptotic symmetries is the BMS4 algebra in
the bosonic sector.
In addition, there are fermionic symmetries parametrized by aneven spinorial function on the 2-sphere.

Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

8 / 25

Infinite-dimensional supersymmetric extension
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebra

The algebra of the asymptotic symmetries is the BMS4 algebra in
the bosonic sector.
In addition, there are fermionic symmetries parametrized by aneven spinorial function on the 2-sphere.
These fermionic symmetries transform as spinors under Lorentz
transformations.

BMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

8 / 25

Infinite-dimensional supersymmetric extension
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations

The algebra of the asymptotic symmetries is the BMS4 algebra in
the bosonic sector.
In addition, there are fermionic symmetries parametrized by aneven spinorial function on the 2-sphere.
These fermionic symmetries transform as spinors under Lorentz
transformations.
Even in infinite number, the fermionic symmetries close,
however, only on the ordinary translations.

Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

8 / 25

Infinite-dimensional supersymmetric extension
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation

The algebra of the asymptotic symmetries is the BMS4 algebra in
the bosonic sector.
In addition, there are fermionic symmetries parametrized by aneven spinorial function on the 2-sphere.
These fermionic symmetries transform as spinors under Lorentz
transformations.
Even in infinite number, the fermionic symmetries close,
however, only on the ordinary translations.
[Q, Q] ∼ γ0 γµ Pµ .

BMS4
superalgebras
Conclusions andcomments

8 / 25

Infinite-dimensional supersymmetric extension
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

9 / 25

Infinite-dimensional supersymmetric extension
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France

We refer to the papers for the details.

Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

9 / 25

Infinite-dimensional supersymmetric extension
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France

We refer to the papers for the details.
Here, we shall discuss only some aspects of Lorentz
representation theory in order to answer the question :Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

9 / 25

Infinite-dimensional supersymmetric extension
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraWe refer to the papers for the details.
Here, we shall discuss only some aspects of Lorentz
representation theory in order to answer the question :From the algebraic point of view (independently of any
dynamical realization), could the superalgebra be different ?

BMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

9 / 25

Infinite-dimensional supersymmetric extension
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations

We refer to the papers for the details.
Here, we shall discuss only some aspects of Lorentz
representation theory in order to answer the question :From the algebraic point of view (independently of any
dynamical realization), could the superalgebra be different ?
The theory of representations of the Lorentz algebra sheds
indeed an instructive light on this question.

Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

9 / 25

Infinite-dimensional supersymmetric extension
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras

We refer to the papers for the details.
Here, we shall discuss only some aspects of Lorentz
representation theory in order to answer the question :From the algebraic point of view (independently of any
dynamical realization), could the superalgebra be different ?
The theory of representations of the Lorentz algebra sheds
indeed an instructive light on this question.
We shall first review general features of irreducible,
infinite-dimensional (but not necessary unitary) representationsof the Lorentz group.

Conclusions andcomments

9 / 25

Lorentz generators
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

The Lorentz algebra so(3, 1) reads[A1 , A2 ] = A3 ,[A2 , A3 ] = A1 ,[A3 , A1 ] = A2⇔

[Ai , Aj ] = ²ijk Ak ,[Ai , Bj ] = ²ijk Bk ,[Bi , Bj ] = −²ijk Ak .
The Ak ’s are the generators of spatial rotations, while the Bk ’s are thegenerators of boosts. They transform as vectors under spatial
rotations. It is useful to define
H3 = iA3 ,H+ = iA1 − A2 ,H− = iA1 + A2

F+ = iB1 − B2 ,F− = iB1 + B2 .

and similarly
F3 = iB3 ,10 / 25

Irreducible representations
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

11 / 25

Irreducible representations
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France

General results on irreducible (infinite-dimensional, notnecessarily unitary) representations of so(3, 1) by Gel’fand,Naimark, Harish-Chandra.

Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

11 / 25

Irreducible representations
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation

General results on irreducible (infinite-dimensional, notnecessarily unitary) representations of so(3, 1) by Gel’fand,Naimark, Harish-Chandra.
Since so(3) is a (compact) subalgebra of the Lorentz algebraso(3, 1), any representation R of so(3, 1) decomposes as a direct
sum of representations Rl of so(3),R = ⊕Rl
where l (the weight/so(3)-spin of Rl ) is an non-negative integer orhalf-integer. The sum can be infinite.

BMS4
superalgebras
Conclusions andcomments

11 / 25

Irreducible representations
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

General results on irreducible (infinite-dimensional, notnecessarily unitary) representations of so(3, 1) by Gel’fand,Naimark, Harish-Chandra.
Since so(3) is a (compact) subalgebra of the Lorentz algebraso(3, 1), any representation R of so(3, 1) decomposes as a direct
sum of representations Rl of so(3),R = ⊕Rl
where l (the weight/so(3)-spin of Rl ) is an non-negative integer orhalf-integer. The sum can be infinite.
If the representation R of so(3, 1) is irreducible, each
representation Rl of so(3) that appears in R occurs at most once,i.e., is non degenerate.

11 / 25

Irreducible representations
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

12 / 25

Irreducible representations
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction

Let l0 ≥ 0 be the lowest so(3)-spin that appears in thedecomposition of R. Then the so(3)-spins are l = l0 , l0 + 1, · · · , witha maximum value l0 + n (n integer) if the representation R is
finite-dimensional, and no upper bound if R is
infinite-dimensional.

Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

12 / 25

Irreducible representations
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

Let l0 ≥ 0 be the lowest so(3)-spin that appears in thedecomposition of R. Then the so(3)-spins are l = l0 , l0 + 1, · · · , witha maximum value l0 + n (n integer) if the representation R is
finite-dimensional, and no upper bound if R is
infinite-dimensional.
In a standard basis ξlm of Rl , one has
H3 ξlm = mξlm ,p
H− ξlm = (l + m)(l − m + 1)ξl,m−1 ,p
H+ ξlm = (l + m + 1)(l − m)ξl,m+1 ,(m = −l, −l + 1, · · · , l − 1, l). These relations are unaffected if werescale all ξlm ’s by the same m-independent factor h(l),ξlm → h(l)ξlm .

12 / 25

Irreducible representations
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

13 / 25

Irreducible representations
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France

The Bi ’s are vector operators, so that by the Wigner-Eckart
theorem, they can only change the so(3)-spin l by ±1, i.e. Bi ξlm isa linear combination of vectors ξl0 m0 with l0 = l − 1, l, l + 1.

Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

13 / 25

Irreducible representations
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras

The Bi ’s are vector operators, so that by the Wigner-Eckart
theorem, they can only change the so(3)-spin l by ±1, i.e. Bi ξlm isa linear combination of vectors ξl0 m0 with l0 = l − 1, l, l + 1.
One finds explicitly, upon suitable normalization of the ξlm ’s,ppl2 − m2 ξl−1,m − al mξl,m − cl+1 (l + 1)2 − m2 ξl+1,mpp
F+ ξlm = cl (l − m)(l − m − 1)ξl−1,m+1 − al (l − m)(l + m + 1)ξl,m+1
p+cl+1 (l + m + 1)(l + m + 2)ξl+1,m+1pp
F− ξlm = −cl (l + m)(l + m − 1)ξl−1,m−1 − al (l + m)(l − m + 1)ξl,m−1
p−cl+1 (l − m + 1)(l − m + 2)ξl+1,m−1
F3 ξlm = cll = l0 , l0 + 1, · · · ,m = −l, −l + 1, · · · , l − 1, l,

Conclusions andcomments

13 / 25

Irreducible representations
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

14 / 25

Irreducible representations
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France

where
il0 l1al =,l(l + 1)

i
cl =
l

s

(l2 − l02 )(l2 − l12 )4l2 − 1Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

14 / 25

Irreducible representations
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France

where
il0 l1al =,l(l + 1)

i
cl =
l

s

(l2 − l02 )(l2 − l12 )4l2 − 1Introduction
Results

for some l1 that can be an arbitrary complex number

Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

14 / 25

Irreducible representations
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France

where
il0 l1al =,l(l + 1)

i
cl =
l

s

(l2 − l02 )(l2 − l12 )4l2 − 1Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation

for some l1 that can be an arbitrary complex number
Thus, any representation of the Lorentz group is determined by apair of numbers (l0 , l1 ) where l0 (the lowest so(3)-spin) is a
non-negative integer or half-integer, and where l1 is an arbitrarycomplex number.

BMS4
superalgebras
Conclusions andcomments

14 / 25

Irreducible representations
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France

where
il0 l1al =,l(l + 1)

i
cl =
l

s

(l2 − l02 )(l2 − l12 )4l2 − 1Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras

for some l1 that can be an arbitrary complex number
Thus, any representation of the Lorentz group is determined by apair of numbers (l0 , l1 ) where l0 (the lowest so(3)-spin) is a
non-negative integer or half-integer, and where l1 is an arbitrarycomplex number.
It is important to realize that l0 and l1 enter symmetrically theformulas giving the coefficients al and cl . However, only l0 is
required to be a non-negative integer or half-integer.

Conclusions andcomments

14 / 25

Irreducible representations
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

15 / 25

Irreducible representations
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity

A representation with minimum so(3)spin l0 contains the weights
l0 , l0 + 1, · · · .

Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

15 / 25

Irreducible representations
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations

A representation with minimum so(3)spin l0 contains the weights
l0 , l0 + 1, · · · .
The representation is finite-dimensional if this series stops.
Assume that it stops at lmax = l0 + n for some integer n. This will
occur if and only if clmax +1 = cl0 +n+1 = 0, which will be the case if
and only if l0 + n + 1 = |l1 |. Thus, a representation isfinite-dimensional if and only if |l1 | − l0 − 1 is a non-negative
integer. |l1 | − 1 is then the highest so(3)-spin occuring in the
representation.

Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

15 / 25

Irreducible representations
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation

A representation with minimum so(3)spin l0 contains the weights
l0 , l0 + 1, · · · .
The representation is finite-dimensional if this series stops.
Assume that it stops at lmax = l0 + n for some integer n. This will
occur if and only if clmax +1 = cl0 +n+1 = 0, which will be the case if
and only if l0 + n + 1 = |l1 |. Thus, a representation isfinite-dimensional if and only if |l1 | − l0 − 1 is a non-negative
integer. |l1 | − 1 is then the highest so(3)-spin occuring in the
representation.
The representation is unitary in one of two cases :BMS4
superalgebras
Conclusions andcomments

15 / 25

Irreducible representations
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation

A representation with minimum so(3)spin l0 contains the weights
l0 , l0 + 1, · · · .
The representation is finite-dimensional if this series stops.
Assume that it stops at lmax = l0 + n for some integer n. This will
occur if and only if clmax +1 = cl0 +n+1 = 0, which will be the case if
and only if l0 + n + 1 = |l1 |. Thus, a representation isfinite-dimensional if and only if |l1 | − l0 − 1 is a non-negative
integer. |l1 | − 1 is then the highest so(3)-spin occuring in the
representation.
The representation is unitary in one of two cases :l1 pure imaginary (including 0), no restriction on l0 (“main (orprincipal) series”) ;BMS4
superalgebras
Conclusions andcomments

15 / 25

Irreducible representations
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras

A representation with minimum so(3)spin l0 contains the weights
l0 , l0 + 1, · · · .
The representation is finite-dimensional if this series stops.
Assume that it stops at lmax = l0 + n for some integer n. This will
occur if and only if clmax +1 = cl0 +n+1 = 0, which will be the case if
and only if l0 + n + 1 = |l1 |. Thus, a representation isfinite-dimensional if and only if |l1 | − l0 − 1 is a non-negative
integer. |l1 | − 1 is then the highest so(3)-spin occuring in the
representation.
The representation is unitary in one of two cases :l1 pure imaginary (including 0), no restriction on l0 (“main (orprincipal) series”) ;l0 = 0, l1 real number ∈ [−1, 1] (“supplementary series”).

Conclusions andcomments

15 / 25

Irreducible representations
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

A representation with minimum so(3)spin l0 contains the weights
l0 , l0 + 1, · · · .
The representation is finite-dimensional if this series stops.
Assume that it stops at lmax = l0 + n for some integer n. This will
occur if and only if clmax +1 = cl0 +n+1 = 0, which will be the case if
and only if l0 + n + 1 = |l1 |. Thus, a representation isfinite-dimensional if and only if |l1 | − l0 − 1 is a non-negative
integer. |l1 | − 1 is then the highest so(3)-spin occuring in the
representation.
The representation is unitary in one of two cases :l1 pure imaginary (including 0), no restriction on l0 (“main (orprincipal) series”) ;l0 = 0, l1 real number ∈ [−1, 1] (“supplementary series”).

Only the finite-dimensional trivial (scalar) representation isunitary (l0 = 0, l1 = ±1)
15 / 25

BMS4 supertranslations
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

16 / 25

BMS4 supertranslations
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity

The first example of an infinite-dimensional representation of
the Lorentz group arising in the asymptotic context is given by
the BMS4 supertranslations.

Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

16 / 25

BMS4 supertranslations
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France

The first example of an infinite-dimensional representation of
the Lorentz group arising in the asymptotic context is given by
the BMS4 supertranslations.
The supertranslations τ(xA ) are described by functions on the
2-sphere ((xA ) ≡ (θ, ϕ)) that transform under the Lorentz group as

Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

16 / 25

BMS4 supertranslations
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France

The first example of an infinite-dimensional representation of
the Lorentz group arising in the asymptotic context is given by
the BMS4 supertranslations.
The supertranslations τ(xA ) are described by functions on the
2-sphere ((xA ) ≡ (θ, ϕ)) that transform under the Lorentz group as

Introduction
Results

XY ,b τ = Y A ∂A τ − ∂A b∂A τ − bτ

Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

16 / 25

BMS4 supertranslations
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France

The first example of an infinite-dimensional representation of
the Lorentz group arising in the asymptotic context is given by
the BMS4 supertranslations.
The supertranslations τ(xA ) are described by functions on the
2-sphere ((xA ) ≡ (θ, ϕ)) that transform under the Lorentz group as

Introduction
Results

XY ,b τ = Y A ∂A τ − ∂A b∂A τ − bτ

Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation

Here, Y A is a rotation Killing vector, BA ≡ −∂A b is the conformal
Killing vector on the 2-sphere associated with the boostibi (xi ∂x∂0 + x0 ∂x∂ i ) and b = bi xr .

BMS4
superalgebras
Conclusions andcomments

16 / 25

BMS4 supertranslations
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France

The first example of an infinite-dimensional representation of
the Lorentz group arising in the asymptotic context is given by
the BMS4 supertranslations.
The supertranslations τ(xA ) are described by functions on the
2-sphere ((xA ) ≡ (θ, ϕ)) that transform under the Lorentz group as

Introduction

XY ,b τ = Y A ∂A τ − ∂A b∂A τ − bτResults
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras

Here, Y A is a rotation Killing vector, BA ≡ −∂A b is the conformal
Killing vector on the 2-sphere associated with the boostibi (xi ∂x∂0 + x0 ∂x∂ i ) and b = bi xr .
It is a multiplier representation of the type
XY ,b τ = Y A ∂A τ − ∂A b∂A τ − kbτConclusions andcomments

with k = 1.
16 / 25

BMS4 supertranslations
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

17 / 25

BMS4 supertranslations
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity

The representation is reducible.

Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

17 / 25

BMS4 supertranslations
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction

The representation is reducible.
The four-dimensional subspace of spherical harmonics with
` = 0 and ` = 1 is invariant and yields the vector representation
with l0 = 0 and l1 = 2.

Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

17 / 25

BMS4 supertranslations
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraThe representation is reducible.
The four-dimensional subspace of spherical harmonics with
` = 0 and ` = 1 is invariant and yields the vector representation
with l0 = 0 and l1 = 2.
The quotient representation is irreducible, infinite-dimensionaland characterized by l0 = 2 and l1 = 0.

BMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

17 / 25

BMS4 supertranslations
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation

The representation is reducible.
The four-dimensional subspace of spherical harmonics with
` = 0 and ` = 1 is invariant and yields the vector representation
with l0 = 0 and l1 = 2.
The quotient representation is irreducible, infinite-dimensionaland characterized by l0 = 2 and l1 = 0.
It is called the “tail” of the finite-dimensional representation with
l0 = 1 and l1 = 2. (Incidentally, it is unitary and belongs to theprincipal series.)

BMS4
superalgebras
Conclusions andcomments

17 / 25

BMS4 supertranslations
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

The representation is reducible.
The four-dimensional subspace of spherical harmonics with
` = 0 and ` = 1 is invariant and yields the vector representation
with l0 = 0 and l1 = 2.
The quotient representation is irreducible, infinite-dimensionaland characterized by l0 = 2 and l1 = 0.
It is called the “tail” of the finite-dimensional representation with
l0 = 1 and l1 = 2. (Incidentally, it is unitary and belongs to theprincipal series.)
The full representation is, however, not completely reducible (it isindecomposable) because the subspace of spherical harmonics
with ` ≥ 2 is not invariant.

17 / 25

BMS4 supertranslations and linearized theory
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

18 / 25

BMS4 supertranslations and linearized theory
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France

It is of interest to observe that the two representations havedifferent origins from the point of view of the free (Pauli-Fierz)
theory.

Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

18 / 25

BMS4 supertranslations and linearized theory
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraIt is of interest to observe that the two representations havedifferent origins from the point of view of the free (Pauli-Fierz)
theory.
The four-dimensional representation is just composed of thestandard spacetime translations, which are rigid symmetries ofthe free theory.

BMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

18 / 25

BMS4 supertranslations and linearized theory
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations

It is of interest to observe that the two representations havedifferent origins from the point of view of the free (Pauli-Fierz)
theory.
The four-dimensional representation is just composed of thestandard spacetime translations, which are rigid symmetries ofthe free theory.
The tail contains the improper gauge symmetries (linearized
diffeomorphisms) of the free theory.

Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

18 / 25

BMS4 supertranslations and linearized theory
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation

It is of interest to observe that the two representations havedifferent origins from the point of view of the free (Pauli-Fierz)
theory.
The four-dimensional representation is just composed of thestandard spacetime translations, which are rigid symmetries ofthe free theory.
The tail contains the improper gauge symmetries (linearized
diffeomorphisms) of the free theory.
They combine into the infinite-dimensional (reducible but notdecomposable) representation of the supertranslations.

BMS4
superalgebras
Conclusions andcomments

18 / 25

BMS4 supertranslations and linearized theory
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras

It is of interest to observe that the two representations havedifferent origins from the point of view of the free (Pauli-Fierz)
theory.
The four-dimensional representation is just composed of thestandard spacetime translations, which are rigid symmetries ofthe free theory.
The tail contains the improper gauge symmetries (linearized
diffeomorphisms) of the free theory.
They combine into the infinite-dimensional (reducible but notdecomposable) representation of the supertranslations.
(All symmetries are improper gauge symmetries (non trivial
diffeomorphisms) in the full theory.)

Conclusions andcomments

18 / 25

Spinor representation
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

19 / 25

Spinor representation
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France

The representation relevant to the fermionic charges is describedby spinor functions χ(θ, ϕ) on the 2-sphere.

Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

19 / 25

Spinor representation
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France

The representation relevant to the fermionic charges is describedby spinor functions χ(θ, ϕ) on the 2-sphere.
These transform under the Lorentz group as

Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

19 / 25

Spinor representation
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France

The representation relevant to the fermionic charges is describedby spinor functions χ(θ, ϕ) on the 2-sphere.
These transform under the Lorentz group as

Introduction
Results
Representationsof the Lorentz
algebra1δξ χ = −ξ⊥ γ0 γm ∂m χ + ∂j ξ⊥ γj γ0 χ + L ξk χ
2

BMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

19 / 25

Spinor representation
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France

The representation relevant to the fermionic charges is describedby spinor functions χ(θ, ϕ) on the 2-sphere.
These transform under the Lorentz group as

Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations1δξ χ = −ξ⊥ γ0 γm ∂m χ + ∂j ξ⊥ γj γ0 χ + L ξk χ
2where ξ⊥ = bi xi (boosts) and ξk is a rotation Killing vector.

Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

19 / 25

Spinor representation
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France

The representation relevant to the fermionic charges is describedby spinor functions χ(θ, ϕ) on the 2-sphere.
These transform under the Lorentz group as

Introduction
Results
Representationsof the Lorentz
algebra1δξ χ = −ξ⊥ γ0 γm ∂m χ + ∂j ξ⊥ γj γ0 χ + L ξk χ
2

BMS4
supertranslations

where ξ⊥ = bi xi (boosts) and ξk is a rotation Killing vector.

Infinitedimensional
spinorrepresentation

The parity of χ is preserved by the Lorentz transformations. The
representation that contains the constant spinors is described byeven χ’s and we take therefore χ to be even.

BMS4
superalgebras
Conclusions andcomments

19 / 25

Spinor representation
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

20 / 25

Spinor representation
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity

The spinor representation is reducible but not decomposable.

Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

20 / 25

Spinor representation
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France

The spinor representation is reducible but not decomposable.
As in the bosonic case, there is a finite-dimensional “head”,
corresponding to an invariant subspace,

Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

20 / 25

Spinor representation
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction

The spinor representation is reducible but not decomposable.
As in the bosonic case, there is a finite-dimensional “head”,
corresponding to an invariant subspace,
here, the spin- 12 representation of the Lorentz groupcharacterized by l0 = 12 and l1 = 23 ,Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

20 / 25

Spinor representation
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraThe spinor representation is reducible but not decomposable.
As in the bosonic case, there is a finite-dimensional “head”,
corresponding to an invariant subspace,
here, the spin- 12 representation of the Lorentz groupcharacterized by l0 = 12 and l1 = 23 ,and an infinite-dimensional tail characterized by the dual values
l0 = 23 and l1 = 12 .

BMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

20 / 25

Spinor representation
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation

The spinor representation is reducible but not decomposable.
As in the bosonic case, there is a finite-dimensional “head”,
corresponding to an invariant subspace,
here, the spin- 12 representation of the Lorentz groupcharacterized by l0 = 12 and l1 = 23 ,and an infinite-dimensional tail characterized by the dual values
l0 = 23 and l1 = 12 .
From the point of view of the subalgebra so(3), the representation
decomposes as the infinite direct sum
D1 ⊕D3 ⊕D5 ⊕···
2

2

2

BMS4
superalgebras
Conclusions andcomments

20 / 25

Spinor representation
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

The spinor representation is reducible but not decomposable.
As in the bosonic case, there is a finite-dimensional “head”,
corresponding to an invariant subspace,
here, the spin- 12 representation of the Lorentz groupcharacterized by l0 = 12 and l1 = 23 ,and an infinite-dimensional tail characterized by the dual values
l0 = 23 and l1 = 12 .
From the point of view of the subalgebra so(3), the representation
decomposes as the infinite direct sum
D1 ⊕D3 ⊕D5 ⊕···
2

2

2

but as a representation of the Lorentz algebra, it isindecomposable.

20 / 25

Spinor representation
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

21 / 25

Spinor representation
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France

The head and the tail have also different origins from the point ofview of the free theory.

Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

21 / 25

Spinor representation
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results

The head and the tail have also different origins from the point ofview of the free theory.
The head comes from the rigid supersymmetries of the freemultiplet ( 23 , 2),

Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

21 / 25

Spinor representation
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraThe head and the tail have also different origins from the point ofview of the free theory.
The head comes from the rigid supersymmetries of the freemultiplet ( 23 , 2),while the tail originates from the improper gauge symmetries ofthe free Rarita-Schwinger theory.

BMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

21 / 25

Spinor representation
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation

The head and the tail have also different origins from the point ofview of the free theory.
The head comes from the rigid supersymmetries of the freemultiplet ( 23 , 2),while the tail originates from the improper gauge symmetries ofthe free Rarita-Schwinger theory.
Both are improper gauge symmetries in the full theory andcombine to provide the infinite-dimensional spinor
representation.

BMS4
superalgebras
Conclusions andcomments

21 / 25

Closure of fermionic symmetries
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

22 / 25

Closure of fermionic symmetries
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France

As mentioned above, the fermionic symmetries close on ordinarytranslations only, without supertranslations.

Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

22 / 25

Closure of fermionic symmetries
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraAs mentioned above, the fermionic symmetries close on ordinarytranslations only, without supertranslations.
Symbolically,[HeadF , HeadF ] = HeadB ,[HeadF , TailF ] = 0,
[TailF , TailF ] = 0,

BMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

22 / 25

Closure of fermionic symmetries
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraAs mentioned above, the fermionic symmetries close on ordinarytranslations only, without supertranslations.
Symbolically,[HeadF , HeadF ] = HeadB ,[HeadF , TailF ] = 0,
[TailF , TailF ] = 0,

BMS4
supertranslations
Infinitedimensional
spinorrepresentation

There is no “square root” of the pure supertranslations.

BMS4
superalgebras
Conclusions andcomments

22 / 25

Closure of fermionic symmetries
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraAs mentioned above, the fermionic symmetries close on ordinarytranslations only, without supertranslations.
Symbolically,[HeadF , HeadF ] = HeadB ,[HeadF , TailF ] = 0,
[TailF , TailF ] = 0,

BMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

There is no “square root” of the pure supertranslations.
Could there be a deformation of this superalgebra such that pure
BMS supertranslations are generated through the bracket of two
fermionic symmetries ?

22 / 25

Closure of fermionic symmetries
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

23 / 25

Closure of fermionic symmetries
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity

The tensor product of two infinite-dimensional spinor
representations is highly reducible.

Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

23 / 25

Closure of fermionic symmetries
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France

The tensor product of two infinite-dimensional spinor
representations is highly reducible.
According to the Jacobi identity, two fermionic symmetries
should close on a subrepresentation of this tensor product.

Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

23 / 25

Closure of fermionic symmetries
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebra

The tensor product of two infinite-dimensional spinor
representations is highly reducible.
According to the Jacobi identity, two fermionic symmetries
should close on a subrepresentation of this tensor product.
Now, the decomposition of the tensor productHeadF ⊗ (HeadF ⊕σ TailF ) can be performed, and contains exactly
once the reducible, indecomposable vector representation (inaddition to another infinite-dimensional representation).

BMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

23 / 25

Closure of fermionic symmetries
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations

The tensor product of two infinite-dimensional spinor
representations is highly reducible.
According to the Jacobi identity, two fermionic symmetries
should close on a subrepresentation of this tensor product.
Now, the decomposition of the tensor productHeadF ⊗ (HeadF ⊕σ TailF ) can be performed, and contains exactly
once the reducible, indecomposable vector representation (inaddition to another infinite-dimensional representation).
So, it would be algebraically consistent to have

Infinitedimensional
spinorrepresentation[HeadF , HeadF ⊕σ TailF ] ∼ HeadB ⊕σ TailB )BMS4
superalgebras

even though this is not the asymptotic algebra found in our case.

Conclusions andcomments

23 / 25

Closure of fermionic symmetries
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations

The tensor product of two infinite-dimensional spinor
representations is highly reducible.
According to the Jacobi identity, two fermionic symmetries
should close on a subrepresentation of this tensor product.
Now, the decomposition of the tensor productHeadF ⊗ (HeadF ⊕σ TailF ) can be performed, and contains exactly
once the reducible, indecomposable vector representation (inaddition to another infinite-dimensional representation).
So, it would be algebraically consistent to have

Infinitedimensional
spinorrepresentation[HeadF , HeadF ⊕σ TailF ] ∼ HeadB ⊕σ TailB )BMS4
superalgebras

even though this is not the asymptotic algebra found in our case.

Conclusions andcomments

The decomposition of TailF ⊗ TailF is harder to compute.

23 / 25

Conclusions and comments
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

24 / 25

Conclusions and comments
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France

One can introduce boundary conditions on the supergravity
fields at spatial infinity in such a way that (in the asymptoticallyflat context)

Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

24 / 25

Conclusions and comments
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France

One can introduce boundary conditions on the supergravity
fields at spatial infinity in such a way that (in the asymptoticallyflat context)
the asymptotic symmetry group is a graded extension of the
BMS4 algebra.

Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

24 / 25

Conclusions and comments
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results

One can introduce boundary conditions on the supergravity
fields at spatial infinity in such a way that (in the asymptoticallyflat context)
the asymptotic symmetry group is a graded extension of the
BMS4 algebra.
Two consistent sets of boundary conditions have been given.

Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

24 / 25

Conclusions and comments
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations

One can introduce boundary conditions on the supergravity
fields at spatial infinity in such a way that (in the asymptoticallyflat context)
the asymptotic symmetry group is a graded extension of the
BMS4 algebra.
Two consistent sets of boundary conditions have been given.
The first yields the “small” graded extension sBMS4 of the BMS4
algebra of Awada et al, with a finite number of fermionic
generators.

Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

24 / 25

Conclusions and comments
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation

One can introduce boundary conditions on the supergravity
fields at spatial infinity in such a way that (in the asymptoticallyflat context)
the asymptotic symmetry group is a graded extension of the
BMS4 algebra.
Two consistent sets of boundary conditions have been given.
The first yields the “small” graded extension sBMS4 of the BMS4
algebra of Awada et al, with a finite number of fermionic
generators.
The second yields a much bigger graded extension SBMS4 of the
BMS4 algebra, with an infinite number of fermionic generators.

BMS4
superalgebras
Conclusions andcomments

24 / 25

Conclusions and comments
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

One can introduce boundary conditions on the supergravity
fields at spatial infinity in such a way that (in the asymptoticallyflat context)
the asymptotic symmetry group is a graded extension of the
BMS4 algebra.
Two consistent sets of boundary conditions have been given.
The first yields the “small” graded extension sBMS4 of the BMS4
algebra of Awada et al, with a finite number of fermionic
generators.
The second yields a much bigger graded extension SBMS4 of the
BMS4 algebra, with an infinite number of fermionic generators.
Both contain the BMS4 algebra and the super-Poincaré algebra,
with [HeadF , HeadF ] = HeadB , but in all cases, these are the only
non-trivial brackets of two fermionic generators : the puresupertranslations have no square root.
24 / 25

Conclusions and comments
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

25 / 25

Conclusions and comments
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations

Since there is no algebraic obstruction to taking the square rootof pure BMS supertranslations, it would be interesting to explore
whether the corresponding deformed superalgebra could be
dynamically realized by deforming the asymptotic conditions
and the asymptotic symmetries.
Same techniques on the sphere at spatial infinity as on thecelestial sphere(s).

Infinitedimensional
spinorrepresentation
BMS4
superalgebras
Conclusions andcomments

25 / 25

Conclusions and comments
Super-BMS
algebras and theasymptotic
structure ofsupergravity at
spatial infinity
Marc Henneaux
Université Librede Bruxelles &Collège de France
Introduction
Results
Representationsof the Lorentz
algebraBMS4
supertranslations
Infinitedimensional
spinorrepresentation
BMS4
superalgebras

Since there is no algebraic obstruction to taking the square rootof pure BMS supertranslations, it would be interesting to explore
whether the corresponding deformed superalgebra could be
dynamically realized by deforming the asymptotic conditions
and the asymptotic symmetries.
Same techniques on the sphere at spatial infinity as on thecelestial sphere(s).

THANK YOU !

Conclusions andcomments

25 / 25