Gauged supergravity and E 10

Gauged supergravity and

E

₁₀

Jakob Palmkvist

Albert-Einstein-Institut

in collaboration with

Eric Bergshoeff, Olaf Hohm, Axel Kleinschmidt,

Hermann Nicolai and Teake Nutma

! ! ! ! !

!

!

Three-dimensional maximal supergravity

is globally invariant under

E

₈

! ! ! ! !

!

!

!

! ! ! ! !

!

!

!

! ! ! ! !

!

!

!

!

Three-dimensional maximal supergravity

Can we unify these symmetries into

E

₁₀

?

! ! ! ! !

!

!

!

! ! ! ! !

!

!

!

!

Three-dimensional maximal supergravity

is globally invariant under

E

₈

and

SL

(2

,

R

)

.

!

(

Damour,

Henneaux,

Nicolai

2002

)

(

Julia

1983

)

Can we unify these symmetries into

E

₁₀

?

What if we promote a subgroup of

E

₈

to

a local symmetry?

! ! ! ! !

!

!

!

! ! ! ! !

!

!

!

!

Three-dimensional maximal supergravity

is globally invariant under

E

₈

and

SL

(2

,

R

)

.

!

!

Gauged maximal supergravity

in three dimensions

!

Gauged maximal supergravity

in three dimensions

!

The

E

₁₀

/K

(E

₁₀

) coset model

Outline

!

Gauged maximal supergravity

in three dimensions

!

The

E

₁₀

/K

(E

₁₀

) coset model

!

Comparison between the two theories

Outline

Consider the bosonic sector of

maximal (

N

= 16) supergravity

in three dimensions:

Consider the bosonic sector of

maximal (

N

= 16) supergravity

in three dimensions:

•

Pure gravity does not propagate

•

Consider the bosonic sector of

maximal (

N

= 16) supergravity

in three dimensions:

•

Pure gravity does not propagate

•

Vectors are dual to scalars

•

•

Consider the bosonic sector of

maximal (

N

= 16) supergravity

in three dimensions:

•

Pure gravity does not propagate

•

Vectors are dual to scalars

⇒

All propagating degrees of freedom

of supergravity are scalars.

•

•

Consider the bosonic sector of

maximal (

N

= 16) supergravity

in three dimensions:

L

=

1

₄

e

R

₋

η

αβ

η

_AB

P

_α

A

P

_β

B

A

,

_B

= 1,

2, . . . ,

248

α,

β

= 0

,

1

,

2

The Maurer-Cartan form of the

gravity sector decomposes into its

symmetric and antisymmetric parts:

The Maurer-Cartan form of the

scalar sector decomposes into

a spinor and the adjoint of

SO

(16):

(

E

−

1

∂

_µ

E

)

A

_→

P

_µ

A

,

Q

_µ

IJ

The Maurer-Cartan form of the

gravity sector decomposes into its

symmetric and antisymmetric parts:

The Maurer-Cartan form of the

scalar sector decomposes into

a spinor and the adjoint of

SO

(16):

248

_→

128

+

120

The Maurer-Cartan form of the

scalar sector decomposes into

a spinor and the adjoint of

SO

(16):

A

,

_B

, . . .

= 1

,

2

, . . . ,

248

The Maurer-Cartan form of the

scalar sector decomposes into

a spinor and the adjoint of

SO

(16):

I, J, . . .

= 1

,

2

, . . . ,

16

A, B, . . .

= 1

,

2

, . . . ,

128

248

_→

128

+

120

The scalar sector is described by an

element

E

of the coset

E

₈

/SO

(16), and

is invariant under the transformations

E

(

x

)

_→

g

E

(

x

)

h

(

x

)

h(x)

_∈

SO

(16)

E

(

x

)

_→

g

(

x

)

E

(

x

)

h

(

x

)

g

(

x

)

_∈

G

_⊆

E

₈

Gauging the theory:

!

Promote a subgroup

G

_⊆

E

₈

to a local symmetry

h(x)

_∈

SO

(16)

E

(

x

)

_→

g

(

x

)

E

(

x

)

h

(

x

)

g

(

x

)

_∈

G

_⊆

E

₈

!

Promote a subgroup

G

_⊆

E

₈

to a local symmetry

Gauging the theory:

!

This is done by dualizing

the scalars to vectors

A

_µ

A

.

Gauging the theory:

•

Replace partial derivatives

with gauge covariant ones

P

_µ

A

= (

E

−

1

∂

_µ

E

)

A

_→

(

E

−

1

_D

_µ

E

)

A

Gauging the theory:

•

Replace partial derivatives

with gauge covariant ones

•

Add a potential term

L

V

P

_µ

A

= (

E

−

1

∂

_µ

E

)

A

_→

(

E

−

1

_D

_µ

E

)

A

•

Gauging the theory:

•

Replace partial derivatives

with gauge covariant ones

•

Add a potential term

L

V

P

_µ

A

= (

E

−

1

∂

_µ

E

)

A

_→

(

E

−

1

_D

_µ

E

)

A

•

We can keep an

E

₈

covariant notation

by using the

embedding tensor

Θ

_MN

,

the projection of the

Killing form

η

_MN

We can keep an

E

₈

covariant notation

by using the

embedding tensor

Θ

_MN

,

the projection of the

Killing form

η

_MN

onto the gauge group

G

_⊆

E

₈

.

We can keep an

E

₈

covariant notation

by using the

embedding tensor

Θ

_MN

,

the projection of the

Killing form

η

_MN

onto the gauge group

G

_⊆

E

₈

.

Thus it has two symmetric

E

₈

indices.

The symmetric product of

We can keep an

E

₈

covariant notation

by using the

embedding tensor

Θ

_MN

,

the projection of the

Killing form

η

_MN

onto the gauge group

G

_⊆

E

₈

.

Thus it has two symmetric

E

₈

indices.

The symmetric product of

two adjoint

E

₈

representations:

We can keep an

E

₈

covariant notation

by using the

embedding tensor

Θ

_MN

,

the projection of the

Killing form

η

_MN

onto the gauge group

G

_⊆

E

₈

.

Θ

_MN

_∈

1

+

3875

The condition for supersymmetry:

Thus it has two symmetric

E

₈

indices.

Θ

_MN

=

η

_MN

θ

+ ˜

Θ

_MN

Θ

_MN

_∈

1

+

3875

We divide the embedding tensor

into its irreducible parts:

˜

Θ

_MN

3875

1

θ

We divide the embedding tensor

into its irreducible parts:

˜

Θ

_MN

3875

1

θ

We divide the embedding tensor

into its irreducible parts:

˜

Θ

_MN

_∈

3875

θ

_∈

1

We divide the embedding tensor

into its irreducible parts:

˜

Θ

_MN

_∈

3875

θ

_∈

1

˜

T

_AB

(

x

) =

E

_A

M

(

x

)

E

_B

N

(

x

) ˜

Θ

_MN

We divide the embedding tensor

into its irreducible parts:

L

V

=

−

e

g

2

112

(3 ˜

T

AB

T

˜

AB

+ ˜

T

A IJ

T

˜

A IJ

L

CS

=

−

g

₄

ε

µ

νρ

Θ

_MN

A

µ

M

∂

ν

A

ρ

N

−

g

₁₂

2

ε

µ

νρ

Θ

_MN

Θ

_PQ

f

MP

_R

A

_µ

N

A

_ν

Q

A

_ρ

R

L

V

=

−

e

g

2

112

(3 ˜

T

AB

T

˜

AB

+ ˜

T

A IJ

T

˜

A IJ

L

CS

=

−

g

₄

ε

µ

νρ

Θ

_MN

A

µ

M

∂

ν

A

ρ

N

−

g

₁₂

2

ε

µ

νρ

Θ

_MN

Θ

_PQ

f

MP

_R

A

_µ

N

A

_ν

Q

A

_ρ

R

L

V

=

−

e

g

2

112

(3 ˜

T

AB

T

˜

AB

+ ˜

T

A IJ

T

˜

A IJ

−

T

˜

_{IJ KL}

T

˜

_{IJ KL}

) + 2

e

g

2

θ

2

To compare with the

E

₁₀

model,

we must choose

. . .

To compare with the

E

₁₀

model,

we must choose

. . .

•

An ADM-like split of the dreibein:

e

_µ

α

=





N

₀

0 0

0





To compare with the

E

₁₀

model,

we must choose

. . .

•

An ADM-like split of the dreibein:

•

A temporal gauge for the vector fields:

e

_µ

α

=





N

₀

0 0

0





e

_m

a

[

e, f

] =

h

[h, f

] =

₋

2f

[

h, e

] = 2

e

[

e, f

] =

h

[h, f

] =

₋

2f

[

h, e

] = 2

e

Matrix realization:

e

=

!

0 1

0 0

"

f

=

!

0 0

1 0

"

h

=

!

1 0

0

₋

1

"

The Lie algebra of

SL(2,

_R

):

A

Kac-Moody algebra

g

of rank

_r

is

generated by

r

copies of

SL

(2

,

R

),

A

Kac-Moody algebra

g

of rank

_r

is

generated by

r

copies of

SL

(2

,

R

),

!

!

i

j

!

!

i

j

A

Kac-Moody algebra

g

of rank

_r

is

generated by

r

copies of

SL

(2

,

R

),

!

!

i

j

!

!

i

j

[

h

_i

, e

_j

] = 0

[

h

_i

, f

_j

] = 0

[

e

_i

, e

_j

] = 0

[

f

_i

, f

_j

] = 0

[

h

_i

, e

_j

] =

₋

e

_j

[

h

_i

, f

_j

] =

f

_j

[

e

_i

, e

_j

]

_"

= 0

[

f

_i

, f

_j

]

_"

= 0

[

h

_i

, h

_j

] = [

e

_i

, f

_j

] = 0

A

Kac-Moody algebra

g

of rank

_r

is

generated by

r

copies of

SL

(2

,

R

),

!

!

i

j

!

!

i

j

[

h

_i

, e

_j

] = 0

[

h

_i

, f

_j

] = 0

[

e

_i

, e

_j

] = 0

[

f

_i

, f

_j

] = 0

A

Kac-Moody algebra

g

of rank

_r

is

generated by

r

copies of

SL

(2

,

R

),

modulo the

Chevalley-Serre relations:

[[

e

_i

, e

_j

]

, e

_j

] = 0

[[f

_i

, f

_j

], f

_j

] = 0

[

h

_i

, e

_j

] =

₋

e

_j

The Kac-Moody algebra

g

is spanned

by all multiple (k

_≥

1) commutators

[

_{· · ·}

[[

e

_i

₁

, e

_i

₂

]

, e

_i

₃

]

, . . . , e

_i

_k

]

[

_{· · ·}

[[

f

_i

₁

, f

_i

₂

]

, f

_i

₃

]

, . . . , f

_i

_k

]

!

For any

i

= 1

,

2

, . . . , r

, we can write

where each subspace

g

_!

is spanned

by all multiple commutators at

level

!

= number of

e

_i

₋

number of

f

_i

.

!

For any

i

= 1

,

2

, . . . , r

, we can write

where each subspace

g

_!

is spanned

by all multiple commutators at

level

!

= number of

e

_i

₋

number of

f

_i

.

g

=

_{· · ·}

+

g

₋

₁

+

g

₀

+

g

₁

+

_{· · ·}

!

For any

i

= 1

,

2

, . . . , r

, we can write

where each subspace

g

_!

is spanned

by all multiple commutators at

level

!

= number of

e

_i

₋

number of

f

_i

.

g

=

_{· · ·}

+

g

₋

₁

+

g

₀

+

g

₁

+

_{· · ·}

!

The

maximal compact subalgebra

K

(

g

)

of a Kac-Moody algebra

g

is invariant

!

The

maximal compact subalgebra

K

(

g

)

of a Kac-Moody algebra

g

is invariant

under the

Chevalley involution

ω

(

e

_i

) =

₋

f

_i

ω

(

f

_i

) =

₋

e

_i

ω

(

h

_i

) =

₋

h

_i

!

The

maximal compact subalgebra

K

(

g

)

of a Kac-Moody algebra

g

is invariant

under the

Chevalley involution

ω

(

e

_i

) =

₋

f

_i

ω

(

f

_i

) =

₋

e

_i

ω

(

h

_i

) =

₋

h

_i

!

It is generated by all elements

e

_i

₋

f

_i

modulo the Chevalley-Serre relations.

where

_Q

_∈

K

(

E

₁₀

) and

_"

_P|Q

_#

= 0.

!

Consider the

E

₁₀

/K

(E

₁₀

) coset model

where

_Q

_∈

K

(

E

₁₀

) and

_"

_P|Q

_#

= 0.

!

The local

K

(

E

₁₀

) invariance makes it

possible to choose the

Borel gauge

:

!

Consider the

E

₁₀

/K

(E

₁₀

) coset model

!

Consider the

E

₁₀

/K

(E

₁₀

) coset model

where

_Q

_∈

K

(

E

₁₀

) and

_"

_P|Q

_#

= 0.

!

The local

K

(

E

₁₀

) invariance makes it

possible to choose the

Borel gauge

:

Level decomposition of

E

₁₀

under

SL

(2

,

R

)

_×

E

₈

up to level

!

= 2:

! ! ! ! !

!

!

!

!

!

Level decomposition of

E

₁₀

under

SL

(2

,

R

)

_×

E

₈

up to level

!

= 2:

Level

SL

(2

,

R

)

_×

E

₈

Components

!

representation

of

_P

+

_Q

0

(

1

_⊕

3

,

1

)

P

_ab

, Q

_ab

(

1

,

248

)

P

A

, Q

IJ

1

(

2

,

248

)

P

_a

A

Level decomposition of

E

₁₀

under

SL

(2

,

R

)

_×

E

₈

up to level

!

= 2:

Level

SL

(2

,

R

)

_×

E

₈

Components

!

representation

of

_P

+

_Q

0

(

1

_⊕

3

,

1

)

P

_ab

, Q

_ab

(

1

,

248

)

P

A

, Q

IJ

1

(

2

,

248

)

P

_a

A

2

(

1

,

1

)

P

Q

_ab

P

A

P

A

P

_ab

Q

_ab

P

A

P

P

_a

A

P

_a

IJ

P

_ab

Q

_ab

P

A

P

P

_a

A

P

_a

IJ

P

_ab

Q

IJ

P

_ab

(

t

)

_≡

P

_ab

(

t,

x

₀

)

Q

_ab

(

t

)

_≡

Q

_ab

(

t,

x

₀

)

P

A

(

t

)

_≡

P

_t

A

(

t,

x

₀

)

Q

IJ

(

t

)

_≡

Q

_t

IJ

(

t,

x

₀

)

P

_a

A

(

t

)

_≡

N

ε

_ab

P

_b

A

(

t,

x

₀

)

P

_a

IJ

(

t

)

_{≡ −}

N

ε

_ab

Q

_b

IJ

(

t,

x

₀

)

P

(

t

)

_{≡ −}

N g

θ

(

t,

x

₀

)

P

AB

(

t

)

_{≡ −}

₂₈

1

N g

T

˜

_AB

(

t,

x

₀

)

The dictionary of the

With this identification,

the equations of motion

coincide, up to

. . .

With this identification,

the equations of motion

coincide, up to

. . .

With this identification,

the equations of motion

coincide, up to

. . .

•

The gauge choices on both sides

With this identification,

the equations of motion

coincide, up to

. . .

•

The gauge choices on both sides

•

Higher order spatial derivatives

With this identification,

the equations of motion

coincide, up to

. . .

•

The gauge choices on both sides

•

Higher order spatial derivatives

•

Higher levels in the decomposition

Open questions:

Open questions:

!

No interpretation of

P

_ab

A

•

Second order spatial gradient?

Open questions:

!

No interpretation of

P

_ab

A

•

Second order spatial gradient?

•

Trombone gauging?

Open questions:

!

No interpretation of

P

_ab

A

•

Second order spatial gradient?

•

Trombone gauging?

!

How the reconcile the Killing form

with the indefinite potential?

Open questions:

!

No interpretation of

P

_ab

A

•

Second order spatial gradient?

•

Trombone gauging?

!

How the reconcile the Killing form

with the indefinite potential?