Kaluza-Klein and Coset Spaces

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Kaluza-Klein and Coset Spaces

Alexander Bols

June 17, 2014

Alexander Bols Kaluza-Klein and Coset Spaces June 17, 2014 1 / 16
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Overview

Compactification on Tori.

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Overview


T2 reduction of supergravity : the resulting scalars and theirsymmetries

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Overview



Extension to Tn.

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Overview



Extension to Tn.

Summary of symmetry groups and their isotropy groups.

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Compactification on tori

Consider gravity in Ddimensions. The metric has D2 components

gij=gij(x1, , xD). (1)



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We can obtain a field theory on a D 1 dimensional space by

compactifying one coordinate on T1.

xD xD+ 2LZ (2)

with L the radius of the torus.

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We can obtain a field theory on a D 1 dimensional space by

compactifying one coordinate on T1.

xD xD+ 2LZ (2)

with L the radius of the torus.

Now the fields can be written as a Fourier series

gij(x1, , xD) =n

g(n)ij (x1, , xD1)e

inxD/L (3)

with L the circumference of the torus.Alexander Bols Kaluza-Klein and Coset Spaces June 17, 2014 3 / 16
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The terms with n= 0 correspond to massive fields with

M n/L. (4)

We assume L to be tiny, so the n = 0 modes can be ignored.

We end up with D2 fields

g(0)ij =g

(0)ij (x1, , xD1) (5)

in D 1 dimensions.



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M n/L. (4)



g(0)ij =g

(0)ij (x1, , xD1) (5)

in D 1 dimensions.

The fields g

(0)

i,D form a vector field.

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M n/L. (4)



g(0)ij =g

(0)ij (x1, , xD1) (5)

in D 1 dimensions.

The fields g

(0)

i,D form a vector field.The field g

(0)D,D is a scalar field.

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M n/L. (4)



g(0)ij =g

(0)ij (x1, , xD1) (5)

in D 1 dimensions.

The fields g

(0)

i,D form a vector field.The field g

(0)D,D is a scalar field.

The remaining (D 1)2 components form a metric tensor for theD 1 dimensional space.

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T2 scalar sector

The same procedure can be repeated to obtain ever lower dimensionaltheories of gravity couples to more and more matter fields.

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T2 scalar sector


We look at supergravity in 11 dimensions.

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T2 scalar sector



Some of those matter fields are scalars. The scalar sector resulting

from compactification on T2 is

L = 1

2()2

1

2()2

1

2e2()2. (6)

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T2 scalar sector



Some of those matter fields are scalars. The scalar sector resulting

from compactification on T2 is

L = 1

2()2

1

2()2

1

2e2()2. (6)

The field only appears in the first term. It is decoupled from theothers and has a shift symmetry R.

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SL(2,R) symmetry

The other two fields are described by

L = 1

2()2

1

2e2()2 =

2()2 (7)

where =+ie

.

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SL(2,R) symmetry


L = 1

2()2

1

2e2()2 =

2()2 (7)

where =+ie

.It can be checked that this Lagrangian is invariant under

a+b

c+d (8)

with ad bc= 1.

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SL(2,R) symmetry


L = 1

2()2

1

2e2()2 =

2()2 (7)

where =+ie

.It can be checked that this Lagrangian is invariant under

a+b

c+d (8)

with ad bc= 1.

These transformations from a group isomorphic to SL(2,R).

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Lagrangian emerges from the symmetry group

The generators of sl(2,R) have 2 2 representation

H=

1 00 1

; E+ =

0 10 0

; E=

0 01 0

. (9)

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H=

1 00 1

; E+ =

0 10 0

; E=

0 01 0

. (9)

Consider the matrix

V=eHeE+ =e/2 e/2

0 e/2. (10)

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H=

1 00 1

; E+ =

0 10 0

; E=

0 01 0

. (9)

Consider the matrix

V=eHeE+ =e/2 e/2

0 e/2. (10)

The Lagrangian can then be written as

L =14

Tr M1M = 12

()2 12e2()2 (11)

wereM = VTV. (12)

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SL(2,R) symmetry is manifest

This Lagrangian is manifestly invariant underV V = V (13)

with SL(2,R).

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This Lagrangian is manifestly invariant under

V V = V (13)

with SL(2,R).

The new matrix V

should encode transformed scalar fields. To seewhat the new scalars are, we put V back in upper triangular form bymultiplying with a uniqueorthogonal matrix

V = OV = OV. (14)

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This Lagrangian is manifestly invariant under

V V = V (13)

with SL(2,R).

The new matrix V

should encode transformed scalar fields. To seewhat the new scalars are, we put V back in upper triangular form bymultiplying with a uniqueorthogonal matrix

V = OV = OV. (14)

Again, the Lagrangian is manifestly invariant under thistransformation.

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Scalar moduli space is coset

The group SL(2,R) represented by acts transitively on the scalarmanifold at a fixed point in spacetime

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But there are many that lead to the same transformed fields. Wehave to compensate with a Orthogonal matrix to get a uniquetransformation connecting any two points on the scalar manifold.

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Therefore, each point on the scalar manifold can be identified withthe unique transformation that maps this point to the origin.

scalar manifold SL(2,R)/O(2). (15)

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Therefore, each point on the scalar manifold can be identified withthe unique transformation that maps this point to the origin.

scalar manifold SL(2,R)/O(2). (15)

Note that O(2) is the maximally compact subgroup ofSL(2,R)

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Generalization to Tn

For compactification on Tn

we get a scalar sector of the form

L = 1

2d

1

2

i


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L = 1

2d

1

2

i


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L = 1

2d

1

2

i


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Dilaton vectors are positive roots

In the Iwasawa decomposition, gHgNgeneralizes our V and gKgeneralizes the orthogonal transformation.


Dil i i
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Vis a representative of the coset G/K


Dil i i
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As in the case ofT2, we find that the scalar Lagrangian can be

written in terms ofV

L =1

4Tr (V#V)1(V#V). (17)


Dil t t iti t
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As in the case ofT2, we find that the scalar Lagrangian can be

written in terms ofV

L =1

4Tr (V#V)1(V#V). (17)

The dilaton vectors bij and aijkarise from the commutators ofelements the Cartan subalgebra and positive root generators. Theyare precisely the positive roots ofG!


F L i t t
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From Lagrangian to symmetry group

We now proceed as follows

Find a set of dilaton vectors that can serve as simple roots ofG.


F L g gi t s t g
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Construct the corresponding Dynkin diagram.


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Construct the corresponding Dynkin diagram.Gis the normal real form corresponding to the diagram.


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Construct the corresponding Dynkin diagram.Gis the normal real form corresponding to the diagram.

Kis the maximally compact subgroup ofG.


The simple roots
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The simple roots

The dilaton vectors have a very simple structure.

The simple roots are bi,i+1 and a123.


The simple roots
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The simple roots


The simple roots are bi,i+1 and a123.

All simple roots have length 2.


The simple roots
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The simple roots


The simple roots are bi,i+1 and a123.All simple roots have length 2.

The bij form a chain

bi,i+1 bi+1,i+2 = 2 (18)


The simple roots
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The simple roots


The simple roots are bi,i+1 and a123.All simple roots have length 2.

The bij form a chain

bi,i+1 bi+1,i+2 = 2 (18)

and a123 connects to b34 only

a123 bi,i+1 = 2i,3 (19)

i.e. for n= 3, the root a123 is disconnected from the diagram,otherwise it connects to b34.


The Dynkin diagrams
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The Dynkin diagramsn Dynkin diagram algebra

2

b12

E1 =A1 =sl

(2,R

)3

b12 b23 a123 E3 =A1 A2 = sl(2,R) sl(3,R)

4

b12 b23

b34 a123 E4 =A4 = sl(5,R)

5

b12 b23

b34 b45

a123

E5=D5

6

b12 b23

b34 b45

b56

a123

E6

7

b12 b23

b34 b45

b56 b67

a123

E7

8

b12 b23

b34 b45

b56 b67

b78

a123

E8


The corresponding groups
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The corresponding groups

(see course notes, Table 3)

n algebra G K

2 A1 GL(2) O(2)3 A1 A2 SL(3) SL(2) SO(3) SO(2)

4 A4 SL(5) SO(5)5 D5 O(5, 5) O(5) O(5)6 E6 E6(+6) USp(8)7 E7 E7(+7) SU(8)

8 E8 E8(+8) SO(16)

These are the global symmetry groups (G) and their isotropy groups (K)of the scalar sectors resulting from compactification on tori Tn.


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Kaluza-Klein and Coset Spaces