Tits-Satake projections of homogeneous specialgeometries

Jan Rosseel (ITF, K. U. Leuven)

Based on: P. Fre, F. Gargiulo, J. R., K. Rulik, M. Trigiante, A. Van Proeyen, hep-th/0606173

DESY, 21/02/2007

Outline

1. Introduction and motivation

2. Introduction and motivation

3. N = 2 homogeneous special geometry

4. The TS-projection for symmetric spaces

5. The TS-projection for general solvable algebrasThe paint groupExtra structure

6. Properties

7. Summary and conclusions

Introduction and motivation

• Time dependent solutions of SUGRA : reduce to 3 dimensions→non-linear sigma model (coupled to gravity)

gij(φ)∂µφi∂µφj (1)

• Purely time dependent solutions, e.o.m.’s are geodesic equations.• E.g. : Maximal SUGRA : in d = 3 the scalars lie in the coset

E8(8)/ SO(16). The e.o.m.’s are integrable. (Fre, Gargiulo, Rulik, Trigiante)• More generally : for maximally non-compact cosets, the resulting e.o.m.’s

are integrable. (Fre, Sorin)• For non-maximally non-compact cosets, this is no longer the case. →

TS-projection gives a way of studying these solutions. Relevant fornon-maximal supergravity.

Introduction and motivation

• Time dependent solutions of SUGRA : reduce to 3 dimensions→non-linear sigma model (coupled to gravity)

gij(φ)∂µφi∂µφj (1)

• Purely time dependent solutions, e.o.m.’s are geodesic equations.• E.g. : Maximal SUGRA : in d = 3 the scalars lie in the coset

E8(8)/ SO(16). The e.o.m.’s are integrable. (Fre, Gargiulo, Rulik, Trigiante)• More generally : for maximally non-compact cosets, the resulting e.o.m.’s

are integrable. (Fre, Sorin)• For non-maximally non-compact cosets, this is no longer the case. →

TS-projection gives a way of studying these solutions. Relevant fornon-maximal supergravity.

Introduction and motivation

• Time dependent solutions of SUGRA : reduce to 3 dimensions→non-linear sigma model (coupled to gravity)

gij(φ)∂µφi∂µφj (1)

• Purely time dependent solutions, e.o.m.’s are geodesic equations.• E.g. : Maximal SUGRA : in d = 3 the scalars lie in the coset

E8(8)/ SO(16). The e.o.m.’s are integrable. (Fre, Gargiulo, Rulik, Trigiante)• More generally : for maximally non-compact cosets, the resulting e.o.m.’s

are integrable. (Fre, Sorin)• For non-maximally non-compact cosets, this is no longer the case. →

TS-projection gives a way of studying these solutions. Relevant fornon-maximal supergravity.

Introduction and motivation

• Time dependent solutions of SUGRA : reduce to 3 dimensions→non-linear sigma model (coupled to gravity)

gij(φ)∂µφi∂µφj (1)

• Purely time dependent solutions, e.o.m.’s are geodesic equations.• E.g. : Maximal SUGRA : in d = 3 the scalars lie in the coset

E8(8)/ SO(16). The e.o.m.’s are integrable. (Fre, Gargiulo, Rulik, Trigiante)• More generally : for maximally non-compact cosets, the resulting e.o.m.’s

are integrable. (Fre, Sorin)• For non-maximally non-compact cosets, this is no longer the case. →

TS-projection gives a way of studying these solutions. Relevant fornon-maximal supergravity.

Introduction and motivation

• Time dependent solutions of SUGRA : reduce to 3 dimensions→non-linear sigma model (coupled to gravity)

gij(φ)∂µφi∂µφj (1)

• Purely time dependent solutions, e.o.m.’s are geodesic equations.• E.g. : Maximal SUGRA : in d = 3 the scalars lie in the coset

E8(8)/ SO(16). The e.o.m.’s are integrable. (Fre, Gargiulo, Rulik, Trigiante)• More generally : for maximally non-compact cosets, the resulting e.o.m.’s

are integrable. (Fre, Sorin)• For non-maximally non-compact cosets, this is no longer the case. →

TS-projection gives a way of studying these solutions. Relevant fornon-maximal supergravity.

Introduction and motivation

• TS-projection : non-maximally non-compact coset→ maximallynon-compact one.

• Solutions of the projected theory are also solutions of the original theory.• Different theories have the same projection→ universality classes of

supergravity theories.• All members of the same class show a similar dynamical behaviour• For supergravity theories with symmetric target spaces, the TS projection is

based on theory of real forms of simple Lie algebras.• We will focus on a certain class of N = 2 supergravity theories, namely

those based on homogeneous special geometries. For these, a Tits-Satakeprojection can still be defined.

• Result A grouping of sugra theories in 7 universality classes.

Introduction and motivation

• TS-projection : non-maximally non-compact coset→ maximallynon-compact one.

• Solutions of the projected theory are also solutions of the original theory.• Different theories have the same projection→ universality classes of

supergravity theories.• All members of the same class show a similar dynamical behaviour• For supergravity theories with symmetric target spaces, the TS projection is

based on theory of real forms of simple Lie algebras.• We will focus on a certain class of N = 2 supergravity theories, namely

those based on homogeneous special geometries. For these, a Tits-Satakeprojection can still be defined.

• Result A grouping of sugra theories in 7 universality classes.

Introduction and motivation

• TS-projection : non-maximally non-compact coset→ maximallynon-compact one.

• Solutions of the projected theory are also solutions of the original theory.• Different theories have the same projection→ universality classes of

supergravity theories.• All members of the same class show a similar dynamical behaviour• For supergravity theories with symmetric target spaces, the TS projection is

based on theory of real forms of simple Lie algebras.• We will focus on a certain class of N = 2 supergravity theories, namely

those based on homogeneous special geometries. For these, a Tits-Satakeprojection can still be defined.

• Result A grouping of sugra theories in 7 universality classes.

Introduction and motivation

• TS-projection : non-maximally non-compact coset→ maximallynon-compact one.

• Solutions of the projected theory are also solutions of the original theory.• Different theories have the same projection→ universality classes of

supergravity theories.• All members of the same class show a similar dynamical behaviour• For supergravity theories with symmetric target spaces, the TS projection is

based on theory of real forms of simple Lie algebras.• We will focus on a certain class of N = 2 supergravity theories, namely

those based on homogeneous special geometries. For these, a Tits-Satakeprojection can still be defined.

• Result A grouping of sugra theories in 7 universality classes.

Introduction and motivation

• TS-projection : non-maximally non-compact coset→ maximallynon-compact one.

• Solutions of the projected theory are also solutions of the original theory.• Different theories have the same projection→ universality classes of

supergravity theories.• All members of the same class show a similar dynamical behaviour• For supergravity theories with symmetric target spaces, the TS projection is

based on theory of real forms of simple Lie algebras.• We will focus on a certain class of N = 2 supergravity theories, namely

those based on homogeneous special geometries. For these, a Tits-Satakeprojection can still be defined.

• Result A grouping of sugra theories in 7 universality classes.

Introduction and motivation

• TS-projection : non-maximally non-compact coset→ maximallynon-compact one.

• Solutions of the projected theory are also solutions of the original theory.• Different theories have the same projection→ universality classes of

supergravity theories.• All members of the same class show a similar dynamical behaviour• For supergravity theories with symmetric target spaces, the TS projection is

based on theory of real forms of simple Lie algebras.• We will focus on a certain class of N = 2 supergravity theories, namely

those based on homogeneous special geometries. For these, a Tits-Satakeprojection can still be defined.

• Result A grouping of sugra theories in 7 universality classes.

N = 2 homogeneous special geometry

• We will focus on homogeneous quaternionic-Kahler manifolds:• Appear as target spaces in theories with 8 supercharges (3d, hypermultiplets

in 4d and 5d).• Can often be related to homogeneous special Kahler (4d) and special real (5d)

manifolds, describing vector multiplet couplings via the inverse c and r-maps.• Appear in physical constructions :

• CY moduli spaces e.g. in T2 × K3 compactifications• brane constructions• orbifold and orientifold compactifications

• Homogeneous : every two points are related by an isometry group G→ of

the form = GH , not necessarily symmetric.

• Homogeneous quaternionic-Kahler manifolds are normal.

M' exp [SolvM ]

N = 2 homogeneous special geometry

• We will focus on homogeneous quaternionic-Kahler manifolds:• Appear as target spaces in theories with 8 supercharges (3d, hypermultiplets

in 4d and 5d).• Can often be related to homogeneous special Kahler (4d) and special real (5d)

manifolds, describing vector multiplet couplings via the inverse c and r-maps.• Appear in physical constructions :

• CY moduli spaces e.g. in T2 × K3 compactifications• brane constructions• orbifold and orientifold compactifications

• Homogeneous : every two points are related by an isometry group G→ of

the form = GH , not necessarily symmetric.

• Homogeneous quaternionic-Kahler manifolds are normal.

M' exp [SolvM ]

N = 2 homogeneous special geometry

• We will focus on homogeneous quaternionic-Kahler manifolds:• Appear as target spaces in theories with 8 supercharges (3d, hypermultiplets

in 4d and 5d).• Can often be related to homogeneous special Kahler (4d) and special real (5d)

manifolds, describing vector multiplet couplings via the inverse c and r-maps.• Appear in physical constructions :

• CY moduli spaces e.g. in T2 × K3 compactifications• brane constructions• orbifold and orientifold compactifications

• Homogeneous : every two points are related by an isometry group G→ of

the form = GH , not necessarily symmetric.

• Homogeneous quaternionic-Kahler manifolds are normal.

M' exp [SolvM ]

N = 2 homogeneous special geometry

• We will focus on homogeneous quaternionic-Kahler manifolds:• Appear as target spaces in theories with 8 supercharges (3d, hypermultiplets

in 4d and 5d).• Can often be related to homogeneous special Kahler (4d) and special real (5d)

manifolds, describing vector multiplet couplings via the inverse c and r-maps.• Appear in physical constructions :

• CY moduli spaces e.g. in T2 × K3 compactifications• brane constructions• orbifold and orientifold compactifications

• Homogeneous : every two points are related by an isometry group G→ of

the form = GH , not necessarily symmetric.

• Homogeneous quaternionic-Kahler manifolds are normal.

M' exp [SolvM ]

N = 2 homogeneous special geometry

• We will focus on homogeneous quaternionic-Kahler manifolds:• Appear as target spaces in theories with 8 supercharges (3d, hypermultiplets

in 4d and 5d).• Can often be related to homogeneous special Kahler (4d) and special real (5d)

manifolds, describing vector multiplet couplings via the inverse c and r-maps.• Appear in physical constructions :

• CY moduli spaces e.g. in T2 × K3 compactifications• brane constructions• orbifold and orientifold compactifications

• Homogeneous : every two points are related by an isometry group G→ of

the form = GH , not necessarily symmetric.

• Homogeneous quaternionic-Kahler manifolds are normal.

M' exp [SolvM ]

N = 2 homogeneous special geometry

• We will focus on homogeneous quaternionic-Kahler manifolds:• Appear as target spaces in theories with 8 supercharges (3d, hypermultiplets

in 4d and 5d).• Can often be related to homogeneous special Kahler (4d) and special real (5d)

manifolds, describing vector multiplet couplings via the inverse c and r-maps.• Appear in physical constructions :

• CY moduli spaces e.g. in T2 × K3 compactifications• brane constructions• orbifold and orientifold compactifications

• Homogeneous : every two points are related by an isometry group G→ of

the form = GH , not necessarily symmetric.

• Homogeneous quaternionic-Kahler manifolds are normal.

M' exp [SolvM ]

N = 2 homogeneous special geometry

• SolvM equipped with extra structure : e.g. there is a quaternionic structure

JαJβ = −δαβ + εαβγJγ ,

such that holonomy ⊂ USp(2)× USp(2n).• Classification according to rank, ranging from 1 to 4 (Alekseevsky, Cortes, de Wit,

Van Proeyen)• E.g. : for rank 4 quaternionic spaces, the algebra is summarized as

p0 p1 p2 p3 X− Y− Z− −J3A = J2J1Aq0 q1 q2 q3 X+ Y+ Z+ J2A

SolvQ =g0 g1 g2 g3 X− Y− Z− J1Ah0 h1 h2 h3 X+ Y+ Z+ A

# 1 1 1 1 q (P + P)Dq+1

N = 2 homogeneous special geometry

• SolvM equipped with extra structure : e.g. there is a quaternionic structure

JαJβ = −δαβ + εαβγJγ ,

such that holonomy ⊂ USp(2)× USp(2n).• Classification according to rank, ranging from 1 to 4 (Alekseevsky, Cortes, de Wit,

Van Proeyen)• E.g. : for rank 4 quaternionic spaces, the algebra is summarized as

p0 p1 p2 p3 X− Y− Z− −J3A = J2J1Aq0 q1 q2 q3 X+ Y+ Z+ J2A

SolvQ =g0 g1 g2 g3 X− Y− Z− J1Ah0 h1 h2 h3 X+ Y+ Z+ A

# 1 1 1 1 q (P + P)Dq+1

N = 2 homogeneous special geometry

• SolvM equipped with extra structure : e.g. there is a quaternionic structure

JαJβ = −δαβ + εαβγJγ ,

such that holonomy ⊂ USp(2)× USp(2n).• Classification according to rank, ranging from 1 to 4 (Alekseevsky, Cortes, de Wit,

Van Proeyen)• E.g. : for rank 4 quaternionic spaces, the algebra is summarized as

p0 p1 p2 p3 X− Y− Z− −J3A = J2J1Aq0 q1 q2 q3 X+ Y+ Z+ J2A

SolvQ =g0 g1 g2 g3 X− Y− Z− J1Ah0 h1 h2 h3 X+ Y+ Z+ A

# 1 1 1 1 q (P + P)Dq+1

Applications

• SolvQ(0, P, P), describing the D3/D7-brane system. (Angelantonj, d’Auria,

Ferrara, Trigiante)

Example

• Low energy dynamics of type IIB theory compactified on aK3× T2/Z2-orientifold, in the presence of n3 D3 and n7 D7 branes andfluxes.

• Low-energy 4− d theory with N = 2 supersymmetry, vector multipletsdescribed by Special Kahler manifold, whose quaternionic version is

p0 p1 p2 p3 Y− Z−

q0 q1 q2 q3 Y+ Z+

SolvQ =g0 g1 g2 g3 Y− Z−

h0 h1 h2 h3 Y+ Z+

# 1 1 1 1 n3 n7

Applications

• SolvQ(0, P, P), describing the D3/D7-brane system. (Angelantonj, d’Auria,

Ferrara, Trigiante)

Example

• Low energy dynamics of type IIB theory compactified on aK3× T2/Z2-orientifold, in the presence of n3 D3 and n7 D7 branes andfluxes.

• Low-energy 4− d theory with N = 2 supersymmetry, vector multipletsdescribed by Special Kahler manifold, whose quaternionic version is

p0 p1 p2 p3 Y− Z−

q0 q1 q2 q3 Y+ Z+

SolvQ =g0 g1 g2 g3 Y− Z−

h0 h1 h2 h3 Y+ Z+

# 1 1 1 1 n3 n7

Applications

• SolvQ(0, P, P), describing the D3/D7-brane system. (Angelantonj, d’Auria,

Ferrara, Trigiante)

Example

• Low energy dynamics of type IIB theory compactified on aK3× T2/Z2-orientifold, in the presence of n3 D3 and n7 D7 branes andfluxes.

• Low-energy 4− d theory with N = 2 supersymmetry, vector multipletsdescribed by Special Kahler manifold, whose quaternionic version is

p0 p1 p2 p3 Y− Z−

q0 q1 q2 q3 Y+ Z+

SolvQ =g0 g1 g2 g3 Y− Z−

h0 h1 h2 h3 Y+ Z+

# 1 1 1 1 n3 n7

The TS-projection for symmetric spaces

• G/H non-maximally noncompact coset→ Lie algebra of G is someappropriate real form GR of a complex algebra G. Orthogonal split

GR = H⊕K, [K, K] ⊂ H

rnc = rank (G/H) ≡ dimHnc ; Hnc ≡ CSAG⋂

K

rnc ≤ rank(G)• Make a splitting of the CSA into a compact and a non-compact part

CSAGR = iHcomp ⊕ Hnc

m mCSAG = Hcomp ⊕ Hnc

This allows to decompose roots in a part transverse and a part parallel toHnc:

α = α⊥ ⊕ α||

• TS-projection at the level of the roots = putting α⊥ = 0. This leads inmany cases to a root system ∆TS of simple type.

The TS-projection for symmetric spaces

• G/H non-maximally noncompact coset→ Lie algebra of G is someappropriate real form GR of a complex algebra G. Orthogonal split

GR = H⊕K, [K, K] ⊂ H

rnc = rank (G/H) ≡ dimHnc ; Hnc ≡ CSAG⋂

K

rnc ≤ rank(G)• Make a splitting of the CSA into a compact and a non-compact part

CSAGR = iHcomp ⊕ Hnc

m mCSAG = Hcomp ⊕ Hnc

This allows to decompose roots in a part transverse and a part parallel toHnc:

α = α⊥ ⊕ α||

• TS-projection at the level of the roots = putting α⊥ = 0. This leads inmany cases to a root system ∆TS of simple type.

The TS-projection for symmetric spaces

• G/H non-maximally noncompact coset→ Lie algebra of G is someappropriate real form GR of a complex algebra G. Orthogonal split

GR = H⊕K, [K, K] ⊂ H

rnc = rank (G/H) ≡ dimHnc ; Hnc ≡ CSAG⋂

K

rnc ≤ rank(G)• Make a splitting of the CSA into a compact and a non-compact part

CSAGR = iHcomp ⊕ Hnc

m mCSAG = Hcomp ⊕ Hnc

This allows to decompose roots in a part transverse and a part parallel toHnc:

α = α⊥ ⊕ α||

• TS-projection at the level of the roots = putting α⊥ = 0. This leads inmany cases to a root system ∆TS of simple type.

The TS-projection for symmetric spaces

• At the level of the solvable algebra one can enumerate the generators ofSolv(GR/H)

Solv(GR/H) =

Hi,Φα` ,Ωαs|I

(2)

• Hi are the non-compact Cartan generators• Φα` are in one-to-one correspondence to restricted roots for which the

projection is one-to-one.• Ωαs|I are corresponding to restricted roots for which the projection is not

one-to-one.

• Step operators corresponding to compact roots + compact CSA formGpaint ⊂ GR

[Gpaint , SolvGR ] ⊂ SolvGR .

• Hi and Φα` are singlets under Gpaint, Ωαs|I form representations underGpaint:

∀X ∈ Gpaint :[X , Ωαs|I

]=

(D[αs][X]

) J

IΩαs|J

The TS-projection for symmetric spaces

• At the level of the solvable algebra one can enumerate the generators ofSolv(GR/H)

Solv(GR/H) =

Hi,Φα` ,Ωαs|I

(2)

• Hi are the non-compact Cartan generators• Φα` are in one-to-one correspondence to restricted roots for which the

projection is one-to-one.• Ωαs|I are corresponding to restricted roots for which the projection is not

one-to-one.

• Step operators corresponding to compact roots + compact CSA formGpaint ⊂ GR

[Gpaint , SolvGR ] ⊂ SolvGR .

• Hi and Φα` are singlets under Gpaint, Ωαs|I form representations underGpaint:

∀X ∈ Gpaint :[X , Ωαs|I

]=

(D[αs][X]

) J

IΩαs|J

The TS-projection for symmetric spaces

• At the level of the solvable algebra one can enumerate the generators ofSolv(GR/H)

Solv(GR/H) =

Hi,Φα` ,Ωαs|I

(2)

• Hi are the non-compact Cartan generators• Φα` are in one-to-one correspondence to restricted roots for which the

projection is one-to-one.• Ωαs|I are corresponding to restricted roots for which the projection is not

one-to-one.

• Step operators corresponding to compact roots + compact CSA formGpaint ⊂ GR

[Gpaint , SolvGR ] ⊂ SolvGR .

• Hi and Φα` are singlets under Gpaint, Ωαs|I form representations underGpaint:

∀X ∈ Gpaint :[X , Ωαs|I

]=

(D[αs][X]

) J

IΩαs|J

The TS-projection for symmetric spaces

• Gpaint contains a subalgebra

G0subpaint ⊂ Gpaint

such that:D[αs]

G0subpaint=⇒ 1︸︷︷︸

singlet

⊕ J︸︷︷︸(m[αs]−1)−dimensional

Singlets span the TS-subalgebra, generating a maximally non-compactcoset.

Example

Special Kahler space corresponding to only D3 branes in K3× T2/Z2compactification:

SU(1, 1)U(1)

× SO(2, 2 + n3)SO(2)× SO(2 + n3)

→ SU(1, 1)U(1)

× SO(2, 3)SO(2)× SO(3)

Gpaint = SO(n3)G0

subpaint = SO(n3 − 1) (3)

The TS-projection for symmetric spaces

• Gpaint contains a subalgebra

G0subpaint ⊂ Gpaint

such that:D[αs]

G0subpaint=⇒ 1︸︷︷︸

singlet

⊕ J︸︷︷︸(m[αs]−1)−dimensional

Singlets span the TS-subalgebra, generating a maximally non-compactcoset.

Example

Special Kahler space corresponding to only D3 branes in K3× T2/Z2compactification:

SU(1, 1)U(1)

× SO(2, 2 + n3)SO(2)× SO(2 + n3)

→ SU(1, 1)U(1)

× SO(2, 3)SO(2)× SO(3)

Gpaint = SO(n3)G0

subpaint = SO(n3 − 1) (3)

The TS-projection for general solvable algebrasThe paint group

• The above procedure works well for symmetric spaces, since it relied on ageometrical projection of a root system. How to generalize tonon-symmetric spaces? → One can give an appropriate generalization ofthe paint group.

• For homogeneous special geometries, the structure of their isometry groups(de Wit, Vanderseypen, Van Proeyen) implies:

Gpaint = SO(q)× Sq(P, P),

where

Sq(P, P) =

SO(P)× SO(P) for q = 0, 1, 7 mod 8U(P) for q = 2, 6 mod 8USp(2P)× USp(2P) for q = 3, 4, 5 mod 8

The TS-projection for general solvable algebrasThe paint group

• The above procedure works well for symmetric spaces, since it relied on ageometrical projection of a root system. How to generalize tonon-symmetric spaces? → One can give an appropriate generalization ofthe paint group.

• For homogeneous special geometries, the structure of their isometry groups(de Wit, Vanderseypen, Van Proeyen) implies:

Gpaint = SO(q)× Sq(P, P),

where

Sq(P, P) =

SO(P)× SO(P) for q = 0, 1, 7 mod 8U(P) for q = 2, 6 mod 8USp(2P)× USp(2P) for q = 3, 4, 5 mod 8

The TS-projection for general solvable algebrasExtra structure

p0 p1 p2 p3 X− Y− Z− −J3A = J2J1Aq0 q1 q2 q3 X+ Y+ Z+ J2A

SolvQ =g0 g1 g2 g3 X− Y− Z− J1Ah0 h1 h2 h3 X+ Y+ Z+ A

# 1 1 1 1 q (P + P)Dq+1

• SolvsubTS : singlets under Gpaint

• vector repr.• spinor repr.• conj. spinor repr.

The TS-projection for general solvable algebrasExtra structure

p0 p1 p2 p3 X− Y− Z− −J3A = J2J1Aq0 q1 q2 q3 X+ Y+ Z+ J2A

SolvQ =g0 g1 g2 g3 X− Y− Z− J1Ah0 h1 h2 h3 X+ Y+ Z+ A

# 1 1 1 1 q (P + P)Dq+1

• SolvsubTS : singlets under Gpaint

• vector repr.• spinor repr.• conj. spinor repr.

The TS-projection for general solvable algebrasExtra structure

p0 p1 p2 p3 X− Y− Z− −J3A = J2J1Aq0 q1 q2 q3 X+ Y+ Z+ J2A

SolvQ =g0 g1 g2 g3 X− Y− Z− J1Ah0 h1 h2 h3 X+ Y+ Z+ A

# 1 1 1 1 q (P + P)Dq+1

• SolvsubTS : singlets under Gpaint

• vector repr.• spinor repr.• conj. spinor repr.

The TS-projection for general solvable algebrasExtra structure

p0 p1 p2 p3 X− Y− Z− −J3A = J2J1Aq0 q1 q2 q3 X+ Y+ Z+ J2A

SolvQ =g0 g1 g2 g3 X− Y− Z− J1Ah0 h1 h2 h3 X+ Y+ Z+ A

# 1 1 1 1 q (P + P)Dq+1

• SolvsubTS : singlets under Gpaint

• vector repr.• spinor repr.• conj. spinor repr.

The TS-projection for general solvable algebrasExtra structure

p0 p1 p2 p3 X− Y− Z− −J3A = J2J1Aq0 q1 q2 q3 X+ Y+ Z+ J2A

SolvQ =g0 g1 g2 g3 X− Y− Z− J1Ah0 h1 h2 h3 X+ Y+ Z+ A

# 1 1 1 1 q (P + P)Dq+1

• SolvsubTS : singlets under Gpaint

• vector repr.• spinor repr.• conj. spinor repr.

The TS-projection for general solvable algebrasExtra structure

• The paint algebra Gpaint contains a subalgebra

G0subpaint ⊂ Gpaint

such that

Qv,s,sG0

subpaint=⇒ 1︸︷︷︸singlet

⊕ Jv,s,s,

• Restricting to the singlets of G0subpaint defines a Lie subalgebra of SolvM:

SolvTS ≡ SolvsubTS ⊕ (4, 1) ⊕ (4, 1) ⊕ (4, 1) ,

we get:[SolvTS , SolvTS] ⊂ SolvTS .

• This gives a way to consistently define a subalgebra of the original algebra: the TS algebra.

The TS-projection for general solvable algebrasExtra structure

• The paint algebra Gpaint contains a subalgebra

G0subpaint ⊂ Gpaint

such that

Qv,s,sG0

subpaint=⇒ 1︸︷︷︸singlet

⊕ Jv,s,s,

• Restricting to the singlets of G0subpaint defines a Lie subalgebra of SolvM:

SolvTS ≡ SolvsubTS ⊕ (4, 1) ⊕ (4, 1) ⊕ (4, 1) ,

we get:[SolvTS , SolvTS] ⊂ SolvTS .

• This gives a way to consistently define a subalgebra of the original algebra: the TS algebra.

The TS-projection for general solvable algebrasExtra structure

• The paint algebra Gpaint contains a subalgebra

G0subpaint ⊂ Gpaint

such that

Qv,s,sG0

subpaint=⇒ 1︸︷︷︸singlet

⊕ Jv,s,s,

• Restricting to the singlets of G0subpaint defines a Lie subalgebra of SolvM:

SolvTS ≡ SolvsubTS ⊕ (4, 1) ⊕ (4, 1) ⊕ (4, 1) ,

we get:[SolvTS , SolvTS] ⊂ SolvTS .

• This gives a way to consistently define a subalgebra of the original algebra: the TS algebra.

Applications : D3/D7 system

Example

p0 p1 p2 p3 Y− Z−

q0 q1 q2 q3 Y+ Z+

SolvQ =g0 g1 g2 g3 Y− Z−

h0 h1 h2 h3 Y+ Z+

# 1 1 1 1 n3 n7

• Gpaint = SO(n3)× SO(n7) G0subpaint = SO(n3 − 1)× SO(n7 − 1)

• −→ TS-projection leads to SolvQ(0, 1, 1) (1 D3-brane and 1 D7-brane).

Applications : D3/D7 system

Example

p0 p1 p2 p3 Y− Z−

q0 q1 q2 q3 Y+ Z+

SolvQ =g0 g1 g2 g3 Y− Z−

h0 h1 h2 h3 Y+ Z+

# 1 1 1 1 n3 n7

• Gpaint = SO(n3)× SO(n7) G0subpaint = SO(n3 − 1)× SO(n7 − 1)

• −→ TS-projection leads to SolvQ(0, 1, 1) (1 D3-brane and 1 D7-brane).

Applications : D3/D7 system

Example

p0 p1 p2 p3 Y− Z−

q0 q1 q2 q3 Y+ Z+

SolvQ =g0 g1 g2 g3 Y− Z−

h0 h1 h2 h3 Y+ Z+

# 1 1 1 1 n3 n7

• Gpaint = SO(n3)× SO(n7) G0subpaint = SO(n3 − 1)× SO(n7 − 1)

• −→ TS-projection leads to SolvQ(0, 1, 1) (1 D3-brane and 1 D7-brane).

Applications : D3/D7 system

Example

p0 p1 p2 p3 Y− Z−

q0 q1 q2 q3 Y+ Z+

SolvQ =g0 g1 g2 g3 Y− Z−

h0 h1 h2 h3 Y+ Z+

# 1 1 1 1 1 n7

• Gpaint = SO(n3)× SO(n7) G0subpaint = SO(n3 − 1)× SO(n7 − 1)

• −→ TS-projection leads to SolvQ(0, 1, 1) (1 D3-brane and 1 D7-brane).

Applications : D3/D7 system

Example

p0 p1 p2 p3 Y− Z−

q0 q1 q2 q3 Y+ Z+

SolvQ =g0 g1 g2 g3 Y− Z−

h0 h1 h2 h3 Y+ Z+

# 1 1 1 1 1 1

• Gpaint = SO(n3)× SO(n7) G0subpaint = SO(n3 − 1)× SO(n7 − 1)

• −→ TS-projection leads to SolvQ(0, 1, 1) (1 D3-brane and 1 D7-brane).

Applications : D3/D7 system

Example

p0 p1 p2 p3 Y− Z−

q0 q1 q2 q3 Y+ Z+

SolvQ =g0 g1 g2 g3 Y− Z−

h0 h1 h2 h3 Y+ Z+

# 1 1 1 1 1 1

• Gpaint = SO(n3)× SO(n7) G0subpaint = SO(n3 − 1)× SO(n7 − 1)

• −→ TS-projection leads to SolvQ(0, 1, 1) (1 D3-brane and 1 D7-brane).

Properties

• Several different manifolds have the same image after projection.• ΠTS preserves the rank.• ΠTS : special homogeneous→ special homogeneous.

ΠTS : very special real→ very special realΠTS : special Kahler→ special KahlerΠTS : quaternionic-Kahler→ quaternionic-Kahler.

• Gpaint is invariant under dimensional reduction

Very Special realr-map=⇒ Special Kahler

c-map=⇒ Quaternionic-Kahler

ΠTS ⇓ ΠTS ⇓ ΠTS ⇓(Very Special real)TS

r-map=⇒ (Special Kahler)TS

c-map=⇒ (Quaternionic-Kahler)TS

Properties

• Several different manifolds have the same image after projection.• ΠTS preserves the rank.• ΠTS : special homogeneous→ special homogeneous.

ΠTS : very special real→ very special realΠTS : special Kahler→ special KahlerΠTS : quaternionic-Kahler→ quaternionic-Kahler.

• Gpaint is invariant under dimensional reduction

Very Special realr-map=⇒ Special Kahler

c-map=⇒ Quaternionic-Kahler

ΠTS ⇓ ΠTS ⇓ ΠTS ⇓(Very Special real)TS

r-map=⇒ (Special Kahler)TS

c-map=⇒ (Quaternionic-Kahler)TS

Properties

• Several different manifolds have the same image after projection.• ΠTS preserves the rank.• ΠTS : special homogeneous→ special homogeneous.

ΠTS : very special real→ very special realΠTS : special Kahler→ special KahlerΠTS : quaternionic-Kahler→ quaternionic-Kahler.

• Gpaint is invariant under dimensional reduction

Very Special realr-map=⇒ Special Kahler

c-map=⇒ Quaternionic-Kahler

ΠTS ⇓ ΠTS ⇓ ΠTS ⇓(Very Special real)TS

r-map=⇒ (Special Kahler)TS

c-map=⇒ (Quaternionic-Kahler)TS

Properties

• Several different manifolds have the same image after projection.• ΠTS preserves the rank.• ΠTS : special homogeneous→ special homogeneous.

ΠTS : very special real→ very special realΠTS : special Kahler→ special KahlerΠTS : quaternionic-Kahler→ quaternionic-Kahler.

• Gpaint is invariant under dimensional reduction

Very Special realr-map=⇒ Special Kahler

c-map=⇒ Quaternionic-Kahler

ΠTS ⇓ ΠTS ⇓ ΠTS ⇓(Very Special real)TS

r-map=⇒ (Special Kahler)TS

c-map=⇒ (Quaternionic-Kahler)TS

Properties

• Several different manifolds have the same image after projection.• ΠTS preserves the rank.• ΠTS : special homogeneous→ special homogeneous.

ΠTS : very special real→ very special realΠTS : special Kahler→ special KahlerΠTS : quaternionic-Kahler→ quaternionic-Kahler.

• Gpaint is invariant under dimensional reduction

Very Special realr-map=⇒ Special Kahler

c-map=⇒ Quaternionic-Kahler

ΠTS ⇓ ΠTS ⇓ ΠTS ⇓(Very Special real)TS

r-map=⇒ (Special Kahler)TS

c-map=⇒ (Quaternionic-Kahler)TS

Properties

• Several different manifolds have the same image after projection.• ΠTS preserves the rank.• ΠTS : special homogeneous→ special homogeneous.

ΠTS : very special real→ very special realΠTS : special Kahler→ special KahlerΠTS : quaternionic-Kahler→ quaternionic-Kahler.

• Gpaint is invariant under dimensional reduction

Very Special realr-map=⇒ Special Kahler

c-map=⇒ Quaternionic-Kahler

ΠTS ⇓ ΠTS ⇓ ΠTS ⇓(Very Special real)TS

r-map=⇒ (Special Kahler)TS

c-map=⇒ (Quaternionic-Kahler)TS

Summary and conclusions

• The TS-projection for symmetric spaces can be extended to homogeneousspecial geometry, due to the notion of paint group.

• This allows one to organize supergravity theories based on these manifoldsin 7 universality classes

• This organization is important, since all members in one class share asimilar dynamical behaviour.

• Moreover, in a lot of cases the dynamics for the TS-projected model aretractable (i.e. e.o.m.’s are sometimes integrable).

• In some cases one can also give a physical meaning to this projection.

Summary and conclusions

• The TS-projection for symmetric spaces can be extended to homogeneousspecial geometry, due to the notion of paint group.

• This allows one to organize supergravity theories based on these manifoldsin 7 universality classes

• This organization is important, since all members in one class share asimilar dynamical behaviour.

• Moreover, in a lot of cases the dynamics for the TS-projected model aretractable (i.e. e.o.m.’s are sometimes integrable).

• In some cases one can also give a physical meaning to this projection.

Summary and conclusions

• The TS-projection for symmetric spaces can be extended to homogeneousspecial geometry, due to the notion of paint group.

• This allows one to organize supergravity theories based on these manifoldsin 7 universality classes

• This organization is important, since all members in one class share asimilar dynamical behaviour.

• Moreover, in a lot of cases the dynamics for the TS-projected model aretractable (i.e. e.o.m.’s are sometimes integrable).

• In some cases one can also give a physical meaning to this projection.

Summary and conclusions

• The TS-projection for symmetric spaces can be extended to homogeneousspecial geometry, due to the notion of paint group.

• This allows one to organize supergravity theories based on these manifoldsin 7 universality classes

• This organization is important, since all members in one class share asimilar dynamical behaviour.

• Moreover, in a lot of cases the dynamics for the TS-projected model aretractable (i.e. e.o.m.’s are sometimes integrable).

• In some cases one can also give a physical meaning to this projection.

Summary and conclusions

• The TS-projection for symmetric spaces can be extended to homogeneousspecial geometry, due to the notion of paint group.

• This allows one to organize supergravity theories based on these manifoldsin 7 universality classes

• This organization is important, since all members in one class share asimilar dynamical behaviour.

• Moreover, in a lot of cases the dynamics for the TS-projected model aretractable (i.e. e.o.m.’s are sometimes integrable).

• In some cases one can also give a physical meaning to this projection.