IIB supergravity & 9-branes

IIB supergravity & 9-branes
Fabio Riccioni
DAMTP
based on work with E. Bergshoeff, M. de Roo and S. Kerstan
IIB supergravity & 9-branes – p. 1/1

Introduction
Spacetime-filling branes are not consistent in IIB.
No supergravity solution.
IIB supergravity & 9-branes – p. 2/1

Introduction
Spacetime-filling branes are not consistent in IIB.
No supergravity solution.
One can write down a κ-symmetric effective action.
Wess-Zumino term for a D9-brane:
[ C10 + C8 ∧ F + C6 ∧ F 2 + ... + C2 ∧ F 4 + ... ]
where
F = F + B
IIB supergravity & 9-branes – p. 2/1

Introduction
Spacetime-filling branes are not consistent in IIB.
No supergravity solution.
One can write down a κ-symmetric effective action.
Wess-Zumino term for a D9-brane:
[ C10 + C8 ∧ F + C6 ∧ F 2 + ... + C2 ∧ F 4 + ... ]
where
F = F + B
Type-I string theory: orientifold projection.
C10 is not projected out, but its overall charge vanishes.
IIB supergravity & 9-branes – p. 2/1

Bosonic field content:
IIB & S-duality
gµν φ Bµν C Cµν Cµνρσ
Scalars parametrise SL(2,R)/SO(2) ≃ SU(1, 1)/U(1).
τ = C + ie−φ transforms as
aτ + b
τ →
cτ + d
a b
c d
∈ SL(2,R)
Perturbative string theory: SL(2,R) completely broken.
Conjecture: non perturbative theory has SL(2,Z) symmetry.
IIB supergravity & 9-branes – p. 3/1

IIB & S-duality
S =
0 1
−1 0
τ → − 1
τ
IIB supergravity & 9-branes – p. 4/1

B2
C2
IIB & S-duality
S =
→
C4 → D3
B6
C6
→
C8 → D7
0 1
−1 0
F1
D1
NS5
D5
τ → − 1
τ
IIB supergravity & 9-branes – p. 4/1

B2
C2
IIB & S-duality
S =
→
0 1
−1 0
F1
D1
τ → − 1
τ
doublet
C4 → D3 singlet
B6
C6
→
NS5
D5
doublet
C8 → D7 ∈ triplet
IIB supergravity & 9-branes – p. 4/1

B2
C2
IIB & S-duality
S =
→
0 1
−1 0
F1
D1
τ → − 1
τ
doublet A α 2
C4 → D3 singlet A4
B6
C6
→
NS5
D5
doublet A α 6
C8 → D7 ∈ triplet A αβ
8
IIB supergravity & 9-branes – p. 4/1

IIB & S-duality
B10
C10
→
NS9
D9
IIB supergravity & 9-branes – p. 5/1

IIB & S-duality
B10
C10
→
NS9
D9
?
IIB supergravity & 9-branes – p. 5/1

IIB & S-duality
B10
C10
→
NS9
D9
Bergshoeff, de Roo, Janssen, Ortin (1999): “democratic
formulation”. RR fields together with their magnetic duals.
?
IIB supergravity & 9-branes – p. 5/1

IIB & S-duality
B10
C10
→
NS9
D9
Bergshoeff, de Roo, Janssen, Ortin (1999): “democratic
formulation”. RR fields together with their magnetic duals.
Susy transformations:
δC2n = fermions + C2n−2 ∧ δB2
?
IIB supergravity & 9-branes – p. 5/1

IIB & S-duality
B10
C10
→
NS9
D9
Bergshoeff, de Roo, Janssen, Ortin (1999): “democratic
formulation”. RR fields together with their magnetic duals.
Susy transformations:
Gauge transformations:
δC2n = fermions + C2n−2 ∧ δB2
δC2n = dΛ2n−1 + H3 ∧ Λ2n−3
?
IIB supergravity & 9-branes – p. 5/1

IIB & S-duality
B10
C10
→
NS9
D9
Bergshoeff, de Roo, Janssen, Ortin (1999): “democratic
formulation”. RR fields together with their magnetic duals.
Susy transformations:
Gauge transformations:
δC2n = fermions + C2n−2 ∧ δB2
δC2n = dΛ2n−1 + H3 ∧ Λ2n−3
F.R. (2004): B10 and C10 do not form a doublet under
SL(2,R).
?
IIB supergravity & 9-branes – p. 5/1

SU(1, 1)-covariant formulation
2-form doublet:
3-form field strengths:
δA α 2 = fermions
F α 3 = dA α 2 , δA α 2 = dΛ α 1
IIB supergravity & 9-branes – p. 6/1

SU(1, 1)-covariant formulation
2-form doublet:
3-form field strengths:
4-form singlet:
5-form field strength:
δA α 2 = fermions
F α 3 = dA α 2 , δA α 2 = dΛ α 1
δA4 = fermions + iǫαβA α 2δA β
2
F5 = dA4 + iǫαβA α 2F β
3 , δA4 = dΛ3 + iǫαβΛ α 1F β
3
Self duality condition: F5 = ∗F5
IIB supergravity & 9-branes – p. 6/1

6-forms
One can define the gauge and supersymmetry
transformation rules for the 6-form doublet Aα 6 , magnetic
dual of the 2-form doublet.
IIB supergravity & 9-branes – p. 7/1

6-forms
One can define the gauge and supersymmetry
transformation rules for the 6-form doublet Aα 6 , magnetic
dual of the 2-form doublet.
Supersymmetry:
δA α 6 = fermions + A4δA α 2 + A α 2δA4 + iǫβγδA β
2 Aγ
2 Aα 2
IIB supergravity & 9-branes – p. 7/1

6-forms
One can define the gauge and supersymmetry
transformation rules for the 6-form doublet Aα 6 , magnetic
dual of the 2-form doublet.
Supersymmetry:
δA α 6 = fermions + A4δA α 2 + A α 2δA4 + iǫβγδA β
2 Aγ
2 Aα 2
Field strength and gauge transformations:
Duality relation:
F α 7 = dA α 6 + A α 2F5 + A4F α 3
δA α 6 = dΛ α 5 + Λ α 1F5 + Λ3F α 3
F α 7 ∼ ∗F α 3
IIB supergravity & 9-branes – p. 7/1

8-forms
Magnetic duals of the scalars: A αβ
8 .
Meessen-Ortin, Dall’Agata-Lechner-Tonin (98): only two
independent field strengths.
ǫαγǫβδV α +V β γδ
−F9 = 0
IIB supergravity & 9-branes – p. 8/1

8-forms
Magnetic duals of the scalars: A αβ
8 .
Meessen-Ortin, Dall’Agata-Lechner-Tonin (98): only two
independent field strengths.
Susy transformation:
ǫαγǫβδV α +V β γδ
−F9 = 0
δA αβ
8 = fermions + Aα6δA β
2
+ ...
IIB supergravity & 9-branes – p. 8/1

8-forms
Magnetic duals of the scalars: A αβ
8 .
Meessen-Ortin, Dall’Agata-Lechner-Tonin (98): only two
independent field strengths.
Susy transformation:
ǫαγǫβδV α +V β γδ
−F9 = 0
δA αβ
8 = fermions + Aα6δA β
2
Field strength and gauge transformations:
F αβ
9
= dAαβ 8 + Aα 6F β
3 + Aα2F β
7
, δAαβ
8
+ ...
= dΛαβ 7 + Λα1F β
7 + Λα5F β
3
IIB supergravity & 9-branes – p. 8/1

8-forms
Magnetic duals of the scalars: A αβ
8 .
Meessen-Ortin, Dall’Agata-Lechner-Tonin (98): only two
independent field strengths.
Susy transformation:
ǫαγǫβδV α +V β γδ
−F9 = 0
δA αβ
8 = fermions + Aα6δA β
2
Field strength and gauge transformations:
F αβ
9
= dAαβ 8 + Aα 6F β
3 + Aα2F β
7
, δAαβ
8
+ ...
= dΛαβ 7 + Λα1F β
7 + Λα5F β
3
Bergshoeff, Gran, Roest (2002): classified all BPS 7-brane
solutions
IIB supergravity & 9-branes – p. 8/1

10-forms
No field strength → No duality constraint.
Only constraint: supersymmetry.
IIB supergravity & 9-branes – p. 9/1

10-forms
No field strength → No duality constraint.
Only constraint: supersymmetry.
We find a doublet A α 10 ,
δA α 10 = fermions
IIB supergravity & 9-branes – p. 9/1

10-forms
No field strength → No duality constraint.
Only constraint: supersymmetry.
We find a doublet A α 10 ,
and a quadruplet A αβγ
10 ,
δA α 10 = fermions
δA αβγ = fermions + A αβ
8 δAγ
2
+ ...
IIB supergravity & 9-branes – p. 9/1

10-forms
No field strength → No duality constraint.
Only constraint: supersymmetry.
We find a doublet A α 10 ,
and a quadruplet A αβγ
10 ,
Gauge transformations:
δA α 10 = fermions
δA αβγ = fermions + A αβ
8 δAγ
2
δA αβγ
10
δA α 10 = dΛ α 9
= dΛαβγ 9 + Λα 1F βγ
9
+ ...
γ
+ Λαβ 7 F3 IIB supergravity & 9-branes – p. 9/1

String frame
To which representation does the RR 10-form C10 belong?
IIB supergravity & 9-branes – p. 10/1

String frame
To which representation does the RR 10-form C10 belong?
We need to go to the string frame and look at the dilaton
dependence of the supersymmetry transformation:
δC10 = e −φ fermions + ...
IIB supergravity & 9-branes – p. 10/1

String frame
To which representation does the RR 10-form C10 belong?
We need to go to the string frame and look at the dilaton
dependence of the supersymmetry transformation:
δC10 = e −φ fermions + ...
We find that the RR 10-form belongs to the quadruplet.
IIB supergravity & 9-branes – p. 10/1

Conclusions
A doublet and a quadruplet of 10-forms.
IIB supergravity & 9-branes – p. 11/1

Conclusions
A doublet and a quadruplet of 10-forms.
What about 9-branes?
IIB supergravity & 9-branes – p. 11/1

Conclusions
A doublet and a quadruplet of 10-forms.
What about 9-branes?
Heterotic theory?
IIB supergravity & 9-branes – p. 11/1

Conclusions
A doublet and a quadruplet of 10-forms.
What about 9-branes?
Heterotic theory?
T-duality and M-theory: in progress.
IIB supergravity & 9-branes – p. 11/1