Introduction
Branes and Wrapping Rules
Generalized Geometry
Perspectives
Introduction
Branes and Wrapping Rules
Generalized Geometry
Perspectives
1 Introduction
2 Branes and Wrapping Rules
3 Generalized Geometry
4 Perspectives
Introduction
Branes and Wrapping Rules
Generalized Geometry
Perspectives
Supergravity Theories
dimension D
amount of supersymmetry N (the number of supercharges)
Introduction
Branes and Wrapping Rules
Generalized Geometry
Perspectives
Kaluza Klein Theories
D=11
D=11 is the maximal dimension that admits a supergravity theory. The theory isunique. There is only a supercharge N = 1.
Bosons Fermions
Fields gMN AMNP ψM
dof 44 84 128
total dof 128 128
Maximal Supergravities
Maximal supergravity theories in D dimensions can be obtained from the elevendimensional theory by compactifing 11-D on a torus T 11−D
gMN → gµν gµi gij
AMNP → Aµνρ Aµνi Aµij Aijk
M,N,P = 0, 1, 2, ...,10
µ, ν, ρ = 0, 1, ...,D − 1
i , j , k = D, ...,10
Introduction
Branes and Wrapping Rules
Generalized Geometry
Perspectives
Maximal Supergravity Theories
D = 7 N = 2
Fields gµν gµi gij Aµνρ Aµνi Aµij Aijk 2ψµ 8χ
dof 14 20 10 10 40 30 4 64 64
KK manifold T 4
scalar coset Sl(5,R)/SO(5)
D N G/H
10(IIA) 2 R+
9 2 Gl(2,R)/SO(2)
8 2 (Sl(2) × Sl(3))/(SO(2) × SO(3))
7 2 Sl(5)/SO(5)
6 4 SO(5, 5)/(SO(5) × SO(5))
5 4 E6(6)/USp(8)
4 8 E7(7)/SU(8)
g → g g g
Introduction
Branes and Wrapping Rules
Generalized Geometry
Perspectives
Branes
Branes in String Theory
Branes appear in string theory as boundaryterms for open strings
Relevant Features
Branes have their own dynamics
Branes are non perturbative objects
Branes appear in sugra theories as classicalsolutions
p-Brane
A p-brane is an object that extends in pspatial directions
Examples
0-brane=point particle
1-brane=string
2-brane=membrane
Introduction
Branes and Wrapping Rules
Generalized Geometry
Perspectives
Branes in Supergravity
Introduction
Branes and Wrapping Rules
Generalized Geometry
Perspectives
Branes in Supergravity
Introduction
Branes and Wrapping Rules
Generalized Geometry
Perspectives
Branes in Supergravity
Are all the G-reps states brane states?
Which is the Lie algebra characterization of brane states?
It is the same for alla the kinds of branes?
Introduction
Branes and Wrapping Rules
Generalized Geometry
Perspectives
Branes in Supergravity
Longest Weight Rule
Brane states correspond to the longest realweights of the representation they live in
b
−4 3α1−
5 3α2
− 13 α
1 + 13 α
2
23 α1 − 2
3 α2
−13α1−
23α2
23α1
+13α2
−13 α
1−
53 α2
b
−13 α
1 +43 α
2
2 3α1+
4 3α2
53α1 +
13α2
53α 1
+4
3α 2
− 43 α1 + 1
3 α2
−43α1
−23α2
2 1
-2 2
1 -3
0 -1
-1 1
-1 -2
-2 0
-3 2
3 -1
1 0
0 2
-2 3
Branes, Weights and Central Charges, Eric A. Bergshoeff, Fabio Riccioni, L.R.
arXiv:1303.0221 [hep-th]
Introduction
Branes and Wrapping Rules
Generalized Geometry
Perspectives
Wrapping Rules
Branes are Non Perturbative Objects
T ≈1
gns
Fundamental string: g0s
Fp-brane IIA/IIB 9 8 7 6 5 4 3
0 2 4 6 8 10 12 14
1 1/1 1 1 1 1 1 1 1
Introduction
Branes and Wrapping Rules
Generalized Geometry
Perspectives
Wrapping Rules
Branes are Non Perturbative Objects
T ≈1
gns
D-brane: g−1s
Dp-brane IIA/IIB 9 8 7 6 5 4 3
0 1/0 1 2 4 8 16 32 64
1 0/1 1 2 4 8 16 32 64
2 1/0 1 2 4 8 16 32 64
3 0/1 1 2 4 8 16 32
4 1/0 1 2 4 8 16
5 0/1 1 2 4 8
6 1/0 1 2 4
7 0/1 1 2
8 1/0 1
9 0/1
Introduction
Branes and Wrapping Rules
Generalized Geometry
Perspectives
Wrapping Rules
Branes are Non Perturbative Objects
T ≈1
gns
S-brane: g−2s
Sp-brane IIA/IIB 9 8 7 6 5 4 3
0 1 12 84
1 1 10 60 280
2 1 8 40 160 560
3 1 6 24 80 240
4 1 4 12 32 80
5 1’/1 1’+1 2’+2 4’+4 8’+8
Introduction
Branes and Wrapping Rules
Generalized Geometry
Perspectives
Wrapping Rules
Branes are Non Perturbative Objects
T ≈1
gns
E-brane: g−3s
Ep-brane IIA/IIB 9 8 7 6 5 4 3
0 64
1 32 448
2 16 192 1344
3 8 80 480
4 4 32 160
5 2 12 48
6 1 4 12
7 0/1 1 2
Introduction
Branes and Wrapping Rules
Generalized Geometry
Perspectives
Wrapping Rules
Branes are Non Perturbative Objects
T ≈1
gns
Introduction
Branes and Wrapping Rules
Generalized Geometry
Perspectives
T-duality
T-duality
T-duality is a symmetry in string theory for a string compactified on a torus of radiusR under the exchange R ↔ α′/R
T-duality is not reproduced in supergravity theories
T-duality generalizes for a d-dimensional toroiadal compactification to an O(d, d)symmetry
T-duality is a stringy symmetry
Introduction
Branes and Wrapping Rules
Generalized Geometry
Perspectives
Double Geometry
T-duality in Supergravity
In order to implement T-duality in d-dimensional supergravity one has to put all theobjects in representations of O(d, d), but its fundamental representation is 2ddimensional
Double Coordinates
Coordinates are doubled {x i} → {x i , x̃i}
the dilaton is in a singlet under the T-duality group while the graviton andKalb-Ramond two-form together define the generalized metric
Hµν =
(
g ij −g ikBkj
Bikgkj gij − Bikg
klBlj
)
generalized diffeomorphisms and Lie derivative
Section Constraints
ηMN∂M∂N ... = 0
Introduction
Branes and Wrapping Rules
Generalized Geometry
Perspectives
G and U-duality
The global symmetry group of supergravity theories G is related to the U-dualitysymmetry in string theories
11-dim supergravity
The global symmetry group G of the D dimensional supergravity is already encoded inthe eleven dimensional theory
It is possible to make already in the elven dimensional supergravity the symmetry G ofa D dimensional theory manifest?
Introduction
Branes and Wrapping Rules
Generalized Geometry
Perspectives
Generalized Geometry
Introduction
Branes and Wrapping Rules
Generalized Geometry
Perspectives
1 Introduction
2 Branes and Wrapping Rules
3 Generalized Geometry
4 Perspectives
Introduction
Branes and Wrapping Rules
Generalized Geometry
Perspectives
The Project
Define the relation between the wrapping rules and thegeneralized geometry construction
Define the possible insights the wrapping rules analysis cangive to the generalized geometry approach
Study the role of the supersymmetry algebra in generalizedgeometry
Extend the generalized geometry construction to lesssupersymmetrical cases, N = 2, using our previous results
Towards a classification of branes in N = 2theories, Eric A. Bergshoeff, Fabio
Riccioni, L.R. arXiv:1402.2557 [hep-th]