Introduction Branes and Wrapping Rules Generalized ... · Introduction Branes and Wrapping Rules Generalized Geometry Perspectives Branes Branesin String Theory Branes appear in string

Introduction

Branes and Wrapping Rules

Generalized Geometry

Perspectives

Introduction



Perspectives

1 Introduction

2 Branes and Wrapping Rules

3 Generalized Geometry

4 Perspectives

Introduction



Perspectives

Supergravity Theories

dimension D

amount of supersymmetry N (the number of supercharges)

Introduction



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



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



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



Perspectives

Branes in Supergravity

Introduction



Perspectives


Introduction



Perspectives


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



Perspectives


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



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



Perspectives

Wrapping Rules


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



Perspectives

Wrapping Rules


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



Perspectives

Wrapping Rules


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



Perspectives

Wrapping Rules


T ≈1

gns

Introduction



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



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



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



Perspectives


Introduction



Perspectives

1 Introduction

2 Branes and Wrapping Rules

3 Generalized Geometry

4 Perspectives

Introduction



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]

Documents

Introduction Branes and Wrapping Rules Generalized ... · Introduction Branes and Wrapping Rules Generalized Geometry Perspectives Branes Branesin String Theory Branes appear in string