Introduction Fundamental Branes and D-branes Solitons Other Branes Open Issues

Branes and Wrapping Rules

Eric Bergshoeff

Groningen University

based on a collaboration with Fabio Riccioni,

arXiv:1102.0934, 1106.0212 and 1108.5067 [hep-th]

Padua, September 6 2011

Introduction Fundamental Branes and D-branes Solitons Other Branes Open Issues

Outline

Introduction

Introduction Fundamental Branes and D-branes Solitons Other Branes Open Issues

Outline

Introduction

Fundamental Branes and D-branes

Introduction Fundamental Branes and D-branes Solitons Other Branes Open Issues

Outline

Introduction

Fundamental Branes and D-branes

Solitons

Introduction Fundamental Branes and D-branes Solitons Other Branes Open Issues

Outline

Introduction

Fundamental Branes and D-branes

Solitons

Other Branes

Introduction Fundamental Branes and D-branes Solitons Other Branes Open Issues

Outline

Introduction

Fundamental Branes and D-branes

Solitons

Other Branes

Open Issues

Introduction Fundamental Branes and D-branes Solitons Other Branes Open Issues

Outline

Introduction

Fundamental Branes and D-branes

Solitons

Other Branes

Open Issues

Introduction Fundamental Branes and D-branes Solitons Other Branes Open Issues

Question

• Branes are massive objects with a number of worldvolume andtransverse directions; they play an important role in stringtheory

• Question : What can we learn about branes usingsupergravity as a low-energy approximation to string theory ?

• New insight : supergravity can be extended with many p-formpotentials not describing physical degrees of freedom

Introduction Fundamental Branes and D-branes Solitons Other Branes Open Issues

SUSY Brane Criterion

• Target space potentials form non-trivial “p-form algebra”

• Construction of gauge-invariant WZ term therefore requireson top of the embedding scalars extra worldvolume potentials

• Criterion: the worldvolume fields must fit into a worldvolumesupermultiplet

Introduction Fundamental Branes and D-branes Solitons Other Branes Open Issues

Classification

According to brane tension T ∼ (gs)α :

α = 0 : Fundamental ; α = −1 : Dirichlet; α = −2 : Solitonic

Introduction Fundamental Branes and D-branes Solitons Other Branes Open Issues

Outline

Introduction

Fundamental Branes and D-branes

Solitons

Other Branes

Open Issues

Introduction Fundamental Branes and D-branes Solitons Other Branes Open Issues

Fundamental Branes

For every compactified direction we have two fundamental 0-branescoming from a wrapped fundamental string and a pp-wave

The corresponding 1-forms transform as a vector B1,A under theT-duality group SO(10− D, 10− D)

Wess-Zumino term : LWZ(D < 10) = B2 + ηABF1,AB1,B

contains “extra scalars” b0,A via F1,A = db0,A + B1,A

Hull, Reid-Edwards (2006-2008)

Introduction Fundamental Branes and D-branes Solitons Other Branes Open Issues

Wrapping rule

wrapped → doubled

unwrapped → undoubled

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

(F0)A and F1

Introduction Fundamental Branes and D-branes Solitons Other Branes Open Issues

D-branes

D-branes transform as spinors under T-duality

LWZ(D ≤ 10) = eF2eF1,AΓA

C

Riccioni + E.B. (2010)

Introduction Fundamental Branes and D-branes Solitons Other Branes Open Issues

Wrapping rule

wrapped → undoubled

unwrapped → undoubled

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...

......

......

......

...

8 1/0 1

9 0/1

spinors (Dp)α, α = 1 · · · 29−D

Introduction Fundamental Branes and D-branes Solitons Other Branes Open Issues

Outline

Introduction

Fundamental Branes and D-branes

Solitons

Other Branes

Open Issues

Introduction Fundamental Branes and D-branes Solitons Other Branes Open Issues

“Standard” Solitons

We first restrict to solitons with more than 2 transvere directions

For every compactified direction we have two solitonic (D-4)-branes

coming from an unwrapped solitonic brane and a KK-monopole

Fundamental braneduality←→ Solitonic brane ⇒ “dual” wrapping rule

wrapped → undoubled

unwrapped → doubled

Introduction Fundamental Branes and D-branes Solitons Other Branes Open Issues

Reduced Solitons

Sp-brane IIA/IIB 9 8 7 6 5 4 3

0 1 12

1 1 10

2 1 8

3 1 6

4 1 4

5 1/1 2

S(D-5)-brane and S(D-4)-braneA

Introduction Fundamental Branes and D-branes Solitons Other Branes Open Issues

“Non-standard” Solitons

there are also solitons with less than or equal to 2 transverse directions

U-duality ⊃ SO(10-D,10-D) × R+

S(D-5)-brane

[S(D-4)-brane]A

[S(D-3)-brane]AB

[S(D-2)-brane]ABC

[S(D-1)-brane]ABCD

Not all non-standard solitons are supersymmetric !

Introduction Fundamental Branes and D-branes Solitons Other Branes Open Issues

Supersymmetric Solitons

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 2 4 8 16

dual wrapping rule gives correct number of susy solitons !

Introduction Fundamental Branes and D-branes Solitons Other Branes Open Issues

Question

where do the extra solitons come from?

Introduction Fundamental Branes and D-branes Solitons Other Branes Open Issues

Other Solutions

string theory contains “generalized” KK monopoles with

4–n transverse 5 spatial WV and n isometry directions for n=0,1,2,3,4

Ort́ın et al. (1998); Englert et al. (2003)

in terms of mixed-symmetry fields : D6+n,n ⇔ (T,p,I) = (4-n,5,n)

B2 → Worldvolume , D6 → Transverse , D8,2 → Isometry

Introduction Fundamental Branes and D-branes Solitons Other Branes Open Issues

Outline

Introduction

Fundamental Branes and D-branes

Solitons

Other Branes

Open Issues

Introduction Fundamental Branes and D-branes Solitons Other Branes Open Issues

What is left ?

The only branes left are non-standard branes with α ≤ −3

Among them the only ones with a wrapping rule are the α = −3 branes

Introduction Fundamental Branes and D-branes Solitons Other Branes Open Issues

“E-Branes”

• E-branes are branes with α = −3

• E-branes transform as tensor-spinors under T-duality

• They require a new wrapping rule:

wrapped → doubled

unwrapped → doubled

Introduction Fundamental Branes and D-branes Solitons Other Branes Open Issues

Counting E-branes

Ep-brane 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 1 1 2

New wrapping rule reproduces precisely # of supersymmetricE-brane configurations !

Introduction Fundamental Branes and D-branes Solitons Other Branes Open Issues

Outline

Introduction

Fundamental Branes and D-branes

Solitons

Other Branes

Open Issues

Introduction Fundamental Branes and D-branes Solitons Other Branes Open Issues

Open Issues

• Is there any (geometric) meaning to the wrapping rules ?

• Are the generalized KK monopoles well-defined ?

• What is the role of E11 and the mixed-symmetry tensors ?

• Is there a relation with generalized geometry/non-geometry ?

• Applications to phenomenology ?