Shan-Shan Xu

University of Science and Technology of China

Remarks on Dp & Dp−2 with each carrying a flux

Interdisciplinary Center for Theoretical Study

Based on J.X.Lu and S.S.Xu ’s work:

arXiv: 0906.0679 [hep-th]

Remarks on Dp & Dp−2 with each carrying a flux

Introduction Dp-brane, , bound states

Boundary state description

The string-level force calculations

The analysis of the amplitudes

the long-range interactions the short distance behavior open string pair production

Summary

2( , )p pD D( , )pF D

Introducion

Dp-brane

1/2 BPS

R-R charge

super Yang-Mills

a p-dimensional dynamical object on which open strings can end

Introducion

),(),( 1221

One loop vacuum amplitudes are given by the Coleman-Weinberg formula, which can be thought of as the sum of the zero point energies of all the

modes:

no force

conformal symmetry:

NS-NS R-R

the tree-level closed string cylinder diagram

the open string one-loop annulus diagram Boundary state

Introduction

bound state(p,q) F-string D string

BPS bound:

bound energy:

almost the total tension of the F-

string!

Introduction

2( , )p pD D

( , )pF D bound state: a Dp-brane with an electric flux,

bound state: a Dp brane with one magnetic flux.

a D0-brane and a Dp-brane The BPS bound: 24 kpkp 4

direct calculation of the interaction:

Boundary stateThe state that describes the creation of closed string from the vacuum is called the boundary state

Boundary state

external flux on the world-volume

The string-level force calculationThe interaction under consideration can be calculated as the vacuum amplitude of the closed string tree-level cylinder diagram via the closed string boundary states.

the closed string propagator

The string-level force calculation

AX

/2 ,X

pB

the ghost contributions are independent of fluxes and are always given as

To calculate and , make a respective unitary transformation of the oscillators in such that the -matrix there completely disappears while ends up with a new with Sp the original S-matrix in this boundary state and T denoting the transpose. This new S-matrix shares the same property as the original Sk satisfying with k = p−2 or p but its determinant is always unity and therefore can always be diagonalized to gives its eigenvalues.

/ ,XpB

2T

p pS S S 2pS

( ) ( )Tk kS S

( , )A

The string-level force calculation

2 2 2 21 1 3 3 3 4 4 4

2 1 2 1 1 12 0 0

4 4 2 2it it it it it it it it

1

2 12

11 ( )n

nq q it

1

32 1 1 2 1 12

1

( )1 1

( )n n

n

itq q q

it

the Class I matrix elements for matter fields are

1

612 1 2

1

( )1 1

2sin ( )n n

n

q itq q

it

2 ie

vacuum amplitude

tz q e 1

cos 22

The string-level force calculation

1 2 1 2 1 2 1 21 1 1 1 3 1 3 2 3 3 4 1 4 2 4 4

1/2 1/2 1/2 +1/2 1 12 0 0

2 2 2 2 2 2it it it it it it it it it it it it

the ClassII matrix elements for matter fields are

vacuum amplitude:

2 ji

j e j=1,2

The string-level force calculation

the ClassIII matrix elements for matter fields are

The string-level force calculation

10 1/ 2 20 1/2

21/2 1 vacuum amplitude: 2 0f

2 0f

2 2 2 2 21 2 1 21 1 3 1 3 2 3 4 1 4 2 4 2 1 2 2 22 0 0 0

2 2it it it it it it it it it it it

The analysis of the amplitudes

t 0tz e

41( ) 2 sin

t

it e

the large-separation limit

12( )t

it e

Y

The analysis of the amplitudes

1/t t t

2 1/ 21 1( ) ( )tit ie t i t it

1/ 2( ) ( )it t it

small separation 0t 0Y t

the tree-level closed string cylinder diagram

the open string one-loop annulus diagram

0 00 1/ 2

divergent

tachyon

The analysis of the amplitudes

00 0i

0

simple poles：

the rate of open string pair production per unit worldvolume in a constant electric flux in the present context is

at least one electric flux (or being electrically dominant) along a NN-direction

0 0

divergent

enhancement factor

The analysis of the amplitudes

1 20 1/ 2,0 1/ 2 divergent tachyon

replusive interaction

The analysis of the amplitudes

10 200 ,0 1/ 2 1 10 2 20,i

simple poles：

10 0 reduce the rate

The analysis of the amplitudes

1 20 1/ 2,0 1

divergent tachyon

1 2 BPS configuration:

10 200 ,0 1 1 10 2 20,i

The analysis of the amplitudes

simple poles：

10 0

20 =1/2

enhancement

2 =0feven

Summary

The amplitudes of 17 flux configurations of Dp&Dp−2

3 structures of the expression

the long-range interactions one replusive forcethree BPS configurations

open string pair production

Further work and applications:

tachyon condensation

more than one flux

low energy brane dynamics

brane inflation

To form a new bound state