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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]
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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
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Introducion
Dp-brane
1/2 BPS
R-R charge
super Yang-Mills
a p-dimensional dynamical object on which open strings can end
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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
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Introduction
bound state(p,q) F-string D string
BPS bound:
bound energy:
almost the total tension of the F-
string!
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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:
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Boundary stateThe state that describes the creation of closed string from the vacuum is called the boundary state
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Boundary state
external flux on the world-volume
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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
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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
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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
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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
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The string-level force calculation
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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
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The analysis of the amplitudes
t 0tz e
41( ) 2 sin
t
it e
the large-separation limit
12( )t
it e
Y
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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
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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
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The analysis of the amplitudes
1 20 1/ 2,0 1/ 2 divergent tachyon
replusive interaction
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The analysis of the amplitudes
10 200 ,0 1/ 2 1 10 2 20,i
simple poles:
10 0 reduce the rate
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The analysis of the amplitudes
1 20 1/ 2,0 1
divergent tachyon
1 2 BPS configuration:
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10 200 ,0 1 1 10 2 20,i
The analysis of the amplitudes
simple poles:
10 0
20 =1/2
enhancement
2 =0feven
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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