JHEP04(2016)172
where ˆg and
ˆ
B are the pullback of the background metric and the NS-NS fluxes and F
is the field strength for the worldvolume gauge field. Here T
D
is the D1-string tension
T
D
= 1/(2πα
′
g
s
), with g
s
being the string coupling. In contrast, remember that the F-
string tension was simply T
F
= 1/2πα
′
. So in weak coupling regime we can write T
D
≫ T
F
.
Also, ǫ
αβ
is the usual antisymmetric tensor with ǫ
01
= 1. Since the mixed flux background
has the constant (or zero) dilaton, the Lagrangian density takes the form,
L = −T
D
q
−det(ˆg +
ˆ
B + 2πα
′
F ) −
1
2
ǫ
αβ
C
αβ
. (2.3)
We choose the following ansatz for a ‘breathing’ mode of the string:
t = ξ
0
, ρ(ξ
0
, ξ
1
) = ρ(ξ
0
) , φ = ξ
1
. (2.4)
The ansatz is like the one for a simple circular F-string motion and can be easliy shown to
be consistent with the equations of motion here. Using the above ansatz we can show,
−det(ˆg +
ˆ
B + 2πα
′
F ) = sinh
2
ρ(cosh
2
ρ − (∂
0
ρ)
2
) − (q cosh
2
ρ − 2πα
′
∂
0
A
φ
)
2
= −det ˆg −F
2
φt
, (2.5)
where we have
det ˆg = −sinh
2
ρ(cosh
2
ρ − (∂
0
ρ)
2
) (2.6)
and
F
φt
= q cosh
2
ρ − 2πα
′
∂
0
A
φ
. (2.7)
Now, we can rewrite the lagrangian density as,
L = −T
D
h
q
−det ˆg −F
2
φt
−
p
1 − q
2
cosh
2
ρ
i
. (2.8)
Conjugate momentum of the Wilson line A
φ
is a quantized constant of the motion given by,
1
2π
Π
φ
=
∂L
∂(∂
0
A
φ
)
=
−2πα
′
T
D
F
φt
q
−det ˆg − F
2
φt
= −n . (2.9)
The integer n is the number of oriented fundamental strings bound to the D-string. The
effective tension of (m, n) circular string is given by,
T
(m,n)
=
q
m
2
T
2
D
+ n
2
T
2
F
. (2.10)
From the above expression and (
2.9) we can write,
T
(1,n)
= T
D
v
u
u
t
1 +
(2πα
′
)
2
T
2
F
F
2
φt
(−det ˆg − F
2
φt
)
. (2.11)
Combining all expression we get the following relation which we will later use to simplify
other expressions,
T
(1,n)
√
−det ˆg
=
T
D
q
−det ˆg −F
2
φt
=
n
2πα
′
F
φt
. (2.12)
– 4 –
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