JHEP02(2018)076
Here, in the second line, we have introduced a canonical Poisson bracket on the membrane
defined by {f, g} =
ab
(∂
a
f)(∂
b
g) for each fixed σ
0
. In terms of the Poisson bracket, the
Chern-Simons term can be written as
Z
C
3
=
µ
6
Z
d
3
σ
ijk
X
i
{X
j
, X
k
}. (2.8)
The gauge fixing condition (2.6) as well as the equation of motion of the auxiliary field
produce the following constraints:
g
µν
˙
X
µ
˙
X
ν
= −
2
ν
2
g
µν
g
ρσ
{X
µ
, X
ρ
}{X
ν
, X
σ
}
g
µν
˙
X
µ
∂
a
X
ν
= 0. (2.9)
From the second constraint, it also follows that
{g
µν
˙
X
µ
, X
ν
} = 0. (2.10)
Thus, the system is reduced to the theory (2.7) with these constraints imposed.
The constraints (2.9) can be explicitly solved in the light cone gauge,
X
+
(σ) = σ
0
. (2.11)
Here, we have defined X
±
by
X
±
=
1
√
2
(X
0
± X
10
). (2.12)
We then consider the Hamilton formalism. We denote by P
µ
the canonical conjugate
momentum of X
µ
. The total light cone momentum is then given by
p
+
=
Z
d
2
σP
+
= 2πνT
M2
, (2.13)
where we have chosen the spacial coordinates such that they have a volume
R
d
2
σ = 4π.
This relates the constant ν to the light cone momentum. The Hamiltonian is given by
H
M2
=
Z
d
2
σ
V
2
2p
+
P
2
A
+
T
2
M2
2
{X
A
, X
B
}
2
+
p
+
2V
2
µ
2
9
(X
i
)
2
+
µ
2
36
(X
a
)
2
−
µT
M2
6
ijk
X
i
{X
j
, X
k
}
, (2.14)
where, V
2
is the volume of the unit sphere, V
2
= 4π. The remaining constraint (2.10) is
written in terms of the transverse components X
A
as
{P
A
, X
A
} = 0. (2.15)
Now, let us consider a vacuum configuration, which minimizes the Hamiltonian (2.14).
Note that the potential for X
i
forms a perfect square,
p
+
µ
2
18V
2
X
i
−
3V
2
T
M2
2µp
+
ijk
{X
i
, X
j
}
2
. (2.16)
– 6 –
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