Student Seminar: Classical and Quantum Integrable Systems

where σ i are the Pauli matrices 9 and
k 0 = m 4
(
λ + 1 )
, k 1 = m (
λ − 1 )
.
λ
4 λ
Show by direct computation that the condition of zero curvature is equivalent to the
Sine-Gordon equation.
The one-parameter family of the flat connections allows one to define the monodromy
matrix T(λ) which is the path-ordered exponential of the Lax component
U(λ):
∫ 2π
T(λ) = P exp dxU(λ) . (4.3)
0
Let us derive the time evolution equation for this matrix. We have
∂ t T(λ) =
=
∫ 2π
0
∫ 2π
0
dx Pe R 2π
x dyU (∂ t U) Pe R x
0 dyU
dx Pe R 2π
x dyU (∂ x V + [V, U]) Pe R x
0 dyU , (4.4)
where in the last formula we used the flatness of L α ≡ (U, V ). The integrand of the
expression we obtained is the total derivative
∂ t T(λ) =
∫ 2π
0
(
dx ∂ x Pe R 2π
x
Thus, we obtained the following evolution equation
dyU V Pe R x
0 dyU )
. (4.5)
∂ t T(λ) = [V (2π, t, λ), T(λ)] . (4.6)
This formula shows that the eigenvalues of T(λ) generate an infinite set of integrals of
motion upon expansion in λ. Thus, the spectral properties of the model are encoded
into the monodromy matrix.
The wording “monodromy” comes from the fact that T(t) represents the monodromy
of a solution of the fundamental linear problem:
9 The Pauli matrices are
σ 1 =
( ) 0 1
, σ 2 =
1 0
Ψ(2π, t) = T(t)Ψ(0, t) .
( )
( )
0 −i
1 0
, σ 3 = . (4.2)
i 0
0 −1
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