Student Seminar: Classical and Quantum Integrable Systems

From the definition, T(λ) is analytic (entire) 10 in λ with an essential singularity at
λ = 0 11 It is easy to find the expansion around λ = ∞:
T(λ) = I + i λ
∫ 2π
dx S(x) − 1 ∫ 2π
dx S(x)
λ 2
∫ x
0
0
0
The development in 1/λ has an infinite radius of convergency.
dy S(y) + · · ·
To find the structure of T(λ) around λ = 0 is more delicate but very important as
it provides the local conserved charges in involution. Let us introduce the so-called
partial monodromy
[ ∫ x ]
T(x, λ) = P exp dy U(y, λ) .
The main point is to note that there exists a local gauge transformation, regular at
λ = 0, such that
T(x, λ) = g(x)D(x)g −1 (0) ,
where D(x) = exp(id(x)σ 3 ) is a diagonal matrix. We can choose g to be unitary,
and, since g is defined up to to a diagonal matrix, we can require that it has a real
diagonal part:
( )
1 1 v
g =
.
(1 + v¯v) 1 2 −¯v 1
Then the differenial equation for the monodromy
∂ x T = UT = − i λ ST
10 In complex analysis, an entire function is a function that is holomorphic everywhere on the
whole complex plane. Typical examples of entire functions are the polynomials, the exponential
function, and sums, products and compositions of these. Every entire function can be represented
as a power series which converges everywhere. Neither the natural logarithm nor the square root
function is entire. Note that an entire function may have a singularity or even an essential singularity
at the complex point at infinity. In the latter case, it is called a transcendental entire function.
As a consequence of Liouville’s theorem, a function which is entire on the entire Riemann sphere
(complex plane and the point at infinity) is constant.
11 Consider an open subset U of the complex plane C, an element a of U, and a holomorphic
function f defined on U − a. The point a is called an essential singularity for f if it is a singularity
which is neither a pole nor a removable singularity. For example, the function f(z) = exp(1/z) has
an essential singularity at z = 0. The point a is an essential singularity if and only if the limit
0
lim f(z)
z→a
does not exist as a complex number nor equals infinity. This is the case if and only if the Laurent
series of f at the point a has infinitely many negative degree terms (the principal part is an
infinite sum). The behavior of holomorphic functions near essential singularities is described by the
Weierstrass-Casorati theorem and by the considerably stronger Picard’s great theorem. The latter
says that in every neighborhood of an essential singularity a, the function f takes on every complex
value, except possibly one, infinitely often.
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