| Abstract: |
| The Hurwitz numbers count the topological types of finite branched
coverings of a compact Riemann surface. The Hurwitz numbers
of the Riemann sphere $\mathbf{C}\mathbf{P}^1$ can be assembled
in a generating function $\tau(t_1,t_2,\ldots)$ that is known
to be a tau function of the KP hierarchy. This tau function
can be extended to a tau function $\tau(s,t_1,t_2,\ldots)$,
$s \in \mathbf{Z}$, of the modified (or discrete) KP hierarchy,
in which $s$ plays the role of a lattice coordinate.
The Lax equations of the modified KP hierarchy,
like the Toda hierarchy, are formulated in the language
of difference operators as
\[
\frac{\partial L}{\partial t_k} = [B_k,L], \quad k = 1,2,\ldots,
\]
where $L$ is a pseudo-difference operator of the form
$L = e^{\partial_s} + u_1 + u_2e^{-\partial_s} + \cdots$,
and $B_k$ is the part $(L^k)_{\ge 0}$ of non-negative powers
of $e^{\partial_s}$ in $L^k$.
Actually, the variable $s$ in the Hurwitz tau function
may be thought of as a continuous variable.
The formal expression $e^{\partial_s}$ of the shift operator
thereby acquires a literal meaning as the exponential
of $\partial_s = \partial/\partial s$. In this interpretation,
the logarithm $\log L$ of $L$ turns out to take
the special form
\[
\log L = \partial_s - ue^{-\partial_s}, \quad
u = u(s,t_1,t_2,\ldots),
\]
and satisfy the Lax equations of the same form as $L$.
This subsystem of the modified KP hierarchy is a large-$N$ limit
of the $N$-step hungry Volterra (aka Bogoyavlensky-Itoh) hierarchy
realized on the fractional lattice $N^{-1}\mathbf{Z} \subset \mathbf{R}$,
hence may be called a {\it continuous} Volterra hierarchy. |
|