Geometry and Integrability in Mathematical Physics GIMP'06
May 15 - 19, 2006, Moscow, Russia
Anton Zabrodin
ITEP, Moscow
L\"owner equations and dispersionless hierarchies
(a joint work with T. Takebe and L.-P. Teo)
The L\"owner equation is a differential equation obeyed by a family of continuously varying univalent conformal maps $G_{\lambda}(w)$ from the exterior of the unit circle onto a domain with a slit formed by a continuously increasing arc of a fixed curve parameterized by a real parameter $\lambda$. Let us normalize the maps so that $G_{\lambda}(w) =e^{\phi}w+ \alpha_0 + \alpha_1 w^{-1}+\ldots$ as $w\to \infty$ with a real $\phi =\phi (\lambda )$, then the (``radial") L\"owner equation reads $$ \frac{\pa G_{\lambda}(w)}{\pa \lambda}=-w\, \frac{\pa G_{\lambda}(w)}{\pa w} \,\, \frac{\sigma (\lambda )+w}{\sigma (\lambda )-w}\,\, \frac{\pa \phi (\lambda )}{\pa \lambda} $$ where $\sigma (\lambda )$ is a continuous real-valued function. There is also a ``chordal" version of the L\"owner equation with the conformal map being normalized in a different way.
The Schramm's discovery of the stochastic L\"owner evolution (SLE) and its spectacular applications to the conformal field theory have inspired a renewed interest in the theory of L\"owner equations. One of its most interesting aspects is the relation to the integrable hierarchies of PDE's first noticed by Gibbons and Tsarev yet before the SLE boom. In short, the L\"owner equations appear to be consistency conditions for reductions of the integrable hierarchies.
Our main result is the following. Given any solution $G_{\lambda}(w)$ to the radial L\"owner equation, the Lax function $\mathcal{L}(w; \boldsymbol{t} )=G_{\lambda}(w)$ solves all the Lax equations of the dToda hierarchy provided the dependence $\lambda =\lambda (\boldsymbol{t})$ is given by the system of hydrodynamic type $\frac{\pa \lambda}{\pa t_n}=\xi_n (\lambda )\, \frac{\pa \lambda}{\pa t_0}$. Here the functions $\xi_n (\lambda )$ are constructed in a canonical way from $\mathcal{L}(w; \boldsymbol{t})$ and $\sigma (\lambda )$. (In the context of the dKP hierarchy, this result was established earlier.) We also prove the converse. Let $\mathcal{L}(w; \boldsymbol{t})$ be the Lax function of the dToda hierarchy. Suppose one has a \emph{one-variable reduction} of the hierarchy, i.e., the Lax function depends on the hierarchical times $\boldsymbol{t}= \{\ldots, t_{-1}, t_0 , t_1 , \ldots \}$ via a single function $\lambda (\boldsymbol{t})$: $\mathcal{L}(w; \boldsymbol{t})=\mathcal{L}(w, \lambda (\boldsymbol{t}))$. Then consistency of this ansatz with the hierarchy implies that the function of two variables $\mathcal{L}(w, \lambda )$ obeys the radial L\"owner equation with some $\sigma (\lambda )$ while the time dependence of $\lambda$ is determined from the hydrodynamic system.
We also clarify the geometric meaning of this result and that of reductions of the dToda hierarchy. This can be understood in terms of the model of normal random matrices in the large $N$ limit. Actually, we use only the eigenvalue integral representation, which is the partition function of $N$ Coulomb charges in 2D in external field. The matrix models are known to be solvable (even at finite $N$) in terms of integrable hierarchies, which become dispersionless as $N\to \infty$. At the same time, the domains subject to conformal maps under study appear in this context as complements of the supports of eigenvalues as $N\to \infty$. This picture provides a transparent geometric meaning of the reductions and conformal maps of slit domains corresponding to them.
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