Conformal Maps and Integrable Hierarchies

Abstract

We show that conformal maps of simply connected domains with an analytic boundary to a unit disk have an intimate relation to the dispersionless 2D Toda integrable hierarchy. The maps are determined by a particular solution to the hierarchy singled out by the conditions known as "string equations". The same hierarchy locally solves the 2D inverse potential problem, i.e., reconstruction of the domain out of a set of its harmonic moments. This is the same solution which is known to describe 2D gravity coupled to c = 1 matter. We also introduce a concept of the τ -function for analytic curves.

HCIZ integral and 2D Toda lattice hierarchy

“…The second relation we use the dispersionless limit of the Fay identity [18], which we shall not prove here and write under the form (see [25]):…”

Towards an Effectivisation of the Riemann Theorem

“…In §1 we reproduce, following [ 3,15,16 ], an important result [ 4 ]: the Riemann maps have a potential F : H → C (termed the tau-function of the curves in [11]) such that w(z, t) = zexp((− 1 2 ∂ 2 0 − ∂ 0 D(z))F (t)) (1.4) and, moreover, F satisfies the differential equations This system of nonlinear differential equations is well known in mathematical physics and in theory of integrable systems as the dispersionless limit of the 2D Toda hierarchy [ 5 ]. The solution F (t) satisfies some additional equation, and it appears in string theory as the "string solution" [ 6 ].…”

“…This formula was obtained first in [ 4 ] by using formulas for conformal maps from an ellipse to the circle. The recursive formulas for coefficients of the Taylor series F give a possibility to estimate the coefficients and to find sufficient convergence conditions for F provided t i ,t i = 0 for i > n. Theorem 4.2.…”

Canonical and Grand Canonical Partition Functions of Dyson Gases as Tau-Functions of Integrable Hierarchies and Their Fermionic Realization

“…Then F 0 is given by 11) which is basically the electrostatic energy of D filled by electric charge with density σ (and with a point-like charge at the origin). In the case U(z,z) = zz, σ = 1/π one obtains the "tau-function of analytic curves" [25] which encodes the integrable structure of parametric families of conformal maps [12] and the Dirichlet boundary value problem in C \ D [13,14]. The analytic continuation of the function F 0 to general values of T k (not necessarily constrained by the condition T −k = −T k ) also has a geometric meaning in terms of pairs of conformal maps [26].…”