String equation--2. Physical solution

TL;DR

This paper investigates special solutions of the string equation related to the Painlevé-1 equation, focusing on their symmetry and asymptotic properties using linear semiclassical methods, with implications for quantum field theory.

Contribution

It develops the linear semiclassical approach for physical solutions of Painlevé-1, connecting Riemann surface theory with Hamiltonian foliations, advancing understanding of these special solutions.

Findings

Linear semiclassics effectively describes physical solutions of P-1.

Connection established between semiclassics on Riemann surfaces and Hamiltonian foliations.

Special symmetry properties of physical solutions are suggested but not yet proved.

Abstract

This paper is a continuation of the paper by S.P.Novikov in Funct.Anal.Appl., v.24(1990), No 4, pp 196-206. String equation is by definition the equation $[L,A]=1$ for the coefficients of two linear ordinary differential operators $L$ and $A$. For the ``double scaling limit'' of the matrix model we always have $L=-\partial_x^2+u(x)$, $A$ is some differential operator of the odd order $2k+1$. In the first nontrivial case $k=1$ we have the Painelev\'e-1 (P-1) equation. Only special real ``separatrix'' solutions of P-1 are important in the quantum field theory. By the conjecture of Novikov these ``physical'' solutions, which are analytically exceptional probably have much stronger symmetry then the other solutions but it is not proved until now. Two asymptotic methods were developed in the previous paper -- nonlinear semiclassics (or the Bogolubov-Whitham averaging method) and the linear semiclassics for the ``Isomonodromic'' method. The nonlinear semiclassics gives a good approximation for the general (``non-physical'') solutions of P-1 but fails in the ``physical'' case. In our paper the linear semiclasics for the ``physical'' solutions of the P-1 equations is studied. In particular connection between the semiclassics on Riemann surfaces and Hamiltonian foliations on these surfaces is established.