Integration over Tropical Plane Curves and Ultradiscretization

Abstract

In this article we study holomorphic integrals on tropical plane curves in view of ultradiscretization. We prove that the lattice integrals over tropical curves can be obtained as a certain limit of complex integrals over Riemann surfaces.

Unquenched flavor and tropical geometry in strongly coupled Chern-Simons-matter theories

“…Notice that the coefficients of (3.19), regarded as an equation for an algebraic curve, become exponentially large or small (or they remain constant). This kind of behavior has been very much studied recently in the mathematical literature and it is known as the tropical limit of the algebraic curve (or the ultradiscretization of the algebraic curve), see for example [27,28]. This limit is only non-trivial if we scale z, y in the same way, where…”

“…In order to do calculations at strong coupling we have to understand what happens to the period integrals in the regime (3.24). It can be shown rigorously (see for example [28]) that, in the limit (3.24), the periods of differentials on the original curve can be computed directly on the tropical curve, and they reduce to simple contour integrals along the two-dimensional diagram in Fig. 2.…”

A geometric realization of the periodic discrete Toda lattice and its tropicalization

“…They also solved the initial value problem to the UD-pTL of genus 1 and constructed its quasi-periodic solutions by means of the ultradiscrete elliptic theta functions. Subsequent steps were made by Inoue -Takenawa [10,11] and Iwao et al [8,15] in terms of tropical geometry [13] and ultradiscretization procedure. They solved the initial value problems to the UD-pTL of arbitrary genus by using tropical hyperelliptic curves, their tropical Jacobians, ultradiscrete theta functions and ultradiscretization of Abelian integrals.…”

Solutions to the ultradiscrete KP hierarchy and its reductions

“…As an analogue of determinant-type solutions, the ultradiscretization of signature-free determinants is discussed in [10] and the relationship between this type of solution and ultradiscrete soliton equations is discussed in [11,12]. The establishment of relationships to other mathematical topics is also studied, for example, algebro-geometrical [13,14,15] and combinatorial theories [16,17,18].…”