On number and evenness of solutions of the $SU(3)$ Toda system on flat tori with non-critical parameters

Abstract

We study the SU(3) Toda system with singular sources, where E τ := C/(Z + Zτ) with Im τ > 0 is a flat torus, δ p k is the Dirac measure at p k , and n i,k ∈ Z ≥0 satisfy ∑ k n 1,k ≡ ∑ k n 2,k mod 3. This is known as the non-critical case and it follows from a general existence result of [3] that solutions always exist. In this paper we prove that (i) The system has at mostWe have several examples to indicate that this upper bound should be sharp. Our proof presents a nice combination of the apriori estimates from analysis and the classical Bézout theorem from algebraic geometry.(ii) For m = 0 and p 0 = 0, the system has even solutions if and only if at least one of {n 1,0 , n 2,0 } is even. Furthermore, if n 1,0 is odd, n 2,0 is even and n 1,0 < n 2,0 , then except for finitely many τ's modulo SL(2, Z) action, the system has exactly n 1,0 +1 2 even solutions. Differently from [3], our proofs are based on the integrability of the Toda system, and also imply a general non-existence result for even solutions of the Toda system with four singular sources.

Monodromy of a generalized Lame equation of third order

“…In this paper, we study the monodromy of a third order linear differential equation of the following form (1.1) y ′′′ (z) − (α℘(z; τ) + B)y ′ (z) + β℘ ′ (z; τ)y(z) = 0, where ℘(z) = ℘(z; τ) is the Weierstrass ℘-function with periods ω 1 = 1 and ω 2 = τ ∈ H := {τ ∈ C|Im τ > 0}, α, β are constants such that the local exponents at the singularity 0 are three distinct integers, denoted by ς 1 < ς 2 < ς 3 , and B ∈ C is a parameter. The ODE (1.1) arises from the subjects of algebraically integrable differential operators on the elliptic curve E τ := C/(Z + τZ) (see [6]) and the SU(3) Toda system on E τ (see [4]). It can also be seen as the third order generalization of the well-known Lamé equation…”

“…where ∆ is the Laplace operator and δ 0 is the Dirac measure at 0. The Toda system is an important integrable system in mathematical physics [1,9,18]; see also [4,11,13] and references therein for the recent development of the Toda system. In a forthcoming paper, we will apply Theorems 1.2-1.4 to prove the following result.…”