High order Fuchsian equations for the square lattice Ising model: $\tilde{\chi}^{(5)}$

TL;DR

This paper analyzes the high order Fuchsian differential equations related to the five-particle contribution to the susceptibility of the square lattice Ising model, revealing factorization properties and connections to elliptic integrals.

Contribution

It demonstrates the factorization of the differential operator for  using calculations from a single prime and links a key factor to the symmetric power of the elliptic integral operator.

Findings

The differential operator for  splits into small order factors.

A fifth order operator is equivalent to the symmetric fourth power of the elliptic integral operator.

The factorization pattern generalizes to all  contributions.

Abstract

We consider the Fuchsian linear differential equation obtained (modulo a prime) for $\tilde{\chi}^{(5)}$, the five-particle contribution to the susceptibility of the square lattice Ising model. We show that one can understand the factorization of the corresponding linear differential operator from calculations using just a single prime. A particular linear combination of $\tilde{\chi}^{(1)}$ and $\tilde{\chi}^{(3)}$ can be removed from $\tilde{\chi}^{(5)}$ and the resulting series is annihilated by a high order globally nilpotent linear ODE. The corresponding (minimal order) linear differential operator, of order 29, splits into factors of small orders. A fifth order linear differential operator occurs as the left-most factor of the "depleted" differential operator and it is shown to be equivalent to the symmetric fourth power of $L_E$, the linear differential operator corresponding to the elliptic integral $E$. This result generalizes what we have found for the lower order terms $\tilde{\chi}^{(3)}$ and $\tilde{\chi}^{(4)}$. We conjecture that a linear differential operator equivalent to a symmetric $(n-1)$-th power of $L_E$ occurs as a left-most factor in the minimal order linear differential operators for all $\tilde{\chi}^{(n)}$'s.