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

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.

33 pages

References in corpus (8)

• Differential Equations for Algebraic Functions
• Experimental mathematics on the magnetic susceptibility of the square lattice Ising model
• High order Fuchsian equations for the square lattice Ising model: ${\tilde{\chi }}^{(5)}$
• The diagonal Ising susceptibility
• Singularities of $n$-fold integrals of the Ising class and the theory of elliptic curves
• Holonomy of the Ising model form factors
• Globally nilpotent differential operators and the square Ising model
• Landau singularities and singularities of holonomic integrals of the Ising class

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