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

Abstract

Abstract. We consider the Fuchsian linear differential equation obtained (modulo a prime) forχ (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 ofχ (1) andχ (3) can be removed fromχ (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 termsχ (3) andχ (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 allχ (n) 's.

Schwarzian conditions for linear differential operators with selected differential Galois groups

“…The fact that being the symmetric cube of an underlying order-two operator verifies automatically the new condition (34) emerging from a compatibility condition of an order-four linear differential operator by pullback is far less obvious. The "parametrization" (35) necessarily yields the Calabi-Yau condition (32) and the new condition (34), and, conversely, (32) and (34) can be parametrized by (35). Our large calculations thus show that the pullback-compatibility of an order-four linear differential operator which verifies the Calabi-Yau condition (32), amounts to saying that this order-four linear differential operator reduces to (the symmetric cube of) an underlying order-two linear differential operator.…”

“…One finds straightforwardly that the coefficients given by (35) verify the Calabi-Yau condition (32), as well as the new condition (34). In this case the differential Galois group is no longer the symplectic differential Galois group Sp(4, C), but actually reduces ‡ to the differential Galois group of the underlying order-two linear differential operator, namely SL(2, C).…”

“…At the last step we consider the identification of the constant terms in D x in L (p) 4 and L (c) 4 . After injecting in this "large" non-linear differential equation, equation (11), the Schwarzian condition (29) with W (x) given by (30), and the Calabi-Yau condition (32), we eventually find that this last "large" equation becomes independent of the pullback y(x) and reduces to a quite simple condition giving s(x) as a polynomial expression in the two coefficients p(x) and q(x) and their derivatives:…”

“…When condition (32) is verified, the order-four linear differential operator L 4 has a symplectic differential Galois group Sp(4, C). Note that if condition (32) is verified, the Calabi-Yau conditions for the pullbacked and conjugated operators L Recall that the Calabi-Yau condition (32) together with the globally nilpotent condition [24] automatically yields L 4 to be conjugated to its adjoint…”

On Christol’s conjecture

“…There is a plethora of multiple integrals in physics: Feynman integrals, lattice Green functions, the summands of the magnetic susceptibility of the 2D Ising model [2,22], that have very specific mathematical properties. These functions are D-finite, i.e., solutions of linear differential operators with polynomial coefficients, and have series expansions with integer coefficients.…”