Singularities of $n$-fold integrals of the Ising class and the theory of elliptic curves

TL;DR

This paper investigates the singularities of multiple integrals related to the Ising model's susceptibility, revealing connections to elliptic curves with complex multiplication and identifying new potential singularities through differential equations.

Contribution

It introduces a method to find and analyze singularities of integrals linked to the Ising model, connecting them to elliptic curves and complex multiplication, and extends understanding of their mathematical structure.

Findings

Singularities correspond to Landau pinch points up to n=6

Reduction of singularities to finite one-dimensional integrals

Identification of elliptic curves with complex multiplication in the singularity structure

Abstract

We introduce some multiple integrals that are expected to have the same singularities as the singularities of the $ n$-particle contributions $\chi^{(n)}$ to the susceptibility of the square lattice Ising model. We find the Fuchsian linear differential equation satisfied by these multiple integrals for $ n=1, 2, 3, 4$ and only modulo some primes for $ n=5$ and $ 6$, thus providing a large set of (possible) new singularities of the $\chi^{(n)}$. We discuss the singularity structure for these multiple integrals by solving the Landau conditions. We find that the singularities of the associated ODEs identify (up to $n= 6$) with the leading pinch Landau singularities. The second remarkable obtained feature is that the singularities of the ODEs associated with the multiple integrals reduce to the singularities of the ODEs associated with a {\em finite number of one dimensional integrals}. Among the singularities found, we underline the fact that the quadratic polynomial condition $ 1+3 w +4 w^2 = 0$, that occurs in the linear differential equation of $ \chi^{(3)}$, actually corresponds to a remarkable property of selected elliptic curves, namely the occurrence of complex multiplication. The interpretation of complex multiplication for elliptic curves as complex fixed points of the selected generators of the renormalization group, namely isogenies of elliptic curves, is sketched. Most of the other singularities occurring in our multiple integrals are not related to complex multiplication situations, suggesting an interpretation in terms of (motivic) mathematical structures beyond the theory of elliptic curves.