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

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 $1+3w+4w^2=0$, 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.

39 pages, 7 figures

References in corpus (5)

• On Rationally Parametrized Modular Equations
• The diagonal Ising susceptibility
• Holonomy of the Ising model form factors
• Landau singularities and singularities of holonomic integrals of the Ising class
• Creation operators and algebraic Bethe ansatz for the elliptic quantum group $E_{\tau ,\eta }(s{o}_3)$

Cited by in corpus (11)

• Experimental mathematics on the magnetic susceptibility of the square lattice Ising model
• A fast algorithm for computing the characteristic polynomial of the p-curvature
• A Fast Algorithm for Computing the p-Curvature
• Lattice Green Functions: the seven-dimensional face-centred cubic lattice
• Ising n-fold integrals as diagonals of rational functions and integrality of series expansions: integrality versus modularity
• Canonical decomposition of linear differential operators with selected differential Galois groups
• From Holonomy of the Ising Model Form Factors to n-Fold Integrals and the Theory of Elliptic Curve
• Factorization of Ising correlations C(M,N) for $\nu =-k$ and M+N odd, $M\le N$, $T<T_c$ and their lambda extensions
• Schwarzian conditions for linear differential operators with selected differential Galois groups (unabridged version)
• The lambda extensions of the Ising correlation functions C(M, N)
• Diagonals of rational functions: from differential algebra to effective algebraic geometry (unabridged version)