Factorization of the Ising model form factors

TL;DR

This paper introduces a method to factorize Ising model form factors analytically, revealing connections to elliptic curves and hypergeometric functions, with explicit results for specific cases and general proofs for n=2,3.

Contribution

The paper develops a general factorization method for Ising model form factors, linking them to elliptic curves and hypergeometric functions, and provides explicit formulas and identities for n=2,3,4.

Findings

Explicit factorization formulas for n=2,3,4 form factors

New quadratic recursion and quartic identities

Connection between form factors and elliptic curve-associated polynomials

Abstract

We present a general method for analytically factorizing the n-fold form factor integrals $f^{(n)}_{N,N}(t)$ for the correlation functions of the Ising model on the diagonal in terms of the hypergeometric functions $_2F_1([1/2,N+1/2];[N+1];t)$ which appear in the form factor $f^{(1)}_{N,N}(t)$. New quadratic recursion and quartic identities are obtained for the form factors for n=2,3. For n= 2,3,4 explicit results are given for the form factors. These factorizations are proved for all N for n= 2,3. These results yield the emergence of palindromic polynomials canonically associated with elliptic curves. As a consequence, understanding the form factors amounts to describing and understanding an infinite set of palindromic polynomials, canonically associated with elliptic curves. From an analytical viewpoint the relation of these palindromic polynomials with hypergeometric functions associated with elliptic curves is made very explicitly, and from a differential algebra viewpoint this corresponds to the emergence of direct sums of differential operators homomorphic to symmetric powers of a second order operator associated with elliptic curve.