J-M. Maillard, CNRS, University of Paris (Pierre et Marie Curie) 

Title:
THE THEORY OF THE 2-D ISING MODEL IS NOTHING BUT THE THEORY OF  
ELLIPTIC CURVES.
(and, more generally, theoretical physics is nothing
  but effective algebraic geometry and their birational transformations)


Abstract:

We recall various multiple integrals with one parameter, related to
the isotropic square Ising model, and corresponding, respectively, to
the $n$-particle contributions of the magnetic susceptibility, to the
(lattice) form factors, to the two-point correlation functions and to
their $\, \lambda$-extensions. The univariate analytic functions
defined by these integrals are holonomic and even {\em G}-functions:
they satisfy Fuchsian linear differential equations. We recall the
explicit forms, found in previous work, of these Fuchsian equations,
as well as their russian-doll and direct sum structures. These linear
differential operators are very selected Fuchsian linear differential
operators, and their remarkable properties have a deep geometrical
origin: they are all globally nilpotent, or, sometimes, even have zero
$\, p$-curvature.  We display miscellaneous examples of globally
nilpotent operators emerging from enumerative combinatorics problems
for which no integral representation is yet known.  Focusing on the
factorized parts of all these operators, we find out that the global
nilpotence of the factors (resp.  $\, p$-curvature nullity)
corresponds to a set of selected structures of algebraic geometry:
elliptic curves, modular curves, curves of genus five, six, \ldots ,
and even remarkable weight-1 modular forms. Noticeably, these
associated weight-1 modular form are also seen in the factors of the
linear differential operator for another $\, n$-fold integral of the
Ising class, $\, \Phi_H^{(3)}$, for the staircase polygons counting,
and in Ap\'ery's study of $\, \zeta(3)$.  {\em G}-functions naturally
occur as solutions of globally nilpotent operators.  In the case where
we do not have {\em G}-functions, but Hamburger functions (one
irregular singularity at $\, 0$ or $\, \infty$) that correspond to the
confluence of singularities in the scaling limit, the $\, p$-curvature
is also found to verify new structures associated with simple
deformations of the nilpotent property.

To sum-up: the theory of the 2-D Ising model is nothing but the theory
of elliptic curves and, more generally, theoretical physics is nothing
  but effective algebraic geometry with a selected role played
by birational transformations.