In one sentence, we are going to present an elliptization of the
results of Beukers and Heckman on the monodromy of mFm-1.

mFm-1 is a solution of the generalized hypergeometric equation
(g.h.g.eq.) which is the closest relative of the famous hypergeometric
equation of Gauss-Riemann. It is known that when all the local
exponents of the g.h.g.eq. are generic real numbers, there exists a
(unique up to a constant multiple) monodromy invariant hermitian form
on the space of solutions H_trig. The m-hypergeometric system
(m-h.g.s.) is a Fuchsian system equivalent to the g.h.g.eq. as a flat
connection. When all its local exponents are generic real numbers,
there exists a (unique up to a constant multiple) complex symmetric
form on the residue space H_o such that the residue matrices are
self-adjoint with respect to it. The formulae for the symmetric product
on H_o and for the hermitian product on H_trig look very similar to
each other despite the different nature of the products. It was our
initial goal to understand reasons for such a similarity.

        It turns out that there exists a Hilbert space H naturally associated
with the problem. This space has three two-parameter families of both
hermitian and complex symmetric forms on it. In particular, it is
hyperkahler. H_trig and H_o are m-dimensional subspaces of H. The
hyperkahler structure on H explains why both H_o and H_trig have
complex symmetric and hermitian forms on them. Moreover, there exists
another m-dimensional subspace H(omega1, omega2) of H such that the
space of solutions of the m-h.g.s. H_trig is its trigonometric limit as
omega2 -> infinity. The residue space H_o is the rational limit of
H(omega1, omega2) as both omega1, omega2 -> infinity. This explains the
similarity between the formulae for the two spaces.

        The main technique used in the paper is to realize solutions of the
m-h.g.s. as fermionic fields. Analytic continuation is then replaced by
the vacuum expectation value pairing.

If time permits, we will discuss applications of the above to the KP
equation and to representation theory.