  In this article, we define a non-commutative deformation of the "symplectic
invariants" of an algebraic hyperelliptical plane curve. The necessary
condition for our definition to make sense is a Bethe ansatz. The commutative
limit reduces to the symplectic invariants, i.e. algebraic geometry, and thus
we define non-commutative deformations of some algebraic geometry quantities.
In particular our non-commutative Bergmann kernel satisfies a Rauch variational
formula. Those non-commutative invariants are inspired from the large N
expansion of formal non-hermitian matrix models. Thus they are expected to be
related to the enumeration problem of discrete non-orientable surfaces of
arbitrary topologies.
