 
Summary: The quantum Euler class and the quantum
cohomology of the Grassmannians
Lowell Abrams
December 4, 1997
Abstract
The Poincar'e duality of classical cohomology and the extension
of this duality to quantum cohomology endows these rings with the
structure of a Frobenius algebra. Any such algebra possesses a canon
ical ``characteristic element;'' in the classical case this is the Euler
class, and in the quantum case this is a deformation of the classical
Euler class which we call the ``quantum Euler class.'' We prove that
the characteristic element of a Frobenius algebra A is a unit if and
only if A is semisimple, and then apply this result to the cases of the
quantum cohomology of the finite complex Grassmannians, and to the
quantum cohomology of hypersurfaces. In addition we show that, in
the case of the Grassmannians, the [quantum] Euler class equals, as
[quantum] cohomology element and up to sign, the determinant of the
Hessian of the [quantum] LandauGinzbug potential.
1 Introduction
In [21], Witten's study of instantons in the context of supersymmetry of
