Properties of The Small Quantum Cup Product
For a, b of pure degree,
and
The small quantum cup product is distributive and Λ-bilinear. The identity element is also the identity element for small quantum cohomology.
The small quantum cup product is also associative. This is a consequence of the gluing law for Gromov-Witten invariants, a difficult technical result. It is tantamount to the fact that the Gromov-Witten potential (a generating function for the genus-0 Gromov-Witten invariants) satisfies a certain third-order differential equation known as the WDVV equation.
An intersection pairing
is defined by
(The subscripts 0 indicate the A = 0 coefficient.) This pairing satisfies the associativity property
Read more about this topic: Quantum Cohomology
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