Quantum Cohomology - Geometric Interpretation

Geometric Interpretation

The only pseudoholomorphic curves in class A = 0 are constant maps, whose images are points. It follows that

in other words,

Thus the quantum cup product contains the ordinary cup product; it extends the ordinary cup product to nonzero classes A.

In general, the Poincaré dual of (a∗b)_A corresponds to the space of pseudoholomorphic curves of class A passing through the Poincaré duals of a and b. So while the ordinary cohomology considers a and b to intersect only when they meet at one or more points, the quantum cohomology records a nonzero intersection for a and b whenever they are connected by one or more pseudoholomorphic curves. The Novikov ring just provides a bookkeeping system large enough to record this intersection information for all classes A.