Quantum Cohomology - Small Quantum Cohomology

Let

be the cohomology of X modulo torsion. Define the small quantum cohomology with coefficients in Λ to be

Its elements are finite sums of the form

The small quantum cohomology is a graded R-module with

The ordinary cohomology H*(X) embeds into QH*(X, Λ) via, and QH*(X, Λ) is generated as a Λ-module by H*(X).

For any two cohomology classes a, b in H*(X) of pure degree, and for any A in, define (ab)A to be the unique element of H*(X) such that

(The right-hand side is a genus-0, 3-point Gromov-Witten invariant.) Then define

This extends by linearity to a well-defined Λ-bilinear map

called the small quantum cup product.

Read more about this topic:  Quantum Cohomology

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