Dubrovin Connection
When the base ring R is C, one can view the evenly-graded part H of the vector space QH*(X, Λ) as a complex manifold. The small quantum cup product restricts to a well-defined, commutative product on H. Under mild assumptions, H with the intersection pairing is then a Frobenius algebra.
The quantum cup product can be viewed as a connection on the tangent bundle TH, called the Dubrovin connection. Commutativity and associativity of the quantum cup product then correspond to zero-torsion and zero-curvature conditions on this connection.
Read more about this topic: Quantum Cohomology
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“The connection between our knowledge and the abyss of being is still real, and the explication must be not less magnificent.”
—Ralph Waldo Emerson (1803–1882)