Symplectic topology | Algebraic geometry | Cohomology theories

Quantum cohomology

In mathematics, specifically in symplectic topology and algebraic geometry, a quantum cohomology ring is an extension of the ordinary cohomology ring of a closed symplectic manifold. It comes in two versions, called small and big; in general, the latter is more complicated and contains more information than the former. In each, the choice of coefficient ring (typically a Novikov ring, described below) significantly affects its structure, as well. While the cup product of ordinary cohomology describes how submanifolds of the manifold intersect each other, the quantum cup product of quantum cohomology describes how subspaces intersect in a "fuzzy", "quantum" way. More precisely, they intersect if they are connected via one or more pseudoholomorphic curves. Gromov–Witten invariants, which count these curves, appear as coefficients in expansions of the quantum cup product. Because it expresses a structure or pattern for Gromov–Witten invariants, quantum cohomology has important implications for enumerative geometry. It also connects to many ideas in mathematical physics and mirror symmetry. In particular, it is ring-isomorphic to symplectic Floer homology. Throughout this article, X is a closed symplectic manifold with symplectic form ω. (Wikipedia).

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Tangent bundle | Cohomology ring | Differential equation | Frobenius manifold | Ideal (ring theory) | Almost complex manifold | Isomorphism | Identity element | Kronecker delta | Torsion subgroup | Connection (mathematics) | Mirror symmetry (string theory) | Closed manifold | Fubini–Study metric | Gromov–Witten invariant | Cup product | Intersection theory | Homology (mathematics) | Mathematics | Enumerative geometry | Pseudoholomorphic curve | Algebraic geometry | Floer homology | Power series | Ring (mathematics) | Vector bundle | Complex number | Curvature | Chern class | Frobenius algebra | Symplectic manifold | Projective plane | Generating function