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.
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.
This allows the quantum cup product, defined below, to record information about pseudoholomorphic curves in X.
, define (a∗b)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.
The only pseudoholomorphic curves in class A = 0 are constant maps, whose images are points.
The Novikov ring just provides a bookkeeping system large enough to record this intersection information for all classes A.
be the Poincaré dual of a line L. Then The only nonzero Gromov–Witten invariants are those of class A = 0 or A = L. It turns out that and where δ is the Kronecker delta.
Then For a, b of pure degree, and The small quantum cup product is distributive and Λ-bilinear.
This is a consequence of the gluing law for Gromov–Witten invariants, a difficult technical result.
This pairing satisfies the associativity property 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.
Commutativity and associativity of the quantum cup product then correspond to zero-torsion and zero-curvature conditions on this connection.
and the Dubrovin connection give U the structure of a Frobenius manifold.
All of the genus-0 Gromov–Witten invariants are recoverable from it; in general, the same is not true of the simpler small quantum cohomology.
To obtain enumerative geometrical information for some manifolds, we need to use big quantum cohomology.
Small quantum cohomology would correspond to 3-point correlation functions in physics while big quantum cohomology would correspond to all of n-point correlation functions.