Gromov–Witten theory

In mathematics, specifically in symplectic topology and algebraic geometry, Gromov–Witten (GW) invariants are rational numbers that, in certain situations, count pseudoholomorphic curves meeting prescribed conditions in a given symplectic manifold. The GW invariants may be packaged as a homology or cohomology class in an appropriate space, or as the deformed cup product of quantum cohomology. These invariants have been used to distinguish symplectic manifolds that were previously indistinguishable. They also play a crucial role in closed type IIA string theory. They are named after Mikhail Gromov and Edward Witten.

The rigorous mathematical definition of Gromov–Witten invariants is lengthy and difficult, so it is treated separately in the stable map article. This article attempts a more intuitive explanation of what the invariants mean, how they are computed, and why they are important.

Consider the following:

Now we define the Gromov–Witten invariants associated to the 4-tuple: (X, A, g, n). Let ${\displaystyle {\overline {\mathcal {M}}}_{g,n}}$ be the Deligne–Mumford moduli space of curves of genus g with n marked points and ${\displaystyle {\overline {\mathcal {M}}}_{g,n}(X,A)}$ denote the moduli space of stable maps into X of class A, for some chosen almost complex structure J on X compatible with its symplectic form. The elements of ${\displaystyle {\overline {\mathcal {M}}}_{g,n}(X,A)}$ are of the form:

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