Gromov-Witten invariants

How many curves of a given degree pass through a given collection of points or lie in a given homology class on a complex or symplectic manifold? 'How many conics through five points in the plane?' (answer: 1) is the toy version; the modern, vastly harder version asks the same on Calabi-Yau threefolds and other spaces. Gromov-Witten invariants are the rigorous answer: they are numbers that 'count' holomorphic curves of a given type, defined so that the count is a deformation invariant — it does not change as you wiggle the geometry — which is what makes it a genuine invariant rather than an accident of position.

Here is the construction in outline. Fix a symplectic manifold (or smooth projective variety) X with a compatible almost-complex structure J. A J-holomorphic curve is a map u from a Riemann surface Sigma of genus g into X satisfying the Cauchy-Riemann equation du . j = J . du — the higher-dimensional analogue of a holomorphic function. To get finite, well-defined counts, you compactify the space of such maps using Gromov's compactness theorem (sequences of J-curves converge to 'stable maps', possibly with bubbled-off spheres and nodal domains); this gives the moduli space of stable maps M-bar_{g,n}(X, beta) of curves of genus g, with n marked points, in homology class beta. The Gromov-Witten invariant is then an integral over (a virtual fundamental class of) this moduli space of cohomology classes pulled back via the evaluation maps at the marked points — concretely, it computes the expected number of genus-g curves in class beta passing through specified cycles. The genus-zero invariants are the rational-curve counts that mirror symmetry predicts; assembling all of them into a generating function gives the quantum cohomology ring, a deformation of ordinary cup product by curve-counting data.

Why they matter: Gromov-Witten invariants are the mathematical content of the curve counts in mirror symmetry, they define quantum cohomology, and they connect symplectic topology, algebraic geometry, and string theory; the same invariants can be computed symplectically (Gromov-Witten) or algebraically (via the moduli of stable maps over C). The honest cautions are substantial. First, GW invariants are in general RATIONAL numbers, not integers — they are weighted counts, where multiple covers and automorphisms of curves contribute fractional amounts, so calling them 'the number of curves' is a useful slogan that is literally false (the integer enumerative count, when it exists, requires extra arguments to extract; the BPS/Gopakumar-Vafa reorganization recovers integers). Second, defining the integral rigorously needs the virtual fundamental class, because the moduli space of stable maps is typically singular and of the wrong dimension — this is serious technology (Li-Tian, Behrend-Fantechi, and in the symplectic category Fukaya-Ono / Hofer-Wysocki-Zehnder polyfolds), and the naive 'just count solutions' is not well-defined. Third, GW invariants in the survey sense are largely a higher-genus and Calabi-Yau frontier with many open problems; this entry is an orientation, not a closed theory.