Stable map

In mathematics, specifically in symplectic topology and algebraic geometry, one can construct the moduli space of stable maps, satisfying specified conditions, from Riemann surfaces into a given symplectic manifold. This moduli space is the essence of the Gromov–Witten invariants, which find application in enumerative geometry and type IIA string theory. The idea of stable map was proposed by Maxim Kontsevich around 1992 and published in Kontsevich (1995).

Because the construction is lengthy and difficult, it is carried out here rather than in the Gromov–Witten invariants article itself.

Fix a closed symplectic manifold ${\displaystyle X}$ with symplectic form ${\displaystyle \omega }$. Let ${\displaystyle g}$ and ${\displaystyle n}$ be natural numbers (including zero) and ${\displaystyle A}$ a two-dimensional homology class in ${\displaystyle X}$. Then one may consider the set of pseudoholomorphic curves

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