The symplectic sum formula for Gromov–Witten invariants

By Eleny-Nicoleta Ionel, Thomas H. Parker. 2000

207 citations recorded by the source corpus.

In the symplectic category there is a 'connect sum' operation that glues symplectic manifolds by identifying neighborhoods of embedded codimension two submanifolds. This paper establishes a formula for the Gromov-Witten invariants of a symplectic sum Z = X#Y in terms of the relative GW invariants of X and Y. Several applications to enumerative geometry are given. Gromov-Witten invariants are counts of holomorphic maps into symplectic manifolds. To define them on a symplectic manifold (X,w) one introduces an almost complex structure J compatible with the symplectic form w and

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2197cae344f924f00e6a7cc1db71e54b7a1fc700

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Logarithmic Geometry and Moduli (2010, background): “Ruan in symplectic geometry [37], see also Ionel-Parker [21, 22].”
Floer cohomology and pencils of quadrics (2010, background): “It follows that the limit disk u∞ has no component contained in C0, and indeed no component which intersects the divisor C0 ∩ C1 (such a component would be glued to a non-constant sphere in the component C0 or it would not be locally smoothable to the nearby fibre; compare to the degeneration formul”
Hamiltonian handleslides for Heegaard Floer homology (2008, background): “This requires closer inspection of the local model for the symplectic sum, and we shall not give the details here (see [6]; also compare [11, Section 2]).”
A mathematical theory of the topological vertex (2004, methodology): “Our theory is based on relative Gromov-Witten theory [26, 18, 19, 24, 25].”
Symplectic hypersurfaces and transversality in Gromov-Witten theory (2007, result): “This paper is clearly related to the work of Ionel and Parker on relative Gromov–Witten invariants [14, 15].”
Minimality and symplectic sums (2006, methodology): “Without reproducing the formula, we recall the essential points, referring readers to [5] for details: in forming the symplectic sum, letting x1 and x2 be complex coordinates in the normal bundles ν1, ν2, one performs the identification by taking x1x2 = λ for some small λ ∈ C ; this results in a sym”
Exploded Manifolds (2009, background): “This case of a family representing a symplectic sum has been studied by Eliashberg Givental and Hofer in [1], Ionel and Parker in [5], Jun Li in [8], and Ruan and A.”