It is a symplectic version of connected summation along a submanifold, often called a fiber sum.
Together the symplectic sum and cut may be viewed as a deformation of symplectic manifolds, analogous for example to deformation to the normal cone in algebraic geometry.
via such that the Euler classes of the normal bundles are opposite: In the 1995 paper that defined the symplectic sum, Robert Gompf proved that for any orientation-reversing isomorphism there is a canonical isotopy class of symplectic structures on the connected sum meeting several conditions of compatibility with the summands
In other words, the theorem defines a symplectic sum operation whose result is a symplectic manifold, unique up to isotopy.
To produce a well-defined symplectic structure, the connected sum must be performed with special attention paid to the choices of various identifications.
is composed with an orientation-reversing symplectic involution of the normal bundles of
(or rather their corresponding punctured unit disk bundles); then this composition is used to glue
The preceding description of the sum of two manifolds then corresponds to the special case where
consists of two connected components, each containing a copy of
Additionally, the sum can be performed simultaneously on submanifolds
Because the sphere is a compact manifold, a symplectic form
, one may projectively complete the normal bundle of
Such identity elements have been used both in establishing theory and in computations; see below.
It is sometimes profitable to view the symplectic sum as a family of manifolds.
and a fibration in which the central fiber is the singular space obtained by joining the summands
(That is, the generic fibers are all members of the unique isotopy class of the symplectic sum.)
Loosely speaking, one constructs this family as follows.
Choose a nonvanishing holomorphic section
of the trivial complex line bundle Then, in the direct sum with
, consider the locus of the quadratic equation for a chosen small
deleted) onto this locus; the result is the symplectic sum
is the symplectic cut of the generic fiber.
An important example occurs when one of the summands is an identity element
This is analogous to deformation to the normal cone along a smooth divisor
In fact, symplectic treatments of Gromov–Witten theory often use the symplectic sum/cut for "rescaling the target" arguments, while algebro-geometric treatments use deformation to the normal cone for these same arguments.
However, the symplectic sum is not a complex operation in general.
The symplectic sum was first clearly defined in 1995 by Robert Gompf.
A number of researchers have subsequently investigated the behavior of pseudoholomorphic curves under symplectic sums, proving various versions of a symplectic sum formula for Gromov–Witten invariants.
Such a formula aids computation by allowing one to decompose a given manifold into simpler pieces, whose Gromov–Witten invariants should be easier to compute.
as a symplectic sum A formula for the Gromov–Witten invariants of a symplectic sum then yields a recursive formula for the Gromov–Witten invariants of