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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, ω) one introduces an almost complex structure J compatible with the symplectic form ω and forms the moduli space of J-holomorphic maps from complex curves into X and the compactified moduli space, called the space of stable maps. One then imposes constraints on the stable maps, requiring the domain to have a certain form and the image to pass through fixed homology cycles in X. When the correct number of constraints is imposed there are only finitely many maps satisfying the constraints; the (oriented) count of these is the corresponding GW invariant. For complex algebraic manifolds these symplectic invariants can also be defined by algebraic geometry, and in important cases the invariants are the same as the curve counts that are the subject of classical enumerative algebraic geometry.In the past decade the foundations for this theory were laid and the invariants were used to solve several long-outstanding problems. The focus now is on finding effective ways of computing the invariants. One useful technique is the method of 'splitting the domain', in which one localizes the invariant to the set of maps whose domain curves have two irreducible components with the constraints distributed between them. This produces recursion relations relating the desired GW invariant to invariants with lower degree or genus. This paper establishes a general formula describing the behavior of GW invariants under the analogous operation of 'splitting the target'. Because we *The research of both authors was partially supported by the NSF. The first author was also supported by a Sloan Research Fellowship. 936ELENY-NICOLETA IONEL AND THOMAS H. PARKER work in the context of symplectic manifolds the natural splitting of the target is the one associated with the symplectic cut operation and its inverse, the symplectic sum.The symplectic sum is defined by gluing along codimension two submanifolds. Specifically, let X be a symplectic 2n-manifold with a symplectic (2n−2)-submanifold V . Given a similar pair (Y, V ) with a symplectic identification between the two copies of V and a complex anti-linear isomorphism between the normal bundles N X V and N Y V of V in X and in Y , we can form the symplectic sum Z = X# V Y . Our main theorem is a 'Symplectic Sum Formula' which expresses the GW invariants of the sum Z in terms of the relative GW invariants of (X, V ) and (Y, V ) introduced in [IP4].The symplectic sum is perhaps more naturally seen not as a single manifold but as a family depending on a 'squeezing parameter'. In Section 2 we construct a f...

We define relative Gromov-Witten invariants of a symplectic manifold relative to a codimension-two symplectic submanifold. These invariants are the key ingredients in the symplectic sum formula of [IP4]. The main step is the construction of a compact space of 'V -stable' maps. Simple special cases include the Hurwitz numbers for algebraic curves and the enumerative invariants of Caporaso and Harris.Gromov-Witten invariants are invariants of a closed symplectic manifold (X, ω). To define them, one introduces a compatible almost complex structure J and a perturbation term ν, and considers the maps f : C → X from a genus g complex curve C with n marked points which satisfy the pseudoholomorphic map equation ∂f = ν and represent a class A = [f ] ∈ H 2 (X). The set of such maps, together with their limits, forms the compact space of stable maps M g,n (X, A). For each stable map, the domain determines a point in the Deligne-Mumford space M g,n of curves, and evaluation at each marked point determines a point in X. Thus there is a mapThe Gromov-Witten invariant of (X, ω) is the homology class of the image for generic (J, ν). It depends only on the isotopy class of the symplectic structure. By choosing bases of the cohomologies of M g,n and X n , the GW invariant can be viewed as a collection of numbers that count the number of stable maps satisfying constraints. In important cases these numbers are equal to enumerative invariants defined by algebraic geometry.In this article we construct Gromov-Witten invariants for a symplectic manifold (X, ω) relative to a codimension two symplectic submanifold V . These invariants are designed for use in formulas describing how GW invariants * The research of both authors was partially supported by the N.S.F. The first author was also supported by a Sloan Research Fellowship. 46ELENY-NICOLETA IONEL AND THOMAS H. PARKER behave under symplectic connect sums along V -an operation that removes V from X and replaces it with an open symplectic manifold Y with the symplectic structures matching on the overlap region. One expects the stable maps into the sum to be pairs of stable maps into the two sides which match in the middle. A sum formula thus requires a count of stable maps in X that keeps track of how the curves intersect V .Of course, before speaking of stable maps one must extend J and ν to the connect sum. To ensure that there is such an extension we require that the pair (J, ν) be 'V -compatible' as defined in Section 3. For such pairs, V is a J-holomorphic submanifold -something that is not true for generic (J, ν). The relative invariant gives counts of stable maps for these special V -compatible pairs. These counts are different from those associated with the absolute GW invariants.The restriction to V -compatible (J, ν) has repercussions. It means that pseudo-holomorphic maps f : C → V into V are automatically pseudo-holomorphic maps into X. Thus for V -compatible (J, ν), stable maps may have domain components whose image lies entirely in V . This creates problems beca...

This paper gives a mathematically rigorous proof of the positive energy theorem using spinors. This completes and simplifies the original argument presented by Edward Witten. We clarify the geometric aspects of this argument and prove the necessary analytic theorems concerning the relevant Dirac operator.

Let Σ be a compact Riemann surface. Any sequence f n : Σ -> M of harmonic maps with bounded energy has a "bubble tree limit" consisting of a harmonic map /o : Σ -> M and a tree of bubblesWe give a precise construction of this bubble tree and show that the limit preserves energy and homotopy class, and that the images of the f n converge pointwise. We then give explicit counterexamples showing that bubble tree convergence fails (i) for harmonic maps f n when the conformal structure of Σ varies with n, and (ii) when the conformal structure is fixed and {/ n } is a Palais-Smale sequence for the harmonic map energy.