# Pseudoholomorphic curve

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In mathematics, specifically in topology and geometry, a pseudoholomorphic curve (or J-holomorphic curve) is a smooth map from a Riemann surface into an almost complex manifold that satisfies the Cauchy–Riemann equation. Introduced in 1985 by Mikhail Gromov, pseudoholomorphic curves have since revolutionized the study of symplectic manifolds. In particular, they lead to the Gromov–Witten invariants and Floer homology, and play a prominent role in string theory.

## Definition

Let $X$ be an almost complex manifold with almost complex structure $J$ . Let $C$ be a smooth Riemann surface (also called a complex curve) with complex structure $j$ . A pseudoholomorphic curve in $X$ is a map $f:C\to X$ that satisfies the Cauchy–Riemann equation

${\bar {\partial }}_{j,J}f:={\frac {1}{2}}(df+J\circ df\circ j)=0.$ Since $J^{2}=-1$ , this condition is equivalent to

$J\circ df=df\circ j,$ which simply means that the differential $df$ is complex-linear, that is, $J$ maps each tangent space

$T_{x}f(C)\subseteq T_{x}X$ to itself. For technical reasons, it is often preferable to introduce some sort of inhomogeneous term $\nu$ and to study maps satisfying the perturbed Cauchy–Riemann equation

${\bar {\partial }}_{j,J}f=\nu .$ A pseudoholomorphic curve satisfying this equation can be called, more specifically, a $(j,J,\nu )$ -holomorphic curve. The perturbation $\nu$ is sometimes assumed to be generated by a Hamiltonian (particularly in Floer theory), but in general it need not be.

A pseudoholomorphic curve is, by its definition, always parametrized. In applications one is often truly interested in unparametrized curves, meaning embedded (or immersed) two-submanifolds of $X$ , so one mods out by reparametrizations of the domain that preserve the relevant structure. In the case of Gromov–Witten invariants, for example, we consider only closed domains $C$ of fixed genus $g$ and we introduce $n$ marked points (or punctures) on $C$ . As soon as the punctured Euler characteristic $2-2g-n$ is negative, there are only finitely many holomorphic reparametrizations of $C$ that preserve the marked points. The domain curve $C$ is an element of the Deligne–Mumford moduli space of curves.

## Analogy with the classical Cauchy–Riemann equations

The classical case occurs when $X$ and $C$ are both simply the complex number plane. In real coordinates

$j=J={\begin{bmatrix}0&-1\\1&0\end{bmatrix}},$ and

$df={\begin{bmatrix}du/dx&du/dy\\dv/dx&dv/dy\end{bmatrix}},$ where $f(x,y)=(u(x,y),v(x,y))$ . After multiplying these matrices in two different orders, one sees immediately that the equation

$J\circ df=df\circ j$ written above is equivalent to the classical Cauchy–Riemann equations

${\begin{cases}du/dx=dv/dy\\dv/dx=-du/dy.\end{cases}}$ ## Applications in symplectic topology

Although they can be defined for any almost complex manifold, pseudoholomorphic curves are especially interesting when $J$ interacts with a symplectic form $\omega$ . An almost complex structure $J$ is said to be $\omega$ -tame if and only if

$\omega (v,Jv)>0$ for all nonzero tangent vectors $v$ . Tameness implies that the formula

$(v,w)={\frac {1}{2}}\left(\omega (v,Jw)+\omega (w,Jv)\right)$ defines a Riemannian metric on $X$ . Gromov showed that, for a given $\omega$ , the space of $\omega$ -tame $J$ is nonempty and contractible. He used this theory to prove a non-squeezing theorem concerning symplectic embeddings of spheres into cylinders.

Gromov showed that certain moduli spaces of pseudoholomorphic curves (satisfying additional specified conditions) are compact, and described the way in which pseudoholomorphic curves can degenerate when only finite energy is assumed. (The finite energy condition holds most notably for curves with a fixed homology class in a symplectic manifold where J is $\omega$ -tame or $\omega$ -compatible). This Gromov compactness theorem, now greatly generalized using stable maps, makes possible the definition of Gromov–Witten invariants, which count pseudoholomorphic curves in symplectic manifolds.

Compact moduli spaces of pseudoholomorphic curves are also used to construct Floer homology, which Andreas Floer (and later authors, in greater generality) used to prove the famous conjecture of Vladimir Arnol'd concerning the number of fixed points of Hamiltonian flows.

## Applications in physics

In type II string theory, one considers surfaces traced out by strings as they travel along paths in a Calabi–Yau 3-fold. Following the path integral formulation of quantum mechanics, one wishes to compute certain integrals over the space of all such surfaces. Because such a space is infinite-dimensional, these path integrals are not mathematically well-defined in general. However, under the A-twist one can deduce that the surfaces are parametrized by pseudoholomorphic curves, and so the path integrals reduce to integrals over moduli spaces of pseudoholomorphic curves (or rather stable maps), which are finite-dimensional. In closed type IIA string theory, for example, these integrals are precisely the Gromov–Witten invariants.

## Related Research Articles In the field of complex analysis in mathematics, the Cauchy–Riemann equations, named after Augustin Cauchy and Bernhard Riemann, consist of a system of two partial differential equations which, together with certain continuity and differentiability criteria, form a necessary and sufficient condition for a complex function to be holomorphic. This system of equations first appeared in the work of Jean le Rond d'Alembert. Later, Leonhard Euler connected this system to the analytic functions. Cauchy then used these equations to construct his theory of functions. Riemann's dissertation on the theory of functions appeared in 1851.

In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, , equipped with a closed nondegenerate differential 2-form , called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology. Symplectic manifolds arise naturally in abstract formulations of classical mechanics and analytical mechanics as the cotangent bundles of manifolds. For example, in the Hamiltonian formulation of classical mechanics, which provides one of the major motivations for the field, the set of all possible configurations of a system is modeled as a manifold, and this manifold's cotangent bundle describes the phase space of the system. Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form. Symplectic geometry has its origins in the Hamiltonian formulation of classical mechanics where the phase space of certain classical systems takes on the structure of a symplectic manifold.

In mathematics, a symplectomorphism or symplectic map is an isomorphism in the category of symplectic manifolds. In classical mechanics, a symplectomorphism represents a transformation of phase space that is volume-preserving and preserves the symplectic structure of phase space, and is called a canonical transformation.

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In mathematics and physics, a Hamiltonian vector field on a symplectic manifold is a vector field defined for any energy function or Hamiltonian. Named after the physicist and mathematician Sir William Rowan Hamilton, a Hamiltonian vector field is a geometric manifestation of Hamilton's equations in classical mechanics. The integral curves of a Hamiltonian vector field represent solutions to the equations of motion in the Hamiltonian form. The diffeomorphisms of a symplectic manifold arising from the flow of a Hamiltonian vector field are known as canonical transformations in physics and (Hamiltonian) symplectomorphisms in mathematics.

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In mathematics, specifically in symplectic topology and algebraic geometry, Gromov–Witten (GW) invariants are rational numbers that, in certain situations, count pseudoholomorphic curves meeting prescribed conditions in a given symplectic manifold. The GW invariants may be packaged as a homology or cohomology class in an appropriate space, or as the deformed cup product of quantum cohomology. These invariants have been used to distinguish symplectic manifolds that were previously indistinguishable. They also play a crucial role in closed type IIA string theory. They are named after Mikhail Gromov and Edward Witten.

In mathematics, Floer homology is a tool for studying symplectic geometry and low-dimensional topology. Floer homology is a novel invariant that arises as an infinite-dimensional analogue of finite-dimensional Morse homology. Andreas Floer introduced the first version of Floer homology, now called Lagrangian Floer homology, in his proof of the Arnold conjecture in symplectic geometry. Floer also developed a closely related theory for Lagrangian submanifolds of a symplectic manifold. A third construction, also due to Floer, associates homology groups to closed three-dimensional manifolds using the Yang–Mills functional. These constructions and their descendants play a fundamental role in current investigations into the topology of symplectic and contact manifolds as well as (smooth) three- and four-dimensional manifolds.

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In the mathematical field of symplectic topology, Gromov's compactness theorem states that a sequence of pseudoholomorphic curves in an almost complex manifold with a uniform energy bound must have a subsequence which limits to a pseudoholomorphic curve which may have nodes or "bubbles". A bubble is a holomorphic sphere which has a transverse intersection with the rest of the curve. This theorem, and its generalizations to punctured pseudoholomorphic curves, underlies the compactness results for flow lines in Floer homology and symplectic field theory.

In mathematics, specifically in symplectic topology and algebraic geometry, a quantum cohomology ring is an extension of the ordinary cohomology ring of a closed symplectic manifold. It comes in two versions, called small and big; in general, the latter is more complicated and contains more information than the former. In each, the choice of coefficient ring significantly affects its structure, as well.

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In mathematics, differential forms on a Riemann surface are an important special case of the general theory of differential forms on smooth manifolds, distinguished by the fact that the conformal structure on the Riemann surface intrinsically defines a Hodge star operator on 1-forms without specifying a Riemannian metric. This allows the use of Hilbert space techniques for studying function theory on the Riemann surface and in particular for the construction of harmonic and holomorphic differentials with prescribed singularities. These methods were first used by Hilbert (1909) in his variational approach to the Dirichlet principle, making rigorous the arguments proposed by Riemann. Later Weyl (1940) found a direct approach using his method of orthogonal projection, a precursor of the modern theory of elliptic differential operators and Sobolev spaces. These techniques were originally applied to prove the uniformization theorem and its generalization to planar Riemann surfaces. Later they supplied the analytic foundations for the harmonic integrals of Hodge (1940). This article covers general results on differential forms on a Riemann surface that do not rely on any choice of Riemannian structure.

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• Dusa McDuff and Dietmar Salamon, J-Holomorphic Curves and Symplectic Topology, American Mathematical Society colloquium publications, 2004. ISBN   0-8218-3485-1.
• Mikhail Leonidovich Gromov, Pseudo holomorphic curves in symplectic manifolds. Inventiones Mathematicae vol. 82, 1985, pgs. 307-347.
• Donaldson, Simon K. (October 2005). "What Is...a Pseudoholomorphic Curve?" (PDF). Notices of the American Mathematical Society . 52 (9): 1026–1027. Retrieved 2008-01-17.