- Pseudoholomorphic curve
In
mathematics , specifically intopology andgeometry , a pseudoholomorphic curve (or "J"-holomorphic curve) is a smooth map from aRiemann surface into analmost complex manifold that satisfies the Cauchy-Riemann equation. Introduced in 1985 byMikhail Gromov , pseudoholomorphic curves have since revolutionized the study ofsymplectic manifold s. In particular, they lead to theGromov-Witten invariant s andFloer homology , and play a prominent role instring theory .Definition
Let be an almost complex manifold with almost complex structure . Let be a smooth
Riemann surface (also called a complex curve) with complex structure . A pseudoholomorphic curve in is a map that satisfies the Cauchy-Riemann equation:Since , this condition is equivalent to:which simply means that the differential is complex-linear, that is, maps each tangent space:to itself. For technical reasons, it is often preferable to introduce some sort of inhomogeneous term and to study maps satisfying the perturbed Cauchy-Riemann equation:A pseudoholomorphic curve satisfying this equation can be called, more specifically, a -holomorphic curve. The perturbation is sometimes assumed to be generated by aHamiltonian (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 , 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 of fixed genus and we introduce marked points (or punctures) on . As soon as the puncturedEuler characteristic is negative, there are only finitely many holomorphic reparametrizations of that preserve the marked points. The domain curve is an element of theDeligne-Mumford moduli space of curves .Analogy with the classical Cauchy-Riemann equations
The classical case occurs when and are both simply the
complex number plane. In real coordinates:and:where . After multiplying these matrices in two different orders, one sees immediately that the equation:written above is equivalent to the classical Cauchy-Riemann equations:Applications in symplectic topology
Although they can be defined for any almost complex manifold, pseudoholomorphic curves are especially interesting when interacts with a
symplectic form . An almost complex structure is said to be -tame if and only if:for all nonzero tangent vectors . Tameness implies that the formula:defines aRiemannian metric on . Gromov showed that, for a given , the space of -tame is nonempty andcontractible . He used this theory to prove anonsqueezing theorem concerning symplectic embeddings of spheres into cylinders.Gromov showed that certain
moduli space s 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 -tame or -compatible). ThisGromov compactness theorem , now greatly generalized usingstable map s, 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 , whichAndreas Floer (and later authors, in greater generality) used to prove the famous conjecture ofVladimir Arnol'd concerning the number of fixed points ofHamiltonian flow s.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 thepath integral formulation ofquantum 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 theA-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 theGromov-Witten invariant s.References
* Dusa McDuff and Dietmar Salamon, "J-Holomorphic Curves and Symplectic Topology", American Mathematical Society colloquium publications, 2004. ISBN 0-8218-3485-1.
* M. Gromov, Pseudo holomorphic curves in symplectic manifolds. Inventiones Mathematicae vol. 82, 1985, pgs. 307-347.
* cite journal
last = Donaldson
first = Simon K.
authorlink = Simon Donaldson
title = What Is...a Pseudoholomorphic Curve?
journal =Notices of the American Mathematical Society
year = 2005
month = October
volume = 52
issue = 9
pages = pp.1026–1027
url = http://www.ams.org/notices/200509/what-is.pdf
format =PDF
accessdate = 2008-01-17
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