Pseudoholomorphic curve

In mathematics, specifically in topology and geometry, a pseudoholomorphic 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, which play a crucial role in type IIA string theory.

Formal 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 o X that satisfies the Cauchy-Riemann equation

Related Topics:
Riemann surface - Complex curve

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:ar partial_{j, J} f := rac{1}{2}(df + J circ df circ j) = 0.

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Since J^2 = -1, this condition is equivalent to

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:J circ df = df circ j,

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which simply means that the differential df is complex-linear. For technical reasons, it is often preferable to introduce some sort of inhomogeneous term u and to study maps satisfying the perturbed Cauchy-Riemann equation

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:ar partial_{j, J} f = u.

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A pseudoholomorphic curve satisfying this equation can be called, more specifically, a (j, J, u)-holomorphic curve.

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Notice that 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. So C is an element of the Deligne-Mumford moduli space of curves. As soon as the punctured Euler characteristic 2 - 2 g - n is negative, there are only finitely many holomorphic reparametrizations of C that preserve the marked points.

Related Topics:
Closed - Deligne-Mumford moduli space of curves - Euler characteristic

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