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 and Floer homology, which play a crucial role in 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 algebraic curve) with complex structure j. A pseudoholomorphic curve in X is a map f : C o X that satisfies the Cauchy-Riemann equation

:ar 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. 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

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

A pseudoholomorphic curve satisfying this equation can be called, more specifically, a (j, J, u)-holomorphic curve. The perturbation u is sometimes assumed to be generated by a Hamiltonian (particularly in Floer theory), but in general it need not be.

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.

=Analogy with the classical Cauchy-Riemann equations=

If we specialize to the case when X and C are both simply the complex number plane, then in real coordinates we can write

:j = J = egin{bmatrix} 0 & -1 \ 1 & 0 end{bmatrix}

and

:df = egin{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

:egin{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 the almost complex structure J interacts with a symplectic form omega. We say that J is omega-tame iff

:omega(v, J v) > 0

for all nonzero tangent vectors v. Tameness implies that the formula

:(v, w) = frac{1}{2}left(omega(v, Jw) + omega(w, Jv) ight)

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 nonsqueezing theorem concerning symplectic embeddings of spheres into cylinders.

Gromov showed that certain moduli spaces of pseudoholomorphic curves (satisfying additional specified conditions) are compact space. 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.

=References=