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.

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}

~ ~ ~ ~ ~ ~ ~ ~ ~ ~