Pseudoholomorphic curve

In symplectic topology, pseudoholomorphic curves (PHK) denote a smooth mapping of a Riemann surface into an almost complex manifold that satisfies the Cauchy-Riemann differential equations. They were introduced by Mikhail Gromow in 1985 and have since revolutionized the study of symplectic manifolds , where they are particularly important for the definition and study of Gromov-Witten invariants and Floer homology . They also play a role in string theory .

definition

Let be a manifold with an almost complex structure and a smooth Riemann surface (corresponding to a complex algebraic curve) with a complex structure . A pseudoholomorphic curve in is a figure representing the Cauchy-Riemann differential equations ${\displaystyle X}$ ${\displaystyle J}$ ${\displaystyle C}$ ${\displaystyle j}$ ${\displaystyle X}$ ${\displaystyle f\colon C\to X}$

${\displaystyle {\bar {\partial }}_{j,J}f:={\frac {1}{2}}(df+J\circ df\circ j)=0.}$

Fulfills. Because of this, this is equivalent to ${\displaystyle J^{2}=-1}$

${\displaystyle J\circ df=df\circ j,}$

Geometrically, this means that the differential is complex-linear, that is, maps every tangent space onto itself. For technical reasons, an inhomogeneous term is often introduced and the disturbed Cauchy-Riemann differential equations are then used ${\displaystyle df}$ ${\displaystyle J}$ ${\displaystyle T_{x}f(C)\subset T_{x}X}$ ${\displaystyle \nu }$

${\displaystyle {\bar {\partial }}_{j,J}f=\nu .}$

educated. The corresponding curves are then called -holomorphic curves . Sometimes one assumes that the perturbation is generated by a Hamilton function (especially in the Floertheory ), but it doesn't have to be. ${\displaystyle (j,J,\nu )}$ ${\displaystyle \nu }$

According to their definition, PHC are always parameterized, but in practice one is also interested in non-parameterized curves, that is, embedded 2-submanifolds of , and "integrated" via the reparameterization degrees of freedom of the area that preserve the structure. In the case of the Gromov-Witten invariants, for example, only closed areas with a fixed gender are considered and marked points (or punctures , i.e. distant points) are introduced . As soon as the dotted Euler characteristic is negative, there are only finitely many holomorphic reparametrizations of that receive the marked points. The curve is an element of the Deligne-Mumford modular space of curves. ${\displaystyle X}$ ${\displaystyle C}$ ${\displaystyle g}$ ${\displaystyle n}$ ${\displaystyle C}$ ${\displaystyle 2-2g-n}$ ${\displaystyle C}$ ${\displaystyle C}$

Analogy with the classical Cauchy-Riemann differential equations

In the classic case, both are equal to the complex number plane. In real coordinates is ${\displaystyle X}$ ${\displaystyle C}$

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

and

${\displaystyle df={\begin{bmatrix}du/dx&du/dy\\dv/dx&dv/dy\end{bmatrix}}}$

whereby . If you multiply these matrices in the two possible orders you can immediately see that the above equation ${\displaystyle f(x,y)=(u(x,y),v(x,y))}$

${\displaystyle J\circ df=df\circ j}$

equivalent to the classical Cauchy-Riemann differential equations:

${\displaystyle {\begin{cases}du/dx=dv/dy\\dv/dx=-du/dy.\end{cases}}}$

Applications in symplectic topology

Although they can be defined for any almost complex manifold, PHK are particularly interesting when associated with a symplectic form . An almost complex structure is -tame ( -tame) if and only if ${\displaystyle J}$ ${\displaystyle \omega }$ ${\displaystyle J}$ ${\displaystyle \omega }$ ${\displaystyle \omega }$

${\displaystyle \omega (v,Jv)>0}$

for all non-zero tangent vectors . "Tameness" has the consequence that ${\displaystyle v}$

${\displaystyle (v,w)={\frac {1}{2}}\left(\omega (v,Jw)+\omega (w,Jv)\right)}$

defines a Riemannian metric on . Mikhail Gromov showed that for a given room the tame room is not empty and contractible . He thus proved his "nonsqueezing theorem" about the symplectic embedding of spheres in cylinders. ${\displaystyle X}$ ${\displaystyle \omega }$ ${\displaystyle \omega }$ ${\displaystyle J}$

Gromov further showed that certain modular spaces of PHK (with certain additional conditions) are compact and described how PHK can degenerate if only finite energy is available. This “Gromov's compactness theorem” - later strongly generalized through the use of stable maps - enables the definition of Gromov-Witten invariants , which count PHK in symplectic manifolds.

Compact modular rooms from PHK are also used to construct the Floer homology , which Andreas Floer used to prove the famous Arnold conjecture.

Applications in physics

In type II superstring theory, one considers “world sheets” - areas of strings that move on 3-dimensional Calabi-Yau manifolds . In the path integral formulation of quantum field theory , one would like to integrate all these surfaces over space (“module space”). However, this space has an infinite number of dimensions and is generally not mathematically accessible, but one can deduce in the so-called A-twist that these surfaces are parameterized by PHK, so that one is dealing with an integration in the finite dimensional module space of the PHK. In the II A string theory these integrals are precisely the Gromov-Witten invariants .

literature

• Dusa McDuff , Dietmar Salamon : J-Holomorphic Curves and Symplectic Topology ( American Mathematical Society. Colloquium publications 52). American Mathematical Society, Providence RI 2004, ISBN 0-8218-3485-1 .
• Mikhail Gromov : Pseudo holomorphic curves in symplectic manifolds. In: Inventiones Mathematicae . 82, 1985, pp. 307-347, online .

Web links

• Donaldson What is a pseudoholomorphic curve , Notices AMS 2005 (English)