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.

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, and integrals over infinite-dimensional spaces of such paths. As of today, these path integrals are not generally mathematically well-defined. 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, which are finite-dimensional. These integrals are Gromov-Witten invariants.

Related Topics:
Calabi-Yau - Path integral - Moduli spaces of pseudoholomorphic curves

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