Surveys in Differential Geometry

Volume 15 (2010)

A new look at the path integral of quantum mechanics

Pages: 345 – 420



Edward Witten (School of Natural Sciences, Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540 U.S.A.)


The Feynman path integral of ordinary quantum mechanics is complexified and it is shown that possible integration cycles for this complexified integral are associated with branes in a two-dimensional A-model. This provides a fairly direct explanation of the relationship of the A-model to quantum mechanics; such a relationship has been explored from several points of view in the last few years. These phenomena have an analog for Chern-Simons gauge theory in three dimensions: integration cycles in the path integral of this theory can be derived from \N=4 super Yang-Mills theory in four dimensions. Hence, under certain conditions, a Chern-Simons path integral in three dimensions is equivalent to an \N=4 path integral in four dimensions.

Published 1 January 2010