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# Surveys in Differential Geometry

## Volume 15 (2010)

### A new look at the path integral of quantum mechanics

Pages: 345 – 420

DOI: http://dx.doi.org/10.4310/SDG.2010.v15.n1.a11

#### Author

#### Abstract

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.