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# Communications in Analysis and Geometry

## Volume 17 (2009)

### Number 5

### Orientability in Yang–Mills theory over nonorientable surfaces

Pages: 903 – 953

DOI: http://dx.doi.org/10.4310/CAG.2009.v17.n5.a3

#### Authors

#### Abstract

The first two authors have constructed a gauge-equivariant Morse stratification on thespace of connections on a principal $U(n)$-bundleover a connected, closed, nonorientable surface $\Si$. This space can be identified with the real locusof the space of connections on the pullback of this bundle over the orientable double coverof $\Si$. In this context, the normal bundles to the Morse strata are real vector bundles.We show that these bundles, and their associated homotopy orbit bundles, are orientablefor any $n$ when $\Si$ is not homeomorphic to the Klein bottle, and for$n\leq 3$ when $\Si$ is the Klein bottle. We also derive similar orientability results whenthe structure group is ${\rm SU}(n)$.