Communications in Analysis and Geometry

Volume 17 (2009)

Number 5

Orientability in Yang–Mills theory over nonorientable surfaces

Pages: 903 – 953



Nan-Kuo Ho (Department of Mathematics, National Tsing Hua University, Taiwan)

Chiu-Chu Melissa Liu (Department of Mathematics, Columbia University)

Daniel Ramras (Department of Mathematical Sciences, New Mexico State University)


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 ${\rm SU}(n)$.

Published 1 January 2009