### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 10 (2014), 083, 11 pages      arXiv:1401.5462      https://doi.org/10.3842/SIGMA.2014.083

### Generalised Chern-Simons Theory and ${\rm G}_2$-Instantons over Associative Fibrations

Henrique N. Sá Earp
Imecc - Institute of Mathematics, Statistics and Scientific Computing, Unicamp, Brazil

Received January 29, 2014, in final form August 07, 2014; Published online August 11, 2014; References updated November 20, 2016

Abstract
Adjusting conventional Chern-Simons theory to ${\rm G}_2$-manifolds, one describes ${\rm G}_2$-instantons on bundles over a certain class of $7$-dimensional flat tori which fiber non-trivially over $T^4$, by a pullback argument. Moreover, if $c_2\neq0$, any (generic) deformation of the ${\rm G}_2$-structure away from such a fibred structure causes all instantons to vanish. A brief investigation in the general context of (conformally compatible) associative fibrations $f:Y^7\to X^4$ relates ${\rm G}_2$-instantons on pullback bundles $f^*E\to Y$ and self-dual connections on the bundle $E\to X$ over the base, a fact which may be of independent interest.

Key words: Chern-Simons; Yang-Mills; ${\rm G}_2$-manifolds; associative fibrations.

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