|
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.
pdf (398 kb)
tex (18 kb)
[previous version:
pdf (397 kb)
tex (18 kb)]
References
-
Bryant R.L., Metrics with holonomy ${\rm G}_2$ or ${\rm Spin}(7)$, Lecture Notes in Math., Vol. 1111, Springer, Berlin, 1985, 269-277.
-
Clarke A., Instantons on the exceptional holonomy manifolds of Bryant and Salamon,
J. Geom. Phys. 82 (2014), 84-97,
arXiv:1308.6358.
-
Donaldson S.K., Floer homology groups in Yang-Mills theory, Cambridge Tracts in Mathematics, Vol. 147, Cambridge University Press, Cambridge, 2002.
-
Donaldson S.K., Thomas R.P., Gauge theory in higher dimensions, in The Geometric Universe (Oxford, 1996), Oxford University Press, Oxford, 1998, 31-47.
-
Harland D., Ivanova T.A., Lechtenfeld O., Popov A.D., Yang-Mills flows on nearly Kähler manifolds and ${\rm G}_2$-instantons, Comm. Math. Phys. 300 (2010), 185-204, arXiv:0909.2730.
-
Harvey R., Lawson Jr. H.B., Calibrated geometries, Acta Math. 148 (1982), 47-157.
-
Ivanova T.A., Popov A.D., Instantons on special holonomy manifolds, Phys. Rev. D 85 (2012), 105012, 10 pages, arXiv:1203.2657.
-
Joyce D.D., Compact manifolds with special holonomy, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2000.
-
McLean R.C., Deformations of calibrated submanifolds, Comm. Anal. Geom. 6 (1998), 705-747.
-
Milnor J.W., Stasheff J.D., Characteristic classes, Princeton University Press, Princeton, N.J., 1974.
-
Sá Earp H.N., Instantons on ${\rm G}_2$-manifolds, Ph.D. Thesis, Imperial College London, 2009.
-
Sá Earp H.N., ${\rm G}_2$-instantons over asymptotically cylindrical manifolds, Geom. Topol., 19 (2015), 61-111, arXiv:1101.0880.
-
Sá Earp H.N., Walpuski T., ${\rm G}_2$-instantons on twisted connected sums, Geom. Topol. 19 (2015), 1263-1285, arXiv:1310.7933.
-
Salamon S., Riemannian geometry and holonomy groups, Pitman Research Notes in Mathematics Series, Vol. 201, Longman Scientific & Technical, Harlow, 1989.
-
Thomas R.P., Gauge theory on Calabi-Yau manifolds, Ph.D. Thesis, Oxford University, 1997.
-
Tian G., Gauge theory and calibrated geometry, I, Ann. of Math. 151 (2000), 193-268, math.DG/0010015.
-
Walpuski T., ${\rm G}_2$-instantons on generalised Kummer constructions, Geom. Topol. 17 (2013), 2345-2388, arXiv:1109.6609.
-
Wang S., A higher dimensional foliated Donaldson theory, I, Asian J. Math. 19 (2015), 527-554, arXiv:1212.6774.
|
|