Abstract
In the first part, we discuss the mathematical structure of the gauge orbit space stratification, including a Tubular Neighbourhood Theorem and a Regularity Theorem. Moreover, we show that every stratum admits a natural Riemannian metric, calculate its volume element and find the corresponding Riemann curvature. In the second part, we present our results on the enumeration of gauge orbit types for the gauge group \(G = \text {SU}(n)\) in detail. The classification is given in terms of certain characteristic classes fulfilling a number of algebraic relations. We also show how the natural partial ordering of strata can be read off from these relations.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
This choice of partial ordering corresponds to comparing the size of the orbits. It is consistent with [103] but not with [388] and several other authors who choose the inverse partial ordering.
- 2.
Clearly, the proposition holds with \(C^\infty \) replaced by any differentiability class.
- 3.
Note that \(\mathrm{N}_{x,\varepsilon }\) is not just the \(\varepsilon \)-disk bundle of \(\mathrm{N}_x\), because orthogonality and length are taken with respect to different metrics.
- 4.
See the remarks on the notion of stratification in Sect. 6.6 of Part I.
- 5.
Let us note that the number of Howe subgroups in a compact Lie group is actually finite. This follows from the fact that any centralizer in a compact Lie group is generated by finitely many elements [92, Chap. 9] and that a compact group action on a compact manifold has a finite number of orbit types [103].
- 6.
We note that the Howe subgroup labelled by \(J=(1, 1|1, 1)\) is the toral subgroup \(\mathrm{U}(1)\) of \(\mathrm{SU}(2)\) and that \(\alpha _{1, 1}\) is just the first Chern class of the corresponding reduction of P. By virtue of this transliteration, Eq. (8.8.8) is consistent with the literature [338].
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2017 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Rudolph, G., Schmidt, M. (2017). The Gauge Orbit Space. In: Differential Geometry and Mathematical Physics. Theoretical and Mathematical Physics. Springer, Dordrecht. https://doi.org/10.1007/978-94-024-0959-8_8
Download citation
DOI: https://doi.org/10.1007/978-94-024-0959-8_8
Published:
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-024-0958-1
Online ISBN: 978-94-024-0959-8
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)