Identification of Gauge Theories
Lemma 8.49. The G 2 -monopole equations in differential equations form (compare to Re- Re-mark 5.6) are given by
9.2 About the Boundary Conditions and Other Remarks
The boundary conditions we chose in this work were dictated by the fact that we wanted to use the Nahm transform. However it would be interesting to understand how the conditions in [MW13] translate to boundary conditions on the space of Nahm data. In particular it would be interesting to understand how the knot topology can be understood in this framework.
While there is a blank check principal bundle reduction in the case of the Nahm equations in Lemma 8.35, the author is unaware of such a reduction for the case of Bogomolny equations.
It seems interesting to take a closer look at the topology of the involved bundles in the transform defined via Theorem8.66 (and extended by Theorem8.70and8.71). In particular it would be interesting to find an explicit example for which the transformed data consists of multiple (different) bundles.
Another question that arose while working on this is if the Nahm transform can be lifted to dimensions 5and7without the necessity of the intermediate step to gen-eralized Seiberg-Witten. For this note that a 5-dimensional manifold with holonomy inSOp4q (evenOp4q) is spin, as is a 7-dimensional manifold with holonomy inG2. It is possible to copy the definitions to this setting, but many analytic questions would need addressing (the index, positivity, ... of the operators). Also the proof of the trans-form fails because ofcross-termsin the general setting. This could maybe be addressed by requiring certain invariance of the involved data.
For the Examples 8.43 and 8.60 it seems interesting to understand the space of aholomorphic maps from a 4-dimensional quaternionic manifold to the spaces MN
andMB.
It might also be interesting to look at the physical implications of the mentioned transform, even though the author’s knowledge does not suffice to make an informed judgment.
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