| Register
\(\newcommand{\Cat}{{\rm Cat}} \) \(\newcommand{\A}{\mathcal A} \) \(\newcommand{\freestar}{ \framebox[7pt]{$\star$} }\)

7. Nahm transform and index bundles

The extension of the Nahm transform to degenerate cases in higher dimensions is a natural problem in gauge theory and special holonomy. Some examples of gauge theories where this formalism would be useful are as follows: $G_{2}$ monopoles, $\mathrm{Spin}(7)$, any gauge theory in dimension $d>4$, non-minimal solutions in lower dimensions, and the special case $d=3$.
    1. Nahm transform and index bundles

          Motivated by the search of sensible generalizations of the Nahm transform in higher dimensions, and in particular the study of $G_{2}$ monopoles, it would be highly desirable to arrive at a definition of a generalized Nahm transform whose construction does not rely on the implementation of vanishing theorems for index bundles.

      Problem 7.1.

      [Mark Stern] In the context of the Nahm transform, index bundles are usually realized by kernel bundles which come with a canonical connection. Is there a way to construct a canonical connection in the situation when both kernel and cokernel are nonzero? In other words, does there exist a canonical connection on the index bundle of a Dirac operator when its kernel has nonconstant dimension?
          The following are two natural approaches that could be considered:

      (a) Canonical connections with singularities along the sets where the dimension jumps. This would require extensive analysis of the set and the connection where the kernel’s dimension jumped.

      (b) Embeddings in auxiliary trivial bundles. Here one can control dimension jumps, but smoothness of the resulting connection remains unclear.
        • Fiberwise Nahm transform and octonionic monopoles

              Similarly to the Nahm transform that relates two reductions of the self-dual Yang–Mills equation in four dimensions (i.e., solutions of the Nahm equation and solutions of the monopole equation of Bogomolny), a relation between solutions of the Haydys–Witten system and octonionic monopoles is expected. A conjectural construction of a fiberwise Nahm transform related to knot invariants is discussed in a paper of Sergey Cherkis [MR3339202], where an octonionic interpretation of Haydys–Witten equations is given as a reduction of Spin(7) instanton equations.

          Problem 7.2.

          [Mark Stern] To apply the fiberwise Nahm transform to construct a Nahm transform relating solutions of the Haydys–Witten system and octonionic monopoles.

              Cite this as: AimPL: Geometry and physics of ALX metrics in gauge theory, available at http://aimpl.org/geomphysalx.