# A proof for physicists of Proposition 5.5 in "Foundations of Differential Geometry Vol 1" by Kobayashi and Nomizu

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My questions are :

1.  How to give a proof for physicists of this proposition?

2.  What could be a physical application of this proposition?

retagged Mar 23, 2019

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A possible application of the the proposition 5.5  in physics occurs in the case of the Kapustin-Witten equation:

In this case  $\phi$ is a tensorial 1-form of type  $ad G$ and $d_{A} \phi$  is given by the proposition 5.5.

It is conjectured that the coefficients of the Jones polynomial of a knot can be computed by counting solutions of the KW equations on a half-space in $R^4$ with the generalized Nahm pole boundary conditions.

The Jones polynomial is a Laurent series $J \left( q \right) =\sum _ na_{{n}}{q}^{n}$,  and the conjecture is that $a_ n$  is an algebraic count of the number of solutions of the KW equations with second Chern class   equal to $n$.

Reference :  https://arxiv.org/pdf/1712.00835.pdf

answered Mar 23, 2019 by (1,130 points)
edited Mar 23, 2019 by juancho
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Other application of the proposition 5.5  is given in the following lemma

with

where:

answered Apr 3, 2021 by (1,130 points)
edited Apr 3, 2021 by juancho

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