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- TL;DR Summary
- What is the relationship between the Chern number in a gauge theory and the number of isolated zeros of a section?

Greetings. I still struggle a little with the mathematics involved in the description of gauge theories in terms of fiber bundles, so please pardon and correct me if you find conceptual errors anywhere in this question. I would like to understand the connection (when it exists) between the number of zeros, counted with multiplicity, and some topological invariants of the theory. To avoid making this question too broad, I will restrict myself to theories in which the only topological invariant is an integer given by the integral of the Chern-form of a bundle (but feel free to share any insight into theories that don't belong to this group, if you want). In the famous Maxwell-Higgs theory, for example, we encounter the result

\begin{equation}

c_1\equiv\frac{1}{2\pi}\int_{M} F = n,

\end{equation}

where ##M## is a compact manifold, usually taken as the two dimensional plane, and ## F ## is the curvature two-form associated with the connection ##iA_{\mu}## that appears in the covariant derivative ##D_{\mu}=\partial_{\mu} + iA_{\mu}##, which acts on a section of the fiber bundle. This also holds for other vortex theories, and a similar result appears in Yang-Mills-Higgs, where the relevant integral is the

I find such relationships between the zeros and topology very interesting, and would like to know more about their meaning (I think this is related to the Poincaré-Hopf and Gauss-Bonet theorems. Could I relate the zeros to the index of a vector field?). I believe they would remain valid in other gauge theories with the same topology in two/three space dimensions, since I've made no mention to the specific form of the action (if anyone could correct this statement or make it more general, please do so, it would be appreciated). I would also like to know if the zeros of sections also play a role in a theory without a scalar field. For example, the 4-D Yang-Mills theory and the Dirac monopole have an integer-valued Chern-number (the second and the first, respectively). There's usually no mention scalar field in such theories, but their nontrivial topology implies the existence of nontrivial sections on the associated bundle. Can I say anything about their zeros?

\begin{equation}

c_1\equiv\frac{1}{2\pi}\int_{M} F = n,

\end{equation}

where ##M## is a compact manifold, usually taken as the two dimensional plane, and ## F ## is the curvature two-form associated with the connection ##iA_{\mu}## that appears in the covariant derivative ##D_{\mu}=\partial_{\mu} + iA_{\mu}##, which acts on a section of the fiber bundle. This also holds for other vortex theories, and a similar result appears in Yang-Mills-Higgs, where the relevant integral is the

*second*Chern-number. Now in such theories we may identify the section with a scalar field ##\phi## in some gauge. Some authors, like Manton and Sutcliffe, mention that this topological charge/degree ##n## corresponds to the number of zeros of ##\phi ##, counted with multiplicity, at least when said zeros are isolated. This has been proven in some cases (see the work of Taubes for example), and it makes sense, since the absence zeros would allow me to write ##F## as a globally defined exact form, which would imply zero flux. One could even show that leaving the points where ##\phi## vanishes out of the integration would make this integral zero, so that, in a sense, it could be said that only the zeros contribute to the flux (this appears as a sum of delta distributions in vortices). In the Yang-Mills-Higgs ##SU(2)## theory, the math is more complicated, but the points where ##\phi## vanish can be seen as sources i.e the locations of magnetic monopoles, which one could count to find the total magnetic flux.I find such relationships between the zeros and topology very interesting, and would like to know more about their meaning (I think this is related to the Poincaré-Hopf and Gauss-Bonet theorems. Could I relate the zeros to the index of a vector field?). I believe they would remain valid in other gauge theories with the same topology in two/three space dimensions, since I've made no mention to the specific form of the action (if anyone could correct this statement or make it more general, please do so, it would be appreciated). I would also like to know if the zeros of sections also play a role in a theory without a scalar field. For example, the 4-D Yang-Mills theory and the Dirac monopole have an integer-valued Chern-number (the second and the first, respectively). There's usually no mention scalar field in such theories, but their nontrivial topology implies the existence of nontrivial sections on the associated bundle. Can I say anything about their zeros?