On the relationship between Chern number and zeros of a section

[Othin]

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 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?

 

[Infrared]

In general the top Chern class of a complex vector bundle is Poincaré dual to the homology class of the zero set of a transverse section. In particular, if the real rank of a complex vector bundle equals the real dimension of the base manifold (for example, a complex line bundle on a surface), then the top Chern number is the number of zeros of a generic section, counted with multiplicity.

where $M$ is a compact manifold, usually taken as the two dimensional plane,

Do you mean $M=S^2$? The plane is of course not compact.

 

[Othin]

Do you mean $M=S^2$? The plane is of course not compact.

Thank you for answering. I should have added that the boundary conditions are strong enough to compactify the plane, which if I understand correctly happens because they restrict the scalar field to be of the form $\phi=e^{in\theta}$ at infinity.

In general the top Chern class of a complex vector bundle is Poincaré dual to the homology class of the zero set of a transverse section. In particular, if the real rank of a complex vector bundle equals the real dimension of the base manifold (for example, a complex line bundle on a surface), then the top Chern number is the number of zeros of a generic section, counted with multiplicity.

That makes sense. I think I've seen a similar result stated as a consequence of the Poincaré-Hopf theorem: that the dimension of the zero locus of the section equals the Euler Characteristic, provided that it is transverse to the zero section, as you mentioned. But how do I know that the section is transverse when I don't have its explicit form (like in the case of solutions of differential equations, which not have a closed form)?

 

[Othin]

I didn't find a way to edit the first post (which is strange, I could swear to have edited questions that were, at the time, older than this one is), so I'll write it as a new reply rather than creating a new thread. Looking up other sources, I realized I might be making a confusion about the meaning of the word "multiplicity" in these theorems. The "multiplicity" they are talking about is given by the local degree of the singularity of the section $\phi/|\phi|$ . Is the multiplicity obtained through this definition a generalization of the multiplicity of a zero (as in the one that appears in the fundamental theorem of algebra, or the one we mention when we talk about the multiplicity of a zero/pole of a holomorphic function)?

If I had a function $\phi: \mathbb{C}\to \mathbb{C}$ than these two multiplicities would be the same by the argument principle . If I have instead $\phi: \mathbb{R}^2\to \mathbb{C}$ (as in the case of a vortex) is that still true? For example, let $\phi=h(r)e^{in\theta}$, where $n>0$ is the first Chern number. If this solution has only one isolated zero at the origin, can I conclude that $\lim_{r\to 0}\frac{f(r)}{r^n}$ if finite and nonzero. Or, if it has a power series, could I say that the leading order term in its expansion about the origin is $\propto r^n$? This is the case for the topological vortex theories I know (like Maxwell-Higgs, Chern-Simons, etc). If these two concepts of multiplicity are related, then the answer above from @Infrared would explain these these coincidence, since that would mean that $f(r)$ has a zero of multiplicity $n$ at the origin (in this algebraic sense), which would explain why the solution has the form $Cr^n$ near the origin.