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If $\lambda>0$ is an eigenvalue of $L^\dagger L$ with associated eigenfunction $u$, the then $Lu$ is an eigenfunction of $L L^\dagger$ with the same eigenvalue. As a consequence, in each of these expressions the contributions of the non-zero eigenvalues cancel in pairs. Only the zero modes contribute to give  {\rm Index}= {\rm dimKer}[L L^\...

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The misunderstanding many have is that topology just the study of topological spaces. It is really also about continuous functions between two topological spaces. If one has an infinite lattice model, where the Hilbert spaces is more-or-less $\ell^2(\mathbb{Z}^d)$, and if the Hamiltonian is periodic (a huge limitation) then one has momentum space that is ...

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Based on OP's elaboration in the comments on their background, I will try to make this answer pedagogical and self-contained, erring ont he side of simplicity in the beginning; however, since OP mentioned some familiarity with topological manifolds and vector bundles, the concluding section will become more sophisticated. Before beginning, however, I'll do ...

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Edit: this was written before J. Murray's excellent answer Saying solid state physics isn't my expertise is an understatement, so I'm not sure my answer is correct, but I'll do my best: According to my understanding, the topological space involved is the electron band structure (see for example this gif from Wikipedia which shows the structure for some ...

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Berry phase is equal to the surface integration of Berry Curvature. In the first case, Berry curvatures are located in the tube, so you move your particle around the tube will collect all the Berry phase and it get quantization result. By the way - if you move the particle into the tube, you also get non-quantized Berry phase. In the spin model, the source ...

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Suppose there exists a point $p\in\mathcal{M}$ for which every past inextendible timelike curve through $p$ intersects $S$, but there exist an (inextendible) null curve through $p$ that does not intersect $S$. Then $p \in \tilde{D}^{+}(S)$, but $p \notin D^{+}(S)$.

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There are some classic results showing, on Lorentzian spacetime, topology changes violate causality (Geroch proved this in the '60s) or cannot satisfy the Einstein equations with non-negative energy density (Tippler proved this in the '70s). A decent review paper on topology change in classical Lorentzian GR: Arvind Borde, "Topology Change in Classical ...

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