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Often $X$ is a coadjoint orbit of a Lie group. These have a natural symplectic structure; see https://en.wikipedia.org/wiki/Symplectic_reduction


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Recall that the (global) conformal group is given by $${\rm Conf}(p,q)~\cong~O(p\!+\!1,q\!+\!1)/\{\pm {\bf 1} \},\tag{1} $$ cf. e.g. this Phys.SE post. Using the embedding $\imath: \mathbb{R}^{p,q}\hookrightarrow \overline{\mathbb{R}^{p,q}}$ into the conformal compactifification $\overline{\mathbb{R}^{p,q}}$, one may show after a short calculation that the ...


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Conceptually, the idea is that planes and spheres are equivalent from the point of conformal geometry. Conformal transformations map {planes, spheres} to {planes, spheres}, and in fact do this transitively -- any object in the set {planes, spheres} can be obtained from any other by a conformal transformation. Inversion is a reflection against a sphere, and ...


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The Chern number you mention is the thing you get when you integrate a particular two-form over a surface. It turns out that this two form represents the first Chern class of the system (the system, in this case, consists of the parameter space and a line bundle describing the relative Berry phase along paths in the parameter space). The most important ...


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Thinking of the sphere $S^n$ as the one-point compactification of $\mathbb{R}^n$, we can consider the stereographic projection from the plane defined by $x^0 = 0$ to the unit sphere $\{x\in \mathbb{R}^n\,:\,|x| = 1\}$. This map is actually defined on $\mathbb{R}^n\cup\{\infty\}$, it takes the point $\infty$ to the north pole of the unit sphere. Moreover, ...


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The director (living on $\mathbb{R}P^1$ )is given only only modulo the interval$ [-\frac{\pi}{2}, \frac{\pi}{2} ]$ , however the winding number has to be computed on the universal covering space $\mathbb{R}$. The same situation happens for $S^1$ parametrized by $[-\pi, \pi]$ but the winding angles can be arbitrary integer multiples of $2 \pi$. Thus in ...


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The QCD Lagrangian that has a CP violating symmetry is $$ {\cal L}~=~-\frac{g^2}{4}F_{ab}F^{ab}~-~\frac{g^2\theta}{4} {F_{ab}}^* F^{ab}~+~\bar\psi(i\gamma^a D_a~-~me^{i\gamma_5\theta})\psi $$ where the angle $\theta$ is the chiral phase and a field that mixes fields. It is even proposed to look for this angle or field in its mixing of electromagnetic $\vec E$...


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Topological terms of all types are always required not to depend on the metric, so their integrals will correspond to topological invariants, which serve as topological charges in quantum field theory. However, it is important to distinguish between two the types of topological terms mentioned in the question, because they lead to different physical ...


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In this case I would rather forget about possible no go theorems on topology changes. They are more related to how we mathematically model the problem, than to the physics. The physical fact is that matter is collapsing in a initially approximately flat space. General relativity is unable to explain in full generality what happen next, indeed the prediction ...


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The singularity for a Schwarzschild metric is a spatial surface. It is a surface where the Weyl curvature diverges. So technically there is no topology change. For the Kerr blackhole the singularity is a ring and things are a bit different. However, the inner horizon is a Cauchy horizon where photons of arbitrary wavelength pile up. So this might in fact be ...


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A p-cycle is a differential form that lives in $ker(\partial_p)$ for the differential $\partial_p$ (in grading $p$), and such a form is nontrivial if it is not in the image of $\partial_{p+1}$. Mathematically we can see this as a cycle that is not the boundary of anything, picture a circle around a torus that bounds no area on the torus. If one has a ...


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I am going to offer a small bone here. I am somewhat interested in the role of Taub-NUT spacetimes, and so contributing will help me to track this in order to read other contributions. Thanks for the Gubser paper. From a more physical perspective I will just throw out something with magnetic monopoles, which are related to Taub-NUT spacetimes that have a ...


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A manifold $M$ is geodesically complete (i.e. has the property of "geodesic completeness") if for every point $P\in M$ and every direction $dx^\mu$ from that point, the infinitesimal line interval $dx^\mu$ away from $P$ may be transported by parallel transport by any amount $K\in(-\infty,+\infty)$ in both directions, i.e. if the geodesics (maximally straight ...



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