Questions tagged [topology]
In mathematics, topology examines the properties of space (such as connectedness and compactness) that are preserved under continuous deformations, such as stretching and bending, but not tearing or gluing.
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Non-simply connectedness of the Moebius group [migrated]
Assuming that a map of the form: $a\in SL(2,\mathbb{C}) \rightarrow m[a]$ with $m[a](z)=\frac{a_1z + a_2}{a_3z+a_4}$ is a group homomorphism, it is easy to show that this mapping is not bijective, ...
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Do the canonical twisting bundles play a role in quantum physics?
$\Bbb {CP}^1$ the Riemann sphere appears in quantum physics as the Bloch sphere, and is the natural home for two-dimensional phenomena such as spin. It is sometimes said that bundle theory is relevant ...
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Configuration space of symmetric rigid bodies in $n$ dimensions
Suppose I have a connected, compact, rigid body $X$ in $\mathbb{R}^n$ with a closed subgroup of rotational symmetries $G\subset SO(n)$. That is, two configurations of $X$ are identical iff they differ,...
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Prove surface is compact in Hawking and Ellis
I am trying to prove proposition 9.2.1 from text book "The large scale structure of spacetime" by Hawking and Ellis at page # 311, last line and got stuck in a topological/geometrical ...
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Penrose's topological argument that forbids negative energy in GR?
From a Sean Caroll's "Mindscape" podcast at 16:19. Penrose says:
The question that I was interested in is could you in any way make
this consistent with Einstein's general relativity ...
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Why are classical gauge theories insensitive to large gauge transformations, while quantum gauge theories are affected by them?
In a previous question regarding large gauge transformations, one of the answers mentions that large gauge transformations are true redundancies for classical gauge theories, while they contain ...
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Why the Chern Number should be calculated in 2D system with energy gap?
I feel that it might be related to the "splitting principle", a concept in complex vector bundles. In quantum mechanics, the first Chern number of a complex vector bundle is always zero，...
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Point group or Space group symmetry on Berry connection [closed]
I want to consider How Berry connection transform by Point group or Space group symmetry and whether is it possible to think selection rule as other quantity.
For <n|O|m> term, we usually ...
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Smooth Manifolds Without Vacuum Solutions?
Are there any smooth 4-manifolds, $M$, whose topology somehow forbids them from admitting vaccuum solutions to GR? This would require that no Lorentzian metrics, $g_{ab}$, definable on $M$ are ...
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Possible Isometries for GR set on a Spatially Asymmetric Manifold
I am studying the possibility of setting GR on rather exotic spatial topologies. In particular I am interested in the case where $(M,g)$ is globally hyperbolic with $M\cong R\times \Sigma$ with $\...
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Interpretation of partition functions on general spacetimes
The partition function of a Euclidean quantum field theory (say, defined by a path integral) on the spacetime $X\times S^1_{\beta}$ (where $S^1_{\beta}$ means a circle of length $\beta$) has the ...
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The torus as "a borderline case" for event horizons in 4d spacetime
In high spacetime dimensions, event horizons can have rather interesting topologies, including black rings. However, in four spacetime dimensions, there is a powerful no-go theorem by Hawking in his ...
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Spinor existence in curved space
In curved spacetime, spinors do not existe because they are not representations of the general linear group $GL(4)$, however, through the tetrad formalism, we can define spinors locally and ...
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Where is the spin structure in the RNS string?
In the RNS formalism of string theory, the fundamental fields are embedding coordinates $X^{\mu}$ and worldsheet fermions $\psi^{\mu}_{\alpha}$ which are spinors in the worldsheet and vectors in the ...
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If space has positive curvature, it its geometry spherical or elliptic?
I don't know much about non-euclidean or differential geometry but I think this question makes sense.
If space has positive curvature, it is a closed universe. If a spaceship left earth in a straight ...
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What does the nontriviality of the Hopf fibration tell us about the global phases of qubit states on the Bloch sphere?
In this paper, discussing the Hopf fibration, the autors state:
Its non trivial character implies that $S_3 \neq S_2 \times S_1$. This non trivial character
translates here into the known failure in ...
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Do stable gravitational geons violate the final state conjecture?
I don't work in general relativity, but it's an area that I often feel curious about. I learned about geons from an answer by John Rennie. Despite the etymology of the term geon (gravitational ...
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How does the wormhole between 2 parallel universes inside a black hole work? (Penrose diagram)
There is a theory that inside of an eternal Schwarzschild black hole two astronauts (assuming they are immortal) from two parallel universes could meet and it is show on a Penrose diagram for an ...
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Prove that Cauchy hypersurface is closed
Witten in https://arxiv.org/abs/1901.03928 at page#18 proved that Cauchy hypersurface S is closed hypersurface by contradiction. He wrote
suppose that p ∈ M is in the closure of S but not in S. Let γ ...
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Showing integral of Berry curvature is invariant under small perturbations of states
Suppose I have a functional $S[f]= \int d^dx L(f(x), f'(x))$, and I want to argue that the functional is topological in some sense. Then what I imagine doing is calculating $S[f+\epsilon g] - S[f]$, ...
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Symmetry properties of spin Berry curvature in time-reversal symmetric systems
I know that the usual charge Berry curvature $\mathbf{\Omega}_n(\mathbf{k})$,
$$
\mathbf{\Omega}_n(\mathbf{k}) = i\sum_{m\ne n}\frac{\langle u_{n,{\mathbf{k}}}| \partial_\mathbf{k} H(\mathbf{k})| u_{...
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Topological magnon bands, chiral edge states and broken time-reversal symmetry
The existence of topological magnon bands was initially proposed within a spin model in which a DM term gaps out the Dirac points (breaks inversion symmetry). At the level of spin wave theory, or a ...
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Occupation state rule for anyons and their existence in 2D and lower
Can somebody please explain the occupation state rule for anyons? For fermions, they follow the pauli exclusion principle, while multiple bosons can occupy the same state. I understand that anyons ...
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Entropy production and anisotropies in compact Bianchi type VIIh "universes"...?
I had a couple of questions concerning this interesting paper where the authors hypothesize how could anisotropic "stresses" be exploited to produce entropy (they indicate that an example of ...
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4d Riemannian Manifold as Part of 4d Lorentzian Manifold
If I'm understanding correctly, this paper talks about the union of an initial space-like hypersurface and a final one after a change in spatial topology as the boundary of a 4d manifold. But is a 4d ...
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