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7
votes
2answers
576 views

Vector Potential for Magnetic field when the field is not in simply-connected region

According to Poincare's Lemma, if $U\subset \mathbb{R}^n$ is a star-shaped set and if $\omega$ is a $k$-form defined in $U$ that is closed, then $\omega$ is exact, meaning that there's some ...
0
votes
0answers
63 views

What is ``thermal" about a thermal quotient of EdS and EAds?

This is in continuation of my previous question and is in reference to this paper. I guess that the authors are interested in $S^n$ and $\mathbb{H}^n$ since these are the Euclideanized versions of ...
1
vote
1answer
243 views

Defining Euclidean global AdS

How does one see that that the Euclidean AdS is the same as the hyperbolic space at the same dimension ie $EAdS_n = \mathbb{H}_n = SO_0(n,1)/SO(n)$? Or is this to be seen as the definition of ...
5
votes
3answers
3k views

What is the shape of a black hole?

I was thinking; what shape does a black hole have?. By 'Shape', I mean its form (e.g, circle , cylinder, sphere, torus, etc..). We usually think of black holes as if they're plugholes (e.g, a flat ...
1
vote
1answer
97 views

How much does the global structure of a Lorentzian spacetime restrict the metric? And vice versa

E.g. if I know that my topology is that of a hyperboloid, how much freedom do I have left for my choice of the metric? And the other way around: if my metric is some conformal factor times the unit ...
6
votes
1answer
271 views

What do we mean when we say the QM wave function is a section of the $U(1)$ bundle?

I have a couple questions here. To keep the discussion simple lets stick to the following case: what is the quantum mechanics of a single particle in the presence of a background EM field, such as ...
8
votes
1answer
1k views

Trivial and Non-trivial topology of band structure

I don't understand the meaning of the expression "trivial topology" or "non-trivial topology" for an electronic band structure. Does anybody have a good explanation?
2
votes
1answer
110 views

Moving the endpoints of a wormhole towards each other

Suppose we have a perfectly safe portal/wormhole and we place the two endpoints facing each other so that a person between them would see an endless corridor (with infinite number of herself). What ...
5
votes
1answer
240 views

Aharonov-Bohm Effect in Torus

I had a very brief introduction to the Aharonov-Bohm effect in class. The lecturer introduced the notion that $H(\Phi=\Phi_0)$ and $H(\Phi=0)$ gives identical energy spectrum and that the Hamiltonians ...
4
votes
1answer
92 views

Are there any restrictions on building the topology of spacetime out of the complement of open balls?

I assume that for a Lorentzian manifold (i.e. with Minkowski signature), the analog of an open ball is the interior of a light cone. My question is motivated by the observation that whereas any point ...
15
votes
1answer
507 views

Does the existence of Higgs imply the existence of Magnetic Monopoles?

I am aware that in theories with spontaneous symmetry breaking, Magnetic Monopoles can exist as topological solitons. Can the same be done with the Standard Model gauge group. I am familiar with the ...
2
votes
1answer
128 views

Topological vs. non-topological noetherian charges

What (if any) is the relationship between the conserved (non-topological) noetherian charges and topological charges? Namely, is there any "generalization" of the Noether's first theorem that includes ...
3
votes
2answers
423 views

Excluding big bang itself, does spacetime have a boundary?

My understanding of big bang cosmology and General Relativity is that both matter and spacetime emerged together (I'm not considering time zero where there was a singularity). Does this mean that ...
9
votes
1answer
305 views

Our Universe Can't be Looped? [duplicate]

With reference to the Twin-Paradox (I am new with this), now information of who has actually aged comes from the fact that one of the twins felt some acceleration. So if universe was like a loop, and ...
4
votes
1answer
123 views

Consequences of Compactness in Physics

If we understand spacetime as a $4$-dimensional manifold $M$, from the point of view of physics what are the consquences of a subset of it being compact? My point here is simple: in math we usually ...
4
votes
2answers
944 views

Are all points in the universe connected?

Is it true that every point in the universe is connected or could be so theoretically? If so how is this mediated? Is it through the quantum nature of the fabric of space or is it through the ...
2
votes
0answers
187 views

Do we expect that the universe is simply-connected? [duplicate]

I heard recently that the universe is expected to be essentially flat. If this is true, I believe this means (by the 3d Poincare conjecture) that the universe cannot be simply-connected, since the ...
3
votes
1answer
95 views

Topological phase transitions - breaking of continuous translational invariance [closed]

I'm relatively new to the theoretical side of physics. I have a question about topology, continuous symmetry breaking and phase transitions. Your help is much appreciated! Ok so I have an infinite ...
4
votes
1answer
282 views

Tangent bundles and $\mathbb{C}P^n$ and $\mathbb{C}^n$

As discussed here the complex projective space $\mathbb{C}P^n$ is the set of all lines on $\mathbb{C}^n$ passing through the origin. It would seem natural to assume that any $\mathbb{C}P^n$ can be ...
1
vote
1answer
357 views

Topology for physicists [duplicate]

Which are the best introductory books for topology, algebraic geometry, manifolds etc, needed for string theory?
5
votes
1answer
169 views

How is the direction of time determined in general relativity?

In special relativity every frame has its own unique time axis, represented in Minkowski diagrams by a fan-out of time vectors that grows infinitely dense as you approach the surface of the light cone ...
5
votes
2answers
580 views

Chiral edge state as topological properity of bulk state

As far as I know, quantum hall effect and quantum spin hall effect has chiral edge state. Chiral edge state is usually closely related with delocalization, since back scattering is forbidden. However, ...
3
votes
2answers
169 views

Proof of quantization of magnetic charge of monopoles using homotopy groups

Suppose we place a monopole at the origin $\{{\bf 0}\}$, and the gauge field is well-definded in region $\mathbb R^3-\{0\}$ which is homomorphic to a sphere $S^2$. Then the total manifold is $U(1)$ ...
4
votes
1answer
1k views

First Chern number, monoples and quantum Hall states

The first Chern number $\cal C$ is known to be related to various physical objects. Gauge fields are known as connections of some principle bundles. In particular, principle $U(1)$ bundle is said to ...
4
votes
1answer
213 views

Gauss-Bonnet theorem in the Hawking/Ellis book

At the page 336 of Hawking, Ellis: The Large Scale Structure of Space-Time, the Gauss-Bonnet theorem is stated as $$\int_H \hat{R}\ d\hat{S} = 2\pi \chi(H) \qquad (1)$$ with $$\hat{R} = R_{abcd} ...
3
votes
0answers
263 views

Is it mathematically possible or topologically allowable for cutouts, or cavities, to exist in a 3-manifold?

A few weeks back, I posted a related question, Could metric expansion create holes, or cavities in the fabric of spacetime?, asking if metric stretching could create cutouts in the spacetime manifold. ...
7
votes
2answers
1k views

Questions about Thouless-Kohmoto-Nightingale-den Nijs (TKNN) paper

I am reading the famous and concise Thouless-Kohmoto-Nightingale-den Nijs (TKNN) paper Quantized Hall Conductance in a Two-Dimensional Periodic Potential, Phys. Rev. Lett. 49, 405–408 (1982), where I ...
9
votes
2answers
360 views

Topology and Quantum mechanics

I have a very simple question. Can we know about the topology of the underlying space-time manifolds from Quantum mechanics calculations? If the Space-time is not simply connected, how can one measure ...
7
votes
1answer
270 views

What is the simplest possible topological Bloch function?

Kohmoto (1985) pointed out in Topological Invariant and the Quantization of the Hall Conductance how TKNN's calcuation of Hall conducance is related to topology, in which topologically nontriviality ...
4
votes
0answers
115 views

Alternate geodesic completions of a Schwarzschild black hole

The Kruskal-Szekeres solution extends the exterior Schwarzschild solution maximally, so that every geodesic not contacting a curvature singularity can be extended arbitrarily far in either direction. ...
22
votes
2answers
591 views

Does a charged or rotating black hole change the genus of spacetime?

For a Reissner–Nordström or Kerr black hole there is an analytic continuation through the event horizon and back out. Assuming this is physically meaningful (various site members hereabouts think ...
4
votes
3answers
407 views

Could metric expansion create holes, or cavities in the fabric of spacetime?

Is it possible for metric expansion to create holes, or cavities in the fabric of spacetime? According to the Schwarzschild metric, the metric expansion of space around a black hole goes to infinity ...
1
vote
1answer
237 views

Proper times of two observers in a three-torus

Consider two observer in a tree-torus space of size $L$. Observer $A$ is at rest, while observer $B$ moves in the $x$-direction with constant velocity $v$. $A$ and $B$ began at the same event, and ...
16
votes
3answers
556 views

Homotopy $\pi_4(SU(2))=\mathbb{Z}_2$

Recently I read a paper using $$\pi_4(SU(2))=\mathbb{Z}_2.$$ Do you have any visualization or explanation of this result? More generally, how do physicists understand or calculate high dimension ...
9
votes
1answer
278 views

Lagrangian for Goldstone mode + topological excitation

The XY-model Hamiltonian is the following, $${\cal H}~=~-J\sum_{\langle i,j\rangle} \cos (\theta_i -\theta_j).$$ The Goldstone mode corresponds to term $(\nabla \theta)^2$ in the effective ...
6
votes
2answers
3k views

How is the topological $Z_2$ invariant related to the Chern number? (e.g. for a topological insulator)

This question relates to the $Z_2$ invariant defined e.g. for topological insulators: Is it correct to relate $Z_2$ = 1 to an odd Chern number and $Z_2$ = 0 to an even Chern number? If yes, is it ...
3
votes
1answer
734 views

Chern number in condensed matter physics

In mathematics, the Chern number is defined in terms of the Chern class of a manifold. What is the exact definition of Chern number in condensed matter physics, i.e. quantum hall system?
3
votes
2answers
382 views

(Co)homology of the universe

In this post let $U$ be the universe considered as a manifold. From what I gather we don't really have any firm evidence whether the universe is closed or open. The evidence seems to point towards it ...
3
votes
1answer
397 views

Is a preferred reference frame of the universe the old aether?

About two years ago I posted a question about a symmetrical twin paradox: Here. Recently a new answer was posted and an intense discussion ensued: Here. One of the points discussed concerns a ...
8
votes
1answer
291 views

7 sphere, is there any physical interpretation of exotic spheres?

Basically an exotic sphere is topologically a sphere, but doesn't look like a one. Or more accurately: homeomorphic but not diffeomorphic to the standard Euclidean n-sphere The first exotic ...
3
votes
5answers
181 views

Why is the world sheet of an open string a cylinder?

I went to a lecture a few weeks ago and was told the following: The world sheet of a closed string is a normal, standing cylinder. The world sheet of an open string is a cylinder on its side. This ...
6
votes
1answer
194 views

Can closed loops evade the spin-statistic theorem in 3 dimensions?

The famous spin-statistics result asserts that there are only bosons and fermions, and that they have integer and integer-and-a-half spin respectively. In two-dimensional condensed matter systems, ...
3
votes
1answer
553 views

Large gauge transformations

I would like to understand what is the importance of large gauge transformations. I read that these gauge transformation cannot be deformed to the identity, but why should we care about that?
2
votes
2answers
154 views

Graph Invariants and Statistical Mechanics

Many intuitive knot invariants including Jones' polynomial are inspired by statistical mechanics. Further profound connections have been explored between knot theory and statistical mechanics. I was ...
2
votes
0answers
38 views

Reference request: Classical Mechanics as an Application to Smooth Manifolds [duplicate]

Possible Duplicate: Classical Mechanics for Mathematician Last time I asked a question, but it does not sound specific. I am currently taking graduate topology class (using Lee's ...
9
votes
1answer
578 views

What is topological degeneracy in condensed matter physics?

What is topological degeneracy in strongly correlated systems such as FQH? What is the difference between topological degeneracy and ordinary degeneracy? Why is topological degeneracy important for ...
9
votes
1answer
175 views

Why are topological solitons present in some phases for lattice models?

Over a spatial continuum, it is easy to see why some topological solitons like vortices and monopoles have to be stable. For similar reasons, Skyrmions also have to be stable, with a conserved ...
1
vote
2answers
202 views

Is a compact universe consistent with the results of (for example) the Michelson-Morley experiment?

If the universe is compact then there is a twin paradox that is resolvable only by selecting a preferred inertial reference frame (arXiv). I was under the impression that the lack of a preferred ...
6
votes
3answers
848 views

Aharonov-Bohm Effect and Flux Quantization in superconductors

Why is the magnetic flux not quantized in a standard Aharonov-Bohm (infinite) solenoid setup, whereas in a superconductor setting, flux is quantized?
3
votes
2answers
368 views

On Aharonov–Bohm effect

Aharonov–Bohm effect in brief is due to some singularities in space. In books it's infinite solenoid most of the time, which makes some regions of space not simply connected. What intrigues me is the ...