The topology tag has no wiki summary.
18
votes
1answer
321 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 ...
9
votes
1answer
173 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 ...
2
votes
1answer
53 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 ...
1
vote
2answers
168 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 ...
5
votes
1answer
142 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
256 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 ...
2
votes
1answer
64 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 ...
6
votes
4answers
913 views
Shape of the universe?
What is the exact shape of the universe? I know of the balloon analogy, and the bread with raisins in it. These clarify some points, like how the universe can have no centre, and how it can expand ...
3
votes
2answers
90 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 ...
13
votes
3answers
341 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 ...
1
vote
0answers
64 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 ...
1
vote
1answer
46 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 ...
9
votes
1answer
105 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 ...
4
votes
2answers
348 views
Does spacetime in general relativity contain holes?
Are there physical models of spacetimes, which have bounded (four dimensional) holes in them?
And do the Einstein equations give restrictions to such phenomena?
Here by holes I mean ...
4
votes
1answer
183 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
1answer
176 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 ...
2
votes
0answers
113 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 ...
4
votes
1answer
236 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
102 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 ...
4
votes
2answers
349 views
Book covering Topology required for physics and applications
I am a physics undergrad, and interested to learn Topology so far as it has use in Physics. Currently I am trying to study Topological solitons but bogged down by some topological concepts. I am not ...
3
votes
4answers
478 views
Topology needed for Differential Geometry [duplicate]
I am a physics undergrad, and need to study differential geometry ASAP to supplement my studies on solitons and instantons. How much topology do I need to know. I know some basic concepts reading from ...
1
vote
1answer
108 views
Topology for physicists [duplicate]
Which are the best introductory books for topology, algebraic geometry, manifolds etc, needed for string theory?
7
votes
1answer
211 views
Quantum dimension in topological entanglement entropy
In 2D the entanglement entropy of a simply connected region goes like
\begin{align}
S_L \to \alpha L - \gamma + \cdots,
\end{align}
where $\gamma$ is the topological entanglement entropy.
$\gamma$ is ...
2
votes
1answer
66 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 $U(1)$ ...
16
votes
1answer
233 views
Why is there no theta-angle (topological term) for the weak interactions?
Why is there no analog for $\Theta_\text{QCD}$ for the weak interaction? Is this topological term generated? If not, why not? Is this related to the fact that $SU(2)_L$ is broken?
2
votes
1answer
85 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} ...
0
votes
0answers
116 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. ...
8
votes
3answers
48 views
Does the complex 3-sphere have a complex structure modulus?
This question has a flavor which is more mathematical than physical, however it is about a mathematical physics article and I suspect my misunderstanding occurs because the precise mathematical ...
11
votes
1answer
47 views
Relationship between Weak Cosmic Censorship and Topological Censorship
The weak cosmic censorship states that any singularity cannot be in the causual past of null infinity (reference).
The topological censorship hypothesis states that in a globally hyperbolic, ...
3
votes
3answers
261 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 ...
7
votes
1answer
171 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 ...
6
votes
2answers
209 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 ...
4
votes
0answers
89 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.
...
1
vote
1answer
81 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 ...
5
votes
0answers
33 views
What is the importance of studying degeneration on $M_g$
Let $M_g$ be the moduli space of smooth curves of genus $g$. Let $\overline{M_g}$ be its compactification; the moduli space of stable curves of genus $g$.
It seems to be important in physics to study ...
9
votes
5answers
196 views
Where do theta functions and canonical Green functions appear in physics
In the beginning of Section 5 in his article, Wentworth mentions a result of Bost and proves it using the spin-1 bosonization formula. This result provides a link between theta functions, canonical ...
2
votes
5answers
128 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 ...
3
votes
1answer
323 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?
5
votes
2answers
433 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 ...
2
votes
2answers
264 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 ...
8
votes
2answers
395 views
Is spacetime simply connected?
As I've stated in a prior question of mine, I am a mathematician with very little knowledge of Physics, and I ask here things I'm curious about/things that will help me learn.
This falls into the ...
8
votes
3answers
811 views
Can spacetime be non-orientable?
This question asks what constraints there are on the global topology of spacetime from the Einstein equations. It seems to me the quotient of any global solution can in turn be a global solution. In ...
29
votes
6answers
1k views
What is known about the topological structure of spacetime?
General relativity says that spacetime is a Lorentzian 4-manifold $M$ whose metric satisfies Einstein's field equations. I have two questions:
What topological restrictions do Einstein's equations ...
4
votes
0answers
141 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 ...
6
votes
1answer
133 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, ...
2
votes
1answer
175 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
125 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 ...
7
votes
1answer
287 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 ...
0
votes
2answers
127 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 ...
4
votes
2answers
365 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?
