The topology tag has no wiki summary.
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 ...
18
votes
1answer
291 views
+300
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 ...
16
votes
1answer
227 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?
13
votes
3answers
339 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 ...
13
votes
7answers
2k views
Applications of Algebraic Topology to physics
I have always wondered about applications of Algebraic Topology to Physics, seeing as am I studying algebraic topology and physics is cool and pretty. My initial thoughts would be that since most ...
11
votes
1answer
46 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, ...
11
votes
2answers
335 views
The entropic cost of tying knots in polymers
Imagine I take a polymer like polyethylene, of length $L$ with some number of Kuhn lengths $N$, and I tie into into a trefoil knot. What is the difference in entropy between this knotted polymer and ...
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 ...
9
votes
1answer
166 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 ...
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 ...
8
votes
3answers
808 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 ...
8
votes
2answers
393 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
1answer
104 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 ...
7
votes
1answer
283 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 ...
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 ...
7
votes
1answer
170 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
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 ...
6
votes
1answer
132 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, ...
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 ...
6
votes
1answer
136 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 ...
5
votes
2answers
425 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 ...
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 ...
4
votes
3answers
182 views
What are some mechanics examples with a globally non-generic symplecic structure?
In the framework of statistical mechanics, in books and lectures when the fundamentals are stated, i.e. phase space, Hamiltons equation, the density etc., phase space seems usually be assumed to be ...
4
votes
2answers
363 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?
4
votes
2answers
344 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
2answers
345 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 ...
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
1answer
181 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, ...
4
votes
1answer
231 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
249 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 ...
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.
...
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 ...
3
votes
1answer
172 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 ...
3
votes
4answers
477 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 ...
3
votes
3answers
259 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 ...
3
votes
1answer
117 views
geometry inside the event horizon
I'm trying to understand intuitively the geometry as it would look to an observer entering the event horizon of a schwarszchild black hole. I would appreciate any insights or corrections to the above.
...
3
votes
2answers
83 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 ...
3
votes
1answer
318 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
0answers
69 views
are pinch-off bubbles valid solutions to general relativity?
are bubbles of spacetime pinching-off allowed solutions to general relativity? With "pinch-off bubble" i really mean a finite 3D volume of space whose 2D boundary decreases until it reaches zero and ...
2
votes
2answers
263 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 ...
2
votes
1answer
98 views
how does nature prevent transient toroidal event horizons?
.. and does it really need to?
Steps to construct a (transient) toroidal event horizon in a asymptotically flat Minkowski spacetime:
1) take a circle of radius $R$
2) take $N$ equidistant points in ...
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
1answer
129 views
The topology of a “closed” universe - is it really closed?
The spatial part of the positive curvature FRW metric has the form
\begin{equation}
ds^2=\frac{dr^2}{1-(r/R)^2}+r^2d\Omega^2
\end{equation}
or
\begin{equation}
ds^2=R^2(d\chi^2+\sin{\chi}^2d\Omega^2)
...
2
votes
1answer
62 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 ...
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 $U(1)$ ...
2
votes
1answer
84 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} ...
2
votes
2answers
246 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 ...
2
votes
5answers
127 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 ...
2
votes
2answers
124 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
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 ...
