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20

All of physics has two aspects: a local or even infinitesimal aspect, and a global aspect. Much of the standard lore deals just with the local and infinitesimal aspects -- the perturbative aspects_ and fiber bundles play little role there. But they are the all-important structure that govern the global -- the non-perturbative -- aspect. Bundles are the ...


15

As far as I understand it there are essentially two ways in which you can study quantum mechanics on a manifold with some curvature. Classically speaking these two ways lead to the same physics but in a quantum mechanical approach they are distinct. The first approach is to think of a particle moving "freely" through three-dimensional space, but subject to ...


14

Just one additional point that I haven't seen mentioned above: if the space-time has non-trivial fundamental group, it won't be seen by an observer at infinity. This is the content of the Topological Censorship Theorem. The implication is that for an asymptotically flat space-time, any interesting topology will be hidden behind the event-horizon. The proof ...


14

Topology is of fundamental importance even to systems in classical mechanics. The configuration space (or phase space) of a generic classical mechanical system is a manifold and manifolds are topological spaces with some extra structure (e.g. a smooth structure in the case of smooth manifolds). At the very start of any classical mechanics problem, you need ...


13

There are a bunch of questions here. Let me try to take them in order: Is it possible that our Universe has the feature that if you travel far enough you return to where you started? Yes. The standard Big-Bang cosmological model is based on the idea that the Universe is homogeneous and isotropic. One sort of homogeneous spacetime has the geometry of a ...


12

In the presence of massless chiral fermions, a $\theta$ term in can be rotated away by an appropriate chiral transformation of the fermion fields, because due to the chiral anomaly, this transformation induces a contribution to the fermion path integral measure proportional to the $\theta$ term Lagrangian. $$\psi_L \rightarrow e^{i\alpha }\psi_L$$ ...


11

Let me first answer your second questions about the physical intuition behind fiber bundles: Fiber bundles ( with compact structure groups) describe internal degrees of freedom such as spin and isospin just as manifolds describe translational degrees of freedom. For example, (a non-trivial fibre bundle is needed to describe the rotation of a neutral spinning ...


10

the answer to the Magellan question for our Universe is actually "No". If the Universe were a static 3-sphere, as Ted Bunn suggests, then you could "swim" around the Universe just like Megallan around the Earth. But an important fact about our Universe is that its size is changing: the size can't stay constant just like the apple can't sit in the middle of ...


10

No. Firstly, weak cosmic censorship can only hold in the generic sense, as there are known examples of nakedly singular space-times. (See, e.g. Christodoulou 1993, and Christodoulou 1999.) Observe in particular that the nakedly singular space-time constructed in the 1993 paper is spherically symmetric with a central axis, and the initial data is ...


9

I don't know the answer, but your intuition is right on -- the fact that the equations are local doesn't mean that there cannot be a constraint on the topology of a global solution. For example, in Euclidean signature, $R_{ij} = g_{ij}$ immediately implies that the scalar curvature is positive, which in turn leads to topological constraints. If the ...


9

First a warning: I don't know much about either algebraic topology or its uses of physics but I know of some places so hopefully you'll find this useful. Topological defects in space The standard (but very nice) example is Aharonov-Bohm effect which considers a solenoid and a charged particle. Idealizing the situation let the solenoid be infinite so that ...


8

Our spacetime cannot be unorientable. That's because the laws of physics describing our spacetime are not left-right symmetric. We say that they break the P-symmetry (parity) or that they are "chiral" (derived from a Greek word for the hand which is either left or right.) For example, a left-handed neutrino would turn into a right-handed neutrino if you ...


8

I believe that Andy and Cumrun didn't want to say that this manifold would have a complex structure modulus in isolation. However, as is clear from the "conifold" setup, the manifold given by $xy-zt=\mu$ is being incorporated into a larger manifold, so this equation only describes the vicinity of some region. When you exploit the fixed asymptotic shape of ...


7

I suppose there are many aspects to look at this from, anna v mentioned how Calabi-Yao manifolds in string theory (might?) have lots of holes, I'll approach the question from a purely General Relativity perspective as far as global topology. Solutions in the Einstein Equations themselves do not reveal anything about global topology except in very specific ...


7

The explicit formula for the diffeomorphism between the affine quadric and the cotangent bundle $T^{*}S^3$ is known. It is given for example in: Hall and Mitchel's Coherent states on spheres (Equation 18). I'll write it here using the same notation of the question for completeness: $\mathbf{z}(\mathbf{x}, \mathbf{p}) = ...


7

(1) What is topological degeneracy in strongly correlated systems such as FQH? From Wiki, http://en.wikipedia.org/wiki/Topological_degeneracy : Topological degeneracy is a phenomenon in quantum many-body physics, that the ground state of a gapped many-body system become degenerate in the large system size limit, and that such a degeneracy cannot be lifted ...


7

The answer to your question is yes, your intuition is 100% correct. It all boils down to the topology of the configuration space $\mathcal C$, mainly the first homotopy group $\pi_1(\mathcal C)$ (which is non-zero in your example). See problem set 1, problem 3 from this course at Oxford. This exercise is precisely about loops in 3+1D! One has to argue that ...


7

It is very hard to visualize these homotopy classes, since they correspond to maps $S^4\rightarrow SU(2)\approx S^3$. The homotopy groups of spheres (and any other space) are typically very difficult to calculate in generality and physicists typically ask mathematicians. But there exist simple results in the so-called "stable range" where there is a regular ...


7

This vector potential can be written in every point on the plane except the origin as: $$ A_x = -\frac{\partial \psi}{\partial y}$$ $$ A_y = \frac{\partial \psi}{\partial x} $$ with $$\psi = \frac{1}{2}\mathrm{log}(x^2+y^2)$$ $A$ is not exact, because $\psi$ is singular at the origin. But this means that the magnetic field is zero at every point except ...


7

Very loosely speaking the reasoning is this. Imagine a two band system in which the fermi sea has one filled band with Chern number $n$ and another system with $N$ filled bands but also with Chern number $n$. Physically they have the same topological properties (for example the same Hall conductance, edge states and so on), but cannot be deformed ...


6

The idea of knot inflation is that of A. Grosberg and it is tricky, but I'll try to explain some of it. Construction by way of scaling analysis is required to tackle this. Suppose the volume of our polymer is given by $R$. Now fit a tube of fixed length, $L$, over the polymer; this tube has diameter, $D$, across its length that is just large enough to ...


6

They require a special discussion because they are different. The (defining) fact that they can't be deformed to the identity means that it is not enough to verify the invariance under infinitesimal gauge transformations: the problem is that the large gauge transformations cannot be obtained by combining many infinitesimal gauge transformations! The modular ...


6

(I will assume in my answer that people have read the discussion on the old question, linked to by the OP.) No, it is not like the aether. It is still true that locally, there is no preferred reference frame. You don't even really need to think about spacetime to see what is going on. Consider a two-dimensional plane, parametrised by $(x,y)$, and roll it ...


6

Since $SU(2)$ is topologically a three-sphere $S^3$, you can begin by investigating the homotopy groups of spheres. Unfortunately, although there are some regular results, such as $\Pi_n(S^n)=\mathbb{Z}$, and $\Pi_m(S^n)=0$ for $m<n$, I don't think there is a single method to calculate $\Pi_m(S^n)$ for $m>n$. Individual results for $m>n$ are ...


6

This question has been posted also at http://mathoverflow.net/questions/115866/homotopy-pi-4su2z-2 with both geometric and algebraic (that was mine!) type of answers. The geometric answers tell of Pontjyagin's method of constructing explicit representations of maps to spheres. The algebraic methods tells of the answer from a general theorem which gives some ...



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