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+50

Why are the quantum observables defined on opens sets a presheaf and not a sheaf?

In local quantum field theory or AQFT one can mathematically describe over each open set $U$ of a spacetime $M$ the quantum states or observables of the theory. This structure is commonly referred as ...
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0answers
27 views

Can $S^4$ be the cotangent bundle of a manifold? [migrated]

I am asking the question because in the classical mechanics book by Arnold, he states that there is a distinguished 1-form on $T^*V $. It seems that there is no such distinguished 1-form on a general ...
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0answers
15 views

Any good reference on Maslov index (or Morse index)?

Any good reference on Maslov index (or Morse index)? I have some basic knowledge of differential geometry, calculus of variation. So is there any good reference for me?
1
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2answers
104 views

Is it plausible for spacetime to be shaped something like a torus? [duplicate]

I have heard three theories for how space-time is shaped, flat, sphere-like, or saddle-like. Flat is the most likely, as all our measurements implies that space time has curvature close to 0. Is it ...
1
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1answer
49 views

Necessity, significance of Spinors

This is an area I am researching at my own pace, general rotations in 3D. I've known about the plate trick for a while as well, and have a very rough understanding of the concept of ...
4
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0answers
96 views

The classification of particles or fields in general spacetime- Is it still meaningful to say spin-0, 1/2 ,1 field in general spacetime? [closed]

In 3+1 dim Minkovski spacetime, the classification of particle or field, that is spin-0, 1/2 , 1..., depends on the representation of the universal covering group of $SO(1,3)$, that is $SL(2,C)$. When ...
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0answers
33 views

Why topological strings have to be closed or infinite?

Let's assume spontaneously broken global $U(1)$ group. During phase transition global topological strings are formed. Why they have to be infinite or closed (there doesn't exist finite strings)?
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0answers
31 views

Aplications of algebraic topology/pure mathematics in nuclear fusion

I am planning on working on an independent research class this fall at a community college. My instructor wants to focus it around pure mathematics/topology/homotopy. I think she has done a phd in ...
2
votes
1answer
51 views

Diagonal part of the configuration space of two indistinguishable quantum particles

Why is the configuration space of two indistinguishable particles given by $\frac{M^n-\Delta}{S_n}$? My question is about the $\Delta$. (Notation: $M$ is the configuration space of 1 particle. $M^n$ ...
4
votes
1answer
120 views

Where does the “Supersymmetry” in Witten's proof of the Morse inequalities come from?

Where does the "Supersymmetry" in Witten's proof of the Morse inequalities (original paper and outline of proof for mathematicians) come from? Hopefully someone can provide an intuitive understanding? ...
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0answers
12 views

Correlation length during phase transitions in early Universe

During phase transitions of the second kind topological defects may form on the bounds of two areas separated by correlation length. In early Universe during phase transitions correlation length ...
1
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0answers
49 views

Compactly generated vs. compactly constructed causality violating region?

I am currently trying to grasp the nuance between a compactly generated future Cauchy horizon (as per Hawking's chronological protection conjecture) and a compactly constructed causality violating ...
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0answers
28 views

Topological configurations and phase transitions

It is known that topological defects might appear only during phase transitions of the first kind, while continuous transitions of the second kind and crossovers don't product them. How to explain ...
0
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0answers
43 views

Axion domain walls and QCD phase transition

Now it is known that QCD phase transition corresponds to crossover. This it seems that no topological defects is produced during phase transition. Do axion domain walls arise during QCD phase ...
0
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0answers
41 views

The bounds of axion domain walls are axion strings?

There are two phase transitions which are important for the axion physics. The first one is Peccei-Quinn phase transition, during which axions arise. The second one is QCD phase transition, at which ...
2
votes
1answer
47 views

Kalb-Ramond action and topological string radiation

Let's have simple scalar $\Phi$ action involves spontaneously symmetry breaking in a form $$ \tag 1 S = \int d^{4}x\left( |\partial_{\mu}\psi|^{2} + \psi^{2}|\partial_{\mu}\theta |^{2} - ...
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0answers
32 views

Global cosmic strings evolution

Recently I've read about axion string. It can be shown that the energy per unit length of the string located along $z$ axis is $$ \mu = 2 \pi f_{a}^{2}\ln\left( \frac{L}{\delta}\right), $$ where $L$ ...
2
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0answers
76 views

Axion strings and spontaneously broken symmetry

I have two question about axion strings: Why their appearance is connected with spontaneously broken symmetry? How to demonstrate that? Why they are stable topological configurations (look to the ...
2
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0answers
34 views

When can a $k$-cycle wrap around a manifold?

According to the paper ``Heterotic and Type I String Dynamics from Eleven Dimensions'' (page 7): Even when the topology is wrong -- for instance on $\mathbb{R}^{11}$ where there is no two-cycle ...
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0answers
34 views

How to simulate “inside-out” geometries of a structure?

How can I simulate the structural deformation of a physical material to find all possible "stable" inside out forms? For example, some dome shaped rubber caps can be pushed inside out, like the ...
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0answers
34 views
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0answers
49 views

Are topological vacua of QCD Lorentz invariant?

Are topological vacua of QCD Lorentz invariant or they mix under boosts?
3
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1answer
66 views

Braiding in 3D Space

In arXiv:1005.0583 the authors wrote that in two dimensional space the configuration space of n particles is multiply-connected and therefore the fundamental group of the configuration space is the ...
3
votes
1answer
72 views

Question regarding moduli space of a Calabi-Yau manifold

On page 132 of "Introduction to Supergravity" by Horiatiu Nastase, the author says: On $M = CY_3$ (Calabi-Yau space) there are $b_3$ topologically nontrivial 3-surfaces, for which we can define a ...
2
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3answers
116 views

$SO(3)$ vs 3-Torus ${(S_1)}^3$

From rigid body rotations point of view, why are $SO(3)$ and 3-Torus not the same. Every rigid rotation is rotation about three axes. So how come $SO(3)$ is not ${(S_1)}^3$? It seems it should be. Is ...
3
votes
1answer
54 views

Manifolds, unit 2-sphere and stereographic projection

I am always passing through this example while reading about manifolds that I don't quite get. It is when describing the unit 2-sphere $S^2$ as an example of a manifold. They say, initially it may ...
2
votes
1answer
121 views

Divergence Theorem, mathematical approach to Gauss's Law?

Let $D$ be a compact region in $\mathbb{R}^3$ with a smooth boundary $S$. Assume $0 \in \text{Int}(D)$. If an electric charge of magnitude $q$ is placed at $0$, the resulting force field is ...
2
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1answer
104 views

Spinors and Möbius strips

I asked this question on Math.SE as I thought the perspective of representation theory might be enlightening. But since the question was provoked by a description of Spinors describing the spin of ...
3
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0answers
62 views

Branes wrapping curves in M-theory. What does it mean?

What does it mean that a M5-branes wraps a holomorphic curve in M-theory? In specific a lot of Vafa's paper involve various branes (not only M5) wrapping some cycles. What does this really mean ...
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0answers
177 views

Can the universe be round but still infinite?

Can the universe still be infinite in space if its curvature is > 1? Is a manifold of positive curvature necessarily compact? Does the Tarski paradox have any bearing on the finite or infinite ...
5
votes
1answer
230 views

Topology of Fermi surface

In The universe in a Helium droplet, Grigory Volovik relates the stability of a fermi surface to topology of a Green function. There he gives the example of a Fermi gas and says that the Green ...
2
votes
1answer
73 views

Rindler and Minkowski space future/past infinity

In my black holes course, we are looking at the Penrose diagram for 1+1 D Minkowski space. My notes don't specifically describe $i^{\pm}$ (future/past timelike infinity) but do say all timelike curves ...
1
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2answers
79 views

Allowable spacetime deformations [closed]

What deformations are possible with spacetime? By 'deformation' I am referring to the kind of change in spacetime caused by the presence of a mass which deforms spacetime sufficiently to deflect ...
3
votes
0answers
70 views

Feynman Path integrals in space with holes in it [closed]

Feynman Path Integrals are a way of calculating the wave function of quantum mechanics. It usually integrates every possible path through all of space. I wonder if there is any study of Feynman path ...
3
votes
1answer
55 views

Can one make a synthetic dimension “curl around” into a cylinder?

A really cool recent proposal, Synthetic Gauge Fields in Synthetic Dimensions. A. Celi et al. Phys. Rev. Lett. 112, 043001 (2014), arXiv:1307.8349, shows how you can simulate a synthetic ...
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0answers
54 views

$\mathbb{Z}_2$ topological insulators which obey inversion symmetry as well

According to Fu & Kane (2006), systems with simultaneous time-reversal invariance and inversion symmetry have their $\mathbb{Z}_2$ topological invariant given by the product of the parity ...
0
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0answers
24 views

2D CFT for nontrivial topology

What is a systematic way to calculate a general $N$-points correlation function of 2D CFT for a nontrivial topology? Piece by piece of this can be found in many different CFT and String Theory ...
3
votes
1answer
156 views

$U(1)$ 5-dimensional Kaluza-Klein topological defects

Five-dimensional Kaluza-Klein theory is well-known to predict that the electromagnetic field can be described as a curled additional dimension over four-dimensional spacetime. That is, you only need ...
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0answers
416 views

Does the existence of instantons imply non-trivial cohomology of spacetime?

Gauge theories are considered to live on $G$-principal bundles $P$ over the spacetime $\Sigma$. For convenience, the usual text often either compactify $\Sigma$ or assume it is already compact. An ...
2
votes
2answers
166 views

What is the connection between geometry of physical space and Hilbert space?

In Quantum Mechanis (QM), the dynamical variables are the (quantized) coordinates $x_j$ and their canonical conjugate $p_j = -i\partial_j$ with the commutation relation $[x_j,p_k]=i\delta_{jk}$ ...
6
votes
2answers
221 views

Is a spinor in some sense connected to space?

Spinors transform under the representation of $SL(2,\mathbb{C})$ which is the double cover of the Lorentz group $SO(1,3)$ - or in the non-relativistic case under $SU(2)$, the double cover of $SO(3)$. ...
2
votes
1answer
149 views

Question about universe expansion

In general relativity, we cannot determine the global structure of the universe (since it is not flat), therefore all measurements and observations are only meaningful locally. In particular, we can ...
0
votes
1answer
57 views

Is a black hole really a hole in space? [closed]

What if when a supernova occurs, instead of it condensing into a singularity it creates enough force to tear a hole into the fabric of space? Is a black hole just what is sounds like, a hole in space? ...
2
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0answers
64 views

Validity of topological thermodynamics?

I've been reading some material by R. Kiehn, developing a topological approach to non-equilibrium thermodynamics through Cartan forms, where the fundamental claim is that irreversible processes are ...
1
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2answers
124 views

Is an achronal set contained in its own causal future?

I use Wald's notation: $I^+$ is the chronological future and $J^+$ is the causal future. My confusion arises from the following passage in Wald (1984): Now, let $S$ be a closed, achronal set ...
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0answers
104 views

Quantum phase space

Classical phase space is defined as a space in which all possible states are represented. Every state corresponds to a unique point in the phase space. On the other hand, in quantum mechanics every ...
3
votes
1answer
70 views

What are the latest findings on the topology and size of the universe?

The paper G. Aslanyan & A.V. Manohar, The Topology and Size of the Universe from the Cosmic Microwave Background, JCAP 06 (2012) 003, arXiv:1104.0015, uses the 7-year WMAP data. Has any ...
1
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0answers
86 views

Euclidean AdS space in Poincaré coordinates

I have read anti-de Sitter (AdS) space and its Euclidean version both in Global and Poincaré coordinates. For Lorentzian case it is clear how one Poincaré patch cover only one half of the whole AdS ...
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0answers
46 views

Topology of a bit

From a math perspective, it seems obvious that the electric field (or voltage which ever) of a bit in a computer, when its in a stable 0, or 1 state, must have a singularity, a set of points where the ...
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0answers
88 views

Manifold for Schwarzschild and Bertotti-Robinson

In short: what is the manifold in discussion for Schwarzschild metric $$ ds^2 = -(1-\frac {2M}r)dt^2 + \frac1{1-\frac{2M}r} dr^2 + r^2 (d\theta^2 + \sin^2 \theta d\phi^2)$$ and Bertotti-Robinson ...