In mathematics, topology examines the properties of space (such as connectedness and compactness) that are preserved under continuous deformations, such as stretching and bending, but not tearing or gluing.

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The expansion of the universe and its edge

I have understood that the universe is expanding at an accelerated rate. I have also thought, perhaps mistakenly, that this expansion means not that the masses in the universe are moving away from ...
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How does a position based force work on a torus?

Consider a 2 particle system in $\Bbb R^2$. Let's say they have some force acting between them. For the sake of argument let that be $F=\mathbf r/|\mathbf r|^3$. This is a simple inverse square system ...
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76 views

Yang-Mills potential and principal bundles

In section 2.7.2 of Bertlmann's "Anomalies in quantum field theory", it is stated that since a non-trivial principal bundle (based on a Lie group $G$) does not admit a global section, the Yang-Mills ...
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45 views

Relation between projective representations, connectivity of a group manifold and number of equivalence classes of paths

Projective representations of a multiply-connected group is defined as $$U(g_1)U(g_2)=c(g_1,g_2)U(g_1g_2)$$ where $c(g_1,g_2)$ is phase. Reading various articles, and this old post of mine, it appears ...
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49 views

Lorentz surfaces, conformal metrics and eigenvalues

From what I understand of Lorentz surfaces (spacetimes of dimension 2), it seems that, according to Kulkarni's theorem, two reasonable enough Lorentz surfaces (I am only interested in surfaces with ...
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26 views

Wave equation for odd spacetimes and source terms

It seems to be quite common practice, when solving the wave equation in spacetimes with odd topologies or horizons, to decompose the solution into a sum of the various origins (or destinations) of the ...
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3answers
88 views

How can I show that inversion is continuously connected to a reflection?

From Ex 3.1 in the TASI lectures on the conformal bootstrap: http://arxiv.org/abs/1602.07982 the problem is the inversion map (with Euclidean signature) $$ I\colon x^\mu \mapsto \frac{x^\mu}{x^2} $$ ...
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61 views

Finding $\pm 2 \pi$ defects in 2-D lattice nematic simulation

I'm working on a Monte Carlo simulation of a two-dimensional nematic system (XY-like model with even-order Legendre polynomial interactions, such that the director angle $\theta$ obeys $\theta \equiv \...
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1answer
35 views

Is CP problem the problem?

I've heard an argument that the question of smallness of QCD $\theta$ parameter is called the problem (namely, strong CP problem), since the other dimensionless couplings (like $\alpha_{s}$), are of ...
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1answer
78 views

Two definitions of topological terms in field theory

I've seen two distinct definitions for "topological" terms in the context of quantum field theory. Topological terms don't depend on the metric $g_{\mu\nu}$. This makes sense since topology is '...
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62 views

Field solution for spacetimes with identified regions

For a spacetime surgery wormhole, we have a manifold such that, for two connected compact sets $D_1$ and $D_2$, we remove $D_1$ and $D_2$ from the manifold and identify their boundaries. According to ...
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51 views

Reference for orbifolds in string- and M-theory

A number of orbifold constructions have been studied heavily in string- and M-theory over the years, establishing various dualities between different theories. Can someone point me to a slightly more ...
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2answers
105 views

What is the singularity of an actual collapsing black hole?

In most general relativity texts, the singularity is treated as a point removed from the manifold, to avoid having to deal with the infinite curvature of the Ricci scalar. But in the case of a more ...
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1answer
36 views

p-cycles and Fluxes

I would like to ask why the existence of a non-trivial p-cycle leads to a non-trivial flux. I would say that e.g. for a five-form $F_{(5)}$ field strength , the flux is: $$\int\limits_{\mathcal{C}^{5}}...
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1answer
81 views

Global Hyperbolicity in spacetime Manifold [closed]

If space time is timelike or null geodesically incomplete but cannot be embedded in a larger spacetime then we say that it has singularity. What does incompleteness means here?
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1answer
30 views

Multi-center Taub-NUT geometry and homologically nontrivial cycles

In the string theory book by Ibanez and Uranga (click here for the Google books excerpt), the four-dimensional multi-center Taub-NUT metric is written as $$ds^2 = \frac{V(\textbf{x})}{4}d\textbf{x}^2 ...
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1answer
65 views

Book on Berry phase and its relation to topology

I am searching for a book covering the Berry phase. Griffith has a good outline, but I would like a bit more detail, especially on the relation to topology. According to this post Ballentine also has ...
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1answer
83 views

Spacetime surgery - why are there unglueable points?

In The time travel paradox by S. Krasnikov (2002), Deutsch-Politzer spacetime is constructed by making two cuts and rejoining the manifold by gluing opposite "banks" of the cuts... omitting the "...
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1answer
173 views

A question on the Chern number and the winding number?

Let $\mid \psi(x,y) \rangle$ be a normalized wavefunction living in a $d$-dimensional Hilbert space and depend on two real parameters $(x,y)$ that belong to a closed surface (e.g., $S^2, T^2$, ...). ...
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31 views

AdS boundary global vs Poincare'

Is the global boundary of AdS the same of the boundary written in Poincare' coordinates?
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1answer
51 views

Left-right topology

Are there non-trivial topological solutions (in particular 't Hooft-Polyakov magnetic monopoles) associated with the (local) breaking \begin{equation} SU(2)_R \times SU(2)_L \times U(1)_{B-L} \to SU(...
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84 views

Derivation of TKNN's main result from Kubo formula

I have a question about a small but meaningful (to me at least) step in the original TKNN paper (http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.49.405). I understand the construction of the ...
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2answers
205 views

What manifold is spacetime?

In General Relativity, spacetime is a $4$-dimensional manifold with one Lorentzian metric tensor defined on it. In the Special Relativity case what manifold is spacetime is quite clear: it is ...
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1answer
79 views

Practical Calculation of Geometric Phase

I'm a graduate student working in the field of quantum chemistry, specifically in the field of non-adiabatic dynamics of molecular systems. I've run into a slight problem in a project that I've ...
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1answer
48 views

Causal structure, time orientability and equivalence classes

Quoting from this Wikipedia article, if $(M,g)$ is a Lorentzian manifold then the tangent vectors at each point in the manifold can be classified into three different types. Using a $(+,-,-,-)$ metric ...
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90 views

Can a rotating black hole have a donut-shaped event horizon? [closed]

It is conjectured that a rotating black hole has at its center a ring-shaped singularity. Thus, at the center of the ring-shaped singularity the gravitational field must be zero (similar to ...
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1answer
75 views

Equivariant cohomology and Mayer-Vietoris sequence

I'm reading this article upon topological field theory and I'm a bit confused about the way he compute equivariant cohomology of $S^2$ wrt $\mathrm{U}(1)$, i.e. $H^\bullet_{S^1}(S^2)$. You can find ...
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12 views

Constant electric background field from theta-term in QCD?

The theta-term of $1+1$ dimensional QED corresponds to a constant electric background field. Does the theta-term of QCD also induce a constant electric background field, which could be measurable in ...
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1answer
64 views

Spacetime manifold surgery: is this result still a valid etc. spacetime?

Given a valid classical GR spacetime manifold $M$ (i.e. 4D, Lorentzian, Hausdorff, paracompact, ?etc.), and $B\subset M$, a closed spatial subset (e.g. a closed ball at fixed $t$) to be excised, [...
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24 views

Book recommendations on geometrical methods for physicists (like Topology, Diff. Geometry) [duplicate]

I would like to obtain a book that has to do with geometrical methods/subjects for physicists. When I say geometrical methods/subjects I mean things like Topology, Differential Geometry, Lie ...
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1answer
77 views

If we live on the surface of Earth then why Earth images shows maps around it? [closed]

If you visits google map and go to earth we see the image as attached below. My question is if the earth is round like sphere ball and if we live on the surface of this ball (point me if i am ...
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1answer
114 views

What do we exactly mean by a “topological object” in physics?

I have been working on topological defects like monopoles, etc. for some time. One think that I have not been able to understand is the physical meaning of the phrase "topological object". I have ...
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2answers
88 views

What's the metric of the Standard Non-Time-Orientable Spacetime

If you've read any spacetime topology, you know that spacetime. It is the amazing rotating lightcone identified after half a rotation. And outside of De Sitter space with some identifications, it is ...
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1answer
76 views

Some questions about gauge theory

Let's talk quantum mechanics. I have a charged particle moving on a sphere. It has a wave function $\psi$. At time $t=0$, there is no magnetic flux piercing the sphere. Instantaneously, I introduce a ...
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56 views

Physical meaning of the Morse functions? [closed]

What is the physical correspondence of the Morse functions in a physical system? Currently I am studying Mirror symmetry but I can not get a physical intuition out of it.
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51 views

Why Dirac monopole is a topological defect in a $U(1)$ gauge theory? [duplicate]

How does $U(1)$ gauge group at long distances, give rise to magnetic monopoles? Also why is it said that Dirac monopole is a topological defect in a compact $U(1)$ gauge theory?
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1answer
61 views

Integrals of Chern class, $c_i$ in YM theories

I am a bit confused with the definition of the 1st (and 2nd by extension) Chern class in YM theories. I understand that in general $c_i \in H^{2i}(M,\mathbb{Z})$ where $M$ is a smooth manifold. Then, ...
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1answer
61 views

Relation between a change in the topological invariant and the closure of the gap

I would like to understand the relation between a change of the topological invariant (e.g. when the Chern number changes from 1 to 2) and the closure of the gap of a condensed matter system. I know ...
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1answer
61 views

Schwarzschild manifold

I am given the following metric $$ds^2 = \frac{dr^2}{1-2m/r} + r^2dS,$$ where $dS$ is the standard metric on the unit sphere $S^2$. I am told that this is isometric to $\mathbb{R}^3$ or (taking its ...
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41 views

Topological terms VEVs and ghosts

Suppose we have the Standard model, and we want to calculate with VEVs of topological susceptibilities of $SU_{L}(2), U_{Y}(1)$ and $SU_{c}(3)$ fields, which have the form $$ \tag 1 \kappa \equiv \...
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113 views

About the $Z_2$ topological invariant

In Kitaev 2001 it is shown that the topological invariant $Z_2$ in a topological superconductor (Class D or BDI, one dimensional) can be defined as $$ (-1)^\nu={\rm sign\, Pf} [ A ]={\rm sign\, Pf}[ \...
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1answer
119 views

What does $L^2(S^1,\mu_H)$ mean?

It's a Hilbert space, $\mu_H$ stands for the Haar measure on $U(1)$, but what does $S^1$ mean? I found it in one of my quantum mechanics books which approaches from a very 'mathematical' way.
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115 views

Examples of topologies that are not metric topologies with relevance in physics?

Can you give me examples of topologies, that are not metric topologies, that are relevant in physics?
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1answer
118 views

Why pseudo-Riemannian metric cannot define a topology?

It is not clear for me why a positive definite metric is necessary to define a topology as noted in some textbooks like the one by Carroll. Does this imply that in cosmology, say through FLRW metric, ...
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52 views

Squashed spheres in general dimension

The point of this question is to help me find references regarding squashed spheres in general dimension. I am interested in the general theory of squashing for arbitrary dimension. All of the ...
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What are the instances of usage of four color theorem in the theory of fractional statistics?

How important is four-color theorem (Hypothesis) in theory of Fractional Statistics?
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Topological susceptibility in QCD and corresponding pole

The topological susceptibility in QCD (here I've used path integral approach, and hence I will neglect all contact terms) is defined as $$ \kappa (p) \equiv \lim_{y \to 0}\int d^{4}x e^{ip(x-y)}Q(x)Q(...
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1answer
102 views

What is the relationship between a brane, a manifold, and a space?

I've read many ways to define manifold; one way is to define it as a type of mathematical space (a type of topological space to be exact). All of the definitions that I've seen for brane, on the ...
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28 views

Is there any reason (other than convenience) to assume the universe is paracompact?

In this discussion on MathOverflow, it is mentioned that the universe, being a Riemannian manifold, must be paracompact. But is there any reason to assume the universe is globally 'small enough'? In ...
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1answer
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Globally defined solutions in bc CFT system

Consider $bc$-system which is 2-dimensional CFT of fermions: $S = \int_\Sigma d^2 z \ b \bar{\partial} c + h.c. $ where $\Sigma$ - 2-dimensional manifold of genus $p$, fields $b, c$ have ...