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Questions tagged [topology]

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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Why every projective representations of $SO(2,1)^{\uparrow}$ is in a one-to-one correspondence with linear representations of $SL(2,\mathbb{R})$? [closed]

We know that if $G$ is a connected group and $\tilde{G}$ its universal group with $H^2 (\tilde{\mathfrak{g}};\mathbb{R})=0$, then (by the Bargmann's theorem) every projective representation of $G$ is ...
Mahtab's user avatar
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Visualization of a gauge field with non-null winding number

In QCD you may add the term $\mathcal{L}_{\theta} = \theta\dfrac{g^2}{16\pi^2} \text{Tr}F\tilde{F}$, which turns out to be a total derivative. Now, it can be proven that the action of this lagrangian ...
Gabriel Ybarra Marcaida's user avatar
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Understanding topological censorship: is something wrong with this example?

Earlier today I was discussing with someone about the possibility of space (not spacetime) being a torus, and how this is different from a sphere. For simplicity, let us assume spacetime is of the ...
Níckolas Alves's user avatar
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Abelian Chern-Simons large gauge transform

My question concerns the $U(1)$ Chern-Simons theory with the action $$S = \frac{k}{2\pi}\int A\wedge \mathrm{d}A.$$ In my lecture, it is stated that: A large gauge transformation involves taking $A\...
shamwowexcitante's user avatar
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Anyons and Elementary particles in 2D

I'm doing my master's degree and I'm starting to learn more about Anyons. I want to understand more deeply why they can exist and how. I've done some research on the internet and found this question ...
Lucas Sievers's user avatar
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How to verify the compatibility condition for Berry's connection?

I am reading Mikio Nakahara's Geometry, Topology and Physics. In Chapter 10, he defines the Berry's connection one-form on the $U(1)$ bundle as $$ \mathcal{A}=\left\langle\mathbf{R}\right|d\left|\...
Zhicheng ZHANG's user avatar
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2 answers
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Solutions to Maxwell's equations with $dF=0$ but $F \neq dA$ -- can the new solutions be summarized by considering only the vacuum equations?

I am trying to learn a bit about differential forms. I saw a question and answer noting that the homogeneous Maxwell equations can be written as $dF=0$. However, as noted there, depending on the ...
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How to numerically calculate Zak phase for SSH3 model?

The k-space hamiltonian of SSH3 model with nearest neighbour hopping is given by H(k)= \begin{bmatrix} 0 & u & w e^{-ika} \\ u & 0 & v \\ w e^{ika} & v & 0 \end{bmatrix} ...
SUMANTA SANTRA's user avatar
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Is the Berry phase at a degeneracy trivial or non-trivial?

This question is about the original paper on Berry phases by M. Berry (1984). There, the Berry phase $\phi_{B}$ is defined as $$ \phi_{B}= \oint_\mathcal{C}\underbrace{ \left\langle n(\mathbf{R}) \...
xabdax's user avatar
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Is there a true one-dimensional object? [closed]

I'm reviewing and expanding my knowledge of dimensions. We live in three spatial dimensions but, apart from volume, we also have the concept of surface and curve. However, if you write a line on paper,...
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2-dimensional connected Lorentz group [closed]

Consider the connected Lorentz group $SO(1,1)^{\uparrow}$. I was wondering if someone could help me about showing that $SO(1,1)^{\uparrow}\cong \mathbb{R}\times \mathbb{Z}_2$. I just need a hint.
Mahtab's user avatar
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What is the manifold topology of a spinning Cosmic String?

Given the following metric which is that of a rotating Cosmic String: $$g=-c^2 dt^2 + d\rho^2 + (\kappa^2 \rho^2 - a^2) d\phi^2 - 2ac d \phi dt + dz^2.$$ can one determine the manifold topology ...
Bastam Tajik's user avatar
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Black holes, singularities and topology in relativity

General relativity is defined on a base manifold which, viewed as a topological space, is simply connected (which means there's no holes). However, we know that inside a black hole there's a ...
Tomás's user avatar
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Original source "GF92" for two drawings of Dirac belt trick

I am trying to track down the original source/artist of these two drawings of the Dirac belt trick (see link below) to use in my thesis (which is in mathematics, but I believe these pictures likely ...
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Tenfold way symmetry classification for systems with pseudomomentum

For classifying Hamiltonians $H(\vec{k})$ of topological insulators/superconductors in the tenfold way, one has to see whether the Hamiltonians obeys (disobeys) symmetries of the following type (let's ...
Dave Force's user avatar
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How does time-reversal symmetry breaking work in a Weyl crystal?

In a Weyl crystal, we can break the TRS (Time-Reversal Symmetry) by introducing FM or AFM atoms, and this breaking occurs due to the emergence of spontaneous magnetization in the material (I think), ...
Gabriel Elyas's user avatar
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1 answer
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Preferred state of motion from topology of certain spacetimes

A (1+1)-dimensional spacetime where the spatial dimension wraps around on itself (so it has the topology of $E^1\times S^1$) has a preferred state of motion, even though it is everywhere locally ...
Matt Dickau's user avatar
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1 answer
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Is there a topological invariance/winding number for non-translation invariance system?

My question is: Is there a topological invariance/winding number for non-translation invariance system? For example, if we modify the interacting parameter in SSH model, such that it depends on the ...
feng lin's user avatar
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The Lebesgue covering dimension of the Cosmic String interval topology

Take the spacetime $(M,g)$ that satisfies Einstein's Field Equations exactly where $g$ is locally: $$g= - c^2 dt^2 + d \rho^2 + (\kappa^2 \rho^2 - a^2) d \phi^2 - 2 ac d\phi dt + dz^2 \ $$ in the ...
Bastam Tajik's user avatar
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2 votes
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How can I construct a projective representation when the group is not simply connected?

S. Weinberg, in his book "The quantum theory of fields", states this theorem (page 83): The phase of any projective representation $U(T)$ of a given group can be chosen so that $\phi =0$ if ...
Mahtab's user avatar
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1 answer
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Why future infinity have no future end points?

I am studying Hawking's area theorem from the book the large scale structure of spacetime by Hawking and Ellis. At the end of page#318, it said: null geodesic generators of future infinity have no ...
Talha Ahmed's user avatar
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How does null infinity differ from ordinary infinity?

Null infinity is the diagonal lines on the edge of a Penrose diagram. It seems to be the place where beams of light go if they never bump into anything, but only light can go there. It appears to be ...
L Turner's user avatar
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Could the universe have a form of a $T^3$-torus?

Cosmological measurements suggest that we live in a flat universe. However, what might be less clear is its topology. So could the flat universe have the form of a $T^3$-torus, i.e. the torus whose ...
Frederic Thomas's user avatar
2 votes
0 answers
134 views

Einstein's gravity Lagrangian invariance under the change of differential structure

I came across an article claiming the appearance of singularities in the energy-momentum tensor $T_{\mu \nu}$ as a result of changing the differential structure: I wonder what symmetry or current (in ...
Bastam Tajik's user avatar
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-4 votes
2 answers
123 views

What model tells us there is nothing outside the universe?

Is there an existing model or theory that shows there is nothing outside of the universe that interacts with anything inside the universe? Or to put it in other words, is there a model or theory ...
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3 votes
1 answer
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Divergence of the Berry connection

Given the Berry connection \begin{equation} \boldsymbol{\mathcal{A}}(\mathbf{R}) = i \langle u(\mathbf{R}) | \nabla_\mathbf{R} | u(\mathbf{R}) \rangle, \end{equation} the Berry curvature can be ...
Lucas Baldo's user avatar
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How is the coupling between order parameter and strain determined from symmetry?

In the article Ehrenfest Relations for Ultrasound Absorption in Sr2RuO4, Sigrist M., Progress of Theoretical Physics, Vol. 107, No. 5, May 2002 A superconductor with proposed p-wave pairing and ...
Nitzan R's user avatar
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Conformal symmetry and group in arbitrary dimensions [duplicate]

As far as i understand, the full symmetry of relativity is conformal symmetry. This is represented by the conformal group $ \operatorname{Conf}(1, 3) $ Of Minkowski spacetime which is $ \mathbb{R}^{1, ...
Tomás's user avatar
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3 votes
2 answers
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Why radial quantization gives different spectrum?

For example we work with 1+1D massless free boson, in canonical quantization we allow creation operators at any momentum so the Hamiltonian has continuous spectrum. But if we conformally map to a ...
Peter Wu's user avatar
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JT gravity metric - solution to the dilaton equations of motion

I am reading Closed universes in two dimensional gravity by Usatyuk1, Wang and Zhao. The question is not too technical, it is about the solutions to the equations of motion that result from the ...
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Understanding Exceptional Points

Exceptional points occur generically in eigenvalue problems that depend on a parameter. By variation of such parameter (usually into the complex plane) one can generically find points where ...
ZHENGYAO HUANG's user avatar
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Are there reasonable models of Earth's surface as the space $\mathbb{R}P^2$? [closed]

Everyone knows that the Earth's surface is a 2-sphere, or a geoid. Flat earthers propose that Earth's surface is a disk or some variation of that, and there is lots of discussion on why its not true ...
Rumpelstiltskin's user avatar
2 votes
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Examples of spacetimes that are asymptotically flat at future timelike infinity

There are interesting non-trivial examples of spacetimes which are asymptotically flat at null and spacelike infinities. For example, the Kerr family of black holes satisfies these conditions. However,...
Níckolas Alves's user avatar
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What is the Chern number of twisted bilayer graphene without hBN substrate?

I am approaching at the study of topological materials and I need help to understand the role of Berry curvature in the twisted bilayer graphene. The raise up of the Moire lattice and consequently the ...
Andrea Zacheo's user avatar
9 votes
1 answer
792 views

Mathematical anatomy of general relativity

I was always told that spacetime in general relativity was a Lorentzian manifold, that is, a Pseudo-Riemannian manifold $ (M, g) $ with metric signature $(+, -, -, -)$ or $(-, +, +, +)$ and that that ...
Tomás's user avatar
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2 votes
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Reason to consider only compact world-sheets in string theory

Generally speaking, the "sum over world-sheets" in string theory involves summing over all possible topologies of compact, orientable and connected, as Polchinski says in page $100$ of his ...
Генивалдо's user avatar
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1 answer
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How can we design the global structure of the phase space?

I want to know how to design a classical mechanical system that has a phase space $M$ with a nontrivial global topology. If I naively consider a system in which the generalized coordinate $q_1,\cdots,...
watahoo's user avatar
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Non-Abelian anomaly: why does non-Hermitian operator have complete basis of eigenvectors?

In section 13.3 of his book [1], Nakahara computes the non-Abelian anomaly for a chiral Weyl fermion coupled to a gauge field by making use of an operator $$ \mathrm{i}\hat{D} = \mathrm{i}\gamma^\mu (\...
xzd209's user avatar
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Relationship between holonomy and fundamental group

In my notes of topological QFT we demonstrated that the holonomy associated with a path in $\mathbb{R}^3$ is invariant under smooth deformation of the path if the connection is flat. Then I wrote: If ...
polology's user avatar
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Is there any explicit theoretical application of Brouwer's Topological Invariance of Dimension theorem?

I'm interested in applications of Brouwer's Topological Invariance of Dimension theorem. I study mathematics but know very little about physics, but I imagined that the Invariance of Dimension theorem ...
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1 answer
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Holes in spacetime

Suppose a take a standard 1+1 Minkowski spacetime. Then I "make a hole in it", in the sense that I remove set of points that satisfy $x^2+t^2 \leq 1$ in some inertial frame. Can the ...
Nick Ormrod's user avatar
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2 answers
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How does symmetry act on general case fermionic operator?

I am trying to understand symmetries and how they work in condensed matter physics to understand some concepts from topology. In general second-qunatized Hamiltionian can be written in the following ...
IhateDonuts's user avatar
1 vote
1 answer
171 views

Understanding Chern-Simons on non-trivial manifold

I am studying abelian Chern-Simons theory on a non-trivial manifold. Could you let me know how accurate my understanding is? Here's what I figured out: The action of $U(1)$ leaves the action invariant ...
polology's user avatar
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de Sitter spacetime, spacetime without boundaries and holographic principle

One of the most intriguing aspects of black hole thermodynamics is that of holography or the holographic principle, namely, that the degrees of freedom of a putative theory of quantum gravity is ...
riemannium's user avatar
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1 vote
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Are loops allowed for paths in the path integral formulation?

In Wikipedia is stated that the quantum-mechanical paths are not allowed to selfintersect (create loops). (It’s stated in the description of the first figure of five paths.) On the other hand, loops ...
Xhorxho's user avatar
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Are there applications of quandles in quantum theory?

I have learned from a knot theorist that quandles were developed in order to more sensitively distinguish between distinct knots. However, they seem to be an interesting algebraic object in themselves....
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How big was the surface of the cosmic background radiation?

The cosmic microwave background radiation is the furthest and oldest visible light in the universe. But the universe has expanded considerably since that light was emitted. At the time that that light ...
John Berryman's user avatar
3 votes
1 answer
135 views

Why can Principal $G$ Bundles be Trivialized when $G = SU(N)$?

Reading about TQFT one usually comes about the fact that over 3-manifolds, Simply Connect Lie Group-bundles can be trivialized, yet it is a bit hard to find a clear answer online. Why is that the case?...
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Null infinity reachable by timelike worldlines?

Usually, Penrose diagrams are marked with points and segments being named past/future timelike infinity $i^{-,+}$, past/future null infinity $\mathscr{I}^{-,+}$ and spacelike infinity $i^0$ -- see for ...
Octavius's user avatar
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3 votes
1 answer
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Quantization of charge from the path integral

Consider a complex scalar field, with the usual Lagrangian: $$ \mathcal{L} = | \partial_{\mu} \phi|^2 - V(|\phi|^2). $$ This theory has a $U(1)$ symmetry, $\phi \to e^{i \alpha} \phi$, and the ...
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