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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How does GR determine the topology of spacetime? [duplicate]

The crux of GR is the action $$ S=\int _\mathcal M d^n x \sqrt{|g|}\,R $$ Varying this and setting $\delta S=0$ gives you the Einstein field equations. However, that only determines the metric, not ...
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Global angular forms

In the study of anomaly inflow due to an M5 brane (see for instance the paper by Freed, Harvey, Minasian and Moore), one regularizes the $\delta$ function near a source as $$\delta^{(5)} \rightarrow ...
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How to visualize a sphere bundle?

In the paper ``Gravitational Anomaly Cancellation for M Theory Fivebranes", the authors consider removing a tubular region of radius $\epsilon$ around the M5 brane (in order to make sense of the three ...
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Non-zero gravitational anomaly

Analogous to the Adler-Jackiw-Bell anomaly of QCD, we have an anomaly in gravity when we consider gravity to be coupled to chiral fermions: \begin{equation} \partial_\nu J^\nu_5\propto R\tilde{R}, \...
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about the Horizontal Lift in a Princiapal Bundle

I'm currently studying Fibre Bundle by Nakahara's book, and I'm a bit confused about the following: Imagine we have a Principal Bundle $P(M,G)$ with open chart {$U_i$} and a local section {$\sigma_i$}...
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Non-time orientable quotient of de Sitter space

Examples of non-time orientable spacetimes are pretty scarce, but it seems the big one is quotients of de Sitter space of the form $dS^n/\pi_1$, where $\pi_1$ is some subgroup of the isometries of de ...
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Two dimensional spacetime and the Gauss Bonnet theorem

Generally two dimensional spacetimes are deemed to be static, as the Gauss Bonnet theorem implies that the Einstein Hilbert action would be a constant independent of $g$. But as far as I can tell, ...
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143 views

Cones with deficit angle 2$\pi$ and euler characteristics

I've managed to confuse myself with cones and deficit angles. Let's consider a conical defect in 2 dimensions. So the metric is the usual one in polar coordinates, $$ ds^2 = dr^2 + r^2 d\phi^2,$$ ...
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Gromov-Witten invariants

I'm a mathematician studying Schubert calculus, and I'm out to compute the Gromov-Witten invariants of the complete flag manifold. Well, I actually already know how to compute them, but only in a way ...
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148 views

Resources for algebraic topology in condensed matter physics

I wanted to know if anyone had any good introductions on algebraic topology for the theoretical physicist? I am particularly interested in applications to condensed matter physics, but would be happy ...
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231 views

Homotopy proof of the lack of foliation of the Gödel metric

A common proof of the lack of foliation of the Gödel universe, apparently mostly copy pasted from Hawking and Ellis, goes thusly : A closed timelike curve must cross a spacelike hypersurface ...
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The universal covering group of a symmetry group [duplicate]

In Weinberg QFT Vol.1, it says one can enlarge the symmetry group $H$ to the universal covering group $C$ such that one obtains a trivial cocycle or $C$ is simply connected whereas $H$ is not. I get ...
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119 views

Gauge group topology

The fundamental difference between spinors and tensors is that spinors are sensitive to the homotopy classes of paths through the rotation group $SO(3)$: \begin{equation} \pi_1(SO(3)) = \mathbb{Z}_2, ...
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Topology of Anti-de Sitter manifold with black hole

I'm interested in understanding the topology of space-time with a black hole. In other words how does having a black hole affect quantities such as the fundamental group, de-Rham cohomologies, Euler ...
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78 views

What does $\mathbb{R}^3$ and $\mathbb{T}^3$ look physically for the Navier-Stokes equation?

What does the Navier-Stokes equation solution according to the Clay Math Institute look like in real life? As in how do you visualize $\mathbb{R}^3$ and $\mathbb{T}^3$ without the math? I actually ...
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56 views

What happens if locally manifold is seen as an Euclidean space? [closed]

I have been trying to understand the definition of a manifold and I have found out that the most common definition can be paraphrased as: A manifold is a space that has a complex "topology" globally ...
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1answer
46 views

Why don't closed strings's world-sheets have boundaries?

I have been told that the world-sheet described by a closed string is a world-sheet without boundaries. On the contrary, the world-sheet described by an open string has boundaries. I do see why the ...
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128 views

How can I derive Fusion-Rules for Anyons?

I am reading Pachos "Introduction to Topological Quantum Computation". Pachos writes that a model for anyons consists of a list of all anyons and a fusion rule for them. Given a model with anyonic ...
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52 views

Euler density of two-dimensional manifolds

I am asking this question after reading this post: What is Euler Density?. For a two dimensional manifold, the Euler density is given by: \begin{equation} E_2=2R_{1212} \end{equation} (note that $...
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81 views

Classical Field Theory Using Geometry

I would like to know if there are good introductory courses on Classical Field Theory taught in a differential geometry approach yet one doesn't need a background in those mathematical subjects but ...
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Concerning topology of BPS states of the M5-brane

My question is about the M5-brane in M-theory. I would like to know whether the BPS states of the M5-brane worldvolume theory (especially the 1/2 BPS and 1/4 BPS ones) are independent of the topology ...
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283 views

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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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?
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181 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 ...
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61 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 simple-...
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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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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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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. $M^n$ ...
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148 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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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 $l_{...
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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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44 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 ...
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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 ...
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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 ...
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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} - \frac{\lambda}...
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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 $L$ ...
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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 "...
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50 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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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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Are topological vacua of QCD Lorentz invariant?

Are topological vacua of QCD Lorentz invariant or they mix under boosts?
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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 ...
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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 ...
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3answers
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$SO(3)$ 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 ...
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1answer
155 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 ...
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165 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 $q\vec{r}/...
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282 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 ...
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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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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 ...
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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 ...