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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Homotopy group of the conformal group [migrated]

I would like to know which are the first three homotopy groups of the conformal group SO(4,2): $$ \pi_n(SO(4,2))=? \quad n=1,2,3 $$
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24 views

Can a balloon in fly without tearing change its surface topology? [on hold]

It is very well known that an inflated balloon in fly without tearing will not change its surface topology, the elastic deformations in its surface will be like "smooth transformations" in its surface ...
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1answer
43 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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Is there a Symmetry-Topology duality? [closed]

Both symmetry breaking and topology "create" information within the Universe. It seems that topology causally affects symmetry and symmetry causally affects topology. Can the two concepts co-exist as ...
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1answer
80 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 ...
3
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62 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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27 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
47 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 ...
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70 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
187 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 ...
3
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1answer
70 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
41 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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84 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
68 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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0answers
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
60 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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0answers
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
65 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
103 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
84 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 ...
3
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0answers
54 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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0answers
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
58 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, ...
5
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1answer
52 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
54 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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39 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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102 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
117 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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1answer
114 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
116 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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51 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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22 views

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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1answer
40 views

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 ...
3
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1answer
93 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
57 views

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 ...
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45 views

Do we always have particle-hole symmetry?

If we write the BdG equation for any model, and double the degrees of freedom (e.g. 4N*4N matrix for a N site chain), then we are guaranteed the particle-hole symmetry. Is there any constraints to do ...
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34 views

Topological susceptibility of electroweak theta-term

Suppose EW theory generating functional: $$ Z[\text{sources}] = \int D(A,\psi,\bar{\psi}, H,H^{\dagger})\text{exp}\bigg[i\int d^{4}x\bigg(-\frac{1}{4g_{EW}^2}F_{EW}^2 + \bar{\psi}(D - m)\psi + ...
12
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1answer
1k views

Equation of a torus

In the recent paper http://arxiv.org/abs/1509.03612, page 37. They say that a torus can be described by the equation $$y^2=x(z-x)(1-x)$$ where $x$ is a coordinate on the base $\mathbb{P}_1$. Could ...
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Three-dimensional flat bands and how to create them?

Consider a three-dimensional non-interacting system with translational invariance. It is easy to create a two-dimensional flat band by putting this system into a uniform magnetic field. The energy ...
3
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0answers
99 views

Homotopy and homology groups in physics [closed]

What is the connection between homotopy and homology groups and physics? When does one want or need to find invariants of manifolds in physics? Not interested in string theory. Narrow down how? Please ...
3
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0answers
51 views

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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14 views

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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50 views

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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44 views

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 ...
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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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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, $$ ds^2 = dr^2 + r^2 d\phi^2,$$ ...