All Questions
Tagged with dg.differential-geometry mp.mathematical-physics
290 questions
4
votes
1
answer
547
views
Question on Lorentzian geometry
I apologize in advance if this is a too basic question.
Let $(M,g)$ be a Lorentzian manifold with signature convention $(-,+,\dots,+)$. Now, lets suppose $X\in\Gamma(TM)$ defines a global time-...
- 1,171
1
vote
0
answers
117
views
Question on globally hyperbolic manifolds and coordinates
Consider a globally hyperbolic Lorentzian manifold $(M,g)$. Then, a well-known result of Bernal-Sánchez (see Theorem 1.1 in arXiv:gr-qc/0401112) states that it can globally be written as
$$M=\mathbb{R}...
- 1,171
9
votes
1
answer
174
views
Is there a relationship between fusion and S^1-equivariance for spinors on loop space?
A while ago (officially in 1987), Witten conjectured that string structures on a manifold $M$ correspond to "$S^1$-equivariant" (or more precisely $\mathrm{Diff}^+(S^1)$-equivariant) spin ...
- 1,969
6
votes
1
answer
334
views
Tensor component calculation
First of all, this question may be more suited for the Math stack exchange site. If anyone finds this question irrelevant here, please transfer to the relevant site.
Recall that in terms of Weyl and ...
- 319
0
votes
0
answers
134
views
Positive mass theorem and Seiberg-Witten equations
Apologies for not a very rigorous question. I came across this PhD thesis by XIAO ZHANG, a student of Yau. From the thesis:
"We also investigate some
basic facts on Spin$^c$ structure on $4$-...
- 954
1
vote
0
answers
170
views
Order isomorphism + manifold homeomorphism => path topology homeomorphism?
Suppose time-oriented spacetimes $(M_1 , g_1)$ and $(M_2, g_2)$ are homeomorphic under their manifold topologies $\mathcal{M}_1$ and $\mathcal{M}_2$ respectively.
Let's call this map $\phi: (M_1, \...
- 612
1
vote
1
answer
361
views
Can the Causal Structure recover the manifold topology for non-chronological spacetimes?
Given a time-oriented spacetime $(M,g)$, a binary relation $\ll$ can be defined on this spacetime where $p \ll q$ for $p, q \in M$ if and only if there exists a time-like path connecting $p$ and $q$.
...
- 612
3
votes
1
answer
367
views
Topology and local isometry, spinning cosmic string
Suppose one is given the spacetime $(M,g)$ where $M$ is a fixed differentiable manifold and $g$ is a Lorentzian metric whose local expression is:
$$g= -(dt + a \, d \phi)^2 + d\rho^2 + \kappa^2 \rho^2 ...
- 612
1
vote
0
answers
111
views
What is the "intrinsic reason" for the failure of Schwarzschild coordinates in general relativity?
It is well known that the Schwarzschild metric fails at r = 2M (in units where c = G = 1) and this is the result of choosing "bad" coordinates. I find this surprising because the coordinates ...
- 463
6
votes
0
answers
127
views
Is there a canonical smooth structure on tame Fréchet orbit type stratifications?
In finite dimension orbit type stratifications, it is known that the orbit space $M/G$ resulting from an action of a proper Lie Group $G$ on a smooth manifold $M$, satisfying a set of certain ...
- 808
5
votes
1
answer
368
views
Systems of (hyperbolic) 2nd order PDEs with lower order constraints
Certain surfaces in mechanics are endowed with the fundamental forms
\begin{align}
\text{I} &= \mathrm{d}u^2+\mathrm{d}v^2+2\cos\gamma\: \mathrm{d}u\: \mathrm{d}v \\
\text{II} &= \alpha\left(\...
- 134
3
votes
0
answers
126
views
On the linearized evolution equations in general relativity
The following puzzles me already for quite some time: In mathematical relativity, especially in the discussion of the Cauchy problem, one usually works in the so-called ADM-Formalism, in which one ...
- 1,429
1
vote
0
answers
99
views
How causal is a strongly causal purely electric spacetime?
Take a generic Lorentzian spacetime $(M, g)$ where $M$ is a time-oriented 4d manifold and $g$ is the Lorentzian metric that is strongly causal and purely electric.
According to this answer:
Is every ...
- 612
8
votes
0
answers
318
views
Flat Maurer-Cartan connection iff flat Berry connection
I am studying two connections on induced representation spaces $\text{Ind}_{H}^{G} \Gamma$, where $H \subseteq G$ are groups, and $\Gamma$ is an irrep of $H$.
The first is the canonical or $H$-...
- 227
3
votes
1
answer
238
views
1D topological defects in $d>3$ spatial dimensions
I am trying to construct a 1D topological defect solution in 4 spatial dimensions, i.e., a solution to some PDE (likely the equations of motion of some Lagrangian) on $\mathbb{R}^{4}$ which is ...
- 147
3
votes
0
answers
186
views
Properties of the stress energy tensor in Wightman formulation of CFT
In various papers that I have been reading about applying the Wightman axioms to conformal field theory, the authors write things like the following about the stress-energy tensor:
$$\int \mathrm{d}x^...
3
votes
0
answers
786
views
Is this set a manifold?
Take a general spacetime that is not strongly causal.
Call this spacetime $(M, g) $ where $M$ is a connected time-oriented manifold and $g$ is the Lorentzian metric that satisfies the Einstein's Field ...
- 612
0
votes
1
answer
525
views
Non-diffeomorphic but homeomorphic (under Lorentzian topology) Lorentzian manifolds
$\newcommand{\lorentzian}{\mathrm{lorentzian}}\newcommand{\lorentzian}{\mathrm{lorentzian}}\newcommand{\diff}{\mathrm{diff}}\newcommand{\manifold}{\mathrm{manifold}}$Take a time-oriented Lorentzian ...
- 612
1
vote
1
answer
265
views
Spin connection vs. Cartan connection
I am studying the tetradic Palatini formalism of general relativity. In this formalism, one usually considers a manifold $M$, which is either non-compact or compact with Euler-characteristic $\chi(M)=...
- 1,171
2
votes
0
answers
142
views
Naked curvature singularity vs Cauchy horizon in stably causal space-time
There is a result
that says (theorem 2.11) that any stably causal space-time $M$ is either a product $\Sigma\times \mathbb{R}$ or the time-like gradient $\nabla f$ of a time function $f:M\rightarrow \...
1
vote
0
answers
59
views
Number of divergence free symmetric two tensor in dimension 4 [duplicate]
In a $4$ dimensional (semi)-Riemannian manifold $(M^{4}, g)$, both Einstein tensor $G= \operatorname{Ric}(g)- \frac{R(g)}{2}g$ and stress-energy tensor $T$ symmetric and divergence-free. Is there any ...
- 319
3
votes
1
answer
333
views
Definitions fundamental forms and their geometric Intuition
Let $(M^{n+1}, g)$ be a Lorentzian manifold (spacetime) that contains a Riemannian/spacelike hypersurface $(\Sigma ^{n},h).$ Then we can define the second fundamental form of the hypersurface in many ...
- 319
34
votes
8
answers
6k
views
Applications of super-mathematics to non-super mathematics
Supergeometry and more broadly supermathematics has been around for few decades. Since its introduction by physicists, there has been an some mathematical interest in them.
Although interesting in its ...
4
votes
0
answers
189
views
Physical intuition for curvature on higher order frame bundles?
$\DeclareMathOperator\SO{SO}$A priori: I apologize if this isn't up to Mathoverflow standards, I've had very little luck getting questions on this subject answered elsewhere.
I'm looking for a physics ...
- 250
10
votes
1
answer
566
views
D'Alembert's Principle: rigorous formulation using notions from modern differential geometry
Is there a rigorous definition of D'Alembert's principle of virtual dynamic work in the language of differential geometry? Some questions I'm hoping to answer are:
How to view the configuration space ...
- 101
6
votes
0
answers
159
views
Nonlinear-PDE arising from flat conformal Chebyshev nets
Consider a flat, simply connected surface endowed with the Riemannian metric $g_0=e^{2\Omega(u,v)}\left(\mathbb{d}^2u +\mathbb{d}^2v \right)$, so that $\Omega(u,v)$ is an arbitrary harmonic function. ...
- 134
1
vote
0
answers
101
views
NSR superstring as a map of supermanifolds
On one hand, I know that the NSR superstring is described by a map $\Phi: \Sigma \to X$, where $\Sigma$ is a supermanifold with local coordinates $(\sigma,\theta)=(\sigma^0,\sigma^1 | \bar{\theta},\...
- 11
3
votes
2
answers
222
views
$2\mathrm{d}$ area maximizing short embeddings
Think of a beach ball on an pool of water or sand.
Let $\left(\mathcal{M}^2,g\right)$ be a surface homeomorphic to a sphere, endowed with a Riemannian metric $g$, and $\left(\mathcal{N}^2,h\right)$ a ...
- 134
2
votes
0
answers
114
views
Why does the solution to pendulum problem with the geometric approach of Jacobi metric does not correspond to the solution with Lagrangian approach? [closed]
When we solve the pendulum problem with EL equation, we get to the differential equation $\ddot{q}+\frac{g}{l}\sin q=0$
but when I apply the substitution $t \rightarrow t\sqrt\frac{g}{l}$ and ...
2
votes
0
answers
71
views
Covariant momenta associated to higher order Lagrangians
Let $\pi:Y\rightarrow X$ be a fibered manifold with fibered coordinates $(U,x^i,y^\rho)$ (whenever local calculations are needed) and $m$ dimensional base $X$ ($\dim X=m$).
Suppose that $L\in\Omega^m_{...
- 2,792
4
votes
0
answers
182
views
What is the natural framework for Lagrangians in QFT?
I wonder what is the natural geometric setting for Lagrangians in QFT, in the case of a general polynomial $P(\phi_i)$ of fields which could be scalars, or spinors etc:
Are there natural, geometrical ...
- 41
2
votes
0
answers
49
views
Lie group and symmetry concept for weak notions of surfaces
I am studying measure-theoretic and functional analytic notions of surfaces such as varifolds and, since my background comes from physics I am wondering whether there is a simiar concept such as Lie ...
- 21
3
votes
1
answer
258
views
Symplectic orbits in projective Hilbert spaces are simply connected
Let $G$ be a connected Lie group and let $(\pi, \mathcal{H})$ be an irreducible unitary representation of $G$ on an infinite-dimensional Hilbert space $\mathcal{H}$. Denote by $\mathcal{H}^{\infty}$ ...
- 131
7
votes
2
answers
282
views
In which dimensions is a strongly causal Lorentzian manifold determined conformally by its causal structure?
Let $M$ be a strongly causal Lorentzian manifold. If $M$ has dimension 4, a theorem of Hawking, King, and McCarthy (see Thm 5) says that $M$ is determined up to conformal isomorphism by its class of ...
- 63.9k
6
votes
0
answers
516
views
Yang–Mills existence and mass gap official statement on Euclidean $\mathbb{R}^4$, why not Minkowski $ \mathbb{R}^{3,1}$?
Yang–Mills existence and mass gap problem is officially stated by Clay Mathematics Institute:
Yang–Mills Existence and Mass Gap.'' Prove that for any compact simple gauge group G, a non-trivial ...
- 10.5k
2
votes
0
answers
111
views
Two identical objects circling the center of mass periodically in general relativity
In Newton's gravity we can have two identical objects circle the center of mass periodically (assuming the surroundings are vacuum).
Is something like this possible in general relativity? Is there an ...
- 31
2
votes
1
answer
182
views
Signs of curvatures of integrals lines of frames with constant principal values
Let $D\subset\mathbb{R}^2$ be a planar domain (maybe simply connected) and consider all the mappings $f:D\to\mathbb{R}^2$ with constant, fixed, positive singular values. Let $E=(E_1,E_2)$ be the ...
- 134
3
votes
0
answers
254
views
What are quantum extremal surfaces from a mathematical viewpoint?
It is said that they are surfaces which locally maximize area and bulk entanglement entropy. It would be great if I could receive some introductory material on it and some prerequisites to understand ...
2
votes
0
answers
132
views
Definition of trace in topological BF-theories
I very important example of topological field theories are "BF-theories", which are usually defined as follows: Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$ and let $\pi:P\to\...
- 1,429
5
votes
1
answer
406
views
Connection on a Hilbert bundle
Is there a well-defined notion of connection on a measurable bundle of Hilbert spaces?
- 103
4
votes
1
answer
899
views
The Yang-Mills Higgs Lagrangian
Let's say we have a principal bundle $(P,B,\pi;G)$ and associated bundle $E=P \times_{(G,\rho)}V$and $Ad(P)=P\times_{(G,Ad)} \mathfrak{g}$ the adjoint bundle. The Yang-Mills-Higgs action (without ...
- 247
5
votes
1
answer
372
views
Spin connection in the tetradic Palatini-formalism of general relativity
$\DeclareMathOperator\SO{SO}$I am trying to understand the tetradic Palatini-formalism of general relativity from a mathematical point of view. I am graduate student and quite new to mathematical ...
- 1,429
0
votes
0
answers
118
views
Dirac operator on a 5 dimensional tangent manifold with a $Spin(3)$-bundle
In p.3 of Witten paper from this Physics Letters B, Volume 117, Issue 5, 18 November 1982, Pages 324-328 Physics Letters B, 117(5), 324–328, he says that about the Dirac equation on a 5-dimensional
...
- 2,147
3
votes
0
answers
327
views
Discrete spectrum of Dirac operator
It is said that if we take the spacetime manifold to be a sphere $S^d$ of large volume so
that the spectrum of Dirac operator $$i\gamma^\mu D_{\mu}$$ is discrete.
For example at least for $d=4$, this ...
- 2,147
2
votes
0
answers
308
views
Maxwell $U(1)$ gauge theory's electric and magnetic sources turned on simultaneously in the classical differential geometry
Question:
How do we couple $U(1)$ electric (E) and magnetic (M) sources simultaneously in the classical differential geometry language, in a $U(1)$ gauge theory based on $U(1)$ gauge bundle and its $U(...
- 10.5k
0
votes
1
answer
160
views
Reference for action-angle coordinates [closed]
Does anyone know a good reference to start studying Action-Angle coordinates?
Thank you in advance !
- 97
2
votes
0
answers
98
views
Topological implications of curvature singularities
In popular articles on astronomy/physics, singularities are typically described as "holes or rips in the fabric of space".
Now algebraic topology has a lot of methods for detecting "...
- 675
12
votes
1
answer
579
views
Alternative approaches to topological QFTs
A while ago I read the paper 'Quantum Field Theory and the Jones Polynomial' by Edward Witten. This article uses a lot of concepts from physics like BRST symmetry and the Chern-Simons action which ...
- 5,146
5
votes
1
answer
382
views
Stabilizer groups of Yang-Mills connections
Let $G$ be a compact Lie group with complexification $G^c$, and consider a principal $G^c$-bundle $P^c \to M$ together with a reduction $P \subseteq P^c$ to $G$. Assume that $M$ is a Riemann surface.
...
- 5,824
6
votes
0
answers
154
views
What is Ryu-Takayanagi Entanglement Entropy?
I have a question about how to think about the Ryu-Takayanagi entanglement entropy mathematically.
For simplicity, let's work in the simplified setting of a time-symmetric slice of $AdS_4$ space -- i....
- 2,478