# Tagged Questions

**0**

votes

**0**answers

105 views

### When the leaves of a distribution are compact

Let $(M,\omega)$, be a symplectic manifolds. A subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex polarization if
$P$ is Lagrangian
P involutive
...

**0**

votes

**1**answer

153 views

### An example for the case tht if the leaves of polarization be non-compact then the polarized sections are not square integrable

Let $(M,\omega)$, be a symplectic manifolds. A subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex polarization if
$P$ is Lagrangian
P involutive
...

**-1**

votes

**1**answer

137 views

### $S^n$ admit a real polarization $D\subset TS^n$?

When the $n$-sphere, $S^n$,admit a real polarization $D\subset TS^n$

**3**

votes

**0**answers

197 views

### Looking for a necessary and sufficient condition for the polarization $\mathbb{P}$ being positive

My question is about positivity of polarization in Geometric quantization theory. Let $\mathbb{P}$, be a complex polarization on symplectic manifold $(M,\Omega)$. For every $m\in M$, we can define a ...

**2**

votes

**1**answer

320 views

### looking for an identity for higher jet bundle $J^kM$?

We know this fact that the first jet Bundle $J^1M$ is diffeomorphic with $T^*M×\mathbb{R}$.i.e,
($J^1M=T^*M×\mathbb{R}$)
Is there something like this identity for higher jet bundle $J^kM$?
I editted ...

**5**

votes

**2**answers

263 views

### Physical meaning of the integral cohomology condition in Souriau-Kostant pre-quantization?

The question is in the title. The form of the condition looks like the Bohr-Sommerfeld quantization formula of angular momentum, is there a link between the two formulas?

**10**

votes

**2**answers

1k views

### What is Kirillov's method good for?

I am planing to study Kirillov's orbit method. I have seen Kirillov's method in several branch of mathematics, for instance, functional analysis, geometry, .... Why is this theory important for ...

**7**

votes

**1**answer

312 views

### How unique is a conformal compactification?

I'm trying to understand the term "conformal compactification" which is often used in physics. I reckon that most places take this to mean a (sometimes specific) compact conformal completion. That is, ...

**6**

votes

**0**answers

331 views

### Would a closed universe with special relativity violate causality? Does the universe have to be simply connected?

This question may be more appropriate for physics.stackexchange.com, but it would be helpful to get feedback from experts in Minkowski geometry.
The classic twin paradox is a false thought experiment ...

**17**

votes

**2**answers

776 views

### How does hyperbolicity of space time affect our lives?

My main research has been in hyperbolic geometry and geometric group theory. I always thought that the only real "application" of my work was that the universe is a 3-manifold.
But recently I found ...

**6**

votes

**1**answer

285 views

### Relation between TQFT and Wilson lines, boundary conditions, surface defects etc

I have been studying (extended) topological quantum field theories (in short TQFTs) from the mathematical point of view and I have no background of the physics point of view. Sometimes I encountered ...

**1**

vote

**2**answers

388 views

### Can a sphere be a phase space?

Put in other words, given an even-dimensional sphere $S^{2k}$: is there a manifold $M$ such that $T^* M$ is diffeomorphic to $S^{2k}$?

**1**

vote

**2**answers

312 views

### Is there a lattice model of E8 manifold?

Background
I'm using physics terminology because I'm not sure what the right mathematical terminology is, perhaps a simplicial complex?
I'm interested, for various physics reasons, in four manifolds ...

**7**

votes

**0**answers

198 views

### What is known about the Yang-Mills stratification over 3-manifolds?

Rade proved in his thesis (Crelle's Journal, 1992, available here: digizeitschriften.de/dms/toc/?PPN=PPN243919689) that if $E\rightarrow M$ is a $U(n)$-bundle over a 3-manifold, then the gradient flow ...

**6**

votes

**4**answers

595 views

### Symplectic structures from Lagrangians?

In Witten's paper 'Quantization of Chern-Simons Gauge Theory with Complex Gauge Group,' he makes the statement (p. 35) that the symplectic form on the moduli space of flat $G_\mathbb{C}$ connections ...

**5**

votes

**0**answers

123 views

### BKS pairing in the SU(2) Chern-Simons theory

I know that usually, the way to compare the Hilbert spaces arising from $SU(2)$ Chern-Simons theory with different Kähler polarizations is via the Hitchin connection. However, it should be possible, I ...

**5**

votes

**2**answers

605 views

### Yang Mills gradient/heat flow on 4-torus

The classic Donaldson-Kronheimer book (Geometry of 4-manifolds) uses the Yang Mills gradient flow (sometimes called heat flow) on $M$ all over the place,
$\frac{d A}{dt} = -\frac{\delta YM(A)}{\delta ...

**3**

votes

**4**answers

612 views

### Where to start with research regarding maslov index/class

Hi,
I am a physicist and currently doing my bachelor thesis about geometric quantization.
In the book by Bates and Weinstein I encountered the Maslov index, which seems to be very important :-).
But ...

**13**

votes

**6**answers

2k views

### Topology of SU(3)

$U(1)$ is diffeomorphic to $S^1$ and $SU(2)$ is to $S^3$, but apparently it is not true that $SU(3)$ is diffeomorphic to $S^8$ (more bellow). Since $SU(3)$ appears in the standard model I would like ...

**7**

votes

**1**answer

514 views

### Interpreting Witten's Asymptotic Expansion of the WRT invariant.

Witten's asymptotic expansion conjecture as described in "Problems on invariants of
knots and 3-manifolds" in Geometry and Topology Monographs, Volume 4 states that
...

**23**

votes

**2**answers

576 views

### What is the right notion of self-dual (two-dimensional) percolation in R^4?

For a lattice in $\mathbb{R}^2$, if we include each edge independently with probability $p$ (i.e. bond percolation), it is well known that there is a critical probability $0 < p_c < 1$ depending ...

**2**

votes

**2**answers

289 views

### Name for the motion of an immersion?

I have an immersion of a 2-simplicial complex S in $\mathbb{R}^3$, and then a piecewise linear motion of that immersion over an interval of time [0,1].
Is there an existing name for the map ...

**45**

votes

**1**answer

4k views

### The mathematical theory of Feynman integrals

It is well known that Feynman integrals are one of the tools that physicists have and mathematicians haven't, sadly.
Arguably, they are the most important such tool. Briefly, the question I'd like to ...

**7**

votes

**1**answer

635 views

### Casson's invariant and the trivial connection contribution to witten's 3-manifold invariant

This question might turn out not to make any sense, but here it is: Witten's (and Reshetikhin and Turaev's) 3-manifold invariant can be "defined" as an integral over the space of connections on the ...

**24**

votes

**5**answers

5k views

### Poincaré Conjecture and the Shape of the Universe

Has the solution of the Poincaré Conjecture helped science to figure out the shape of the universe?