6
votes
0answers
125 views

Characterize spin cobordism invariants in dimer models

The paper by Cimasoni and Reshetikhin http://arxiv.org/abs/math-ph/0608070 shows that one can map problems about spin structures on a Riemann surface into problems about dimmer configurations on a ...
1
vote
0answers
97 views

Combinatorial Pin structure

David Cimasoni and Nicolai Reshetikhin have a paper on the combinatorial description of spin structure http://arxiv.org/abs/math-ph/0608070, where it shows the equivalence of spin structure to the ...
0
votes
0answers
110 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
1answer
157 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
1answer
141 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
0answers
202 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
1answer
334 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
2answers
282 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?
11
votes
2answers
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
1answer
332 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
0answers
339 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
2answers
790 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
1answer
295 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
2answers
410 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
2answers
342 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
0answers
207 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
5answers
656 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
0answers
135 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 ...
6
votes
2answers
638 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
4answers
631 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 ...
15
votes
7answers
3k 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 ...
22
votes
1answer
2k views

Math and Wormholes

Hopefully, MathOverflow is the correct place for this. I had a student approach me and ask me what kinds of mathematics goes into the study of wormholes. She specifically asked whether there is any ...
7
votes
1answer
525 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
2answers
581 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
2answers
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 ...
47
votes
1answer
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
1answer
648 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
5answers
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?