**68**

votes

**0**answers

5k views

### Dropping three bodies

Consider the usual three-body problem with Newtonian
$1/r^2$ force between masses. Let the three masses start off at rest,
and not collinear. Then they will become collinear a finite time ...

**21**

votes

**0**answers

979 views

### Correspondence between eigenvalue distributions of random unitary and random orthogonal matrices

In the course of a physics problem (arXiv:1206.6687), I stumbled on a curious correspondence between the eigenvalue distributions of the matrix product $U\bar{U}$, with $U$ a random unitary matrix and ...

**19**

votes

**0**answers

284 views

### Infinitely many planets on a line, with Newtonian gravity

(I previously asked essentially this on physics.stackexchange, but was actually
hoping for answers with something closer to a proof than what I got there.)
Suppose we have a unit mass planet at each ...

**13**

votes

**0**answers

389 views

### Catalan objects associated to a univariate polynomial

Given a monic degree $n$ polynomial $f(z)$ with no double roots, and a phase $0\leq \theta < \pi$, there are natural constructions which associate to this data:
a noncrossing matching on $2n$ ...

**11**

votes

**0**answers

325 views

### Which limit to take as a key applied math decision

The Borel-Kolmogorov paradox refers to situations where non-uniqueness in the notion of conditioning on a set of measure zero leads to apparent contradictions. As a formal matter, one requires ...

**11**

votes

**0**answers

685 views

### conformal blocks for beginners

I have given now a couple of talks that involve conformal blocks bundles on the moduli stack $\overline{\mathcal{M}}_{g,n}$, in front of a public of algebraic geometers but not specialists of the ...

**9**

votes

**0**answers

175 views

### “extended TQFT” versus “TQFT with defects”

There are two ways in which higher categories appear in topological field theory: in extended TFTs and in TFTs with defects. How are these appearances related?
According to the Atiyah-Segal axioms, a ...

**9**

votes

**0**answers

439 views

### The open problem of nth quantization

In trying to explain a quote by E. Nelson, "First quantization is a mystery, but second quantization is a functor!" Baez points out what follows (full text available in this week find; I'm also ...

**8**

votes

**0**answers

227 views

### State of the art of BPS and Donaldson-Thomas invariants for toric Calabi-Yau threefolds

I am trying to understand what has been done with regards to computing BPS invariants and Donaldson-Thomas type invariants of Calabi-Yau threefolds. To make the question more focused, let's say that I ...

**8**

votes

**0**answers

323 views

### Equivariant quantum cohomology and K-theory of vector bundles over a Grassmannian

Consider a sum of m copies of the tautological bundle over the Grassmannian of n-planes in complex k-dimensional vector space. There is an obvious action of an (m+k)-dimensional torus T on the total ...

**8**

votes

**0**answers

369 views

### Parametrisations for Null Temperature Functions: nonuniqueness of solutions to the Heat Equation

Disclaimer I expect this is a highly open problem, but maybe I'm wrong and someone has come up with some answers besides those given here. In any case, all information appreciated, thanks!
Definition ...

**8**

votes

**0**answers

689 views

### triangulated/derived categories in Physics and algebraic geometry

Why do physicists care about the triangulated/derived categories?
I mean what are the problems we want to approach using the machinery of triangulated/derived categories. e.g. in homological mirror ...

**7**

votes

**0**answers

126 views

### What's the appropriate notion of a Unitary representation of a Lie algebra?

Here Lie algebras/groups are real. The most straightforward definition might be:
Def: A representation $\rho:\mathfrak{g} \rightarrow \mathfrak{gl}(V)$ is unitary if $V$ is equipped with a Hermitian ...

**7**

votes

**0**answers

310 views

### Curvature as infinitesimal holonomy

Let $P \to M$ be a principal $G$-bundle, assume as much regularity as you want (compact $G$ or compact base manifold, ect). Via parallel transport, a connection $A$ on $P$ gives rise to the holonomy ...

**7**

votes

**0**answers

231 views

### The space-time dimension of the N-superstring theory?

Let $\mathfrak{W}$ be the Lie algebra generated by $d_{n} = ie^{in\theta}\frac{d}{d\theta}$ and $\mathfrak{Vir} = \mathfrak{W} \oplus C \mathbb{C}$ its central extension:
$$
...

**7**

votes

**0**answers

69 views

### Explicit expression of WZ term for orthogonal groups

Consider the Wess Zumino term on the the space $W=I\times D$, where D is a two dimensional disk disk and $I$ is an interval, $[0,1]$, say, i.e.,
$$
\int_{I\times D} \langle(u^{-1} \, du)^3\rangle
$$
...

**7**

votes

**0**answers

216 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

**0**answers

171 views

### Can one classify irreducible unitary representations of the Weyl algebra?

I saw in this MO post: Is there a machinery describing all the irreducible representations ? that classifying irreducible representations of the Weyl algebra is essentially intractable. My question is ...

**6**

votes

**0**answers

150 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 ...

**6**

votes

**0**answers

150 views

### Density of odd and even eigenstates of an integral operator

Consider an integral operator $(Kf)(x)=\int_{-1}^{1}K(x-y)f(y)dy$, where the kernel $K(-x)=K(x)$ is an even function.
Let $\lambda_n$ be the ordered eigenvalues of $K$ and $f_n(x)$ the ...

**6**

votes

**0**answers

219 views

### Are there exactly solvable CFTs?

I am wondering if there are CFTs such that n-point correlation functions in them of the fields (may be the primaries or of some notion of twist fields) is exactly known.
Are there such?
Aren't ...

**6**

votes

**0**answers

152 views

### Spectral theory for Dirac Laplacian on a funnel

I would like to study the spectral theory of the Dirac Laplacian on a non-compact quotient of the hyperbolic plane by a discrete group (I am particularly interested in the simple case where the ...

**6**

votes

**0**answers

348 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 ...

**6**

votes

**0**answers

255 views

### Birth-Death Process associated with Orthogonal Polynomials

I have read in various places the following objects are related:
orthogonal polynomials
birth-death processes
Lattice paths
continued fractions
After a lot of searching online, I found sketches ...

**6**

votes

**0**answers

782 views

### Harmonic maps into compact Lie groups

Consider locally minimizing harmonic maps from D-dimensional Euclidean space into a compact Lie group G. When $D=3$ the general regularity theory due to Schoen-Uhlenbeck,
Schoen, Richard; Uhlenbeck, ...

**5**

votes

**0**answers

177 views

### Noncommutative geometry and line length

I would like to understand, in some formal sense, the relation between the Dirac operator and the line length introduced by Connes in noncommutative geometry. If $D$ is the Dirac operator, he sets $ds ...

**5**

votes

**0**answers

94 views

### Stationary point processes with arbitrarily slow decorrelation

A point process $P$ (a probability measure on simple, locally finite point configurations $\mathcal{C}$ on $\mathbb{R}$ - I'm restricting to the one-dimensional setting) is stationary when ...

**5**

votes

**0**answers

73 views

### Short examples that are/are not quantum-ergodic

Are there any considerably short examples of manifolds that are/aren't quantum ergodic, or quantum unique ergodic?
Note that a (compact) Riemannian manifold is said to be quantum ergodic if ...

**5**

votes

**0**answers

210 views

### Elementary proof of lack of phase transition in Ising models with external fields

I have a question about the phase transitions in the Ising model in the presence of a (constant) external magnetic field. I will state the question on $\mathbb Z^2$ for simplicity. A definition of the ...

**5**

votes

**0**answers

146 views

### Energy barriers between Hadamard matrices

Hadamard matrices may be characterized as $n\times n$ real orthogonal matrices $U$ that achieve the lowest possible "energy" as defined by the (scaled and shifted) entry-wise 1-norm:
$$
E(U)=n^2 ...

**5**

votes

**0**answers

306 views

### unitary equivalence

Let $U$ be the bilateral shift operator in $l^2(Z)$, and let $V$ stand for a rotation on an irrational angle $\alpha$ in $L^2(T, \mu)$, where $T$ is a circle with a rotation-invariant Lebesgue measure ...

**5**

votes

**0**answers

154 views

### Given that a conditional measure is Gaussian, how bad can the original measure be?

Let $X$ and $Y$ be Banach spaces, and let $\varphi : X \to Y$ be a continuous linear map. Suppose that $\mathbb P$ is a probability measure on $X$ which satisfies the continuous disintegration ...

**5**

votes

**0**answers

379 views

### Jones Polynomial and Quantum Field Theory

I am trying Witten's paper but unable to re-produce the computations presented in the paper.
I tried few things on internet but all these tutorials explicitly don't show the calculations and refer to ...

**5**

votes

**0**answers

181 views

### Generalization of the non-existence of a monostatic planar body

Domokos, Papadopulos, and Ruina showed that there does not exist a convex planar rigid body of uniform density which has only
one orientation of stable equilibrium and one orientation of unstable ...

**5**

votes

**0**answers

141 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

**0**answers

144 views

### what books to read to quickly understand adiabatic approximation

Hi group, I'm a theoretical ecologist with fairly adequate training in applied math (ODE, linear algebra, applied probability, some PDEs). In my current work, I've encountered the use of adiabatic ...

**5**

votes

**0**answers

256 views

### Quantum dynamics on varieties: asymptotic quantum trace distance on SHL varieties

The Question Asked
Definition: the Second-Hand Lion trace distance $D_k$
Let $\mathcal{M}^{(kk)}_r$ be the set of $k\times k$ complex matrices of rank $\le r$ having unit trace norm. Then the ...

**5**

votes

**0**answers

446 views

### Lebesgue measure on Frechet space?

It is well known that there are no Lebesgue measures on infinite-dimensional Banach spaces (see e.g. http://en.wikipedia.org/wiki/There_is_no_infinite-dimensional_Lebesgue_measure). However, I ...

**4**

votes

**0**answers

79 views

### Ground State Degeneracy of 2+1D U(1) Chern Simons Theory?

I am a physics graduate student trying to understand more mathematical aspects of gauge theories.
How can I understand ground state degeneracy of a simple Chern Simons Theory: 2+1D U(1) $S= \int_M ...

**4**

votes

**0**answers

98 views

### First return time in an interval for N particles rotating on the circle at constant random speeds

Here is my problem: draw N velocities $v_1,v_2,\dots,v_n$ in $[-\pi,\pi]^N$ from some measure (Haar measure of uniform independent for simplicity) and make $N$ particles rotate around the circle with ...

**4**

votes

**0**answers

211 views

### Local version of a slice (for a Lie group action)

Let $\Upsilon: G \times M \to M$ be a smooth action of a Lie group $G$ on a manifold $M$.
Isenberg and Marsden (1982) define a slice at $m \in M$ as a submanifold $S \subseteq M$ containing $m$ such ...

**4**

votes

**0**answers

155 views

### Quantum Drinfeld-Sokolov reduction for a module

There is a well-established procedure for quantizing the Drinfeld-Sokolov reduction for an affine Lie algebra. In particular, this paper of de Boer and Tjin describes an algorithm to produce the ...

**4**

votes

**0**answers

143 views

### semiclassical proof of Wigner semicircle

In Terence Tao's discussion of the Gaussian Unitary Ensemble, he derives the Dyson and Airy kernels. The GUE is the probability distribution of the eigenvalues of a random Hermitian matrix.
\[ \int ...

**4**

votes

**0**answers

103 views

### Differences of Numbers of Helicity States in 4-dimensional Strings

The question whether the states in $D=2m + 2$ dimensional string theory,
which carry a representation of $SO(2m)$, span spaces which carry
representations of $SO(2m+1)$ seems hopelessly complicated.
...

**4**

votes

**0**answers

104 views

### Is there a maximal finite depth infinite index irreducible subfactor?

A subfactor $N \subset M $ is irreducible if $N' \cap M = \mathbb{C} $.
It's maximal if it admits no non-trivial intermediate subfactors $N \subset P \subset M $.
It's cyclic if its lattice of ...

**4**

votes

**0**answers

267 views

### Which orbits of a separable representation of the infinite unitary group are closed?

Consider a separable irreducible unitary representation of $U(\mathcal{H})$ in the Hilbert space $V$. Assume that $\mathcal{H}$ is separable. My question is the following:
Is it true that all ...

**4**

votes

**0**answers

438 views

### Open question: non-commutative site following Grothendieck, Quillen, Connes and Crane for quantum gravity.

This is an open question and it's to find out who is interested in this kind of thing, who can benefit from thinking about this. It is very brief but hopefully will only be unclear to people who are ...

**4**

votes

**0**answers

117 views

### FRT construction in the case of super algebras

I'm looking on papers which are talking about the super quantum algebra osp(2|1). I want to understand how one applies the FRT construction in the case of osp(2|1).
Of course there is a super ...

**4**

votes

**0**answers

341 views

### Correlation functions of complex operators

One defines the "scaling dimension" (as opposed to "engineering dimension") of an operator $\cal{O}$ as $[\cal{O}]$ such that if $\cal{O}(t^{-1}x) = t^{[\cal{O}]}\cal{O}(x)$ then the Lagrangian in ...

**4**

votes

**0**answers

173 views

### Bundle of Dirac (Elliptic) operator on $T^2$

I am a physicist with weak math background and am reading a paper (pdf) on the electronic structure of graphene(wiki) (a 2D carbon sheet with hexagonal lattice structure). In the paper they claim it ...