**1**

vote

**0**answers

275 views

### What is the characteristic functional for Brownian motion on a sphere?

I'm a physicist, somewhat familiar with stochastic processes, but I'm a little unsure of what follows. What I basically have is a complicated quantity involving a vector that is equivalent to ...

**6**

votes

**2**answers

1k views

### Mathematical study of Mpemba effect?

It has been known since the days of Aristotle and Descartes that under certain circumstances warm water freezes faster than cold water. This effect is now commonly known as the Mpemba effect, named ...

**0**

votes

**0**answers

145 views

### Metalinear frame bundle on sphere or $\mathbb{C}P^n$

Let $M$ be a smooth manifold. A complex metalinear frame bundle $\tilde F(P)\to M$ of a rank $n$ complex vector bundle $P\to M$ is a principal $ML(n,\mathbb{C})$-bundle together with a covering map ...

**0**

votes

**0**answers

160 views

### RG flow and Ricci flow

It looks like the Laplace operator in the nonlinear sigma model (say the Polyakov action) is different from the Laplace-Beltrami operator, how can one get the Ricci flow as a low order approximation ...

**0**

votes

**1**answer

163 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
...

**2**

votes

**1**answer

254 views

### Expected value of $(1 - X)^{-2} $ over Haar measure of the unitary group, $X \in U(N)$

Let $\lambda_1, \dots, \lambda_n$ be the eigenvalues of a random Unitary matrix. I am interested in the expected value:
$$\mathbb{E}_{X \in U(N)}\left[ \prod_{i=1}^n \frac{1}{(1 - ...

**1**

vote

**1**answer

279 views

### The space of holomorphic sections are finite dimensional?

I start my question with a definition and some motivation.
Let $M$ be a symplectic manifold. A subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex ...

**2**

votes

**1**answer

84 views

### An example to show that when $P$ is a complex polarization the subbundle $P+ \bar P$ is not necessarly involutive

I start my question with some motivation. A subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex polarization if
$P$ is Lagrangian
P involutive
dim$P\cap\bar ...

**-1**

votes

**1**answer

145 views

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

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

**5**

votes

**0**answers

217 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 ...

**7**

votes

**2**answers

268 views

### What is the definition of picture changing operation?

What is the definition of picture changing operation?
What is a standard reference where it is defined - not just used?

**5**

votes

**0**answers

149 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 ...

**2**

votes

**0**answers

259 views

### How to perform this matrix integral?

Edit: some backgrouds added.
In quiver matrix model which is reviewed DV or CKR, the path integral reduce to the matrix integral
$$Z \sim \int \prod_{i=1}^r d\Phi_i \prod_{<a,b>} dQ_{ab} ...

**0**

votes

**1**answer

223 views

### On the Geroch's argument

During the study of Geroch's argument to prove positive mass theorem, I faced a problem explained below:
Suppose $(M,g_{\mu \nu})$ is a four dimensional Lorentzian Manifold and $\Sigma$ is a ...

**2**

votes

**3**answers

231 views

### Positivity of the Coulomb energy in two dimensions

In dimensions $d\geq 3$ the Coulomb energy is always non-negative (since the Fourier transform of $\frac{1}{|\cdot|^{d-2}}$ is non-negative). What can one say about positivity properties of the ...

**2**

votes

**1**answer

199 views

### Probability measures on $L^p$

Let $(X,\mathcal X,\mu)$ be a fixed measure space, and suppose that $\mu$ is stationary and ergodic with respect to the (left) action of a topological group $G$. Stationarity means that $\mu = g_* \mu ...

**3**

votes

**1**answer

191 views

### Question about the Aganagic-Vafa A-brane

According to Aganagic-Vafa (hep-th/0012041) and Fang-Liu (arXiv:1103.0693), for a semi-projective toric Calabi-Yau 3-manifold $X$, the Aganagic-Vafa A-brane $L_{AV}\subset X$ is defined by the ...

**6**

votes

**1**answer

226 views

### Understanding the intermediate field method for the $\phi^4$ interaction

In Rivasseau's and Wang's How to Resum Feynman Graphs, on page 11 they illustrate the intermediate field method for the $\phi^4$ interaction and represent Feynman graphs as ribbon graphs. I had to ...

**1**

vote

**2**answers

518 views

### Uniqueness on square root of complex Line Bundle

Let $L$ be a line bundle over a compex manifold $X$, a square-root of $L$ is a line bundle $M$ such that $M^{\otimes2}=L$. My question is when the square-root of Line Bundle is unique?

**3**

votes

**0**answers

105 views

### Where is there a treatment of double field theory other than in local coordinates?

The n-lab seems to lack a treatment of double field theory. Where is there a treatment other than in local coordinates? Or at least one which identifies the coordinates as local coordinates for a ...

**0**

votes

**1**answer

251 views

### direct proof that schrodinger's equation kernel corresponds to delta-function initial value [closed]

I want to show directly, that the kernel for the n-dimensional free linear schrodinger equation, if taken to time t=0, is dirac's $\delta $ function. I can show that the integral is constant, but it ...

**4**

votes

**1**answer

376 views

### Quantization of symplectic vector space and choice of lagrangian subspaces

My question is related to Geometric Quantization.
I don't undrestand the philosophy of following assertion
If $(V,\omega)$ be a symplectic vector space then the quantizations of
$V$ ...

**2**

votes

**0**answers

193 views

### Looking for good conferences / workshops on applications of renormalization group methods [closed]

I am looking for conferences and/or workshops, where people working on different problems using renormalization group methods come together to share their results and experience.
As I have noted, ...

**1**

vote

**0**answers

115 views

### Brownian particles in a box: the probability that a sphere (of some radius) centered on a particle only contains one particle for a duration of time

Imagine I have a set of $(s_1,...,s_N) \in S$ Brownian particles in a box of sidelength $L$, each with the same coefficient of diffusion $D$. We fix one particle at the center of the box, and draw a ...

**28**

votes

**5**answers

973 views

### are there natural examples of classical mechanics that happens on a symplectic manifold that isn't a cotangent bundle?

I'm curious about just how far the abstraction to a symplectic formalism can be justified by appeal to actual physical examples. There's good motivation, for example, for working over an arbitrary ...

**3**

votes

**0**answers

115 views

### What does the renormalization group flow corresponding to a turbulent subrange with a broad band forcing look like?

In a renormalization group analysis of turbulent flows, such as for example done by Barbi and Münster here who derive an action for the Navier-Stokes equations, insert it into the Wilson equation, and ...

**3**

votes

**0**answers

210 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 ...

**0**

votes

**1**answer

277 views

### Symplectic structure on $Sym^kG^{\mathbb{C}} $

Let $G$ be a compact Lie group, and let $G^\mathbb{C}$ be its complexification.
I am looking for a symplectic structure (without use of coordinates) on
$$
Sym^kG^{\mathbb{C}},
$$
PS:Here ...

**2**

votes

**1**answer

355 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 ...

**4**

votes

**0**answers

213 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 ...

**1**

vote

**1**answer

351 views

### pre-quantization of Jet bundle

We know that the notion of Jet bundle $J^kM×\mathbb{R}$, is generalization of cotangent bundle. What is the prequantization of $J^kM×\mathbb{R}$?

**6**

votes

**0**answers

151 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 ...

**1**

vote

**0**answers

91 views

### H-flux by any other name

There are more than a few papers referring to H-flux and/or H-twist etc.
Is there anywhere a survey relating these variants?

**3**

votes

**1**answer

473 views

### a question about geometric quantization background

I don't understand why for geometric description of a regular system, we take always the classical phase space as a symplectic manifold?

**3**

votes

**0**answers

302 views

### A gentle introduction to CFT [closed]

1) Which is the definition of a conformal field theory?
2) Which are the physical prerequisites one would need to start studying conformal field theories?
(i.e Does one need to know supersymmetry? ...

**11**

votes

**4**answers

1k views

### Can the equation of motion with friction be written as Euler-Lagrange equation, and does it have a quantum version?

My (non-expert) impression is that many physically important equations of motion can be obtained as Euler-Lagrange equations. For example in quantum fields theories and in quantum mechanics quantum ...

**3**

votes

**1**answer

169 views

### Does fixing the reparameterization invariance of the string action correspond to some kind of orbifolding?

Does fixing the reparameterization invariance of the string action, for example by choosing the light-cone gauge
$$
X^{+} = \beta\alpha' p^{+}\tau
$$
$$
p^{+} = \frac{2\pi}{\beta} P^{\tau +}
$$
...

**4**

votes

**1**answer

88 views

### Understanding the diffraction limit in the context of being provided perfect information on an intensity distribution

As per http://scienceworld.wolfram.com/physics/AiryDisk.html, let the intensity distribution given by diffraction around a circular aperture be proportional to:
$I(r) \propto [\frac{J_1(r)}{r}]^2$
...

**5**

votes

**1**answer

413 views

### Why does closed string theory have only one dilaton field instead of $22$? [closed]

Looking at $5D$ Kaluza-Klein theory, the Kaluza-Klein metric is given by
$$
g_{mn} = \left(
\begin{array}{cc}
g_{\mu\nu} & g_{\mu 5} \\
g_{5\nu} & g_{55} \\
\end{array}
\right)
$$
...

**18**

votes

**1**answer

803 views

### Why is there a connection between enumerative geometry and nonlinear waves?

I'm not 100% sure that this question is appropriate for this site. If it's not, please tell me and I'll delete it.
Recently I encountered in a class the fact that there is a generating function of ...

**8**

votes

**5**answers

308 views

### Observables and dimensional analysis

Here is a simple question about physical units that I hope has a simple satisfying answer. In mathematically sophisticated treatments of both quantum and classical physics one often speaks of an ...

**1**

vote

**1**answer

177 views

### What's the asymptotic behavior of this function at large distance? [closed]

This question is based on some Physics motivation. Define a distance function $f(\mathbf{r})=\int_{\Omega }d^2k\int_{\Omega }d^2q \cos[(\mathbf{k}-\mathbf{q})\cdot\mathbf{r}]$, where ...

**7**

votes

**0**answers

233 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:
$$
...

**6**

votes

**0**answers

226 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 ...

**11**

votes

**1**answer

834 views

### Is this error in this paper of Langlands fixable?

The FQS criterion for the Virasoro algebra was discovered by Friedan, Qiu and Shenker (1), but the mathematicians found their proof insufficient, so that, FQS (2) and Langlands (3), published in the ...

**2**

votes

**1**answer

123 views

### Absent 2nd order terms in deformation quantization of Poisson manifolds

I am reading Kontsevich' famous paper on deformation quantization of Poisson manifolds. In section 1.4.2 on page 4 he gives the general formula for the star product associated to a Poisson structure ...

**2**

votes

**1**answer

365 views

### A question about flag variety of $SL(n,\mathbb{C})$

We know that the flag variety $SL(2,\mathbb{C})/B$ which $B$ is Borel subgroup, can be identified with $\mathbb{P^1}$, What can we say about $SL(n,\mathbb{C})/B$ which $B$ is Borel subgroup of ...

**4**

votes

**1**answer

198 views

### How does one calculate homotopy classes for group coset spaces?

Inspired by Witten's Wess-Zumino term arguments, I'm curious to know how one calculates homotopy classes more generally for coset spaces. In the above example the coset is $G/H=(SU(3)_L\times ...

**6**

votes

**1**answer

288 views

### Calogero-Moser system: relationship between dual variables and the KKS construction

This is a question about the relationship between two ways of viewing the Calogero-Moser system.
$$\ddot x_i=2\sum_{j\neq i}\frac{1}{(x_i-x_j)^3}\qquad i=1,\ldots N$$
By introducing the $N$ ...

**52**

votes

**1**answer

3k views

### What is the amplituhedron?

The paper ”Scattering Amplitudes and the Positive Grassmannian” by Nima Arkani-Hamed, Jacob L. Bourjaily, Freddy Cachazo, Alexander B. Goncharov, Alexander Postnikov, and Jaroslav Trnka, introduces ...