# Tagged Questions

**9**

votes

**1**answer

357 views

### Vector bundles, Higgs bundles and the Langlands program

This question is somewhat vaguely structured. But, I hope someone can make it more precise (or) it is indeed possible to answer it in the form that I am stating it.
Background : I recently chanced ...

**5**

votes

**1**answer

273 views

### Is there a specific geometric meaning why fractional charges are allowed in SU(N) gauge theories?

So in the standard model of particle physics, there exist particles with fractional charge. What this means geometrically is as follows: We are given a smooth manifold with a principal $U(1)$ bundle ...

**3**

votes

**1**answer

210 views

### Do cyclic product vectors generatating irreducible representation of a Lie group come from a unique orbit?

Consider a Hilbert space $\mathcal{H}$ which is a carrier space of a unitary, irreducible and strongly continuous representation $\Pi$ of a Lie group $G$. Let $\Pi\otimes \Pi$ denote the corresponding ...

**4**

votes

**1**answer

105 views

### Closed form for 3j-symbol ratios

I am working on the spherical harmonic decomposition of cosmic microwave background maps, therefore I often deal with functions that are proportional to Wigner 3J symbols/Clebsch–Gordan ...

**1**

vote

**0**answers

64 views

### What are the differences and relations between R matrices solutions of Quantum Yang-Baxter equations and set-theoretical solutions of QYBE?

What are the differences and relations between R matrices solutions of Quantum Yang-Baxter equations and set-theoretical solutions of QYBE? Is it possible to write set-theoretical solutions of Quantum ...

**1**

vote

**0**answers

179 views

### quantization for integral coadjoint orbits

I am looking for a referrence for proof of following statement.
Let $G$ be a semi-simple Lie group and $\mathfrak{g}$ be its Lie algebra and $\alpha_k$ be simple roots of $\mathfrak{g}$ and $\mu_0$ ...

**10**

votes

**3**answers

402 views

### Resource for learning quantum mechanics from the viewpoint of representation theory

Quantum mechanics is deeply connected with representation theory. Therefore, I'm looking for a textbook or article which presents quantum mechanics in a representation theoretic manner. Could anyone ...

**2**

votes

**1**answer

241 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 - ...

**7**

votes

**0**answers

224 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:
$$
...

**5**

votes

**0**answers

207 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

819 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

**0**answers

83 views

### The condition of maximality in branching rules of $SO$ group representations

Let the highest weight of a $SO(2n+1)$ representation be given as $(m_1,m_2,...,m_n)$ ($m_1\geq m_2 \geq .. \geq m_n \geq 0$) and the highest weight of a $SO(2n)$ representation be $(s_1,s_2,...,s_n)$ ...

**4**

votes

**1**answer

302 views

### About using the character formula for $SO(2n)$.

I have known of the following equation for characters of a $SO(2n)$ representation with highest weights $(h_1,...,h_n)$ and for $(t_1,t_2,..,t_n,t_1^{-1},t_2^{-1},..,t_n^{-1})$ being the eigenvalues ...

**10**

votes

**2**answers

390 views

### the spectrum of the Laplacian and Dirac operator on $S^3$

A paper on supersymmetry in 3-dimensions uses results on the spectra of elliptic operators on $S^3$:
The eigenvalues of the vector Laplacian on divergenceless vector
fields is $(\ell + 1)^2$ ...

**3**

votes

**0**answers

136 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 ...

**3**

votes

**0**answers

148 views

### $\mathbb Z/2$-orbifolds in Virasoro representations, CFTs, VOAs

Suppose that ${\rm Vir}_c$ is a rational Virasoro algebra with central charge $c$.
Then ${\rm Vir}_c$ has finitely many irreducible modules $M_h$, parametrised by the highest weights $h$.
Furthermore ...

**3**

votes

**2**answers

238 views

### Integration over special unitary group

It is known that for $SU(N)$
$$
\int \chi_{\mu_1}(UV_1)\chi_{\mu_2}(U^{-1}V_2)\, dU = \delta_{\mu_1\mu_2}\frac{\chi_{\mu_1}(V_1V_2)}{\dim(\mu_1)}
$$
where $dU$ is Haar measure on $SU(N)$ normalized ...

**8**

votes

**1**answer

249 views

### Is there a version of supersymmetry for homogeneous spaces?

The notion of "supersymmetry" that I am aware of proceeds as follows. One fixes a spacetime $\mathbb R^n$ and signature; I will write $\mathrm{SO}(n)$ for the corresponding group of orthogonal ...

**4**

votes

**0**answers

85 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.
...

**0**

votes

**0**answers

110 views

### Semi-Standard Young Tableaux: Do Diagrams for $O(2m)$ combine to Diagrams from $O(2m+1)$?

Let $n_\lambda^K$ be the number all semi-standard Young tableaux of size
$K$ with Ferrers diagrams diagram $\lambda$
(i.e. the number of all fillings of $\lambda$ with natural numbers with weakly ...

**4**

votes

**1**answer

483 views

### Dijkgraaf-Witten TQFT vs. Representation Theory?

From what I had read, group characters can be "glued" together in a topological fashion and there is something to this effect in the paper by Dijkgraaf and Witten. TQFT seems to be a topological ...

**2**

votes

**2**answers

420 views

**1**

vote

**0**answers

194 views

### How to correctly name “irreducible subrepresentation of an indecomposable representation”

I am studying the representation theory of some infinite dimensional algebras, for example infinite dimensional Lie-algebras, Kac-Moody algebras, W-algebras. These algebras arise as symmetry algebras ...

**4**

votes

**1**answer

161 views

### q-deformed group characters

In a paper by Yuji Tachikawa, I found a q-deformed "2d Yang-Mills paritition function for a cylinder". Here it is (adapted):
$$ Z(q, x_L, x_R) = \mu(q, x_L)^{-1/2} \langle x_L | \bigg\[
\sum_{R ...

**4**

votes

**1**answer

190 views

### Well defined Tensoring of spectral triples

Hi,
I have a misunderstanding that I am hoping is really quite trivial. I will give my question directly and context below for those that need/want it.
Question: In connes standard model he takes ...

**8**

votes

**1**answer

355 views

### A q,t-extension of Plancherel Measure thru Yang-Mills Theory ?

Buried in the physics paper by Nekrasov and Okounkov, a strange identity is proven:
$$ \prod_{n > 0} (1 - q^n)^{\mu^2-1} = \sum_{\mathbf{k}} q^{|\mathbf{k}|} \prod_{\square \in k} \left( 1 - ...

**2**

votes

**1**answer

510 views

### Conformal Killing spinors

In general I would like to know about the significance of conformal Killing spinors (especially keeping in mind supersymmetric theories on curved space-time).
If $\epsilon$ and the $\bar{\epsilon}$ ...

**7**

votes

**4**answers

340 views

### Hamiltonians which commute both as operators and as connections

This is something which I suspect is written up in introductory books on mathematical physics if I knew where to look. Suppose I have some parameters $t_1$, ..., $t_k$ ranging over a neighborhood in ...

**6**

votes

**4**answers

442 views

### level 2,3 characters of affine su(2)

Does anyone know where I can find an explicit formula to compute the level 2 or level 3
characters of affine $su(2)$? I have found several sources that give a formula to compute the
level 1 ...

**5**

votes

**1**answer

472 views

### Is there a generalization of Schur - Weyl duality and plethysm for direct product of special unitary groups?

Consider the semisimple compact group $K=SU(N_1)\times SU(N_2) \times \ldots \times SU(N_S)$ acting naturally on $\mathcal{H}=\mathcal{H}_1 \otimes \mathcal{H}_2 \otimes \ldots \otimes \mathcal{H}_S$, ...

**1**

vote

**1**answer

509 views

### very very basic question on semi-simple Lie algebras

I have a very basic question on Lie algebras. I'm doing particle physics, and a lot of emphasis seems to be placed on the weight diagrams of simple Lie algebras. But these simple Lie algebras are ...

**8**

votes

**1**answer

643 views

### Symmetric tensor product of bosonic/fermionic Hilbert space

Consider two representation of the group $SU(n)$: $Sym^k(\mathbb{C}^n)$ and $\wedge^k\mathbb{C}^n$ ($k\leq n$) and take their symmetric tensor products: $Sym^2(Sym^k(\mathbb{C}^n))$, ...

**1**

vote

**2**answers

569 views

### Why the Gell-Mann matrices in the SU(3)-model need to be trace orthogonal ?

Thank you Cristi Stoica for your answer to the previous post of this question. Your hint is to the point I think. We should look at the requirements to construct the corresponding root system.
My ...

**0**

votes

**1**answer

976 views

### Why the Gell-Mann matrices in the SU(3)-model need to be trace orthogonal?

Why the Gell-Mann matrices in the SU(3)-model need to be trace orthogonal?

**2**

votes

**2**answers

2k views

### What is the structure of $SO(3)$ and its Lie Algebra? [closed]

First I want to give you some background how the question arised, before actually asking it.
Recently, in the context of quantum mechanics, I thought about the group $SO(3)$ and its Lie Algebra ...

**3**

votes

**3**answers

765 views

### Fractional Quantum Hall Effect - Mathematics

Just to include something that starts to answer my own question Topological Quantum Computation Lecture notes covers a lot of the Mathematics of the Fractional Quantum Hall effect, or topological ...

**6**

votes

**1**answer

731 views

### Conformal Field Theory and Langlands

I'm a Mathematics masters student currently
studying some aspects of TQFT. I'm interested in Langlands, mainly
because it sounds oppressive! Is anyone familiar with any links between
CFT and ...

**7**

votes

**1**answer

503 views

### Spinor space to Euc. vector space: does there exist a universal bilinear map?

Let $S$ be a spin representation of the Euclidean
spin group $Spin(d)$ and let ${\mathbb R}^d$
be Euclidean $d$-space with $Spin(d)$ action on it
in the canonical way, via the 2:1 cover to $SO(d)$.
...

**15**

votes

**7**answers

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 ...

**2**

votes

**1**answer

220 views

### Characterization of the weight orbit in the projective space via second order Casimir.

This is the spin-off of the question I previously asked.
First, let me remind you some notation from that question:
$G_0$ - compact, simply connected Lie group giving rise (by complexification) ...

**1**

vote

**2**answers

621 views

### Sum relation for Clebsch-Gordan-Coefficients?

In the context of (numerically) calculating reduced density matrices in the Lipkin-Meshkov-Glick model (a model introduced to describe atomic nuclei, which has however found many other applications as ...

**3**

votes

**1**answer

670 views

### highest weight orbit characterization (reformulated and extended )

Edit 1: I think that the question was not stated clearly enough so modified it a little
Edit 2:
I thought over the physics that lies behind this question which led me to reformulation of the original ...

**6**

votes

**2**answers

474 views

### Killing form vs its counterpart in a given represenation

Let $\mathfrak{g}$ be a semi-simple Lie algebra and let $\phi:\mathfrak{g}\rightarrow\mathfrak{gl}(V)$ be its finite-dimensional complex irreducible representation. You can define two non-degenerate ...

**20**

votes

**3**answers

993 views

### What was Casimir's precise role in describing the center of the universal enveloping algebra of a semisimple Lie algebra?

This question is prompted by a recent MO question on explicit computations of Weyl group invariants for certain exceptional simple Lie algebras:
37602. Like some others who started graduate study in ...

**3**

votes

**0**answers

201 views

### Characters of Kac-Moody algebra from orbifold

In a Wess–Zumino–Witten model on some Lie group G, the character of a particular integrable representation is the same as the specialized character from the corresponding Kac-Moody algebra. Suppose ...

**5**

votes

**1**answer

831 views

### Generalization of Schur's lemma (Update)

I am not a mathematician nor physicist. I just know the basics of the representation theory. In my research, I realized that there is an orthogonality relation between the unitary group matrix ...

**4**

votes

**2**answers

576 views

### Categorifications of the Fibonacci Fusion Ring arising from Conformal Field Theory

I was reading about realizations of the "Fibonacci" fusion ring $X \otimes X = X \oplus 1$ in Fusion Categories of Rank 2 by Victor Ostrik. Apparently, there are two of them and they arise in various ...

**22**

votes

**6**answers

2k views

### Examples of applications of the Borel-Weil-Bott theorem?

In "Quantum field theory and the Jones polynomial" (Comm. Math. Phys. 1989 vol. 121 (3) pp. 351-399), Witten writes:
A representation Ri of a group G should be seen as a quantum object. This ...