Representations of SU(2) and tensors on SU(2)

Question

I have only recently started exploring this region of homogeneous spaces and its geometry and the question is born from that and given the beginner state of my exploration the questions might sound ill-framed.

All this is motivated by trying to understand how the spectrum of the Dirac Operator is obtained on homogeneous spaces which in physics is a very natural situation.

1. Can anyone explain to me how connections on SU(2) can be labeled by the irreducible representations of SU(2) (what in physics is called "spin") ?

2. Accordingly how are laplacians on SU(2) labelled by irreducible representations of SU(2)?

3. Literature seem to state that symmetric, traceless and divergence less tensors on $S^3$ are also labeled by irreducible representations of SU(2). How is that?

4. What is the precise definition of a "tensor harmonic" on a homogeneous space? And how do "tensor harmonics" on SU(2) give a basis for expanding sections of the spin-bundle on SU(2)?

5. SU(2) can be thought of as the homogeneous space $SU(2)\times SU(2)/SU(2)$ under the diagonal action and then how does sections of this principle bundle give a basis for tensor valued functions on SU(2)?

6. I am adding another point here. In the above bundle with the diagonal action the projection map to the base space is $(g_1,g_2) \mapsto g_1g_2 ^{-1}$ which will have as inverse images orbits of the above diagonal action. And hence one sees the fibers of this principle bundle structure of this homogeneous space.

   Now some calculations in the literature seem to tell me that choosing a section in this bundle with respect to the above projection map somehow canonically defines me vielbiens in the base $SU(2)$.

   The point being that given the usual diagonal metric on $S^3$ one could have guessed the vielbiens upto a sign. But somehow it seems that the choice of a section fixes the signs.

   Can anyone kindly explain how the choice of a section over $S^3$ (the fields obviously lying in the bundle-space $SU(2)\times SU(2)$) gives a "natural" vielbien on the base $S^3$?

   One of the popular sections here are called "thermal sections" in physics.

Answer 1

You may wish to take a look at this survey article by Christian Bär about the spectrum of the Dirac operator. It contains a section (1.1.2.2) on homogeneous spaces. For the particular case of three dimensions, there is also his earlier paper.

Answer 2

A cop-out answer is to read Chapters 22-3 (maybe 21 also) of Lam's Topics In Contemporary Mathematical Physics. These chapters are viewable in Google Books at

https://books.google.com/books?id=bXq_M3_t1voC&dq=Topics+in+Contemporary+Mathematical+Physics

The treatment is accessible to undergraduates. I have included notes and errata below.

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Chapter 21 p. 223: NB. Regarding equation (21.19), note that the Young diagrams $\{6,5,4,1\}$, $\{6,4,4,2\}$, and $\{6,4,4,1,1\}$ aren't included on the RHS because they have more than $n = 3$ rows.

p. 228. For Corollary 21.1, see also Theorem 19.19 in Fulton and Harris. At the bottom of the page, other legitimate Young diagrams with two rows should be included.

p. 229. At the top of the page, legitimate Young diagrams of the form (e.g.) $\{r_1, r_2,1\}$ should be included. Regarding the first full paragraph, see section 19.5 of Fulton and Harris for Weyl's ``associated" diagrams.

p. 230: The reference to equation (20.71) should be to (20.72); just below, note that $\dim([\mu_1]) = \tbinom{\mu_1 + n - 1}{n - 1}$ equals the expressions given for $n=3$. The ``Completely" after equation (21.50) should be decapitalized. It may be worth highlighting (or elaborating on) the reference to spin.

Chapter 22 p. 234: In the first line of equation (22.3), the term $S_{2j}(e_1^{(2j)})$ should be $S_{2j}(e_1^{2j})$. In the text immediately following, some kind of reference should be made along the lines of the sentence beginning ``Note that in the above equation..." in the paragraph after equation (22.9).

p. 235: NB. $u' \equiv \left( \begin{smallmatrix} u^1 & u^2 \end{smallmatrix} \right) \left( \begin{smallmatrix} a & b \\ -\overline{b} & a \end{smallmatrix} \right)$, cf. (22.22).

p. 237: In equations (22.17) and (22.19), factorial symbols need to accompany all the terms under the square root. The reference to equation (22.25) near the end of the page should be to (22.15).

p. 238: In equations (22.24-5), factorial symbols need to be added for the terms in the square roots.

p. 239: quqntum" should read quantum".

p. 240: NB. $a = e^{-i\phi/2}$ and $b=0$ implies equation (22.30).

p. 241: There should be a mention to the effect that there exists $\pi$ such that the eighth equality of equation (22.42) holds.

p. 242: Factorial symbols need to go in the square root in equation (22.47).

p. 245: In the penultimate sentence of the middle paragraph, a reference could profitably be made to (22.35).

p. 246: I skipped these exercises, but the theorem numbers appear to be low by 1.

Chapter 23 p. 249: Regarding the first sentence of the first full paragraph, note that $P^1$ is invariant under $SO(3)$, and the invariance of $P^k$ follows. Equation (23.13) should be $H^k \equiv ker(\nabla^2)|_{P^k}$.

p. 250: NB. At the bottom of the page, draw a correspondence $2\theta \rightsquigarrow \phi$. Note that \begin{equation} D(R_3(2\theta)) \cdot \begin{cases} x \pm iy \\ z \end{cases} = \begin{cases} (x \cos 2\theta - y \sin 2\theta) \pm i(x \sin 2\theta + y \cos 2\theta) \\ z \end{cases} = \begin{cases} (x \pm iy)e^{\pm 2i\theta} \\ z \end{cases}. \end{equation}

p. 251: The unnumbered equation and its preceding text is not quite right: see (the results of) exercise 23.2.

p. 252: Regarding the comment for p. 251 and exercise 23.2, note that $\frac{x \pm iy}{r} = e^{\pm i\phi}\sin \theta$ and $\frac{z}{r} = \cos \theta$. It follows that $Y_2^0 = \sqrt{\frac{5}{16\pi}}\left( 2 \left(\frac{z}{r}\right)^2 - \left(\frac{x+iy}{r}\right)\left(\frac{x-iy}{r}\right) \right), \quad Y_2^1 = -\sqrt{\frac{15}{8\pi}} \left(\frac{z}{r}\right)\left(\frac{x+iy}{r}\right), \quad Y_2^2 = -\sqrt{\frac{15}{32\pi}} \left(\frac{x+iy}{r}\right)^2;$ $Y_3^0 = \sqrt{\frac{7}{16\pi}}\left( 2 \left(\frac{z}{r}\right)^3 - 3\left(\frac{z}{r}\right)\left(\frac{x+iy}{r}\right)\left(\frac{x-iy}{r}\right) \right), \quad Y_3^1 = -\sqrt{\frac{21}{64\pi}} \left(\frac{x+iy}{r}\right) \left( 4 \left(\frac{z}{r}\right)^2 - \left(\frac{x+iy}{r}\right)\left(\frac{x-iy}{r}\right) \right),$ $Y_3^2 = \sqrt{\frac{105}{32\pi}} \left(\frac{z}{r}\right)\left(\frac{x+iy}{r}\right)^2, \quad Y_3^3 = -\sqrt{\frac{35}{64\pi}} \left(\frac{x+iy}{r}\right)^3.$

p. 253: Before equation (23.44) reference equations (18.5-6) and discussion preceding them; also see (23.52-3). For equation (23.46), include $J^3|El0\rangle = 0$. The reference to (22.42) should be to (22.41).