S5 in orthotoric coordinates

We will write the metric for $S^5$ first as a cone over $CP^2$ written in orthotoric coordinates.

(1)
\begin{align} ds^2 = (d\psi - A)^2 + ds^2_{CP^2}\ , \end{align}

where $A= (\xi+\eta) d\theta_1 + (1+\xi\eta)d\theta_2$ and

(2)
\begin{align} ds^2_{CP^2} = (\xi-\eta) \left(\frac{d\xi^2}{F(\xi)}-\frac{d\eta^2}{G(\eta)}\right) + \frac1{(\xi-\eta)}\Big(F(\xi) \big[d\theta_1+\eta d\theta_2\big]^2 - G(\eta)\big[d\theta_2+\xi d\theta_1\big]^2 \Big)\ , \end{align}

and $F(x)=G(x)=\lambda x(1-x^2)$. The angles $\theta_1$ and $\theta_2$ have periodicity $2\pi$ and $\psi$ has periodicity $4\pi$ (need to check this!). The variables $(\xi,\eta)$ are restricted to take values in the interior of the right-handed triangle with vertices $(0,0)$, $(1,0)$ and $(1,1)$ in the $\xi\eta$-plane.

Order the coordinates as $x^\mu=(\xi,\eta,\theta_1,\theta_2,\psi)$. Consider the following basis of one-forms

(3)
\begin{eqnarray} e^1 &=& \left(\frac{\xi-\eta}{F(\xi)}\right)^{1/2}\ d\xi\ . \\ e^2 &=& \left(\frac{\eta-\xi}{G(\eta)}\right)^{1/2}\ d\eta\ . \\ e^3 &=& \left(\frac{\xi-\eta}{F(\xi)}\right)^{-1/2}\ \big[d\theta_1 + \eta d\theta_2\big]\ . \\ e^4 &=& \left(\frac{\eta-\xi}{G(\eta)}\right)^{-1/2}\ \big[\xi d\theta_1 + d\theta_2\big]\ . \\ e^5 &=& d\psi -(\xi+\eta) d\theta_1 - (1+\xi\eta)d\theta_2 \end{eqnarray}

The metric for $S^5$ is then given by

(4)
\begin{align} ds^2 = e^a \delta_{ab} e^b= \big[{e^a}_\mu \delta_{ab} {e^b}_\nu\big]\ dx^\mu dx^\nu\ , \end{align}

${e^a}_\mu$ are the vielbein. Let ${E^\mu}_a$ be the inverse vielbein ie., it is the matrix inverse of the vielbein: ${e^a}_\mu {E^\mu}_b=\delta_b^a$. Then, consider the vector fields $E_a \equiv E^\mu_a \partial_\mu$

(5)
\begin{eqnarray} E_1 &=& \left(\frac{\xi-\eta}{F(\xi)}\right)^{-1/2}\ \partial_\xi \\ E_2 &=& \left(\frac{\eta-\xi}{G(\eta)}\right)^{-1/2}\ \partial_\eta \\ E_3 &=&\frac1{1-\xi\eta} \left(\frac{\xi-\eta}{F(\xi)}\right)^{1/2}\ \partial_{\theta_1} + \frac\eta{\xi\eta-1} \left(\frac{\eta-\xi}{G(\eta)}\right)^{1/2} \ \partial_{\theta_2}\\ E_4 &=&\frac\xi{\xi\eta-1} \left(\frac{\xi-\eta}{F(\xi)}\right)^{1/2}\ \partial_{\theta_1} + \frac1{1-\xi\eta} \left(\frac{\eta-\xi}{G(\eta)}\right)^{1/2} \ \partial_{\theta_2}\\ E_5 &=& \frac{\xi+\eta+\xi^2\eta}{\xi\eta-1} \left(\frac{\xi-\eta}{F(\xi)}\right)^{1/2}\ \partial_{\theta_1} + \frac{\eta^2}{1-\xi\eta} \left(\frac{\eta-\xi}{G(\eta)}\right)^{1/2} \ \partial_{\theta_2}+\partial_\psi \end{eqnarray}

Spectrum of the scalar Laplacian on the n-sphere

The spectrum of the scalar Laplacian on $n$-sphere, $S^n$, is given by

(6)
\begin{align} k(k+n-1)\quad,\quad k=0,1,2,\ldots \end{align}

with multiplicity

(7)
\begin{align} \binom{k+n}{n} - \binom{k+n-2}{n}\ . \end{align}

Note that this number is the dimension of the fully symmetric and traceless rank-$k$ tensor of $SO(n+1)$.

  1. A. Terras, Harmonic Analysis on Symmetric Space and Applications. I , Springer, 1985.
  2. I. Vardi, Determinants of Laplacians and multiple gamma functions, SIAM J. Math. Anal. 19 (1988), 493–507.

Spectrum of the scalar Laplacian on $T^{1,1}$

The metric on $T^{1,1}$ is of the form

(8)
\begin{align} ds^2_ {T^{1,1}} = \frac19 \left(d\psi + \sum_{i=1}^2 \cos\theta_id\varphi_i\right)^2 +\frac16 \sum_{i=1}^2 \Big(d\theta_i^2 + \sin^2\theta_i d\varphi_i^2\Big) \end{align}

where $(\theta_i,\varphi_i)$ are coordinates on two $S^2$'s. The metric has a $SU(2)\times SU(2)\times U(1)$ isometry.

The scalar Laplacian has eigenfunctions in the representation $(k/2,k/2)$ of the two $SU(2)$'s with eigenvalue $3\big(k(k + 2) - \tfrac{k^2}4\big)$.

  1. S.S. Gubser, Einstein manifolds and conformal field theories, Phys. Rev. D59, (1999) 025006.
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