On the smoothness of multi center coplanar black hole and membrane horizons

Abstract

We study the differentiability of the metric and other fields at any of the horizons of multi center Reissner–Nordstrom black hole solutions in $d \ge 5$ and of multi center M2 brane solutions. The centers are distributed in a plane in transverse space, hence termed coplanar. We construct the Gaussian null co-ordinate system for the neighborhood of a horizon by solving the geodesic equations in expansions of (appropriate powers of) the affine parameter. Organizing the harmonic functions that appear in the solution in terms of what can be called generalized Gegenbauer polynomials is key to obtaining the solution to the geodesic equations in a compact and manageable form. We then compute the metric and other fields in the Gaussian null co-ordinate system and find that the differentiability of the coplanar solution is identical to the differentiability of the collinear solution (centers distributed on a line in transverse space). We end the paper with a conjecture on the degree of smoothness of the most general multi center solution, the one with centers distributed arbitrarily.

Appendices

Solution to geodesic equations in 5d

Here we provide the solutions to the geodesic equations described in (2.1.1). All the generalized Gegenbauer polynomials ${\mathcal {G}}_n$’s appearing in this “Appendix” are the ones relevant for $d = 5$, i.e. constructed out of five dimensional Gegenbauer polynomials.

$$\begin{aligned} r(\lambda )= & {} \sqrt{2}\mu _1^{1/4}\,\lambda ^{1/2} +\frac{1}{2\sqrt{2}\mu _1^{1/4}}{\mathcal {G}}_0 \,\lambda ^{3/2} +\frac{2}{5}{\mathcal {G}}_1\,\lambda ^2 \nonumber \\&-\,\frac{1}{48\sqrt{2} \mu _1^{3/4}}\left[ 3\, {\mathcal {G}}_0^2 - 32\mu _1\, {\mathcal {G}}_2\right] \lambda ^{5/2}-\frac{2}{35 \mu _1^{1/2}} \left[ {\mathcal {G}}_0 \,{\mathcal {G}}_1 \right. \nonumber \\&\left. -\,10 \mu _1\,{\mathcal {G}}_3 \right] \lambda ^3+\frac{1}{1600 \sqrt{2} \mu _1^{5/4}} \left[ 25\, {\mathcal {G}}_0^3 - 48 \mu _1\,{\mathcal {G}}_1^2 -2800 \mu _1\, (\partial _\Theta {\mathcal {G}}_1)^2\right. \nonumber \\&- \,2800 \mu _1\,\csc ^2\Theta \,(\partial _\Phi {\mathcal {G}}_1)^2 \left. +1600\mu _1^2\, {\mathcal {G}}_4 \right] \lambda ^{\frac{7}{2}} \nonumber \\&+ \,\frac{1}{126 \mu _1}\left[ 3\,{\mathcal {G}}_0^2 G_1 + 12\mu _1\,\mathcal{G}_0 G_3 -175 \mu _1 \, \partial _\Theta {\mathcal {G}}_1 \partial _\Theta {\mathcal {G}}_2 \right. \nonumber \\&-\,175 \mu _1 \csc ^2 \Theta \, \partial _\Phi {\mathcal {G}}_1 \partial _\Phi {\mathcal {G}}_2 \nonumber \\&\left. +\, 112 \mu _1^2\, \mathcal{G}_5 \right] \lambda ^4 -\frac{1}{1{,}612{,}800 \sqrt{2} \mu _1^{7/4}} \left[ 7875\, {\mathcal {G}}_0^4 -33{,}600\mu _1\,{\mathcal {G}}_0^2 {\mathcal {G}}_2 \right. \nonumber \\&- \,42{,}048\mu _1\, {\mathcal {G}}_0 {\mathcal {G}}_1^2 - 35{,}840\mu _1^2\, {\mathcal {G}}_2^2 \nonumber \\&- \,7{,}656{,}768\mu _1\,{\mathcal {G}}_0 (\partial _\Theta {\mathcal {G}}_1)^2 - 7{,}656{,}768\mu _1\,\csc ^2 \Theta \,\mathcal{G}_0 (\partial _\Phi {\mathcal {G}}_1)^2\nonumber \\&-\, 184{,}320\mu _1^2\, {\mathcal {G}}_1 {\mathcal {G}}_3 - 564{,}480\mu _1^2\, {\mathcal {G}}_0 {\mathcal {G}}_4 \nonumber \\&\left. +\, 967{,}680\mu _1^2\, (\partial _\Theta {\mathcal {G}}_2)^2 + 967{,}680\mu _1^2\, \csc ^2 \Theta (\partial _\Phi {\mathcal {G}}_2)^2\right. \nonumber \\&+ 2{,}322{,}432\mu _1^2\, \partial _\Theta {\mathcal {G}}_1 \partial _\Theta \mathcal{G}_3 \nonumber \\&\left. + \,2{,}322{,}432\mu _1^2\, \csc ^2 \Theta \partial _\Phi {\mathcal {G}}_1 \partial _\Phi {\mathcal {G}}_3 - 2{,}580{,}480 \mu _1^3\, {\mathcal {G}}_6 \right] \lambda ^{\frac{9}{2}}\nonumber \\&+ \,\frac{1}{57{,}750 \mu _1^{3/2}} \left[ -600\, {\mathcal {G}}_0^3\,+ 294 \mu _1\, {\mathcal {G}}_1^3 \right. \nonumber \\&+\, 1500\mu _1 \,{\mathcal {G}}_0 \, {\mathcal {G}}_1 \, {\mathcal {G}}_2 + 750\mu _1 \,{\mathcal {G}}_0^2\, {\mathcal {G}}_1 {\mathcal {G}}_3 + 279{,}825\mu _1\, {\mathcal {G}}_1 \, (\partial _\Theta \mathcal{G}_1)^2 \nonumber \\&+\, 279{,}825\mu _1\, \csc ^2 \Theta \,{\mathcal {G}}_1 \, (\partial _\Phi {\mathcal {G}}_1)^2 \nonumber \\&+\, 208{,}775\mu _1\, {\mathcal {G}}_0\, \partial _\Theta {\mathcal {G}}_1\,\partial _\Theta {\mathcal {G}}_2 + 208{,}775\mu _1\, \cos ^2 \Theta \, {\mathcal {G}}_0\, \partial _\Phi \mathcal{G}_1\,\partial _\Phi {\mathcal {G}}_2 \nonumber \\&-\, 34{,}125 \mu _1\, ( \partial _\Theta {\mathcal {G}}_1)^2 \, \partial ^2_\Theta {\mathcal {G}}_1 \nonumber \\&-\,34{,}125 \mu _1\, \csc ^4 \Theta \,( \partial _\Phi {\mathcal {G}}_1)^2 \, \partial ^2_\Phi {\mathcal {G}}_1 - 68{,}250 \mu _1\, \csc ^2\Theta \, \partial _\Theta {\mathcal {G}}_1\, \partial _\Phi \mathcal{G}_1\,\partial ^2_{\Theta \Phi } {\mathcal {G}}_1 \nonumber \\&+\, 34{,}125\mu _1\, \cot \Theta \, \csc ^2 \Theta \, (\partial _\Phi \mathcal{G}_1)^2 \, \partial _\Theta {\mathcal {G}}_1 + 6000\mu _1^2\, {\mathcal {G}}_2\, {\mathcal {G}}_3 \nonumber \\&+\, 12{,}600\mu _1^2\, {\mathcal {G}}_1\, {\mathcal {G}}_4 + 28{,}000\mu _1^2\, {\mathcal {G}}_0\, {\mathcal {G}}_5 \nonumber \\&-\, 47{,}250 \mu _1^2\, \partial _\Theta {\mathcal {G}}_1\, \partial _\Theta {\mathcal {G}}_4 - 47{,}250 \mu _1^2\, \csc ^2 \Theta \, \partial _\Phi {\mathcal {G}}_1\, \partial _\Phi {\mathcal {G}}_4 \nonumber \\&-\, 38{,}850 \mu _1^2 \, \partial _\Theta {\mathcal {G}}_2\, \partial _\Theta {\mathcal {G}}_3 \nonumber \\&\left. -\, 38{,}850 \mu _1^2 \, \csc ^2 \Theta \, \partial _\Phi {\mathcal {G}}_2\, \partial _\Phi {\mathcal {G}}_3 + 84{,}000 \mu _1^3\,{\mathcal {G}}_7 \right] \lambda ^5 + \cdots \end{aligned}$$

(A.1)

$$\begin{aligned} \theta (\lambda )= & {} \Theta + \frac{\sqrt{2} }{\mu _1^{1/4}} \partial _\Theta {\mathcal {G}}_1\,\lambda ^{3/2} + \frac{3}{4} \partial _\Theta {\mathcal {G}}_2 \,\lambda ^2 \nonumber \\&-\frac{1}{10 \sqrt{2} \mu _1^{3/4}} \left[ 17 \,{\mathcal {G}}_0 \partial _\Theta {\mathcal {G}}_1 - 8 \mu _1\, \partial _\Theta {\mathcal {G}}_3 \right] \lambda ^{5/2} + \frac{1}{20 \mu _1^{1/2}} \left[ -31\, {\mathcal {G}}_1 \partial _\Theta {\mathcal {G}}_1 \right. \nonumber \\&+\, 5\, \partial _\Theta {\mathcal {G}}_1 \, \partial ^2_\Theta {\mathcal {G}}_1 - 10 \, {\mathcal {G}}_0\, \partial _\Theta {\mathcal {G}}_2 + 15\, \cot \Theta \, \csc ^2 \Theta \, (\partial _\Phi {\mathcal {G}}_1)^2 \nonumber \\&\left. +\, 5\, \csc ^2 \Theta \, \partial _\Phi {\mathcal {G}}_1\, \partial ^2_{\Theta \Phi }{\mathcal {G}}_1 + 10 \mu _1\, \partial _\Theta {\mathcal {G}}_4 \right] \lambda ^3 \nonumber \\&+ \frac{1}{560 \sqrt{2} \mu _1^{5/4}} \left[ 937\, {\mathcal {G}}_0^2\, \partial _\Theta {\mathcal {G}}_1 - 1568\mu _1\, {\mathcal {G}}_2 \, \partial _\Theta {\mathcal {G}}_1 + 192 \mu _1\, \partial _\Theta {\mathcal {G}}_1\,\partial ^2_\Theta {\mathcal {G}}_2 \right. \nonumber \\&-\,576\mu _1\, {\mathcal {G}}_1\, \partial _\Theta {\mathcal {G}}_2 - 208\mu _1\, \mathcal{G}_0 \, \partial _\Theta {\mathcal {G}}_3 \nonumber \\&+\, 72 \mu _1\, \partial _\Theta {\mathcal {G}}_2\, \partial ^2_\Theta {\mathcal {G}}_1 + 72 \mu _1\, \csc ^2\Theta \, \partial _\Phi {\mathcal {G}}_2 \, \partial ^2_{\Theta \Phi } {\mathcal {G}}_1 \nonumber \\&+ 576 \mu _1\, \cot \Theta \, \csc ^2\Theta \, \partial _\Phi {\mathcal {G}}_1\, \partial _\Phi \mathcal{G}_2 \nonumber \\&\left. +\, 192 \mu _1\,\csc ^2 \Theta \, \partial _\Phi {\mathcal {G}}_1 \partial ^2_{\Theta \Phi }{\mathcal {G}}_2 + 384 \mu _1^2\, \partial _\Theta {\mathcal {G}}_5 \right] \lambda ^{7/2} + \cdots \end{aligned}$$

(A.2)

$$\begin{aligned} \phi (\lambda )= & {} \Phi + \frac{\sqrt{2}}{\mu _1^{1/4}} \csc ^2\Theta \,\partial _\Phi {\mathcal {G}}_1\,\lambda ^{3/2} + \frac{3}{4} \csc ^2 \Theta \, \partial _\Phi {\mathcal {G}}_2 \,\lambda ^2 \nonumber \\&-\frac{1}{10 \sqrt{2} \mu _1^{3/4}} \csc ^2 \Theta \left[ 17 \,{\mathcal {G}}_0 \partial _\Phi {\mathcal {G}}_1 - 8 \mu _1\, \partial _\Phi {\mathcal {G}}_3 \right] \lambda ^{5/2} \nonumber \\&+ \frac{1}{20 \mu _1^{1/2}} \csc ^2 \Theta \left[ -31\, {\mathcal {G}}_1 \, \partial _\Phi {\mathcal {G}}_1+ 5\, \partial _\Theta {\mathcal {G}}_1 \partial ^2_{\Theta \Phi }{\mathcal {G}}_1 -10\, {\mathcal {G}}_0 \,\partial _\Phi {\mathcal {G}}_2 \right. \nonumber \\&+ \,5 \csc ^2 \Theta \, \partial _\Phi {\mathcal {G}}_1\, \partial ^2_\Phi {\mathcal {G}}_1 \left. - 40 \cot \Theta \, \partial _\Phi {\mathcal {G}}_1\, \partial _\Theta {\mathcal {G}}_1+ 10 \mu _1\, \partial _\Phi \mathcal{G}_4\right] \lambda ^3 \nonumber \\&+ \frac{1}{560 \sqrt{2} \mu _1^{5/4}} \csc ^2 \Theta \left[ 937\, {\mathcal {G}}_0^2\,\partial _\Phi {\mathcal {G}}_1 - 1568\mu _1\, {\mathcal {G}}_2\, \partial _\Phi {\mathcal {G}}_1 \right. \nonumber \\&+\, 192 \mu _1\, \partial _\Theta {\mathcal {G}}_1\, \partial ^2_{\Theta \Phi }{\mathcal {G}}_2 - 576 \mu _1\, {\mathcal {G}}_1 \, \partial _\Phi {\mathcal {G}}_2 -208 \mu _1\, {\mathcal {G}}_0\,\partial _\Phi {\mathcal {G}}_3 \nonumber \\&+\, 72 \mu _1\, \partial _\Theta {\mathcal {G}}_2\, \partial ^2_{\Theta \Phi } {\mathcal {G}}_1 + 72 \mu _1\,\csc ^2 \Theta \, \partial _\Phi {\mathcal {G}}_2\, \partial ^2_{\Phi }{\mathcal {G}}_1 - 960\mu _1\, \cot \Theta \, \partial _\Phi {\mathcal {G}}_2\, \partial _\Theta {\mathcal {G}}_1 \nonumber \\&-\, 720\mu _1\, \cot \Theta \, \partial _\Phi {\mathcal {G}}_1\, \partial _\Theta {\mathcal {G}}_2\nonumber \\&\left. +\, 192\mu _1\, \csc ^2 \Theta \, \partial _\Phi \mathcal{G}_1 \, \partial ^2_{\Phi } {\mathcal {G}}_2 + 384 \mu _1^2 \, \partial _\Phi {\mathcal {G}}_5 \right] \lambda ^{7/2} + \cdots \end{aligned}$$

(A.3)

$$\begin{aligned} T(\lambda , \Theta , \Phi )= & {} - \frac{\mu _1}{4}\,\lambda ^{-1} + \frac{3\mu _1^{1/2}}{4} \, {\mathcal {G}}_0 \log \lambda + \frac{8 \sqrt{2}\mu _1^{3/4}}{5} \, {\mathcal {G}}_1 \lambda ^{1/2} \nonumber \\&+ \frac{1}{48} \left[ 33 {\mathcal {G}}_0^2 + 80 \mu _1\,{\mathcal {G}}_2 \right] \lambda \nonumber \\&+ \frac{2\sqrt{2}\mu _1^{1/4}}{35}\,\left[ 19\,\mathcal{G}_0\, {\mathcal {G}}_1 + 20\, \mu _1\, {\mathcal {G}}_3 \right] \lambda ^{3/2} \nonumber \\&+\frac{1}{400 \mu _1^{1/2}} \left[ 25\, {\mathcal {G}}_0^3 + 363 \mu _1\, {\mathcal {G}}_1^2 + 750 \mu _1\, {\mathcal {G}}_0\, {\mathcal {G}}_2 \right. \nonumber \\&\left. +\, 575 \mu _1 \, (\partial _\Theta {\mathcal {G}}_1)^2 + 575 \mu _1 \,\csc ^2 \Theta \, (\partial _\Phi {\mathcal {G}}_1)^2 + 700\mu _1^2\, {\mathcal {G}}_4 \right] \lambda ^2 \nonumber \\&+ \frac{1}{630 \sqrt{2} \mu _1^{1/4}} \left[ 345 \,{\mathcal {G}}_0^2\, {\mathcal {G}}_1 + 2016\mu _1\, {\mathcal {G}}_1 \, {\mathcal {G}}_2 \right. \nonumber \\&+ \,2136\mu _1\, {\mathcal {G}}_0\, {\mathcal {G}}_3 + 1736 \mu _1\, \partial _\Theta {\mathcal {G}}_1 \, \partial _\Theta \mathcal{G}_2 + 1736 \mu _1\, \csc ^2 \Theta \, \partial _\Phi {\mathcal {G}}_1 \, \partial _\Phi {\mathcal {G}}_2 \nonumber \\&\left. +\, 1792\mu _1^2\, {\mathcal {G}}_5 \right] \lambda ^{5/2} \nonumber \\&+ \frac{1}{100{,}800 \mu _1} \left[ -1575 {\mathcal {G}}_0^4 + 71{,}400 \mu _1\, {\mathcal {G}}_0^2\,{\mathcal {G}}_2 + 65{,}124\mu _1 \,\mathcal{G}_0\, {\mathcal {G}}_1^2 \right. \nonumber \\&+ \,2772\mu _1\, {\mathcal {G}}_0\, (\partial _\Theta {\mathcal {G}}_1)^2 + 2772\mu _1\, \csc ^2\Theta \,\mathcal{G}_0\, (\partial _\Phi {\mathcal {G}}_1)^2 \nonumber \\&+ \,143{,}360\mu _1^2\, {\mathcal {G}}_2^2 + 293{,}760\mu _1^2\, {\mathcal {G}}_1 \,{\mathcal {G}}_3 + 317{,}520\mu _1^2\, \mathcal{G}_0\, {\mathcal {G}}_4 \nonumber \\&+\, 60{,}480 \mu _1^2\, (\partial _\Theta {\mathcal {G}}_2)^2 + 60{,}480 \mu _1^2\, \csc ^2 \Theta (\partial _\Phi {\mathcal {G}}_2)^2\nonumber \\&+ 185{,}472 \mu _1^2\, \partial _\Theta {\mathcal {G}}_1\, \partial _\Theta {\mathcal {G}}_3 \nonumber \\&\left. +\, 185{,}472 \mu _1^2\, \csc ^2 \Theta \, \partial _\Phi {\mathcal {G}}_1\, \partial _\Phi {\mathcal {G}}_3 + 241{,}920 \mu _1^3\, {\mathcal {G}}_6 \right] \lambda ^3 \nonumber \\&+\frac{1}{231{,}000 \sqrt{2} \mu _1^{3/4}} \left[ -18{,}525\, \mathcal{G}_0^3\, {\mathcal {G}}_1 + 105{,}264 \mu _1\, {\mathcal {G}}_1^3 \right. \nonumber \\&+ \,696{,}000\mu _1\, {\mathcal {G}}_0 \,{\mathcal {G}}_1\, {\mathcal {G}}_2 + 397{,}500 \mu _1 \,{\mathcal {G}}_0^2\, {\mathcal {G}}_3 - 49{,}200\mu _1\, {\mathcal {G}}_1 \, (\partial _\Theta {\mathcal {G}}_1)^2 \nonumber \\&-\, 49{,}200\mu _1\, \csc ^2\Theta \, \mathcal{G}_1 \, (\partial _\Phi {\mathcal {G}}_1)^2 + 204{,}000 \mu _1\, (\partial _\Theta {\mathcal {G}}_1)^2\, \partial ^2_\Theta {\mathcal {G}}_1 \nonumber \\&+\,204{,}000 \mu _1\, \csc ^4\Theta \,(\partial _\Phi {\mathcal {G}}_1)^2\, \partial ^2_\Phi {\mathcal {G}}_1 + 241{,}000 \mu _1\, {\mathcal {G}}_0\, \partial _\Theta {\mathcal {G}}_1\, \partial _\Theta {\mathcal {G}}_2 \nonumber \\&+\, 241{,}000 \mu _1\,\csc ^2 \Theta \, {\mathcal {G}}_0\, \partial _\Phi {\mathcal {G}}_1\, \partial _\Phi {\mathcal {G}}_2\nonumber \\&+ 408{,}000\mu _1\, \csc ^2 \Theta \, \partial _\Phi {\mathcal {G}}_1\, \partial _\Theta {\mathcal {G}}_1\, \partial ^2_{\Theta \Phi }{\mathcal {G}}_1\nonumber \\&-\, 204{,}000 \mu _1 \cot \Theta \, \csc ^2 \Theta \, (\partial _\Phi \mathcal{G}_1)^2\,\partial _\Theta {\mathcal {G}}_1 + 1{,}200{,}000\mu _1^2\, \mathcal{G}_2\, {\mathcal {G}}_3 \nonumber \\&+\, 1{,}252{,}800\mu _1^2 \, {\mathcal {G}}_1\, {\mathcal {G}}_4 + 1{,}376{,}000\mu _1^2\, {\mathcal {G}}_0 \, {\mathcal {G}}_5 + 648{,}000 \mu _1^2 \, \partial _\Theta {\mathcal {G}}_1\, \partial _\Theta {\mathcal {G}}_4 \nonumber \\&+\, 648{,}000 \mu _1^2 \,\csc ^2\Theta \,\partial _\Phi \mathcal{G}_1\, \partial _\Phi {\mathcal {G}}_4 + 348{,}000 \mu _1^2\, \partial _\Theta {\mathcal {G}}_2\, \partial _\Theta {\mathcal {G}}_3 \nonumber \\&\left. +\, 348{,}000 \mu _1^2\, \csc ^2 \Theta \, \partial _\Phi {\mathcal {G}}_2\, \partial _\Phi {\mathcal {G}}_3 + 960{,}000\mu _1^3\, {\mathcal {G}}_7 \right] \lambda ^{7/2} + \cdots \end{aligned}$$

(A.4)

Solution to geodesic equations in $d \ge 6$

Here we provide the solutions to the geodesic equations described in (2.2.1). All the generalized Gegenbauer polynomials ${\mathcal {G}}_n$’s appearing in this “Appendix” are the ones relevant for generic $d > 5$, i.e. constructed out of $d > 5$ dimensional Gegenbauer polynomials.

$$\begin{aligned} r(\lambda )= & {} (d-3)^{1/{d-3}}\, \mu _1^{\frac{d-4}{(d-3)^2}}\, \lambda ^{1/{d-3}}\nonumber \\&+ \sum _{l=-3}^{d-7} \left[ \frac{d-4}{2d+l-3}\,(d-3)^{\frac{l+4}{d-3}}\, \mu _1^{\frac{d(l+3)-4l-13}{(d-3)^2}}\,{\mathcal {G}}_{l+3} \, \left( \lambda ^{1/{d-3}} \right) ^{d+l+1} \right] \nonumber \\&+ \frac{d-4}{24\, (d-3)^2} \, (d-3)^{\frac{1}{d-3}}\,\mu _1^{-\frac{d-2}{(d-3)^2}}\,\left[ (d-2) (d-6) \,{\mathcal {G}}_0^2 \right. \nonumber \\&\left. +\, 8 (d-3)^2 \mu _1 \,{\mathcal {G}}_{d-3} \right] (\lambda ^{1/{d-3}})^{2 d - 5} \nonumber \\&+ \frac{d-4}{2 (d-3) (2 d-5) (3 d - 8)} (d-3)^{\frac{2 }{d-3}}\mu _1^{-\frac{2}{(d-3)^2}} \nonumber \\&\left[ (d-1) \left( d^2-8 d+14\right) {\mathcal {G}}_0\, {\mathcal {G}}_1 \right. \nonumber \\&\left. + \,2 (d-3)^2 (2 d-5) \mu _1\,{\mathcal {G}}_{d-2} \right] (\lambda ^{1/{d-3}})^{2d - 4} + \cdots \end{aligned}$$

(B.1)

$$\begin{aligned} \theta (\lambda )= & {} \Theta + \sum _{l = 0}^{d-4} \frac{d-2}{(d+l-2)\,(l+1)}\,(d-3)^{\frac{l+1}{d-3}}\, \mu _1^{\frac{(d-4)l-1}{(d-3)^2}}\,\nonumber \\&\partial _\Theta {\mathcal {G}}_{l+1}\,(\lambda ^{1/{d-3}})^{l+d-2} + \cdots \end{aligned}$$

(B.2)

$$\begin{aligned} \phi (\lambda )= & {} \Phi + \sum _{l = 0}^{d-4} \frac{d-2}{(d+l-2)\,(l+1)}\,(d-3)^{\frac{l+1}{d-3}}\, \mu _1^{\frac{(d-4)l-1}{(d-3)^2}}\,\nonumber \\&\csc ^2 \Theta \, \partial _\Phi {\mathcal {G}}_{l+1}\,(\lambda ^{1/{d-3}})^{l+d-2} + \cdots \end{aligned}$$

(B.3)

$$\begin{aligned} T(\lambda , \Theta , \Phi )= & {} - \frac{1}{(d-3)^2}\,\mu _1^{2/{d-3}}\, \lambda ^{-1} \left[ 1 - (d -2)\,\mu _1^{-1/{d-3}}\,{\mathcal {G}}_0 \, \lambda \,\log \lambda \right. \nonumber \\&\left. - \sum _{l=1}^{d-4} \frac{2 d + 2 l - 4}{l\,(2 d + l - 6)}\,(d-3)^{2+ \frac{l}{d-3}} \,\mu _1^{\frac{l(d-4)-d+3}{(d-3)^2}}\,{\mathcal {G}}_l \, \lambda ^{\frac{d + l -3}{d-3}} + \cdots \right] \nonumber \\ \end{aligned}$$

(B.4)

Solution to geodesic equations in 11d

Here we provide the solutions to the geodesic equations described in (3.1). All the generalized Gegenbauer polynomials ${\mathcal {G}}_n$’s appearing in this “Appendix” are the ones relevant for the eleven dimensional membrane harmonic function, i.e. constructed out of $d = 9$ dimensional Gegenbauer polynomials.

$$\begin{aligned} r(\lambda )= & {} \sqrt{2}\mu _1^{1/{12}}\lambda ^{1/2} + \frac{1}{2\sqrt{2}\mu _1^{5/{12}}}{\mathcal {G}}_0\,\lambda ^{7/2} + \frac{8}{27 \mu _1^{1/3}}\,{\mathcal {G}}_1 \,\lambda ^4 \nonumber \\&+\frac{4\sqrt{2} }{15 \mu _1^{1/4}} {\mathcal {G}}_2\, \lambda ^{9/2} + \frac{16}{33 \mu _1^{1/6}} {\mathcal {G}}_3\, \lambda ^5\nonumber \\&+ \frac{4 \sqrt{2}}{9 \mu _1^{1/{12}}} {\mathcal {G}}_4\, \lambda ^{{11}/2} + \frac{32}{39} \mathcal{G}_5\, \lambda ^6 + \frac{1}{252 \sqrt{2} \mu _1^{11/12}} \left[ - 119 \, {\mathcal {G}}_0^2 + 384 \mu _1 {\mathcal {G}}_6 \right] \lambda ^{{13}/2}\nonumber \\&+ \frac{32}{405 \mu _1^{5/6}} \left[ -11 {\mathcal {G}}_0 \,{\mathcal {G}}_1 + 18\mu _1\, {\mathcal {G}}_7 \right] \lambda ^7 \nonumber \\&+ \frac{\sqrt{2}}{127{,}575 \mu _1^{3/4}} \left[ -102{,}060\, {\mathcal {G}}_0 \,{\mathcal {G}}_2 - 51{,}625\, {\mathcal {G}}_1^2 - 2673 \, (\partial _\Theta {\mathcal {G}}_1)^2 \right. \nonumber \\ \end{aligned}$$

$$\begin{aligned}&\left. - \,2673 \, \csc ^2 \Theta \, (\partial _\Phi {\mathcal {G}}_1)^2 + 170{,}100\mu _1\, \mathcal{G}_8 \right] \lambda ^{{15}/2} \nonumber \\&+\frac{4}{58{,}905 \mu _1^{2/3}} \left[ - 21{,}700\, {\mathcal {G}}_0\, {\mathcal {G}}_3 - 22{,}176\, {\mathcal {G}}_1 \,{\mathcal {G}}_2 - 935\, \partial _\Theta {\mathcal {G}}_1\, \partial _\Theta {\mathcal {G}}_2 \right. \nonumber \\&\left. -\, 935 \csc ^2 \Theta \, \partial _\Phi {\mathcal {G}}_1\, \partial _\Phi {\mathcal {G}}_2 + 36{,}960 \mu _1 \,{\mathcal {G}}_9 \right] \lambda ^8 \nonumber \\&+ \frac{2 \sqrt{2}}{467{,}775 \mu _1^{7/12}}\left[ -165{,}396 \,{\mathcal {G}}_2^2 - 327{,}600\, {\mathcal {G}}_1\, {\mathcal {G}}_3- 317{,}625 \mathcal{G}_0\, {\mathcal {G}}_4 \right. \nonumber \\&-\, 5775 (\partial _\Theta {\mathcal {G}}_2)^2 - 11{,}440 \, \partial _\Theta {\mathcal {G}}_1\, \partial _\Theta {\mathcal {G}}_3 \nonumber \\&\left. -\, 5775 \csc ^2 \Theta (\partial _\Phi {\mathcal {G}}_2)^2 - 11{,}440 \csc ^2\Theta \, \partial _\Phi {\mathcal {G}}_1\, \partial _\Phi {\mathcal {G}}_3 + 554{,}400 \mu _1\, {\mathcal {G}}_{10}\right] \lambda ^{{17}/2} \nonumber \\&+\frac{8}{23{,}108{,}085 \mu _1^{1/2}}\left[ - 7{,}665{,}840\, {\mathcal {G}}_2\, {\mathcal {G}}_3 - 7{,}527{,}520 \, {\mathcal {G}}_1\, {\mathcal {G}}_4 - 7{,}234{,}920 \, {\mathcal {G}}_0\, {\mathcal {G}}_5 \right. \nonumber \\ \end{aligned}$$

$$\begin{aligned}&-\, 225{,}225\, \partial _\Theta {\mathcal {G}}_2\, \partial _\Theta {\mathcal {G}}_3 - 220{,}077 \, \partial _\Theta {\mathcal {G}}_1\, \partial _\Theta {\mathcal {G}}_4 - 225{,}225\, \csc ^2 \Theta \, \partial _\Phi {\mathcal {G}}_2\, \partial _\Phi {\mathcal {G}}_3 \nonumber \\&\left. -\, 220{,}077 \,\csc ^2 \Theta \, \partial _\Phi {\mathcal {G}}_1\, \partial _\Phi {\mathcal {G}}_4 + 12{,}972{,}960 \mu _1 \,{\mathcal {G}}_{11} \right] \lambda ^9 \nonumber \\&+ \,\frac{1}{1{,}248{,}647{,}400 \sqrt{2} \mu _1^{17/12}}\left[ 1{,}534{,}988{,}455 \,{\mathcal {G}}_0^3 - 3{,}133{,}428{,}480 \mu _1\, {\mathcal {G}}_3^2 \right. \nonumber \\&-\, 6{,}215{,}489{,}280 \mu _1\, {\mathcal {G}}_2\, {\mathcal {G}}_4 - 6{,}056{,}117{,}760 \mu _1\, {\mathcal {G}}_1\, {\mathcal {G}}_5\nonumber \\&- 5{,}771{,}525{,}760 \mu _1\, {\mathcal {G}}_0\, {\mathcal {G}}_6 \nonumber \\ \end{aligned}$$

$$\begin{aligned}&-\, 78{,}524{,}160 \mu _1\, (\partial _\Theta {\mathcal {G}}_3)^2 - 155{,}387{,}232 \mu _1 \, \partial _\Theta {\mathcal {G}}_2\, \partial _\Theta {\mathcal {G}}_4 \nonumber \\&-\, 149{,}140{,}992 \mu _1\, \partial _\Theta {\mathcal {G}}_1\, \partial _\Theta {\mathcal {G}}_5 - 78{,}524{,}160 \mu _1\, \csc ^2 \Theta \, (\partial _\Phi {\mathcal {G}}_3)^2\nonumber \\&-\, 155{,}387{,}232 \mu _1 \, \csc ^2 \Theta \,\partial _\Phi {\mathcal {G}}_2\, \partial _\Phi {\mathcal {G}}_4 - 149{,}140{,}992 \mu _1\, \csc ^2 \Theta \, \partial _\Phi {\mathcal {G}}_1\, \partial _\Phi {\mathcal {G}}_5 \nonumber \\&\left. +\, 10{,}655{,}124{,}480 \mu _1^2\, {\mathcal {G}}_{12}\right] \lambda ^{{19}/2} \nonumber \\&+ \frac{16}{42{,}567{,}525 \mu _1^{4/3}} \left[ 9{,}214{,}205\, {\mathcal {G}}_0^2 \, {\mathcal {G}}_1 - 12{,}612{,}600 \mu _1\, \mathcal{G}_3\, {\mathcal {G}}_4\right. \nonumber \\&\left. - 12{,}418{,}560 \mu _1\, {\mathcal {G}}_2\, {\mathcal {G}}_5\right. \nonumber \\&-\, 12{,}012{,}000 \mu _1\, {\mathcal {G}}_1\, {\mathcal {G}}_6- 11{,}351{,}340 \mu _1 \, {\mathcal {G}}_0\,{\mathcal {G}}_7 - 272{,}415 \mu _1\, \partial _\Theta {\mathcal {G}}_3\, \partial _\Theta {\mathcal {G}}_4 \nonumber \\ \end{aligned}$$

$$\begin{aligned}&+\, 266{,}175 \mu _1 \,\partial _\Theta {\mathcal {G}}_2\, \partial _\Theta {\mathcal {G}}_5 - 249{,}678 \mu _1\, \partial _\Theta {\mathcal {G}}_1\, \partial _\Theta {\mathcal {G}}_6 \nonumber \\&-\, 272{,}415 \mu _1\, \csc ^2\Theta \, \partial _\Phi {\mathcal {G}}_3\, \partial _\Phi {\mathcal {G}}_4 + 266{,}175 \mu _1 \, \csc ^2 \Theta \, \partial _\Phi {\mathcal {G}}_2\, \partial _\Phi \mathcal{G}_5 \nonumber \\&-\, 249{,}678 \mu _1\, \csc ^2 \Theta \, \partial _\Phi {\mathcal {G}}_1\, \partial _\Phi {\mathcal {G}}_6 \left. + 21{,}621{,}600 \mu _1^2\, {\mathcal {G}}_{13} \right] \lambda ^{10} \nonumber \\&+ \frac{1}{120{,}405{,}285 \sqrt{2} \mu _1^{5/4}} \left[ 781{,}539{,}759\,{\mathcal {G}}_0^2\, {\mathcal {G}}_2 + 788{,}124{,}337 \, {\mathcal {G}}_0 \, {\mathcal {G}}_1^2 \right. \nonumber \\&+\, 29{,}930{,}553 {\mathcal {G}}_0\, ( \partial _\Theta {\mathcal {G}}_1)^2 + 29{,}930{,}553 \mathcal{G}_0\, \csc ^2 \Theta \, ( \partial _\Phi {\mathcal {G}}_1)^2\nonumber \\ \end{aligned}$$

$$\begin{aligned}&- 542{,}702{,}160 \mu _1\, {\mathcal {G}}_4^2 \nonumber \\&-\, 1{,}077{,}753{,}600 \mu _1 \, {\mathcal {G}}_3\, \mathcal{G}_5 - 1{,}054{,}145{,}664 \mu _1\, {\mathcal {G}}_2 \, {\mathcal {G}}_6\nonumber \\&- 1{,}012{,}467{,}456 \mu _1\, {\mathcal {G}}_1\, {\mathcal {G}}_7 \nonumber \\&-\, 948{,}647{,}700 \mu _1\, {\mathcal {G}}_0 \, {\mathcal {G}}_8 - 10{,}216{,}206 \mu _1\, (\partial _\Theta {\mathcal {G}}_4)^2\nonumber \\&- 20{,}217{,}600 \mu _1\, \partial _\Theta {\mathcal {G}}_3\, \partial _\Theta {\mathcal {G}}_5 \nonumber \\&-\, 19{,}459{,}440 \mu _1\, \partial _\Theta {\mathcal {G}}_2\, \partial _\Theta {\mathcal {G}}_6 - 17{,}729{,}280 \mu _1\, \partial _\Theta {\mathcal {G}}_1\, \partial _\Theta {\mathcal {G}}_7 \nonumber \\ \end{aligned}$$

$$\begin{aligned}&-\, 10{,}216{,}206 \mu _1\, \csc ^2 \Theta \, (\partial _\Phi {\mathcal {G}}_4)^2 - 20{,}217{,}600 \mu _1\, \csc ^2 \Theta \, \partial _\Phi {\mathcal {G}}_3\, \partial _\Phi \mathcal{G}_5 \nonumber \\&-\, 19{,}459{,}440 \mu _1\, \csc ^2 \Theta \, \partial _\Phi \mathcal{G}_2\, \partial _\Phi {\mathcal {G}}_6- 17{,}729{,}280 \mu _1\, \csc ^2 \Theta \, \partial _\Theta {\mathcal {G}}_1\, \partial _\Theta {\mathcal {G}}_7 \nonumber \\&\left. +\, 1{,}868{,}106{,}240 \mu _1^2 \,{\mathcal {G}}_{14} \right] \lambda ^{{21}/2} + \cdots \end{aligned}$$

(C.1)

$$\begin{aligned} \theta (\lambda )= & {} \Theta + \frac{4 \sqrt{2}}{35 \mu _1^{5/12}}\partial _\Theta {\mathcal {G}}_1 \, \lambda ^{7/2} + \frac{1}{6 \mu _1^{1/3}} \partial _\Theta {\mathcal {G}}_2\,\lambda ^4 + \frac{8 \sqrt{2} }{63 \mu _1^{1/4}} \partial _\Theta \mathcal{G}_3\,\lambda ^{9/2} \nonumber \\&+\frac{1}{5 \mu _1^{1/6}} \partial _\Theta {\mathcal {G}}_4\, \lambda ^5 + \frac{16 \sqrt{2}}{99 \mu _1^{1/{12}}} \partial _\Theta {\mathcal {G}}_5\, \lambda ^{{11}/2} \nonumber \\&+ \frac{4}{15} \partial _\Theta {\mathcal {G}}_6\, \lambda ^6 + \frac{2 \sqrt{2}}{2145 \mu _1^{11/12}} \left[ - 241\, {\mathcal {G}}_0\,\partial _\Theta \mathcal{G}_1 + 240 \mu _1 \,\partial _\Theta {\mathcal {G}}_7\right] \lambda ^{{13}/2} \nonumber \\&+ \frac{2}{33{,}075 \mu _1^{5/6}} \left[ -6790 \, {\mathcal {G}}_1\, \partial _\Theta {\mathcal {G}}_1 - 5775\, {\mathcal {G}}_0\, \partial _\Theta {\mathcal {G}}_2 + 90 \,\partial _\Theta {\mathcal {G}}_1 \, \partial ^2_\Theta {\mathcal {G}}_1 \right. \nonumber \\&\left. + \,90\, \csc ^2\Theta \, \partial _\Phi {\mathcal {G}}_1 \, \partial ^2_{\Theta \Phi }{\mathcal {G}}_1 + 126 \cot \Theta \, \csc ^2 \Theta \, (\partial _\Phi {\mathcal {G}}_1)^2 + 6300 \mu _1 \, \partial _\Theta {\mathcal {G}}_8 \right] \lambda ^7 \nonumber \\&+\frac{2 \sqrt{2}}{184{,}275 \mu _1^{3/4}} \left[ - 34{,}776 \, {\mathcal {G}}_2 \, \partial _\Theta {\mathcal {G}}_1 - 29{,}680\, {\mathcal {G}}_1\, \partial _\Theta {\mathcal {G}}_2 - 25{,}650\, {\mathcal {G}}_0\, \partial _\Theta {\mathcal {G}}_3 \right. \nonumber \\&+ 315 \, \partial _\Theta {\mathcal {G}}_2\, \partial ^2_\Theta {\mathcal {G}}_1 + 432 \, \partial _\Theta {\mathcal {G}}_1 \, \partial ^2_\Theta {\mathcal {G}}_2 + 315\, \csc ^2 \Theta \, \partial _ \Phi \mathcal{G}_2\, \partial ^2_{\Theta \Phi } {\mathcal {G}}_1 \nonumber \\&\left. + 432 \csc ^2 \Theta \, \partial _\Phi {\mathcal {G}}_1\, \partial ^2_{\Theta \Phi }{\mathcal {G}}_2 + 1008\, \cot \Theta \, \csc ^2 \Theta \, \partial _\Phi \mathcal{G}_1\, \partial _\Phi {\mathcal {G}}_2\right. \nonumber \\&\left. + 30{,}240 \mu _1\, \partial _\Theta {\mathcal {G}}_9\right] \lambda ^{{15}/2} \nonumber \\&+ \frac{1}{582{,}120 \mu _1^{2/3}}\left[ - 406{,}224 \, {\mathcal {G}}_3\, \partial _\Theta {\mathcal {G}}_1 - 347{,}424\, {\mathcal {G}}_2\, \partial _\Theta {\mathcal {G}}_2 - 301{,}840 \, {\mathcal {G}}_1\, \partial _\Theta {\mathcal {G}}_3 \right. \nonumber \\&-\, 263{,}340 \, {\mathcal {G}}_0\, \partial _\Theta {\mathcal {G}}_4 + 2640 \, \partial _\Theta {\mathcal {G}}_3\, \partial ^2_\Theta {\mathcal {G}}_1 + 3465 \, \partial _\Theta {\mathcal {G}}_2\, \partial ^2_\Theta {\mathcal {G}}_2 \nonumber \\&+\, 4752\, \partial _\Theta {\mathcal {G}}_1 \, \partial ^2_\Theta {\mathcal {G}}_3 + 2640 \, \csc ^2 \Theta \, \partial _\Phi {\mathcal {G}}_3\, \partial ^2_{\Theta \Phi } {\mathcal {G}}_1 + 3465\, \csc ^2 \Theta \, \partial _{\Phi }\mathcal{G}_2\, \partial ^2_{\Theta \Phi }{\mathcal {G}}_2 \nonumber \\&+\, 4752\, \csc ^2 \Theta \, \partial _\Phi {\mathcal {G}}_1\, \partial ^2_{\Theta \Phi } {\mathcal {G}}_3 + 4620\, \cot \Theta \, \csc ^2 \Theta \, (\partial _\Phi {\mathcal {G}}_2)^2\nonumber \\&\left. + \,9504\, \cot \Theta \csc ^2 \Theta \, \partial _\Phi {\mathcal {G}}_1\, \partial _\Phi {\mathcal {G}}_3 + 332{,}640 \mu _1 \, \partial _\Theta {\mathcal {G}}_{10} \right] \lambda ^8 \nonumber \\&+ \frac{4 \sqrt{2}}{883{,}575 \mu _1^{7/12}}\left[ -143{,}220 \,\mathcal{G}_4\,\partial _\Theta {\mathcal {G}}_1 - 122{,}640\, {\mathcal {G}}_3 \partial _\Theta {\mathcal {G}}_2 \right. \nonumber \\&-\, 106{,}920 \, \mathcal{G}_2\, \partial _\Theta {\mathcal {G}}_3 - 93{,}940\, {\mathcal {G}}_1\, \partial _\Theta {\mathcal {G}}_4 - 82{,}390 \, {\mathcal {G}}_0\, \partial _\Theta {\mathcal {G}}_5 \nonumber \\&+\, 693 \, \partial _\Theta {\mathcal {G}}_4 \, \partial ^2_\Theta {\mathcal {G}}_1 + 880\, \partial _\Theta {\mathcal {G}}_3\, \partial ^2_\Theta {\mathcal {G}}_2 \nonumber \\&+\, 1155\, \partial _\Theta {\mathcal {G}}_2\, \partial ^2_\Theta {\mathcal {G}}_3 + 1584 \, \partial _\Theta {\mathcal {G}}_1\, \partial ^2_\Theta {\mathcal {G}}_4 + 693\, \csc ^2 \Theta \, \partial _\Phi {\mathcal {G}}_4\, \partial ^2_{\Theta \Phi }{\mathcal {G}}_1 \nonumber \\&+\, 880\, \csc ^2 \Theta \, \partial _\Phi {\mathcal {G}}_3\, \partial ^2_{\Theta \Phi } \mathcal{G}_2 + 1155\, \csc ^2 \Theta \, \partial _\Phi {\mathcal {G}}_2\, \partial ^2_{\Theta \Phi } {\mathcal {G}}_3 \nonumber \\&+\, 1584\, \csc ^2 \Theta \, \partial _\Phi {\mathcal {G}}_1\, \partial ^2_{\Theta \Phi } \mathcal{G}_4 + 2640\, \cot \Theta \, \csc ^2 \Theta \, \partial _\Phi \mathcal{G}_2\, \partial _\Phi {\mathcal {G}}_3 \nonumber \\&+\, 2772 \, \cot \Theta \, \csc ^2 \Theta \, \partial _\Phi {\mathcal {G}}_1\, \partial _\Phi {\mathcal {G}}_4 \left. + 110{,}880 \mu _1 \, \partial _\Theta {\mathcal {G}}_{11} \right] \lambda ^{{17}/2} + \cdots \end{aligned}$$

(C.2)

$$\begin{aligned} \phi (\lambda )= & {} \Phi + \frac{4 \sqrt{2}}{35 \mu _1^{5/12}}\csc ^2\Theta \,\partial _\Phi {\mathcal {G}}_1\, \lambda ^{7/2} \nonumber \\&+ \frac{1}{6 \mu _1^{1/3}} \csc ^2\Theta \,\partial _\Phi {\mathcal {G}}_2\, \lambda ^4 + \frac{8 \sqrt{2} }{63 \mu _1^{1/4}}\csc ^2\Theta \, \partial _\Phi {\mathcal {G}}_3\, \lambda ^{9/2} \nonumber \\&+\,\frac{1}{5 \mu _1^{1/6}} \csc ^2\Theta \, \partial _\Phi {\mathcal {G}}_4\, \lambda ^5 + \frac{16 \sqrt{2}}{99 \mu _1^{1/{12}}} \csc ^2\Theta \, \partial _\Phi {\mathcal {G}}_5\, \lambda ^{{11}/2} \nonumber \\&+ \frac{4}{15} \csc ^2\Theta \,\partial _\Phi {\mathcal {G}}_6\, \lambda ^6 + \frac{2 \sqrt{2}}{2145 \mu _1^{11/12}}\csc ^2\Theta \, \left[ - 241\, {\mathcal {G}}_0 \,\partial _\Phi {\mathcal {G}}_1 \right. \nonumber \\&\left. +\, 240 \mu _1 \,\partial _\Phi {\mathcal {G}}_7 \right] \lambda ^{{13}/2} + \frac{2}{33{,}075 \mu _1^{5/6}} \csc ^2 \Theta \,\left[ - 6790\, {\mathcal {G}}_1\, \partial _\Phi {\mathcal {G}}_1 - 5775\, {\mathcal {G}}_0 \, \partial _\Phi \mathcal{G}_2 \right. \nonumber \\&+\, 90 \, \partial _\Theta {\mathcal {G}}_1\, \partial ^2_{\Theta \Phi } {\mathcal {G}}_1 + 90\, \csc ^2\Theta \, \partial _\Phi {\mathcal {G}}_1\, \partial ^2_\Phi {\mathcal {G}}_1 - 432 \, \cot \Theta \, \partial _\Phi {\mathcal {G}}_1\, \partial _\Theta {\mathcal {G}}_1 \nonumber \\ \end{aligned}$$

$$\begin{aligned}&\left. +\, 6300 \mu _1 \, \partial _\Phi {\mathcal {G}}_8 \right] \lambda ^7 + \frac{2 \sqrt{2}}{184{,}275 \mu _1^{3/4}} \csc ^2 \Theta \, \left[ - 34{,}776\, {\mathcal {G}}_2\,\partial _\Phi {\mathcal {G}}_1 \right. \nonumber \\&-\, 29{,}680 \, {\mathcal {G}}_1\,\partial _\Phi {\mathcal {G}}_2 - 25{,}650\, {\mathcal {G}}_0 \, \partial _\Phi {\mathcal {G}}_3 + 315\, \partial _\Theta {\mathcal {G}}_2\, \partial ^2_{\Theta \Phi }{\mathcal {G}}_1\nonumber \\&+\, 432 \, \partial _\Theta \mathcal{G}_1 \, \partial ^2_{\Theta \Phi } {\mathcal {G}}_2 + 315 \, \csc ^2 \Theta \, \partial _\Phi {\mathcal {G}}_2\, \partial ^2_\Phi {\mathcal {G}}_1 \nonumber \\&+\, 432\, \csc ^2 \Theta \, \partial _\Phi {\mathcal {G}}_1\, \partial ^2_{\Phi }{\mathcal {G}}_2 - 1872 \, \cot \Theta \, \partial _\Phi {\mathcal {G}}_2\, \partial _\Theta {\mathcal {G}}_1\nonumber \\&\left. -\, 1638 \, \cot \Theta \, \partial _\Phi {\mathcal {G}}_1 \, \partial _\Theta \mathcal{G}_2 + 30{,}240 \mu _1 \, \partial _\Phi {\mathcal {G}}_9 \right] \lambda ^{{15}/2}\nonumber \\ \end{aligned}$$

$$\begin{aligned}&+ \frac{1}{582{,}120 \mu _1^{2/3}}\,\csc ^2\Theta \left[ -406{,}224\, {\mathcal {G}}_3\, \partial _\Phi {\mathcal {G}}_1 - 347{,}424\, {\mathcal {G}}_2 \, \partial _\Phi {\mathcal {G}}_2 \right. \nonumber \\&-\, 301{,}840 \, {\mathcal {G}}_1 \, \partial _\Phi {\mathcal {G}}_3 - 263{,}340 \, {\mathcal {G}}_0 \, \partial _\Phi {\mathcal {G}}_4 + 2640\, \partial _\Theta {\mathcal {G}}_3\, \partial ^2_{\Theta \Phi }{\mathcal {G}}_1 \nonumber \\&+\, 3465 \, \partial _\Theta \mathcal{G}_2\, \partial ^2_{\Theta \Phi }{\mathcal {G}}_2 + 4752\, \partial _\Theta {\mathcal {G}}_1\, \partial ^2_{\Theta \Phi } {\mathcal {G}}_3 + 2640 \, \csc ^2 \Theta \, \partial _\Phi {\mathcal {G}}_3 \, \partial ^2_{\Phi } {\mathcal {G}}_1\nonumber \\&+\, 3465\, \csc ^2 \Theta \, \partial _\Phi \mathcal{G}_2 \, \partial ^2_{\Phi } {\mathcal {G}}_2 + 4752 \, \csc ^2 \Theta \, \partial _\Phi {\mathcal {G}}_1\, \partial ^2_{\Phi } {\mathcal {G}}_3 \nonumber \\ \end{aligned}$$

$$\begin{aligned}&-\, 19{,}008 \, \cot \Theta \, \partial _\Phi {\mathcal {G}}_3 \, \partial _\Theta \mathcal{G}_1 - 16{,}170 \, \cot \Theta \, \partial _\Phi {\mathcal {G}}_2 \, \partial _\Theta {\mathcal {G}}_2 \nonumber \\&\left. - \,14{,}784\, \cot \Theta \, \partial _\Phi {\mathcal {G}}_1\, \partial _\Theta {\mathcal {G}}_3 + 332{,}640 \mu _1\, \partial _\Phi {\mathcal {G}}_{10} \right] \lambda ^8\nonumber \\&+ \frac{4 \sqrt{2}}{883{,}575 \mu _1^{7/12}} \csc ^2 \Theta \,\left[ - 143{,}220\, {\mathcal {G}}_4\, \partial _\Phi {\mathcal {G}}_1 - 122{,}640 \, {\mathcal {G}}_3\, \partial _\Phi {\mathcal {G}}_2 \right. \nonumber \\&-\, 106{,}920 \, {\mathcal {G}}_2 \, \partial _\Phi {\mathcal {G}}_3 - 93{,}940\, {\mathcal {G}}_1 \, \partial _\Phi {\mathcal {G}}_4 - 82{,}390\, {\mathcal {G}}_0 \, \partial _\Phi {\mathcal {G}}_5 \nonumber \\&+\, 693 \, \partial _\Theta {\mathcal {G}}_4\, \partial ^2_{\Theta \Phi }{\mathcal {G}}_1 + 880\, \partial _\Theta {\mathcal {G}}_3\, \partial ^2_{\Theta \Phi }{\mathcal {G}}_2 + 1155\, \partial _\Theta \mathcal{G}_2 \, \partial ^2_{\Theta \Phi }{\mathcal {G}}_3 \nonumber \\ \end{aligned}$$

$$\begin{aligned}&+\, 1584 \, \partial _\Theta {\mathcal {G}}_1\, \partial ^2_{\Theta \Phi }{\mathcal {G}}_4 + 693 \,\csc ^2 \Theta \, \partial _\Phi {\mathcal {G}}_4\, \partial ^2_{ \Phi }{\mathcal {G}}_1 \nonumber \\&+\, 880\, \csc ^2 \Theta \, \partial _\Phi {\mathcal {G}}_3\, \partial ^2_{\Phi }{\mathcal {G}}_2 + 1155\,\csc ^2 \Theta \, \partial _\Phi {\mathcal {G}}_2 \, \partial ^2_{ \Phi }{\mathcal {G}}_3 \nonumber \\&+\, 1584 \, \csc ^2 \Theta \,\partial _\Phi {\mathcal {G}}_1\, \partial ^2_{\Phi }{\mathcal {G}}_4 - 5940 \, \cot \Theta \, \partial _\Phi {\mathcal {G}}_4\, \partial _\Theta {\mathcal {G}}_1 \nonumber \\&-\, 4950\, \cot \Theta \, \partial _\Phi {\mathcal {G}}_3 \, \partial _\Theta \mathcal{G}_2 - 4400 \, \cot \Theta \, \partial _\Phi {\mathcal {G}}_2 \, \partial _\Theta {\mathcal {G}}_3 \nonumber \\&\left. -\, 4158\, \cot \Theta \, \partial _\Phi {\mathcal {G}}_1\, \partial _\Theta {\mathcal {G}}_4 + 110{,}880 \mu _1 \, \partial _\Phi {\mathcal {G}}_{11} \right] \lambda ^{{17}/2} + \cdots \end{aligned}$$

(C.3)

$$\begin{aligned} T(\lambda , \Theta , \Phi )= & {} -\frac{\mu _1^{1/3}}{4}\lambda ^{-1} + \frac{7 }{12 \mu _1^{1/6}} {\mathcal {G}}_0\,\lambda ^2 + \frac{64 \sqrt{2} }{135 \mu _1^{1/{12}}} {\mathcal {G}}_1 \lambda ^{5/2} + \frac{4}{5} \mathcal{G}_2\,\lambda ^3 \nonumber \\&+ \frac{160 \sqrt{2} \mu _1^{1/{12}}}{231} \mathcal{G}_3 \, \lambda ^{7/2} + \frac{11 \mu _1^{1/6}}{9} \mathcal{G}_4\,\lambda ^4 \nonumber \\&+ \frac{128\sqrt{2} \mu _1^{1/4}}{117} \mathcal{G}_5 \, \lambda ^{9/2} + \frac{1}{1260 \mu _1^{2/3}} \left[ -259 \,{\mathcal {G}}_0^2 + 2496 \mu _1\, {\mathcal {G}}_6 \right] \lambda ^5 \nonumber \\&+\frac{224 \sqrt{2}}{4455 \mu _1^{7/12}}\left[ -7\, {\mathcal {G}}_0\, \mathcal{G}_1 + 36 \mu _1\, {\mathcal {G}}_7 \right] \lambda ^{{11}/2}\nonumber \\&+\frac{1}{255{,}150 \mu _1^{1/2}}\left[ -78{,}925 \,{\mathcal {G}}_1^2 - 153{,}090\, {\mathcal {G}}_0\, {\mathcal {G}}_2 + 13{,}851 \, (\partial _\Theta {\mathcal {G}}_1)^2 \right. \nonumber \\&\left. +\, 13{,}851 \, \csc ^2 \Theta \, (\partial _\Phi {\mathcal {G}}_1)^2 + 850{,}500 \mu _1\, {\mathcal {G}}_8 \right] \lambda ^6 \nonumber \\&+ \frac{16 \sqrt{2}}{765{,}765 \mu _1^{5/12}} \left[ - 25{,}872\, {\mathcal {G}}_1\, {\mathcal {G}}_2 - 24{,}325 \, {\mathcal {G}}_0 \, {\mathcal {G}}_3 + 4114\, \partial _\Theta {\mathcal {G}}_1\, \partial _\Theta {\mathcal {G}}_2 \right. \nonumber \\&\left. +\, 4114\, \csc ^2 \Theta \, \partial _\Phi {\mathcal {G}}_1\, \partial _\Phi {\mathcal {G}}_2 \right. \nonumber \\&\left. +\, 147{,}840 \mu _1\, {\mathcal {G}}_9 \right] \lambda ^{{13}/2} + \frac{2}{3{,}274{,}425 \mu _1^{1/3}}\left[ -790{,}944\, {\mathcal {G}}_2^2\right. \nonumber \\&\left. - 1{,}537{,}200 \, {\mathcal {G}}_1\, {\mathcal {G}}_3 \right. \nonumber \\&-\, 1{,}397{,}550\, {\mathcal {G}}_0 \,{\mathcal {G}}_4 + 109{,}725 \,(\partial _\Theta {\mathcal {G}}_2)^2 \nonumber \\&+\, 233{,}200\, \partial _\Theta {\mathcal {G}}_1\, \partial _\Theta {\mathcal {G}}_3 + 109{,}725 \,\csc ^2 \Theta \, (\partial _\Phi {\mathcal {G}}_2)^2 \nonumber \\&\left. +\, 233{,}200\,\csc ^2 \Theta \, \partial _\Phi {\mathcal {G}}_1\, \partial _\Phi {\mathcal {G}}_3 + 9{,}424{,}800 \mu _1 \, {\mathcal {G}}_{10}\right] \lambda ^7 \nonumber \\&+ \frac{16 \sqrt{2}}{115{,}540{,}425 \mu _1^{1/4}} \left[ -6{,}191{,}640\, {\mathcal {G}}_2\, {\mathcal {G}}_3 - 5{,}845{,}840 \, {\mathcal {G}}_1 \, {\mathcal {G}}_4 \right. \nonumber \\&- \,5{,}114{,}340 \, \mathcal{G}_0 \, {\mathcal {G}}_5 + 791{,}505\, \partial _\Theta {\mathcal {G}}_2 \, \partial _\Theta {\mathcal {G}}_3 + 880{,}308\, \partial _\Theta {\mathcal {G}}_1 \, \partial _\Theta {\mathcal {G}}_4 \nonumber \\&+\, 791{,}505\, \csc ^2\Theta \, \partial _\Phi {\mathcal {G}}_2 \, \partial _\Phi {\mathcal {G}}_3 + 880{,}308\, \csc ^2 \Theta \, \partial _\Phi {\mathcal {G}}_1 \, \partial _\Phi {\mathcal {G}}_4 \nonumber \\&\left. + 38{,}918{,}880 \mu _1 \, {\mathcal {G}}_{11} \right] \lambda ^{{15}/2} \nonumber \\&+ \frac{1}{624{,}323{,}700 \mu _1^{7/6}}\left[ 196{,}931{,}735\, {\mathcal {G}}_0^3 - 484{,}553{,}160 \mu _1\, {\mathcal {G}}_3^2 \right. \nonumber \\&-\,943{,}422{,}480 \mu _1\, {\mathcal {G}}_2 \, {\mathcal {G}}_4 - 863{,}736{,}720 \mu _1\, {\mathcal {G}}_1\, {\mathcal {G}}_5 \nonumber \\&-\, 721{,}440{,}720 \mu _1 \, {\mathcal {G}}_0\, {\mathcal {G}}_6 + 55{,}306{,}680 \mu _1 \, (\partial _\Theta \mathcal{G}_3)^2 \nonumber \\&+\, 115{,}846{,}731 \mu _1\, \partial _\Theta {\mathcal {G}}_2\, \partial _\Theta {\mathcal {G}}_4 + 133{,}429{,}296 \mu _1\, \partial _\Theta {\mathcal {G}}_1\, \partial _\Theta {\mathcal {G}}_5 \nonumber \\&+\, 55{,}306{,}680 \mu _1 \, \csc ^2 \Theta \, (\partial _\Phi \mathcal{G}_3)^2 + 115{,}846{,}731 \mu _1\, \partial _\Phi {\mathcal {G}}_2\, \partial _\Phi {\mathcal {G}}_4 \nonumber \\&\left. + \,133{,}429{,}296 \mu _1\, \partial _\Phi \mathcal{G}_1\, \partial _\Phi {\mathcal {G}}_5 + 6{,}326{,}480{,}160 \mu _1^2 \,{\mathcal {G}}_{12} \right] \lambda ^8 \nonumber \\&+ \frac{8 \sqrt{2}}{723{,}647{,}925 \mu _1^{13/12}}\left[ 77{,}322{,}245\, \mathcal{G}_0^2 \,{\mathcal {G}}_1 - 126{,}126{,}000 \mu _1\, {\mathcal {G}}_3\, {\mathcal {G}}_4 \right. \nonumber \\&-\, 119{,}528{,}640 \mu _1\, {\mathcal {G}}_2 \, {\mathcal {G}}_5 - 105{,}705{,}600 \mu _1\, {\mathcal {G}}_1\, {\mathcal {G}}_6 \nonumber \\&-\, 83{,}243{,}160 \mu _1\,\mathcal{G}_0\, {\mathcal {G}}_7 + 13{,}427{,}700 \mu _1\, \partial _\Theta {\mathcal {G}}_3\, \partial _\Theta {\mathcal {G}}_4 \nonumber \\&+\, 14{,}578{,}200 \mu _1 \, \partial _\Theta {\mathcal {G}}_2 \, \partial _\Theta {\mathcal {G}}_5 + 17{,}256{,}096 \mu _1\, \partial _\Theta {\mathcal {G}}_1\, \partial _\Theta {\mathcal {G}}_6\nonumber \\&+\, 13{,}427{,}700 \mu _1\, \csc ^2\Theta \, \partial _\Phi {\mathcal {G}}_3\, \partial _\Phi {\mathcal {G}}_4 \nonumber \\&+\, 14{,}578{,}200 \mu _1 \, \csc ^2 \Theta \, \partial _\Phi {\mathcal {G}}_2 \, \partial _\Phi \mathcal{G}_5\nonumber \\&+ 17{,}256{,}096 \mu _1\, \csc ^2 \Theta \, \partial _\Phi \mathcal{G}_1\, \partial _\Phi {\mathcal {G}}_6 \nonumber \\&\left. + 864{,}864{,}000 \mu _1^2\, \mathcal{G}_{13} \right] \lambda ^{{17}/2} + \cdots \end{aligned}$$

(C.4)

Regularity condition

We begin with (3.35), the condition that the f, g, h need to satisfy to have the metric to be non-singular on the horizon:

$$\begin{aligned}&f^2 (q_6^2 - z_2 z_3) + g^2 (q_4^2 - z_3 z_1) + h^2 (q_2^2 - z_1 z_2) + 2 f g (q_4 q_6 - q_2 z_3) \nonumber \\&\quad - 2 g h (q_2 q_4 - q_6 z_1) + 2 f h (q_6 q_2 - q_4 z_2) \ne 0. \end{aligned}$$

(D.1)

This is the determinant of the following $4\times 4$ matrix (the metric for the “AdS” part), evaluated at the horizon i.e. at $\lambda =0$.

$$\begin{aligned} M = \left( \begin{array}{cccc} 0 &{}\quad -g &{}\quad -h &{}\quad f \\ -g &{}\quad z_2 &{}\quad q_6 &{}\quad q_2 \\ -h &{}\quad q_6 &{}\quad z_3 &{}\quad q_4 \\ f &{}\quad q_2 &{}\quad q_4 &{}\quad z_1 \\ \end{array} \right) \end{aligned}$$

(D.2)

The determinant of M, given by (D.1) can be written as a quadratic form

$$\begin{aligned} \det (M) = -F^{T}S F \end{aligned}$$

(D.3)

with

$$\begin{aligned} F= \left( \begin{array}{ccc} -g&-h&f \end{array} \right) , \qquad S = \left( \begin{array}{ccc} A &{} \quad D &{}\quad G \\ D &{}\quad E &{}\quad H \\ G &{}\quad H &{}\quad I \\ \end{array} \right) . \end{aligned}$$

(D.4)

Here matrix S is the adjugate (or classical adjoint) corresponding to lower $3\times 3$ block in M. It’s a singular matrix. We can see why this matrix appears as follows. If $g_{ab}$ is a symmetric matrix then following identity holds

$$\begin{aligned} \det \left( \begin{array}{ccc} x &{} \quad v_b\\ w_a &{}\quad g_{ab} \\ \end{array} \right) = (x -v_b g^{bc}w_{c}) \det [g_{ab}]. \end{aligned}$$

(D.5)

Applying this identity to matrix M, we get that $\det [M] = -F^{T}S F$. Notice that lower $3\times 3$ block in M i.e. $g_{ab}$ is a singular matrix whose inverse $g^{ab}$ is not defined.

$$\begin{aligned} g= \left( \begin{array}{ccc} z_2 &{}\quad q_6 &{}\quad q_2 \\ q_6 &{}\quad z_3 &{}\quad q_4 \\ q_2 &{}\quad q_4 &{}\quad z_1 \\ \end{array} \right) = \left( \begin{array}{ccc} f_X &{}\quad g_X &{}\quad h_X \\ f_Y &{}\quad g_Y &{}\quad h_Y \\ f_v&{}\quad g_v &{}\quad h_v \\ \end{array} \right) \left( \begin{array}{ccc} -f_X &{}\quad -f_Y &{}\quad -f_v \\ g_X &{}\quad g_Y &{}\quad g_v \\ h_X &{}\quad h_Y &{}\quad h_v \\ \end{array} \right) \end{aligned}$$

(D.6)

Here subscript X, Y, v below f, g, h denotes partial derivative with respect to that variable. Since there is a functional relation between f, g and h, the determinant of these matrices vanish individually. To put it another way, take various partial derivatives of relation $f^2 -g^2 -h^2 =1$ and demand a non-trivial solution for the resulting linear equations. But $g^{ab}Det[g_{ab}]$ still gives the adjugate matrix S. We can think of first working with a non-zero $\lambda$ so that inverse is well defined and then taking the limit $\lambda \rightarrow 0$.

We will analyze this matrix S in detail. Matrix elements are as follows:

$$\begin{aligned} A= & {} z_1z_3 -q_{4}^{2}, \quad D= q_2q_4 - z_1 q_6, \quad E= z_1z_2 -q_{2}^{2} \nonumber \\ I= & {} z_2z_3 -q_{6}^{2}, \quad H= q_2q_6 - z_2 q_4, \quad G= q_6q_4 - z_3q_2 \ \end{aligned}$$

(D.7)

Elements of S satisfy some identities which can be easily checked:

$$\begin{aligned} AE -D^2 = z_1 \det [g] , \quad IA- G^2 = z_3 \det [g], \quad DG-AH= q_4 \det [g] \nonumber \\ EI-H^2 = z_2 \det [g], \quad DH-GE= q_2 \det [g] , \quad GH-ID= q_6 \det [g] \nonumber \\ Az_2 + Dq_6 + Gq_2 = \det [g], \quad Dz_2 + Eq_6 + Hq_2 =0 , \quad Gz_2 + Hq_6 + Iq_2 =0\nonumber \\ \end{aligned}$$

(D.8)

In our case $\det [g]=0$ and so some of the above identities simplify. Eigenvalues $\alpha$ of S are given by the characteristic equation

$$\begin{aligned} \det [S] -c_2 \alpha + Tr(S)\alpha ^2 -\alpha ^3 =0 \end{aligned}$$

(D.9)

where Cayley-Hamilton theorem gives

$$\begin{aligned} c_2 = \frac{1}{2}\left( (Tr(S))^2 - Tr(S^2)\right) = AE + AI + EI -(D^2+G^2+H^2). \end{aligned}$$

(D.10)

Since $\det [S]=0$ and by the above identities, $c_2=0$, we have only one non-zero eigenvalue $\alpha = Tr(S)$. Define 3-vectors

$$\begin{aligned} \vec {K} = \nabla g \times \nabla h,\quad \vec {L} = \nabla f \times \nabla g,\quad \vec {M} = \nabla f \times \nabla h \end{aligned}$$

(D.11)

In terms of these, we can write, after little bit of algebra,

$$\begin{aligned} z_2z_3 -q_6^2= & {} K_v^2 -L_v^2 -M_v^2,\quad z_1z_3 -q_4^2 = K_X^2 -L_X^2 -M_X^2,\nonumber \\ z_2z_1 -q_2^2= & {} K_Y^2 -L_Y^2 -M_Y^2. \end{aligned}$$

(D.12)

So the non-zero eigenvalue

$$\begin{aligned} \alpha = Tr(S) =A+ E+ I = \vec {K}\cdot \vec {K} -\vec {L}\cdot \vec {L}-\vec {M}\cdot \vec {M} \end{aligned}$$

(D.13)

Using the condition (D.1), we can write $\vec {L} = \frac{-h}{f}\vec {K}$ and $\vec {M} = \frac{g}{f}\vec {K}$ since $\nabla f = \frac{g}{f}\nabla g + \frac{h}{f}\nabla h$. Using this, we see that $\alpha = \frac{\vec {K}\cdot \vec {K}}{f^2}$. So if $\vec {K}$ is zero then all eigenvalues are zero and hence the quadratic form vanishes. We can determine the eigenvector corresponding to $\alpha$. This is given as a solution to

$$\begin{aligned} \left( \begin{array}{ccc} A &{}\quad D &{}\quad G \\ D &{}\quad E &{}\quad H \\ G &{}\quad H &{}\quad I \\ \end{array} \right) \left( \begin{array}{ccc} x_1&\quad x_2&\quad x_3 \end{array} \right) = \alpha \left( \begin{array}{ccc} x_1&\quad x_2&\quad x_3 \end{array} \right) \end{aligned}$$

(D.14)

Since A, E, I have same sign, we can write $D=\sqrt{AE}$, $G=\sqrt{AI}$ and $H=\sqrt{EI}$. Then eigenvector is $(\sqrt{A},\sqrt{E},\sqrt{I}$. If we use the matrix of eigenvectors to diagonalize S then quadratic form can be written as

$$\begin{aligned} \det [M]= & {} (f\sqrt{A} -g\sqrt{E} - h\sqrt{I})^2 = (f\sqrt{(-q_6^2 + z_2 z_3)} \nonumber \\&- g\sqrt{(-q_4^2 + z_1 z_3)} -h\sqrt{(-q_2^2 + z_2 z_1)})^2. \end{aligned}$$

(D.15)

Now we know that a quadratic form which doesn’t change sign (since $\det [M]$ is always positive, this condition is satisfied for our case) can only be zero on the null space of the matrix S. So we can have determinant M zero only if F belongs to null space of S. We check that unless f, g, h are zero, this is not the case. An example of eigenvector with zero eigenvalue, using identities (D.8) is $(z_2,q_6,q_2)$.

So the only way for determinant of M to vanish is if $\vec {K} = \nabla g \times \nabla h =0$. We can see that if f, g, h are constants or if all are functions of only one variable then $\det [M]$ would vanish and metric would be singular. Some possible choices of f, g, h are given in (3.37).