i([\tfrac{1}{2},0]_{\ell+\frac{5}{2}}^{\frac{\ell+1}{2},\frac{\ell}{2},+\frac{1}{2}})$
$\displaystyle=-\frac{qp(x^{\ell}+\cdots+y^{\ell})}{(1-q)(1-p)}$
$\displaystyle(\ell=0,1,2,\ldots).$ (217)

By summing up all contributions, we obtain

$\displaystyle f_{\rm vector}^{AdS_{5}\times
S^{3}}=\frac{xy(1-z)}{(1-q)(1-p)(1-x)(1-y)}.$ (218)

###  B.4 Vector multiplet on $AdS_{3}\times S^{5}$

The fixed locus of O7-flip (112)

$\displaystyle\widetilde{U}_{O7}=e^{\pi i(J_{2}+A)}$ (219)

is $AdS_{3}\times S^{5}$. Let us use the notation

$\displaystyle[J_{1}]_{H}^{R_{SU(4)},J_{2}}$ (220)

to represent the quantum numbers. $R_{SU(4)}$ is given by the Dynkin labels.
The mode expansion in $AdS_{3}\times S^{5}$ gives the conformal
representations with the following primaries.

$\displaystyle\begin{array}[]{ccccc}&&[-1]_{\ell+3}^{[0,\ell-2,0],0}\\\
{}\hfil&[-\frac{1}{2}]_{\ell+\frac{5}{2}}^{[0,\ell-2,1],+\frac{1}{2}}&&[-\frac{1}{2}]_{\ell+\frac{5}{2}}^{[1,\ell-2,0],-\frac{1}{2}}\\\
{}[0]_{\ell+2}^{[0,\ell-1,0],+1}&&[0]_{\ell+2}^{[1,\ell-2,1],0}&&[0]_{\ell+2}^{[0,\ell-1,0],-1}\\\
{}\hfil&\underline{[+\frac{1}{2}]_{\ell+\frac{3}{2}}^{[1,\ell-1,0],+\frac{1}{2}}}&&[+\frac{1}{2}]_{\ell+\frac{3}{2}}^{[0,\ell-1,1],-\frac{1}{2}}\\\
{}\hfil&&\underline{[+1]_{\ell+1}^{[0,\ell,0],0}}\end{array}$ (226)

Dynkin labels $[R_{1},R_{2},R_{3}]$ correspond to the following $R$-charges.

$\displaystyle(R_{x},R_{y},R_{z})=R_{1}(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2})+R_{2}(1,0,0)=R_{3}(\tfrac{1}{2},\tfrac{1}{2},-\tfrac{1}{2})$
(227)

Underlined conformal representations contribute to the index.

$\displaystyle i([+1]_{\ell+1}^{[0,\ell,0],0})$
$\displaystyle=\frac{q(x^{\ell}+\cdots+y^{\ell}+\cdots+z^{\ell})}{1-q}$
$\displaystyle(\ell=0,1,2,\ldots),$ $\displaystyle
i([+\tfrac{1}{2}]_{\ell+\frac{3}{2}}^{[1,\ell-1,0],+\frac{1}{2}})$
$\displaystyle=-\frac{pq(x^{\ell-1}+\cdots+y^{\ell-1}+\ldots+z^{\ell-1})}{(1-q)}$
$\displaystyle(\ell=1,2,3,\ldots).$ (228)

Summing up all contributions, we obtain

$\displaystyle f_{\rm vector}^{AdS_{3}\times
S^{5}}=\frac{q(1-p)}{(1-q)(1-x)(1-y)(1-z)}.$ (229)

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