3d $\mathcal N=4$ rank-zero mirror symmetry, TQFT interfaces, and Zagier duality of Nahm sums

Abstract

Starting from the $(E_8,T_1)$ Nahm sum for the vacuum character of $T_{10}M_{\rm eff}(11,2)$, we construct a three-dimensional (3d) $\mathcal N=2$ $U(1)^8$ Chern--Simons matter theory $\mathcal T$ with level matrix $C_{E_8}$. Its $A$-twisted vacuum half-index reproduces the character exactly, while four distinguished Wilson loop half-indices agree with the remaining characters to the orders checked. Applying particle--vortex duality and gauging yields a second 3d $\mathcal N=2$ Chern--Simons matter theory $\mathcal T^\vee$ with level matrix $C_{E_8}^{-1}$. Exact superconformal-index matching supports the claim that the two theories flow to 3d $\mathcal N=4$ rank-zero mirror SCFTs. For the $(\mathcal D,D_c)$ boundary condition, the mirror map between the $A$-twisted Wilson loop half-indices of $\mathcal T$ and the $B$-twisted Wilson loop half-indices of $\mathcal T^\vee$ coincides componentwise with the Zagier transformation, realizing a Zagier duality between the complete $(E_8,T_1)$ and $(T_1,E_8)$ Nahm systems. Using the Bethe-equation formalism, we show that the corresponding $A/B$-twisted TQFT data agree up to orientation reversal and the $E_8$ framing phase $e^{-2\pi \mathrm{i}/3}$. The duality interface gives the torus bilinear $Z_{\mathcal I}(\tau)=\sum_i\chi_i(\tau)\chi_i^\vee(\tau)=\chi_{(E_8)_1}(\tau)$, where $\{\chi_i\}$ and $\{\chi_i^\vee\}$ are the corresponding five-component character bases. Thus Zagier duality is realized as part of a concrete 3d mirror and interface structure.