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 $T_{10}{M}_{\mathrm{eff}}(11,2)$, 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.

70 pages