Rethinking mirror symmetry as a local duality on fields

We introduce an algorithm to piecewise dualize linear quivers into their mirror dual. The algorithm uses two basic duality moves and the properties of the $S$-wall which can all be derived by iterative applications of Seiberg-like dualities.

Figure 1

On the left the $3d$ $T_{\sigma }^{\rho }[SU(N)]$ theory, round nodes denote gauge groups and square boxes flavor groups all of $U(n)$ type. Lines connecting them represent chiral fields. On the right the $4d$ $E_{\rho }^{\sigma }[USp(2N)]$ theory (up to singlets), now all nodes denote groups of $USp(2n)$ type. The ranks $N_i$, $M_j$ are given in terms of the partitions $\rho$, $\sigma$. As explained in [10] reducing the $E_{\rho }^{\sigma }[USp(2N)]$ to $3d$ and turning two deformations that first break the $USp(2n)$ groups to $U(n)$ and then give mass to the fields of the saw (the $USp(2N_i)\times SU(2)$ bifundamentals) we flow to the $T_{\rho }^{\sigma }[SU(N)]$ theory.

Figure 2

The $FE[USp(2N)]$ quiver and its compact representation displaying the manifest and emergent $USp(2N)$ symmetries and the Abelian $U(1{)}_c$ symmetry fugacity. The powers of $t$ and $c$ encode the charges of the fields under $U(1{)}_t$ and $U(1{)}_c$. The crosses denote gauge singlets flipping the diagonal mesons. On the right-hand side (rhs) the asymmetric $S$-wall.

Figure 3

Gluing two $S$-walls yields the identity wall.

Figure 4

On the left, the basic move for the dualization of a bifundamental block including the $SU(2{)}_v$ fundamental chirals. On the right, the basic move for the dualization of a block of $2L$ fundamentals including an Identity wall. For chiral fields the powers of fugacities represent the charges under the corresponding Abelian symmetries and the R-charges are encoded in $(pq{)}^{\frac{R}{2}}$.

Figure 5

An example of the dualization algorithm. Dashed lines denote singlets transforming under the non-Abelian global symmetry that are present from the beginning, while dotted lines denote singlets that are produced during the dualization. The alternating dashed-dotted line at step 4 denotes a pair of chirals with charges respectively 0 and 1 under $U(1{)}_t$ only. We also specify the fugacities of the $i$th gauge node as ${\overrightarrow{z}}^{(i)}$ to facilitate the comprehension of how the block decomposition is performed. We don’t draw singlets uncharged under the non-Abelian symmetries except for the initial and final frame, where they are represented by crosses.