3D mirror symmetry from $S$-duality

We consider type IIB $\mathrm{SL}(2,\mathbb{Z})$ symmetry to relate the partition functions of different 5d supersymmetric Abelian linear quiver Yang-Mills theories in the $\mathrm{\Omega }$-background and squashed ${\mathbb{S}}^5$ background. By Higgsing $S$-dual theories, we extract new and old 3d mirror pairs. Generically, the Higgsing procedure yields 3d defects on intersecting spaces, and we derive new hyperbolic integral identities expressing the equivalence of the squashed ${\mathbb{S}}^3$ partition functions with additional degrees of freedom on the ${\mathbb{S}}^1$ intersection.

Figure 1

From left to right, the diagrams correspond to 5d U(1) gauge theories with 2 hypers, a free theory, and $\mathrm{U}(1{)}^2$ quiver theory with one bifundamental hyper multiplet.

Figure 2

$S$-duality between the two $(p,q)$-webs.

Figure 3

The brane moves of a simple type of Higgsing applied to a 5d linear quiver gauge theory. The NS5s fill the 012348 directions, while the D5s fill the 012347 directions, hence 78 represents the $(p,q)$-plane (here we are slightly simplifying the picture). The D3s are stretched along the 6 direction and fill also the 012 and/or 034 directions, hence they all share a common direction and are supported on two orthogonal planes inside the 5-brane world volumes.

Figure 4

The quiver structure of the 5d/3d/1d theory describing the 5d theory in the presence of intersecting codimension 2 defects. The hyper or chiral multiplets supported on spheres of different dimensions are indicated by their colors.

Figure 5

Another duality that can be obtained by integrating over the FI parameter, viewed as the “mass” for the topological U(1) symmetry.

Figure 6

Quiver world volume theories of intersecting codimension 2 defects following from Higgsing twice. The purple arrows denote bi-fundamental 1d chiral multiplets, while the blue dotted lines denote 1d Fermi multiplets.

Figure 7

A large Young diagram with subdiagrams $Y^{\mathrm{L}}$ and $Y^{\mathrm{R}}$.

Figure 8

Refined topological vertices.

Figure 9

DIM intertwining operators.

Figure 10

Action of the $S$ element in $\mathrm{SL}(2,\mathbb{Z})$ (for $N=0$).

Figure 11

The third triality frame for the resolved conifold.