3D-partition functions on the sphere: exact evaluation and mirror symmetry

TL;DR

We develop a localisation-based framework to compute exact $S^3$ partition functions for 3d $\mathcal{N}=4$ generalised quiver theories, expressing them in terms of building blocks such as $T(SU(N))$ tails and $T_N$ blocks. A central result is the closed form for $\mathcal{Z}^{T(SU(N))}$, which makes mirror symmetry manifest and enables finite-$N$ computations of star-shaped quivers and non-Lagrangian blocks. We perform non-perturbative checks of mirror symmetry across families of Lagrangian theories and, assuming mirror symmetry, derive the partition function of the non-Lagrangian $T_N$ theories (including explicit $T_3$). The work also demonstrates a 2d TQFT-like gluing structure for punctured spheres, validated by associativity relations and dual gluing routes, with potential implications for Wilson-loop observables in these 3d theories.

Abstract

We study N = 4 quiver theories on the three-sphere. We compute partition functions using the localisation method by Kapustin et al. solving exactly the matrix integrals at finite N, as functions of mass and Fayet-Iliopoulos parameters. We find a simple explicit formula for the partition function of the quiver tail T(SU(N)). This formula opens the way for the analysis of star-shaped quivers and their mirrors (that are the Gaiotto-type theories arising from M5 branes on punctured Riemann surfaces). We provide non-perturbative checks of mirror symmetry for infinite classes of theories and find the partition functions of the TN theory, the building block of generalised quiver theories.

Figures (12)

• Figure 1: $U(N)$ with $K$ flavours.
• Figure 2: The $U(1)$ theory with $N$ flavours.
• Figure 3: $SU(2)$ with $K$ flavours.
• Figure 4: the $D_k$ quiver
• Figure 5: $SU(2)\times U(1)^N$.