$q$-Virasoro modular double and 3d partition functions

Abstract

We study partition functions of 3d $\mathcal{N}=2$ U(N) gauge theories on compact manifolds which are $S^1$ fibrations over $S^2$. We show that the partition functions are free field correlators of vertex operators and screening charges of the $q$-Virasoro modular double, which we define. The inclusion of supersymmetric Wilson loops in arbitrary representations allows us to show that the generating functions of Wilson loop vacuum expectation values satisfy two SL(2,$\mathbb{Z}$)-related commuting sets of $q$-Virasoro constraints. We generalize our construction to 3d $\mathcal{N}=2$ unitary quiver gauge theories and as an example we give the free boson realization of the ABJ(M) model.

v1: 59 pages, 3 figures. v2: corrected few typos, references added. v3: corrected few typos, published version

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• 3d Mirror Symmetry from S-duality
• Solving q-Virasoro constraints
• Rectangular superpolynomials for the figure-eight knot
• Intersecting Defects and Supergroup Gauge Theory
• Quantization of Harer-Zagier formulas
• Bootstrapping the $S^5$ partition function
• Quiver ${\text{W}}_{{\epsilon }_1,{\epsilon }_2}$ algebras of 4d $\mathcal{N}=2$ gauge theories
• Exact SUSY Wilson loops on $S^3$ from $q$-Virasoro constraints
• Harer-Zagier formulas for knot matrix models
• Lens partition function, pentagon identity and star-triangle relation
• Virasoro Constraint for Uglov Matrix Model
• Intersecting Surface defects and 3d Superconformal indices
• On Bailey pairs for $\mathcal{N}=2$ supersymmetric gauge theories on $S_b^3/{\mathbb{Z}}_r$