$\hat{Z}$ at large $N$: from curve counts to quantum modularity

Abstract

Reducing a 6d fivebrane theory on a 3-manifold $Y$ gives a $q$-series 3-manifold invariant $\hat{Z}(Y)$. We analyse the large-$N$ behaviour of $F_K=\hat{Z}(M_K)$, where $M_K$ is the complement of a knot $K$ in the 3-sphere, and explore the relationship between an $a$-deformed ($a=q^N$) version of $F_K$ and HOMFLY-PT polynomials. On the one hand, in combination with counts of holomorphic annuli on knot complements, this gives an enumerative interpretation of $F_K$ in terms of counts of open holomorphic curves. On the other, it leads to closed form expressions for $a$-deformed $F_K$ for $(2,2p+1)$-torus knots. They suggest a further $t$-deformation based on superpolynomials, which can be used to obtain a $t$-deformation of ADO polynomials, expected to be related to categorification. Moreover, studying how $F_K$ transforms under natural geometric operations on $K$ indicates relations to quantum modularity in a new setting.

42 pages, 4 figures

References in corpus (10)

• The Refined Topological Vertex
• On the characteristic and deformation varieties of a knot
• Large N Duality, Mirror Symmetry, and a Q-deformed A-polynomial for Knots
• Refined Chern-Simons Theory and Topological String
• BPS states, knots and quivers
• Rozansky-Witten geometry of Coulomb branches and logarithmic knot invariants
• Large color $R$-matrix for knot complements and strange identities
• 3d-3d correspondence for mapping tori
• Multi-cover skeins, quivers, and 3d $\mathcal{N}=2$ dualities
• Quivers for 3-manifolds: the correspondence, BPS states, and 3d $\mathcal{N}$=2 theories

Cited by in corpus (19)

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• Seiberg-like dualities for orthogonal and symplectic 3d $\mathcal{N}=2$ gauge theories with boundaries
• Knots, Perturbative Series and Quantum Modularity
• Knots and Their Related $q$-Series
• Harer-Zagier formulas for knot matrix models
• BPS Invariants for a Knot in Seifert Manifolds
• Quiver diagonalization and open BPS states
• Revisiting the Melvin-Morton-Rozansky Expansion, or There and Back Again
• Knot Complement, ADO-Invariants and their Deformations for Torus Knots
• Quantum Modular $\hat{Z}{}^G$-Invariants
• On knots, complements, and 6j-symbols
• Three Dimensional Topological Field Theories and Nahm Sum Formulas
• Inverted state sums, inverted Habiro series, and indefinite theta functions
• Permutohedra for knots and quivers
• Refined and Generalized $\hat{Z}$ Invariants for Plumbed 3-Manifolds
• A supergroup series for knot complements
• On the ${\Delta }_a$ invariants in non-perturbative complex Chern-Simons theory
• $c_{\mathrm{eff}}$ from Resurgence at the Stokes Line