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3d Modularity
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3d Modularity
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We find and propose an explanation for a large variety of modularity-related symmetries in problems of 3-manifold topology and physics of 3d $\mathcal{N}=2$ theories where such structures a priori are not manifest. These modular structures include: mock modular forms, $SL(2,\mathbb{Z})$ Weil representations, quantum modular forms, non-semisimple modular tensor categories, and chiral algebras of logarithmic CFTs.
Forward citations
Cited by 10 Pith papers
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Orientation Reversal and the Chern-Simons Natural Boundary
Resurgence provides a unique analytic continuation across natural boundaries for Chern-Simons q-series that matches 3-manifold orientation reversal via Mordell integral decompositions.
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A supergroup series for knot complements
Defines the three-variable superalgebra series F_K(y,z,q) for knot complements, derives its surgery relation to hat Z(q), and computes examples for torus knots.
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Fermionic extensions of $W$-algebras via 3d $\mathcal{N}=4$ gauge theories with a boundary
Constructs fermionic extensions of W-algebras W^{-N+1}(sl_N, f_sub) via BRST cohomology in 3d N=4 abelian gauge theories and explicitly computes the N=3 case as an extension of the Bershadsky-Polyakov algebra.
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3d-3d correspondence for knot complements with finite and large $N$
The SU(N) homological block in inverted-Habiro form equals a half-index of an explicit 3d N=2 theory for the figure-eight and the two trefoil knots, with a conjectural all-knot extension.
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Weak-Strong Resurgence Duality
Establishes explicit resurgent duality between zero-radius weak-coupling and infinite-radius strong-coupling expansions, illustrated on Airy/Pearcey integrals and applied to phi^4 Dyson-Schwinger equations and Gross-N...
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On Uniqueness of Mock Theta Functions
Mock theta functions admit a unique resurgent continuation across their natural boundary, with the continuation fixed by their Mordell-Appell integrals via rotated Laplace contours.
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On non-relativistic integrable models and 4d SCFTs
Generalized Schur indices of N=2 class S theories are expressed using eigenfunctions of non-relativistic elliptic Calogero-Moser models, with extensions claimed for N=1 SCFTs via limits of models like Inozemtsev.
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3d-3d correspondence and abelian flat connection
The homological block of a knot complement is realized as a half-index of a 3d N=2 theory via a contour enclosing z=q^k poles, and the same integral at z=q^-k poles gives the colored Jones polynomial.
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$c_{\rm eff}$ from Resurgence at the Stokes Line
Resurgent cyclic orbits' algebraic structure plus the leading q-series term determines the asymptotic growth exponent of dual q-series coefficients, which equals an effective central charge c_eff in a related 3d N=2 QFT.
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Quantum invariants of 3-manifolds and links: a review
This is a survey, not a new result: it reviews the q-series invariants Zhat, F_K, F_L and their supergroup analogues, collecting known conjectures, theorems, and examples.
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