Abstract
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,ℤ) Weil representations, quantum modular forms, non-semisimple modular tensor categories, and chiral algebras of logarithmic CFTs.
Highlights
Introduction and summaryThis work relies on the interplay between different fields of research, including topology, physics and number theory
We find and propose an explanation for a large variety of modularity-related symmetries in problems of 3-manifold topology and physics of 3d N = 2 theories where such structures a priori are not manifest
The central object is a certain family of infinite q-series “Zb(q)”, which plays the role of supersymmetric indices, topological invariants, and quantum modular forms in physics, topology, and number theory respectively
Summary
This work relies on the interplay between different fields of research, including topology, physics and number theory. One of our main results in this paper is that q-series invariants Za(M3) have a “hidden structure,” namely the structure of a projective SL(2, Z) representation, distinct from the role(s) modular group played in this context so far [4, 6] This new structure leads to powerful predictions:. The resulting q-series expressions produced by our physical/topological “factory” turn out to be false theta functions, and their relevance to our problems lies in their quantum modular structure. Supersymmetry allows to define a protected quantity, it helps to compute it, via localization techniques in the regime of weak coupling This leads to an expression for the half-index in terms of the contour integral (in the complexified Cartan of the gauge group): Za =. Where the two factors in the integrand, F3d(x) and Θ(2ad)(x), correspond to the contributions of 3d theory and 2d boundary degrees of freedom, respectively
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