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Periods, the meromorphic 3D-index and the Turaev--Viro invariant

The 3D-index of Dimofte-Gaiotto-Gukov is an interesting collection of $q$-series with integer coefficients parametrised by a pair of integers and associated to a 3-manifold with torus boundary. In this note, we explain the structure of the asymptotic expansions of the 3D-index when $q=e^{2πiτ}$ and $τ$ tends to zero (to all orders and with exponentially small terms included), and discover two phenomena: (a) when $τ$ tends to zero on a ray near the positive real axis, the horizontal asymptotics of the meromorphic 3D-index match to all orders with the asymptotics of the Turaev-Viro invariant of a knot, in particular explaining the Volume Conjecture of Chen-Yang from first principles, (b) when $τ\to 0$ on the positive imaginary axis, the vertical asymptotics of the 3D-index involves periods of a plane curve (the $A$-polynomial), as opposed to algebraic numbers, explaining some predictions of Hodgson-Kricker-Siejakowski and leading to conjectural identities between periods of the $A$-polynomial of a knot and integrals of the Euler beta-function.