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

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• Stavros Garoufalidis (Max Planck Institute for Mathematics)
• Thursday 15 September 2022, 09:00-10:00
• No Room Required.

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AR2W01 - Physical resurgence: On quantum, gauge, and stringy

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 talk we explain the structure of the asymptotic expansions of the 3D-index when $q=e^{2\pi i\tau}$ and $\tau$ tends to zero (to all orders and with exponentially small terms included), and discover two phenomena: (a) when $\tau$ 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 $\tau \to 0$ 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. Joint work with Campbell Wheeler.

This talk is part of the Isaac Newton Institute Seminar Series series.

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