A Proof of a Conjecture of Gukov–Pei–Putrov–Vafa

In the context of 3-manifolds, determining the asymptotic expansion of the Witten–Reshetikhin–Turaev invariants and constructing the topological field theory that provides their categorification remain important unsolved problems. Motivated by solving these problems, Gukov–Pei–Putrov–Vafa refined the Witten–Reshetikhin–Turaev invariants from a physical point of view. From a mathematical point of view, we can describe that they introduced new q-series invariants for negative definite plumbed manifolds and conjectured that their radial limits coincide with the Witten–Reshetikhin–Turaev invariants. In this paper, we prove their conjecture. In our previous work, the author attributed this conjecture to the holomorphy of certain meromorphic functions by developing an asymptotic formula based on the Euler–Maclaurin summation formula. However, it is challenging to prove holomorphy for general plumbed manifolds. In this paper, we address this challenge using induction on a sequence of trees obtained by repeating “pruning trees,” which is a special type of the Kirby moves.