A proof of a conjecture of Gukov-Pei-Putrov-Vafa

TL;DR

This paper proves a conjecture linking Witten-Reshetikhin-Turaev invariants to homological blocks for plumbed 3-manifolds, using an inductive approach on tree structures to establish holomorphy.

Contribution

It provides the first proof of the conjecture for a broad class of 3-manifolds by developing an inductive method to handle holomorphy challenges.

Findings

Witten-Reshetikhin-Turaev invariants are limits of homological blocks.

The proof applies to plumbed 3-manifolds with negative definite linking matrices.

Inductive pruning of trees is key to establishing holomorphy.

Abstract

In this paper, we prove Gukov-Pei-Putrov-Vafa's conjecture that the Witten-Reshetikhin-Turaev invariants are radial limits of homological blocks, which are $ q $-series introduced by them for plumbed $ 3 $-manifolds with negative definite linking matrices. In our previous work, the author attributed this conjecture to the holomorphy of certain rational 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."