Yuya Murakami’s research while affiliated with Kyushu University and other places

We introduce L-functions attached to negative-definite plumbed manifolds as the Mellin transforms of homological blocks. We prove that they are entire functions and their values at s=0 are equal to the Witten–Reshetikhin–Turaev invariants by using asymptotic techniques developed by the author in the previous papers. We also prove linear relations between special values at negative integers of some L-functions, which are common generalizations of Hurwitz zeta functions, Barnes zeta functions and Epstein zeta functions.

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.

In this article, for any Seifert fibered homology 3-sphere, we introduce homological blocks with simple Lie algebra and prove that its radial limits are identified with the Witten--Reshetikhin--Turaev invariants. To prove it, we develop an asymptotic formula and a vanishing result of asymptotic coefficients.