Witten-Reshetikhin-Turaev invariants and indefinite false theta functions for plumbing indefinite H-graphs

TL;DR

This paper constructs indefinite false theta functions as potential homological blocks for certain plumbed 3-manifolds, extending previous work to non-weakly negative definite cases and confirming the case of the Poincaré homology sphere.

Contribution

It introduces indefinite false theta functions as new candidates for homological blocks in non-weakly negative definite plumbed 3-manifolds, and proves their equivalence to known blocks in the Poincaré sphere case.

Findings

Constructed indefinite false theta functions for specific 3-manifolds.

Proved the indefinite theta function matches the homological block for the Poincaré sphere.

Extended the theory of homological blocks beyond weakly negative definite cases.

Abstract

Gukov--Pei--Putrov--Vafa conjectured the existence of $ q $-series whose radial limits are Witten--Reshetikhin--Turaev invariants and called them homological blocks. For weakly negative definite plumbed 3-manifolds, Gukov--Pei--Putrov--Vafa and Gukov-Manolescu constructed homological blocks. In this paper, we construct indefinite false theta functions which are candidates of homological blocks for some plumbed $ 3 $-manifolds which are not weakly negative definite. Moreover we prove that, for the Poincar\'{e} homology sphere, our indefinite false theta function coincides with the original homological block.