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

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 Poincaré homology sphere, our indefinite false theta function coincides with the original homological block.