Modular transformations of homological blocks for Seifert fibered homology 3-spheres. (English) Zbl 07809222
This article is about the \(q\)-series invariants of 3-manifolds, which are also known as Gukov-Pei-Putrov-Vafa (GPPV) invariants, \(\widehat{Z}\)-invariants, or homological blocks. They were first conjectured in [S. Gukov et al., J. High Energy Phys. 2017, No. 7, Paper No. 71, 82 p. (2017; Zbl 1380.58018)] and subsequently were defined for negative-definite plumbed three-manifolds in [S. Gukov et al., J. Knot Theory Ramifications 29, No. 2, Article ID 2040003, 85 p. (2020; Zbl 1448.57020)]. In particular, these authors considered the Witten-Reshetikhin-Turaev (WRT)-invariant \(\tau^{SU(2)}_k[M(\Gamma);q]\) for a plumbed three-manifold \(M(\Gamma)\) and positive integer level \(k\). They performed the analytic continuation of the WRT-invariant (\(q\rightarrow q\)) to get the \(q\)-series invariant of 3-manifolds. It is worth noting that the convergence of \(q\)-series led to the choice of \(|q|<1\), which subsquently forces us to be in the subset of plumbed 3-manifold that is, negative-definite plumbed 3-manifolds. Further, these \(q\)-series or homological blocks are interesting objects in quantum topology as they provide a fresh perspective on the categorification problem of WRT invariants. Moreover, homological blocks can also be seen as the vast generalization of \(q\)-series obtained in [R. Lawrence and D. Zagier, Asian J. Math. 3, No. 1, 93–107 (1999; Zbl 1024.11028); K. Hikami, Int. J. Math. 16, No. 6, 661–685 (2005; Zbl 1088.57013); K. Hikami, Commun. Math. Phys. 268, No. 2, 285–319 (2006; Zbl 1147.58023)]. In addition to all these developments, the modular properties of homological blocks were first discussed in [M. C. N. Cheng et al., J. High Energy Phys. 2019, No. 10, Paper No. 10, 95 p. (2019; Zbl 1427.81118)].
The paper under review studies the modular properties of homological blocks associated to a Seifert fibered integral homology 3-sphere. In particular, the authors have given the explicit form of modular transformation formulas in Theorem 5.5. This result is achieved by rewriting the corresponding homological blocks in terms of false theta functions and then applying their modular properties to derive the final outcome. Further, as an application of their result, the authors also provide asymptotic expansion formulas of the WRT invariants in Theorem 5.10.
The paper under review studies the modular properties of homological blocks associated to a Seifert fibered integral homology 3-sphere. In particular, the authors have given the explicit form of modular transformation formulas in Theorem 5.5. This result is achieved by rewriting the corresponding homological blocks in terms of false theta functions and then applying their modular properties to derive the final outcome. Further, as an application of their result, the authors also provide asymptotic expansion formulas of the WRT invariants in Theorem 5.10.
Reviewer: Sachin Chauhan (Mumbai)
MSC:
| 57K16 | Finite-type and quantum invariants, topological quantum field theories (TQFT) |
| 57K31 | Invariants of 3-manifolds (including skein modules, character varieties) |
| 58J28 | Eta-invariants, Chern-Simons invariants |
| 11F37 | Forms of half-integer weight; nonholomorphic modular forms |
Citations:
Zbl 1380.58018; Zbl 1448.57020; Zbl 1024.11028; Zbl 1088.57013; Zbl 1147.58023; Zbl 1427.81118References:
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