# Witten–Reshetikhin–Turaev Function for a Knot in Seifert Manifolds

@article{Fuji2020WittenReshetikhinTuraevFF, title={Witten–Reshetikhin–Turaev Function for a Knot in Seifert Manifolds}, author={Hiroyuki Fuji and Kohei Iwaki and Hitoshi Murakami and Yuji Terashima}, journal={arXiv: Geometric Topology}, year={2020} }

In this paper, for a Seifert loop (i.e., a knot in a Seifert three-manifold), first we give a family of an explicit function $\Phi(q; N)$ whose special values at roots of unity are identified with the Witten-Reshetikhin-Turaev invariants of the Seifert loop for the integral homology sphere. Second, we show that the function $\Phi(q; N)$ satisfies a $q$-difference equation whose classical limit coincides with a component of the character varieties of the Seifert loop. Third, we give an…

## 6 Citations

### Modular transformations of homological blocks for Seifert fibered homology $3$-spheres

- Mathematics
- 2021

In this article, for any Seifert fibered homology 3-sphere, we give explicit modular transformation formulas of homological blocks introduced by Gukov-Pei-Putrov-Vafa. Moreover, based on the modular…

### Resurgence analysis of quantum invariants of Seifert fibered homology spheres

- MathematicsJournal of the London Mathematical Society
- 2022

For a Seifert fibered homology sphere X$X$ , we show that the q$q$ ‐series invariant Ẑ0(X;q)$\hat{\operatorname{Z}}_0(X;q)$ , introduced by Gukov–Pei–Putrov–Vafa, is a resummation of the Ohtsuki…

### Witten-Reshetikhin-Turaev invariants and homological blocks for plumbed homology spheres

- Mathematics
- 2022

. In this paper, we prove a conjecture by Gukov–Pei–Putrov–Vafa for a wide class of plumbed 3-manifolds. Their conjecture states that Witten–Reshetikhin–Turaev (WRT) invariants are radial limits of…

### Witten-Reshetikhin-Turaev Invariants, Homological Blocks, and Quantum Modular Forms for Unimodular Plumbing H-Graphs

- MathematicsSymmetry, Integrability and Geometry: Methods and Applications
- 2022

. Gukov–Pei–Putrov–Vafa constructed q -series invariants called homological blocks in a physical way in order to categorify Witten–Reshetikhin–Turaev (WRT) invariants and conjectured that radial…

### Quantum phase transition and Resurgence: Lessons from 3d N = 4 SQED

- PhysicsProgress of Theoretical and Experimental Physics
- 2021

We study a resurgence structure of a quantum field theory with a phase transition to uncover relations between resurgence and phase transitions. In particular, we focus on three-dimensional N = 4…

### Resurgent Analysis for Some 3-manifold Invariants

- Mathematics
- 2020

We study resurgence for some 3-manifold invariants when $G_{\mathbb{C}}=SL(2, \mathbb{C})$. We discuss the case of an infinite family of Seifert manifolds for general roots of unity and the case of…

## References

SHOWING 1-10 OF 56 REFERENCES

### Witten–Reshetikhin–Turaev Invariants of¶Seifert Manifolds

- Mathematics
- 1999

Abstract:For Seifert homology spheres, we derive a holomorphic function of K whose value at integer K is the sl2 Witten–Reshetikhin–Turaev invariant, ZK, at q= exp 2πi/K. This function is expressed…

### Reshetikhin-Turaev invariants of Seifert 3-manifolds for classical simple Lie algebras, and their asymptotic expansions

- Mathematics
- 2002

We derive formulas for the Reshetikhin-Turaev invariants of all oriented Seifert manifolds associated to an arbitrary complex finite dimensional simple Lie algebra $\mathfrak g$ in terms of the…

### ON THE QUANTUM INVARIANT FOR THE BRIESKORN HOMOLOGY SPHERES

- Mathematics
- 2005

We study an exact asymptotic behavior of the Witten–Reshetikhin–Turaev SU(2) invariant for the Brieskorn homology spheres Σ(p1, p2, p3) by use of properties of the modular form following a method…

### Quantum invariants, modular forms, and lattice points II

- Mathematics
- 2006

We study the SU(2) Witten-Reshetikhin-Turaev (WRT) invariant for the Seifert fibered homology spheres with M-exceptional fibers. We show that the WRT invariant can be written in terms of…

### Quantum invariant, modular form, and lattice points

- Mathematics
- 2005

We study the Witten--Reshetikhin--Turaev SU(2) invariant for the Seifert manifold with 4-singular fibers. We define the Eichler integrals of the modular forms with half-integral weight, and we show…

### Fivebranes and 3-manifold homology

- Mathematics
- 2016

A bstractMotivated by physical constructions of homological knot invariants, we study their analogs for closed 3-manifolds. We show that fivebrane compactifications provide a universal description of…

### Localization for Wilson Loops in Chern-Simons Theory

- Mathematics
- 2009

We reconsider Chern-Simons gauge theory on a Seifert manifold M , which is the total space of a nontrivial circle bundle over a Riemann surface Σ, possibly with orbifold points. As shown in previous…

### Quantum invariants of Seifert 3–manifolds and their asymptotic expansions

- Mathematics
- 2002

We report on recent results of the authors concerning calculations of quantum invariants of Seifert 3–manifolds. These results include a derivation of the Reshetikhin–Turaev invariants of all…

### Resurgent Analysis for Some 3-manifold Invariants

- Mathematics
- 2020

We study resurgence for some 3-manifold invariants when $G_{\mathbb{C}}=SL(2, \mathbb{C})$. We discuss the case of an infinite family of Seifert manifolds for general roots of unity and the case of…

### Resurgence analysis of quantum invariants of Seifert fibered homology spheres

- MathematicsJournal of the London Mathematical Society
- 2022

For a Seifert fibered homology sphere X$X$ , we show that the q$q$ ‐series invariant Ẑ0(X;q)$\hat{\operatorname{Z}}_0(X;q)$ , introduced by Gukov–Pei–Putrov–Vafa, is a resummation of the Ohtsuki…