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Hikami, Kazuhiro

doi:10.1070/rd2005v010n04abeh000328

Cited by 37 publications

(11 citation statements)

References 23 publications

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“…This fact plays an important role in explicit calculations in sections 4.2 and 4.3. An analogous property for WRT invariants was studied earlier in [10][11][12][13][14][15][16].…”

Section: Introductionmentioning

confidence: 68%

Ẑ invariants at rational τ

Kucharski

2019

J. High Energ. Phys.

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Ẑ invariants of 3-manifolds were introduced as series in q = e 2πiτ in order to categorify Witten-Reshetikhin-Turaev invariants corresponding to τ = 1/k. However modularity properties suggest that all roots of unity are on the same footing. The main result of this paper is the expression connecting Reshetikhin-Turaev invariants withẐ invariants for τ ∈ Q. We present the reasoning leading to this conjecture and test it on various 3-manifolds.

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“…This fact plays an important role in explicit calculations in sections 4.2 and 4.3. An analogous property for WRT invariants was studied earlier in [10][11][12][13][14][15][16].…”

Section: Introductionmentioning

confidence: 68%

Ẑ invariants at rational τ

Kucharski

2019

J. High Energ. Phys.

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“…figure 1) to Seifert manifolds with three singular fibres. The relation between false theta functions and the WRT invariants for this family of three-manifolds was first pointed out in [19] and later extensively studied in [20][21][22]. Here, we are interested in their quantum invariants Z a (M 3 ).…”

Section: False Mock and Three-manifoldsmentioning

confidence: 99%

“…Note that the q-hypergeometric series converges both for |q| < 1 and |q| > 1. It can be shown that [21]…”

Section: Lemma 26 ([10]mentioning

confidence: 99%

Three-manifold quantum invariants and mock theta functions

Cheng

Ferrari

Sgroi

2019

Phil. Trans. R. Soc. A.

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Mock modular forms have found applications in numerous branches of mathematical sciences since they were first introduced by Ramanujan nearly a century ago. In this proceeding we highlight a new area where mock modular forms start to play an important role, namely the study of three-manifold invariants. For a certain class of Seifert three-manifolds, we describe a conjecture on the mock modular properties of a recently proposed quantum invariant. As an illustration, we include concrete computations for a specific three-manifold, the Brieskorn sphere Σ (2, 3, 7). This note is partially based on the talk by the first author in the conference "Srinivasa Ramanujan: in celebration of the centenary of his election as FRS" held at the Royal Society in 2018.

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“…[80]. This mock theta function gives the WRT invariant τ N of the Seifert manifold M (2, 3, 4) [94] in the limit q → e 2πi/N…”

Section: Example: (3 4) Minimal Model and Order 3 Mock Theta Functionsmentioning

confidence: 99%

BPS indices, modularity and perturbations in quantum K-theory

Jockers

Mayr²,

Ninad³

et al. 2022

J. High Energ. Phys.

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We study a perturbation family of $$ \mathcal{N} $$ N = 2 3d gauge theories and its relation to quantum K-theory. A 3d version of the Intriligator-Vafa formula is given for the quantum K-theory ring of Grassmannians. The 3d BPS half-index of the gauge theory is connected to the theory of bilateral hypergeometric q-series, and to modular q-characters of a class of conformal field theories in a certain massless limit. Turning on 3d Wilson lines at torsion points leads to mock modular behavior. Perturbed correlators in the IR regime are computed by determining the UV-IR map in the presence of deformations.

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Abstract: ABSTRACT. We establish a correspondence between the SU(2) Witten-Reshetikhin-Turaev invariant for the Seifert manifold M (p 1 , p 2 , p 3 ) and Ramanujan's mock theta functions.

Cited by 37 publications

References 23 publications

Ẑ invariants at rational τ

Ẑ invariants at rational τ

Three-manifold quantum invariants and mock theta functions

BPS indices, modularity and perturbations in quantum K-theory

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