3-Manifolds and VOA Characters

Cheng, Miranda C. N.; Chun, Sungbong; Feigin, Boris; Ferrari, Francesca; Gukov, Sergei; Harrison, Sarah M.; Passaro, D.

doi:10.48550/arxiv.2201.04640

Cited by 8 publications

(16 citation statements)

References 74 publications

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“…While preparing the manuscript, we found that [21] appeared, which overlaps with parts of results in this paper.…”

Section: Note Addedsupporting

confidence: 78%

BPS invariants for a Knot in Seifert manifolds

Chung

2022

J. High Energ. Phys.

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We calculate homological blocks for a knot in Seifert manifolds when the gauge group is SU(N). We obtain the homological blocks with a given representation of the gauge group from the expectation value of the Wilson loop operator by analytically continuing the Chern-Simons level. We also obtain homological blocks with the analytically continued level and representation for a knot in the Seifert integer homology spheres.

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“…While preparing the manuscript, we found that [21] appeared, which overlaps with parts of results in this paper.…”

Section: Note Addedsupporting

confidence: 78%

BPS invariants for a Knot in Seifert manifolds

Chung

2022

J. High Energ. Phys.

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“…3. Our notion explains conceptually the improved analyticity properties conjectured in relation to the structure of exact and perturbative invariants of Chern-Simons theory [22,23,24,32,33], as well as the work of Gukov et al relating logarithmic CFTs and VOAs to Chern-Simons theory [10]. 4.…”

Section: Definition and Propertiesmentioning

confidence: 72%

“…This system appeared in [25, Eqn. (10)] and [22,Eqn. (134)], and it is q-holonomic of rank 2 in the variables (t, x).…”

Section: State Integralmentioning

confidence: 99%

Modular $q$-holonomic modules

Garoufalidis¹,

Wheeler²

2022

Preprint

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We introduce the notion of modular q-holonomic modules whose fundamental matrices define a cocycle with improved analyticity properties and show that the generalised q-hypergeometric equation, as well as three key q-holonomic modules of complex Chern-Simons theory are modular. This notion explains conceptually recent structural properties of quantum invariants of knots and 3-manifolds, and of exact and perturbative Chern-Simons theory [22,23,24,33], and in addition provides an effective method to solve the corresponding linear q-difference equations. An alternative title of our paper, emphasising the equations rather than the modules, is: Modular linear q-difference equations Contents 1. Introduction 2 1.1. Summary 2 1.2. Motivation 3 1.3. Definition and properties 3 1.4. The generalised q-hypergeometric equation is modular 5 1.5. Modular q-holonomic modules in Chern-Simons theory 7 2. A review of linear q-difference equations 12 2.1. Preliminaries 12 2.2. An algorithm for a fundamental matrix 13 2.3. The q-Borel and the q-Laplace transforms 14 2.4. The slash operator and proof of Theorem 1.2 17 2.5. Duality 20 2.6. Categorical aspects 23 3. Heine's q-hypergeometric functions 24 3.1. Solutions 24 3.2. Self duality 25 3.3. State integral 28 3.4. Resonance 29 4. Proof of Theorems 1.5 and 1.6 30 4.1. The q-Pochhammer symbol 30 4.2. The Appell-Lerch sums 32

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“…Much earlier, in [23], Kashaev invariants of torus knots T (p, p ′ ) were shown to be related to the Eichler integrals of Vir(p, p ′ ) = W 2 (p, p ′ ) minimal model characters. There are even more relations between certain quantum invariants of 3-manifolds and characters of certain logarithmic VOAs found recently, [10]. Finally, we mention [14] where HOMFLYPT polynomials of torus knots were related to characters of certain rational Cherednik algebras.…”

Section: Introductionmentioning

confidence: 79%

Coloured $\mathfrak{sl}_r$ invariants of torus knots and characters of $\mathcal{W}_r$ algebras

Kanade¹

2022

Preprint

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Let p < p ′ be a pair of coprime positive integers. In this note, generalizing Morton's work in the case of sl2, we give a formula for the slr Jones invariants of torus knots T (p, p ′ ) coloured with the finite-dimensional irreducible representations Lr(nΛ1). When r ≤ p, we show that appropriate limits of the shifted (non-normalized, framing dependent) invariants calculated along Lr(nrΛ1) are essentially the characters of certain minimal model principal W algebras of type A, namely, Wr(p, p ′ ), up to some factors independent of p and p ′ but depending on r. In particular, these limits are essentially modular. We expect these limits to be the 0-tails of corresponding sequences of invariants. At the end, we formulate a conjecture on limits for p < r.

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3-Manifolds and VOA Characters

Cited by 8 publications

References 74 publications

BPS invariants for a Knot in Seifert manifolds

BPS invariants for a Knot in Seifert manifolds

Modular $q$-holonomic modules

Coloured $\mathfrak{sl}_r$ invariants of torus knots and characters of $\mathcal{W}_r$ algebras

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