arXiv subjects

Campbell Wheeler

Publications and source records attributed to Campbell Wheeler.

13 recordsLinked to original sources

arXiv2026

A lift of the colored Jones polynomial of a knot

Habiro lifted the Witten-Reshetikhin-Turaev invariant of an integer homology 3-sphere (a complex-valued function on the set of complex roots of unity) to an element of the Habiro ring. We lift the colored Jones polynomial of a knot, with Alexander polynomial $\Delta(t)$, to the recently introduced Habiro ring of the \'etale map $\mathbb{Z}[t^{\pm 1}]\to \mathbb{Z}[t^{\pm 1},\Delta(t)^{-1}]$ (with Frobenius lifts $t\mapsto t^p$ for all primes $p$). This implies the existence of a loop expansion at roots of unity (confirming a conjecture of Habiro), and a lift of power series invariants of Ohtsuki for 3-manifolds with Betti number 1 to a Habiro ring. Our results have natural extensions to the skein module of a knot complement, and they suggest a natural lift of the colored Jones polynomial colored by representations of a simple Lie (super) algebra.

math.GT↗

arXiv2025

Explicit classes in Habiro cohomology

We propose a cycle description of the Habiro cohomology of a smooth variety $X$ over the spectrum $B$ of an \'etale $Z[\lambda]$-algebra and construct explicit nontrivial cycles using either the Picard-Fuchs equation on $X/B$ of a hypergeometric motive, or a push-forward of elements of the Habiro ring of $X/B$. In particular, we give explicit classes for 1-parameter Calabi--Yau families. The $q$-hypergeometric origin of our cycles imply that they generate $q$-holonomic modules that define $q$-deformations of the classical Picard-Fuchs equation. We illustrate our theorems with three examples: the Legendre family of elliptic curves, the $A$-polynomial curve of the figure eight knot, and for the quintic three-fold, whose $q$-Picard Fuchs equation appeared in its genus $0$-quantum $K$-theory. Our methods give a unified treatment of quantum $K$-theory and complex Chern-Simons theory around higher dimensional critical loci.

math.AG↗

arXiv2024

The Habiro ring of a number field

We introduce the Habiro ring of a number field $\mathbb{K}$ and modules over it graded by $K_3(\mathbb{K})$. Elements of these modules are collections of power series at each complex root of unity that arithmetically glue with each other after applying a Frobenius endomorphism, and after dividing at each prime by a collection of series that depends solely on an element of the Bloch group. The main theorems of this paper concern number fields, their algebraic $K$-theory and its regulator maps (Borel, $p$-adic and \'etale), whereas the explicit collections of series are defined by a careful algebraic analysis of the infinite Pochhammer symbol at roots of unity. The origin of the above mentioned power series comes from perturbative Chern--Simons theory and by expansions of the admissible series of Kontsevich--Soibelman, both ultimately related to the infinite Pochhammer symbol. This link suggests that some Donaldson-Thomas invariants have arithmetic meaning and that some elements of the Habiro ring of a number field have enumerative meaning. Added subsection 1.1 explaining what the paper is about and subsection 1.8 explaining the relation to perturbative complex Chern-Simons theory.

math.NT↗

arXiv2024

Summability for State Integrals of hyperbolic knots

We prove conjectures of Garoufalidis-Gu-Mari\~no that perturbative series associated with the hyperbolic knots $4_1$ and $5_2$ are resurgent and Borel summable. In the process, we give an algorithm that can be used to explicitly compute the Borel-Laplace resummation as a combination of state integrals of Andersen-Kashaev. This gives a complete description of the resurgent structure in these examples and allows for explicit computations of Stokes constants.

math.GT↗

arXiv2023

Quantum Modularity for a Closed Hyperbolic 3-Manifold

This paper proves quantum modularity of both functions from $\mathbb{Q}$ and $q$-series associated to the closed manifold obtained by $-\frac{1}{2}$ surgery on the figure-eight knot, $4_1(-1,2)$. In a sense, this is a companion to work of Garoufalidis-Zagier, where similar statements were studied in detail for some simple knots. It is shown that quantum modularity for closed manifolds provides a unification of Chen-Yang's volume conjecture with Witten's asymptotic expansion conjecture. Additionally we show that $4_1(-1,2)$ is a counterexample to previous conjectures of Gukov-Manolescu relating the Witten-Reshetikhin-Turaev invariant and the $\widehat{Z}(q)$ series. This could be reformulated in terms of a "strange identity", which gives a volume conjecture for the $\widehat{Z}$ invariant. Using factorisation of state integrals, we give conjectural but precise $q$-hypergeometric formulae for generating series of Stokes constants of this manifold. We find that the generating series of Stokes constants is related to the 3d index of $4_1(-1,2)$ proposed by Gang-Yonekura. This extends the equivalent conjecture of Garoufalidis-Gu-Mari\~no for knots to closed manifolds. This work appeared in a similar form in the author's Ph.D. Thesis.

math.GT↗

arXiv2023

Perturbative invariants of cusped hyperbolic 3-manifolds

We prove that a formal power series associated to an ideally triangulated cusped hyperbolic 3-manifold (together with some further choices) is a topological invariant. This formal power series is conjectured to agree to all orders in perturbation theory with two important topological invariants of hyperbolic knots, namely the Kashaev invariant and the Andersen--Kashaev invariant (also known as the state-integral) of Teichmüller TQFT.

math.GT↗

arXiv2023

Resurgence, Habiro elements and strange identities

We prove resurgence properties for the Borel transform of a formal power series associated to elements in the Habiro ring that come from radial limits of partial theta series via strange identities. As an application, we prove a conjecture in quantum topology due to Costin and Garoufalidis for two families of torus knots.

math.NT↗

arXiv2022

Periods, the meromorphic 3D-index and the Turaev--Viro invariant

The 3D-index of Dimofte-Gaiotto-Gukov is an interesting collection of $q$-series with integer coefficients parametrised by a pair of integers and associated to a 3-manifold with torus boundary. In this note, we explain the structure of the asymptotic expansions of the 3D-index when $q=e^{2πiτ}$ and $τ$ tends to zero (to all orders and with exponentially small terms included), and discover two phenomena: (a) when $τ$ tends to zero on a ray near the positive real axis, the horizontal asymptotics of the meromorphic 3D-index match to all orders with the asymptotics of the Turaev-Viro invariant of a knot, in particular explaining the Volume Conjecture of Chen-Yang from first principles, (b) when $τ\to 0$ on the positive imaginary axis, the vertical asymptotics of the 3D-index involves periods of a plane curve (the $A$-polynomial), as opposed to algebraic numbers, explaining some predictions of Hodgson-Kricker-Siejakowski and leading to conjectural identities between periods of the $A$-polynomial of a knot and integrals of the Euler beta-function.

math.GT↗

arXiv2022

Around the combinatorial unit ball of measured foliations on bordered surfaces

The volume $\mathscr{B}_Σ^{\rm comb}(\mathbb{G})$ of the unit ball -- with respect to the combinatorial length function $\ell_{\mathbb{G}}$ -- of the space of measured foliations on a stable bordered surface $Σ$ appears as the prefactor of the polynomial growth of the number of multicurves on $Σ$. We find the range of $s \in \mathbb{R}$ for which $(\mathscr{B}_Σ^{\rm comb})^{s}$, as a function over the combinatorial moduli spaces, is integrable with respect to the Kontsevich measure. The results depends on the topology of $Σ$, in contrast with the situation for hyperbolic surfaces where Arana-Herrera and Athreya (arXiv:1907.06287) recently proved an optimal square-integrability.

math.GT↗

arXiv2022

Modular $q$-holonomic modules

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, 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.

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arXiv2021

Resurgence of Chern-Simons theory at the trivial flat connection

Some years ago, it was conjectured by the first author that the Chern-Simons perturbation theory of a 3-manifold at the trivial flat connection is a resurgent power series. We describe completely the resurgent structure of the above series (including the location of the singularities and their Stokes constants) in the case of a hyperbolic knot complement in terms of an extended square matrix of $(x,q)$-series whose rows are indexed by the boundary parabolic $\text{SL}_2(\mathbb{C})$-flat connections, including the trivial one. We use our extended matrix to describe the Stokes constants of the above series, to define explicitly their Borel transform and to identify it with state-integrals. Along the way, we use our matrix to give an analytic extension of the Kashaev invariant and of the colored Jones polynomial and to complete the matrix valued holomorphic quantum modular forms as well as to give an exact version of the refined quantum modularity conjecture of Zagier and the first author. Finally, our matrix provides an extension of the 3D-index in a sector of the trivial flat connection. We illustrate our definitions, theorems, numerical calculations and conjectures with the two simplest hyperbolic knots.

math.GT↗

arXiv2021

On the Kontsevich geometry of the combinatorial Teichmüller space

For bordered surfaces S, we develop a complete parallel between the geometry of the combinatorial Teichmüller space $T_S^{comb}$ equipped with Kontsevich symplectic form $ω_K$, and then the usual Weil-Petersson geometry of Teichmüller space $T_S$. The basis for this is an identification of $T_S^{comb}$ with a space of measured foliations with transverse boundary conditions. We equip $T_S^{comb}$ with an analog of the Fenchel-Nielsen coordinates (defined similarly as Dehn-Thurston coordinates) and show they are Darboux for $ω_K$ (analog of Wolpert formula). We then set up the geometric recursion of Andersen-Borot-Orantin to produce mapping class group invariants functions on $T_S^{comb}$ whose integration with respect to Kontsevich volume form satisfy topological recursion. Further we establish an analog of Mirzakhani-McShane identities, and provide applications to the study of the enumeration of multicurves with respect to combinatorial lengths and Masur-Veech volumes. The formalism allows us to provide uniform and completely geometric proofs of Witten's conjecture/Kontsevich theorem and Norbury's topological recursion for lattice point count in the combinatorial moduli space, parallel to Mirzakhani's proof of her recursion for Weil-Petersson volumes. We strengthen results of Mondello and Do on the convergence of hyperbolic geometry to combinatorial geometry along the rescaling flow, allowing us to flow systematically natural constructions on the usual Teichmüller space to their combinatorial analogue, such as a new derivation of the piecewise linear structure of $T_S^{comb}$ originally obtained in the work of Penner, as the limit under the flow of the smooth structure of $T_S$.

math.DG↗

arXiv2020

Topological recursion for Masur-Veech volumes

We study the Masur-Veech volumes $MV_{g,n}$ of the principal stratum of the moduli space of quadratic differentials of unit area on curves of genus $g$ with $n$ punctures. We show that the volumes $MV_{g,n}$ are the constant terms of a family of polynomials in $n$ variables governed by the topological recursion/Virasoro constraints. This is equivalent to a formula giving these polynomials as a sum over stable graphs, and retrieves a result of \cite{Delecroix} proved by combinatorial arguments. Our method is different: it relies on the geometric recursion and its application to statistics of hyperbolic lengths of multicurves developed in \cite{GRpaper}. We also obtain an expression of the area Siegel--Veech constants in terms of hyperbolic geometry. The topological recursion allows numerical computations of Masur--Veech volumes, and thus of area Siegel--Veech constants, for low $g$ and $n$, which leads us to propose conjectural formulas for low $g$ but all $n$. We also relate our polynomials to the asymptotic counting of square-tiled surfaces with large boundaries.

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