Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Summability for State Integrals of hyperbolic knots

Veronica Fantini, Campbell Wheeler

Abstract

We prove conjectures of Garoufalidis-Gu-Mariño 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.

Summability for State Integrals of hyperbolic knots

Abstract

We prove conjectures of Garoufalidis-Gu-Mariño that perturbative series associated with the hyperbolic knots and 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.

Paper Structure

This paper contains 42 sections, 23 theorems, 131 equations, 32 figures.

Table of Contents

  1. Introduction
  2. Deforming the volume of a three-manifold
  3. State integrals
  4. Proving Borel--Laplace summability
  5. From thimbles to state integrals
  6. Existence of thimbles
  7. A relative homology for state integrals contours
  8. The surface $X_{\epsilon,M}$
  9. Behaviour at infinity
  10. The relative homology
  11. A state integral basis
  12. The class of the thimble $\mathcal{C}_{p_0,\vartheta}$
  13. A mix of state integrals and steepest descent contours.
  14. Asymptotics of state integrals
  15. A steepest descent algorithm
  16. ...and 27 more sections

Key Result

Theorem 1.1

The series $\Phi_{4_1}$ and $\Phi_{5_2}$ are Borel--Laplace summable and their resummations are equal to combinations of state integrals.

Figures (32)

  • Figure 1: Domain of $\mathrm{D}_{3\pi/4}(z)$ given by $\mathbb C\backslash((\mathbb Z_{\geq0}+(i-1)\mathbb R_{\leq0})\cup(\mathbb Z_{<0}+(i-1)\mathbb R_{\geq0}))$.
  • Figure 2: The two conventions for the branch cuts of the function $V$.
  • Figure 3: This figure depicts in orange the analytic continuation of the set $\mathcal{C}_{(-5/6,0),\vartheta}$ for $\vartheta=1.6650$ and $(A,B)=(1,2)$. The blue lines are parallel to the lines $\sqrt{-i e^{i\vartheta}}$ and $\sqrt{i e^{i\vartheta}}$. The red crosses are the branch points of $V$.
  • Figure 4: The $m$-sheet of $\Sigma_{\epsilon,M}^+$ in the $z_{m,+}$ coordinates.
  • Figure 5: The projection of $\Sigma_{\epsilon,M}^+\cap\Sigma_{\epsilon,M}^{-}$ on the $m$-sheet in the $z_{m,+}$ coordinates.
  • ...and 27 more figures

Theorems & Definitions (49)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • proof
  • Theorem 1.4
  • Remark 1.5
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • ...and 39 more