The Resurgent Structure of Quantum Knot Invariants.

Area of Science:

• Quantum field theory
• Knot theory
• Mathematical physics

Background:

• Asymptotic expansions of quantum knot invariants in complex Chern-Simons theory yield divergent formal power series.
• Resurgent functions provide a framework for understanding such divergent series.

Purpose of the Study:

• To conjecture that these divergent series are resurgent functions.
• To determine the Stokes automorphism of these series using q-series and q-difference equations.
• To connect a distinguished entry of these matrices to the 3D index and BPS state counting for hyperbolic knots.

Main Methods:

• Analyzing asymptotic expansions in complex Chern-Simons theory.
• Applying the theory of resurgent functions.
• Solving linear q-difference equations to find fundamental solutions.
• Computing q-series with integer coefficients.
• Comparing theoretical predictions with numerical computations for specific knots.

Main Results:

• The study conjectures that formal power series arising from quantum knot invariants are resurgent functions.
• A specific pair of q-series matrices is proposed as the Stokes automorphism.
• These matrices are explicitly determined by solutions to linear q-difference equations.
• A connection is conjectured between a matrix entry and the Dimofte-Gaiotto-Gukov 3D index for hyperbolic knots, implying BPS state counting.
• Conjectures are illustrated and validated for the (2,3) and (2,5) torus knots.

Conclusions:

• The research proposes a novel framework connecting quantum knot invariants, resurgent functions, and q-difference equations.
• The findings suggest a deep relationship between topological invariants and quantum field theory computations, potentially revealing BPS state counting mechanisms.
• Explicit examples for torus knots support the validity of the proposed conjectures.