Analytic Continuation Of Chern-Simons Theory

TL;DR

The paper develops a rigorous framework for analytically continuing Chern–Simons theory to complex coupling, using Morse theory and Lefschetz thimbles to define convergent integration cycles that track Stokes phenomena across parameter space. By promoting the conjugate connection to an independent variable, the author builds a finite‑dimensional analogy and then applies it to the CS path integral, showing how flat connections act as critical points and how their associated thimbles determine the analytic structure. For knots, the approach yields a controlled description of the colored Jones polynomial under continuation, clarifying the roles of abelian and nonabelian flat connections and connecting to the volume conjecture in the figure‑eight case. The work also sketches a deeper 4D interpretation in terms of twisted N=4 super Yang–Mills theory and hints at a link to Khovanov homology, suggesting a unified geometric framework for knot invariants beyond integer CS levels.

Abstract

The title of this article refers to analytic continuation of three-dimensional Chern-Simons gauge theory away from integer values of the usual coupling parameter k, to explore questions such as the volume conjecture, or analytic continuation of three-dimensional quantum gravity (to the extent that it can be described by gauge theory) from Lorentzian to Euclidean signature. Such analytic continuation can be carried out by rotating the integration cycle of the Feynman path integral. Morse theory or Picard-Lefschetz theory gives a natural framework for describing the appropriate integration cycles. An important part of the analysis involves flow equations that turn out to have a surprising four-dimensional symmetry. After developing a general framework, we describe some specific examples (involving the trefoil and figure-eight knots in S^3). We also find that the space of possible integration cycles for Chern-Simons theory can be interpreted as the "physical Hilbert space" of a twisted version of N=4 super Yang-Mills theory in four dimensions.

Figures (15)

• Figure 1: The complex $\lambda$ plane, with the real axis from left to right. The hatched regions are the three "good" regions in which the Airy integrand vanishes at infinity. The contour ${\mathcal{C}}_i$, $i=1,2,3$, connects the $i^{th}$ good region to the $i+1^{th}$.
• Figure 2: This figure illustrates the behavior of flow lines when one crosses a Stokes ray. Sketched are the downward flows from two critical points $p_\sigma$ and $p_\tau$. The lines are sketched as flowing downward, in the direction of smaller $h$ (though it is not literally true that $h$ corresponds to the height in the figure, as this function has no critical points). The behavior at a Stokes ray is depicted in (b); there is a downward flow from $p_\sigma$ to $p_\tau$. The flows "before" and "after" crossing the Stokes ray are depicted in (a) and (c). The downward flowing cycle ${\mathcal{J}}_\tau$ from the lower critical point is unaffected by the Stokes ray. As for the cycle ${\mathcal{J}}_\sigma$ defined by flow from the upper critical point, it is ill-defined in (b) and jumps by ${\mathcal{J}}_\sigma\to{\mathcal{J}}_\sigma+{\mathcal{J}}_\tau$ between (a) and (c).
• Figure 3: The complex ${\mathcal{I}}$ plane is sketched here with $h={\mathrm{Re}}\,{\mathcal{I}}$ running vertically. The exponents ${\mathcal{I}}_\tau$ and ${\mathcal{I}}_\sigma$ of two critical points $p_\tau$ and $p_\sigma$ have arguments between $\pi$ and $3\pi/2$. If ${\mathrm{Arg}}\,{\mathcal{I}}_\tau>{\mathrm{Arg}}\,{\mathcal{I}}_\sigma$, then at a Stokes ray with ${\mathrm{Im}}\,{\mathcal{I}}_\tau={\mathrm{Im}}\,{\mathcal{I}}_\sigma$, one has $h_\tau>h_\sigma$. This ensures that the coefficient ${\mathfrak n}_\tau$ is unaffected in crossing the Stokes line, though ${\mathfrak n}_\sigma$ may jump. The analytically continued integral will grow exponentially when $\lambda$ is varied so that ${\mathrm{Arg}}\,{\mathcal{I}}_\tau$ exceeds $3\pi/2$.
• Figure 4: Qualitative behavior of Lefschetz thimbles for the Airy function for $\epsilon>0$ (a) and $\epsilon<0$ (b). In case $(b)$, $\epsilon$ has been given a small imaginary part, since the negative $\epsilon$ axis is a Stokes curve.
• Figure 5: The two lines $\epsilon=r\exp(\pm\pi i/2)$, with real $r$, are Stokes lines in the sense that flows between the critical point $p$ and the critical orbit ${\mathcal{O}}$ are possible on these lines. Jumping, however, occurs only in crossing the line $\epsilon=r\exp(-\pi i/2)$, shown here as a solid line. There is no jumping of Lefschetz thimbles in crossing the dotted line.