Complex Contours, Stokes Chambers, and Regularization

A localized path integral is not defined by its meromorphic integrand alone. One must specify a middle-dimensional cycle in the complexified Euclidean field space, how that cycle decomposes into steepest-descent thimbles, what happens to zero and negative modes, and which spectral cut and local counterterms define the determinant. These choices can change when parameters cross Stokes walls or poles, even though the formal $Q$-cohomology is unchanged.

Required background. Localization loci and one-loop determinants supplies the finite-dimensional integrand and its omitted kernels. The method of steepest descent supplies saddle contours and asymptotic expansions.

Helpful background. Branches, sheets, continuation, and monodromy supplies spectral cuts and analytic continuation.

The Euclidean path integral is a cycle

After continuation to Euclidean signature, fields that were related by Lorentzian conjugation are generally independent complex variables. The schematic expression

$$Z=\int_\Gamma\mathcal D\Phi\;e^{-S_E[\Phi]}$$

therefore includes essential data $\Gamma$: a middle-dimensional relative homology cycle along which the integral converges. A Lorentzian $i\epsilon$ prescription may determine $\Gamma$ by analytic continuation; in other problems reflection positivity selects a real slice; in cohomological theories one may instead define the observable directly by thimbles. These are different constructions and should not be conflated.

The localizing symmetry must preserve the cycle. Infinitesimally, $Q$ must be tangent to $\Gamma$ or the change of variables must be deformable back to it without crossing singularities or changing its endpoints at infinity. Positivity of $(QV)_{\rm bos}$ is then checked on this particular cycle.

Picard–Lefschetz decomposition

In a finite-dimensional complex manifold $X$ with holomorphic action $S(z)$, assume for the moment that the critical points $z_\sigma$ are nondegenerate. With a Hermitian metric, use downward gradient flow

$$\frac{dz^i}{d\tau} =-g^{i\bar j}\overline{\frac{\partial S}{\partial z^j}}.$$

Along it,

$$\frac{d}{d\tau}\operatorname{Re}S =-\left\lVert\partial S\right\rVert^2, \qquad \frac{d}{d\tau}\operatorname{Im}S=0.$$

The stable middle-dimensional manifold that flows to $z_\sigma$ is a convergent thimble $\mathcal J_\sigma$ when oriented toward regions where $\operatorname{Re}S\to+\infty$. Under suitable tameness assumptions,

$$[\Gamma]=\sum_\sigma n_\sigma[\mathcal J_\sigma], \qquad n_\sigma\in\mathbb Z,$$

where $n_\sigma$ is the intersection number with the dual upward cycle. Thus solving the saddle equations finds candidate thimbles but does not determine which ones occur; that information comes from the original cycle.

A Stokes wall occurs when critical values have aligned phases,

$$\operatorname{Im}(S_\sigma-S_\tau)=0,$$

and a connecting flow exists. The thimble basis and integers $n_\sigma$ jump together so that the underlying cycle stays fixed. Witten develops this framework for analytically continued gauge theory in Witten 2011, §§2–3. In an infinite-dimensional QFT this is a powerful construction but not a universal existence theorem; compactness of flows and singular strata must be controlled model by model.

Worked example: a complex Gaussian

Let $a=|a|e^{i\theta}\ne0$ and

$$I(a)=\int_{\Gamma_a}dz\;e^{-az^2/2}.$$

The real axis converges only when $\operatorname{Re}a>0$. A steepest-descent cycle through the saddle $z=0$ is

$$z=e^{-i\theta/2}x, \qquad x\in\mathbb R,$$

because $az^2=|a|x^2$. Hence

$$I(a)=e^{-i\theta/2}\sqrt{\frac{2\pi}{|a|}} =\sqrt{\frac{2\pi}{a}}.$$

The square-root branch is fixed by continuously transporting the cycle, not by the formal determinant $a^{-1/2}$ alone. After $\theta\to\theta+2\pi$, the oriented thimble reverses and the square root changes sign. This elementary monodromy is the finite-dimensional version of a determinant phase.

Zero modes, negative modes, and poles

If the critical set is a manifold rather than isolated points, Picard–Lefschetz theory is Morse–Bott: integrate tangent zero modes as collective coordinates and construct thimbles only in normal directions. If the Hessian acquires a new zero eigenvalue, the Gaussian approximation is nonuniform; one needs an effective integral retaining that mode. A negative eigenvalue on a proposed real slice signals that it is not a convergent steepest-descent direction, not that its absolute value may simply be inserted into a square root.

After localization the effective integrand is often meromorphic. Deforming a contour across a pole adds a residue, and pinching poles can obstruct continuation. A pole may signal a massless mode that was incorrectly integrated out, so the correct response can be to restore that mode rather than to choose a residue after seeing the desired answer.

Spectral cuts and local counterterms

For a dimensionful positive operator $K$, one may define

$$\zeta_K(s)=\sum_{\lambda\ne0}\lambda^{-s}, \qquad \log\det'\!\left(\frac K{\mu^2}\right) =-\zeta_K'(0)-2\zeta_K(0)\log\mu.$$

For complex eigenvalues, $\lambda^{-s}=e^{-s\log\lambda}$ requires a spectral cut. Moving the cut across eigenvalues changes the phase. For self-adjoint first-order operators that asymmetry is encoded by an eta invariant. A regulator must preserve the same gauge and $Q^2$ symmetries used in the localization argument; otherwise its anomalous variation must be included.

Two symmetry-preserving regulators can differ by a local supersymmetric counterterm. The ambiguity is restricted by dimension, background multiplet, and preserved supercharges, but it is not always zero. In four-dimensional new-minimal backgrounds, the available counterterms and the stronger constraints from two supercharges of opposite chirality are classified by Assel, Cassani, and Martelli 2014, §§4–5.

Accordingly, a reproducible localized result states:

• the original Euclidean cycle or analytic-continuation prescription;
• thimble orientations and Stokes chamber;
• all poles crossed in reaching the final contour;
• treatment of bosonic, fermionic, and gauge zero modes;
• spectral cut, renormalization scale, and determinant phase;
• allowed finite counterterms and normalization conditions.

Exercises

1. Gaussian convergence sectors. For fixed $a$, find the asymptotic directions $z=re^{i\varphi}$ along which $e^{-az^2/2}$ decays.

Solution

Decay requires $\operatorname{Re}(ae^{2i\varphi})>0$, or $\cos(\theta+2\varphi)>0$. The steepest directions through the saddle have $\theta+2\varphi=0\pmod{2\pi}$, giving $\varphi=-\theta/2\pmod\pi$.

2. A determinant zero. What happens to the Gaussian approximation as $a\to0$ in the worked example?

Solution

The width grows like $|a|^{-1/2}$ and the integral diverges on the transported noncompact cycle. The mode is becoming a true zero mode, so it must be retained in an effective integral or stabilized by higher-order terms; a primed determinant alone cannot supply its measure.

References

• Assel, Benjamin, Davide Cassani, and Dario Martelli. “Supersymmetric Counterterms from New Minimal Supergravity.” Journal of High Energy Physics 2014, no. 11 (2014): 135. doi:10.1007/JHEP11(2014)135. Open preprint.
• Witten, Edward. “Analytic Continuation of Chern–Simons Theory.” In Chern–Simons Gauge Theory: 20 Years After, 347–446. AMS/IP Studies in Advanced Mathematics 50. Providence, RI: American Mathematical Society, 2011. Open preprint.

Next step

Contours acquire additional chamber and anomaly data in the presence of boundaries, cutting and gluing, and Jeffrey–Kirwan residues.