Resurgence theory, ghost-instantons, and analytic continuation of path integrals

TL;DR

This work extends resurgence theory to path integrals with both real and complex saddles, showing that ghost-instantons influence large-order perturbation through precise trans-series relations. By analyzing a zero-dimensional prototype and a one-dimensional quantum-mechanical model, it demonstrates how Lefschetz thimbles and analytic continuation yield distinct, non-analytic quantum phases tied to saddle structure. The key result is that perturbative series around the vacuum are governed by both real and complex saddles, with exact resurgence relations linking high-order vacuum terms to low-order fluctuations around nonperturbative saddles. These insights illuminate non-perturbative dynamics in QFT, string theory, and quantum gravity, offering a framework to understand phase transitions and the global analytic structure of path integrals.

Abstract

A general quantum mechanical or quantum field theoretical system in the path integral formulation has both real and complex saddles (instantons and ghost-instantons). Resurgent asymptotic analysis implies that both types of saddles contribute to physical observables, even if the complex saddles are not on the integration path i.e., the associated Stokes multipliers are zero. We show explicitly that instanton-anti-instanton and ghost--anti-ghost saddles both affect the expansion around the perturbative vacuum. We study a self-dual model in which the analytic continuation of the partition function to negative values of coupling constant gives a pathological exponential growth, but a homotopically independent combination of integration cycles (Lefschetz thimbles) results in a sensible theory. These two choices of the integration cycles are tied with a quantum phase transition. The general set of ideas in our construction may provide new insights into non-perturbative QFT, string theory, quantum gravity, and the theory of quantum phase transitions.

Figures (10)

• Figure 1: Doubly periodic structure of sd$^2(z|m)$ in the complex $z$ plane. The real and imaginary periods are $2\mathbb K(m)$ and $2i\mathbb K^\prime(m)$ which define the fundamental torus. Left: In the $d=0$ case, studied in Section \ref{['sec:zero']}, the perturbative $(A)$, real $(B)$ and imaginary $(C)$ saddles are at $z_i=\{0,\mathbb K, i\mathbb K^\prime\}$, respectively. Right: In the $d=1$ QM example, studied in Section \ref{['sec:one']}, instantons and ghost-instantons are functions which extrapolate between $0\rightarrow 2\mathbb K(m)$ and $0\rightarrow 2i \mathbb K^\prime(m)$, respectively, and $[ {\mathcal{I}} \bar{\mathcal{I}} ]$ is a real bounce, and $[ {\mathcal{G}} \bar{\mathcal{G}} ]$ is a complex bounce.
• Figure 2: Upper:${a_n^{\rm actual}(m)}/{ a_n^{\rm naive}(m)}$: The ratio of the actual vacuum perturbation series coefficients to the "naive" prediction (\ref{['eq:d0-naive']}) for the large order growth which does not include the effect of complex saddle $C$. The different curves refer to different values of the elliptic parameter $m$: $m=0$ (blue circles), $m=\frac{1}{4}$ (red squares), $m=0.49$ (gold diamonds), and $m=0.51$ (green triangles). As $m$ approaches $1/2$ from below the agreement breaks down rapidly, showing that the contribution of the saddle $B$ by itself is not sufficient to capture the large order growth. Lower: The ratio ${a_n^{\rm actual}(m)}/{ a_n^{(A)}(m)}$, including the contributions to $a_n^{(A)}(m)$ from both the real and complex saddles, ($B$ and $C$) to leading order, as in (\ref{['eq:d0-leading1']}). The different curves refer to different values of the elliptic parameter $m$: $m=0$ (blue circles), $m=\frac{1}{4}$ (red squares), $m=0.49$ (gold diamonds), and $m=0.51$ (green triangles). Note the dramatically improved agreement, especially for $m\geq 1/2$.
• Figure 3: The complex Borel $u$-plane structure for various values of $m$ for the zero-dimensional prototype. The circles denote branch cut singularities. The singularities on $\mathbb{R}^+$ and $\mathbb{R}^-$ are due to $B$ (real) and $C$ (purely imaginary) saddles. For $0<m<1$, $\arg(g^2) =\theta=0$ and $\theta=\pi$ direction in the coupling constant plane are Stokes lines.
• Figure 4: The ratio ${a_n^{\rm actual}(m)}/{ a_n^{(A)}(m)}$, including the contributions to $a_n^{(A)}(m)$ from both the real and complex saddles, ($B$ and $C$) beyond leading order, as in (\ref{['eq:d0-large-order']}). These plots are all for $m=0.51$ (those for $m=0.49$ are indistinguishable) to leading (blue circles), sub-leading (red squares) and sub-sub leading (gold diamonds) order. At the sub-sub-leading order the agreement is almost perfect even for the low order terms.
• Figure 5: Schematic representation of the resurgent relation (\ref{['eq:d0-large-order']}) between the large orders of perturbative fluctuations about the vacuum saddle $A$ given in (\ref{['power_series']}, \ref{['eq:zero-closed']}) (represented as the large circle), and low orders of fluctuations about the instanton and ghost instanton saddles $B$ and $C$ given in (\ref{['eq:bc_expansions']}) (represented as the small circles), respectively.