Nonperturbative aspects of ABJM theory

TL;DR

The paper develops a nonperturbative analysis of ABJM theory by leveraging the all-genus matrix-model free energy and its spectral-curve structure. It proposes a general prescription that instanton actions are linear combinations of periods in special geometry and applies it to ABJM to identify three distinct instantons across its moduli space; these instantons govern the large-genus growth and provide concrete predictions for Borel summability. At strong coupling, the dominant instanton is shown to correspond to a Euclidean D2-brane wrapping ${\mathbb RP}^3\subset {\mathbb CP}^3$, with a first-principles DBI+CS calculation reproducing the leading action, thereby linking nonperturbative transport in the matrix model to explicit D-brane configurations. The results illuminate the interplay between matrix models, topological strings, and M-theory, and open avenues to explore nonperturbative effects in broader local Calabi–Yau geometries and their holographic duals.

Abstract

Using the matrix model which calculates the exact free energy of ABJM theory on S^3 we study non-perturbative effects in the large N expansion of this model, i.e., in the genus expansion of type IIA string theory on AdS4xCP^3. We propose a general prescription to extract spacetime instanton actions from general matrix models, in terms of period integrals of the spectral curve, and we use it to determine them explicitly in the ABJM matrix model, as exact functions of the 't Hooft coupling. We confirm numerically that these instantons control the asymptotic growth of the genus expansion. Furthermore, we find that the dominant instanton action at strong coupling determined in this way exactly matches the action of an Euclidean D2-brane instanton wrapping RP^3.

Figures (4)

• Figure 1: The topology of the contours $\mathcal{C}_w$, $\mathcal{C}_c$, $\mathcal{C}_s$ for a vicinity of the orbifold point in the moduli space.
• Figure 2: In this figure we depict the absolute value of the three instanton actions in the orbifold or weakly coupled frame. On the left side, the horizontal axis represents the positive real axis of the $\kappa$ variable. The curve in green, which vanishes at the origin, is $|A^{(w)}_{w}(\kappa)|$, while the blue and red lines represent $|A^{(w)}_{c}(\kappa)|$ and $|A^{(w)}_{s}(\kappa)|$, respectively. Notice that, when $\kappa$ is large (i.e. the strong coupling region), the smallest action in absolute value is $A^{(w)}_{s}(\kappa)$. On the right side, the horizontal axis represents the imaginary axis of the $\kappa$ variable. The conifold action $A^{(w)}_{c}(\kappa)$ vanishes at $\kappa_c=-4{\rm i}$, and therefore dominates the large order behavior near that point.
• Figure 3: In these figures, the dots represent the sequence (\ref{['rseq']}) for values of $\lambda$ in the strong coupling region: $\lambda\approx 1.2838$ (left) and $\lambda \approx 4.6687$ (right). The action is then $A_s(\lambda)$, given in (\ref{['stronga']}). The continuous line represents the oscillatory behavior in the r.h.s. of (\ref{['cosibe']}), where the angle is the one associated to the strong coupling action $\theta_s(\lambda)$.
• Figure 4: In these figures, the dots represent the fifth Richardson transform of the sequence (\ref{['qseq']}) for values of $\lambda$ along the negative imaginary axis $\lambda\approx -0.1386 \, {\rm i}$ (left) and $\lambda \approx -0.0620 \, {\rm i}$ (right), and for the conifold and the weak coupling action, respectively. They converge quite rapidly to unity, verifying in this way that the proposed instanton actions control the large order behavior of the genus $g$ free energies.