Multi-Cut Saddles in Matrix Models

• Multi-cut saddles are solutions where the eigenvalue distribution splits into multiple disjoint intervals, encoding instanton effects and tunneling phenomena.
• Resolvent methods and spectral curve analysis precisely determine the endpoints of these cuts, unveiling phase transitions and critical points in matrix models.
• Applications in GWW, CS, and ABJM models demonstrate that multi-cut configurations are key to understanding the full nonperturbative transseries in large-N theories.

Multi-cut saddles are large-$N$ solutions to the saddle-point equations of matrix models in which the eigenvalue distribution support is split over multiple disjoint intervals ("cuts") in the complex plane. These configurations play an essential role in revealing the underlying nonperturbative structure of prominent gauge-theoretic matrix models, particularly the Gross-Witten-Wadia (GWW), pure Chern-Simons (CS), and ABJM models. Multi-cut saddles encode instanton-like effects, eigenvalue tunneling, and phase transitions, and are critical for understanding the exact nonperturbative expansions in large-$N$ matrix models.

1. Large-$N$ Multi-Cut Saddle Formalism

The multi-cut saddle-point structure emerges in the large-$N$ limit of matrix models, where the partition function reduces to a functional integral over the normalized eigenvalue density $\rho(x)$, subject to Coulombic interactions and model-specific single-particle potentials. For the GWW model, the partition function in terms of eigenangles $\{\theta_i\}$ is

$$\mathcal{Z}_N = \prod_{i=1}^N \int_{-\pi}^{\pi} d\theta_i \prod_{i<j} \sin^2\left(\frac{\theta_i-\theta_j}{2}\right) \exp \left[ \frac{2N}{\lambda} \sum_{i=1}^N \cos\theta_i \right]$$

with the normalized density

$$\rho(\theta) = \frac{1}{N} \sum_{i=1}^N \delta(\theta - \theta_i), \qquad \int_{-\pi}^{\pi} \rho(\theta) d\theta = 1,$$

and effective action

$$S[\rho] = \frac{1}{\lambda} \int_{-\pi}^{\pi} \rho(\theta) \cos\theta \, d\theta - \frac{1}{2} \int \int \rho(\theta) \rho(\phi) \ln \left[ \sin^2\left(\frac{\theta - \phi}{2}\right) \right] d\theta d\phi.$$

Analogous constructions exist for pure CS and ABJM models, replacing the trigonometric kernel with hyperbolic or mixed kernels appropriate to their gauge group and localization geometry (Álvarez et al., 2016, Morita et al., 2017).

2. Resolvent Methods and Spectral Curves

The analytic structure of multi-cut solutions is encoded by the resolvent function, a key tool for reconstructing $\rho(x)$ in the continuum limit. Defining, for GWW,

$$\omega(z) = \int_\gamma \frac{\rho(z') |dz'|}{z - z'}$$

where $\gamma$ is the union of cuts in the complex $z$-plane ($z = e^{-i\theta}$), one obtains the large-$N$ saddle condition that the holomorphic potential $W_{\mathrm{h}}(z)$ is constant on each cut. The “loop equation” expresses $y(z)^2$ as a rational function:

$$y(z)^2 = \frac{(z-a_1)(z-a_2)(z-b_1)(z-b_2)}{\lambda^2 z^4}$$

where the locations of the branch points $a_i, b_i$ specify the endpoints of the arcs or intervals supporting $\rho(z)$. In the pure CS case, the resolvent $W(z)$ features a similar spectral curve, and its discontinuity across each cut yields the density by

$$\rho(x) = -\frac{1}{2\pi i} [W(x + i0) - W(x - i0)], \quad x \in \text{cut}.$$

The saddle-point equations admit solutions with support on one, two, or more cuts, each corresponding to a distinct configuration in the complex plane (Álvarez et al., 2016, Morita et al., 2017).

3. Classification of Multi-Cut Configurations

Multi-cut solutions are classified by the number of disjoint intervals ($s$-cuts), their location, and the filling fractions $q_i$ indicating the proportion of total eigenvalue density supported on each cut. The main cases for GWW and CS/ABJM models are summarized below:

Matrix Model	# Cuts ($s$)	Support Structure	Parameter Regime
GWW	1	Arc(s) on unit circle	$q=1$, $0<\lambda<2$ or $q=1$, $\lambda>2$
GWW	2	Arc + real interval	$0<\lambda<2$, $0
GWW	3	Circle + real intervals	$\lambda>2$, $0
CS/ABJM	$s\geq 1$	Disjoint intervals (in $u$-plane)	All $\lambda$

Endpoint locations are precisely determined by algebraic relations involving the coupling $\lambda$ and filling fractions. In ABJM matrix models, an infinite family of multi-cut solutions arises, parametrized by integer shifts, and can be classified by their eigenvalue support and associated branch points (Álvarez et al., 2016, Morita et al., 2017).

4. Gap-Closing, Instanton Actions, and Nonperturbative Sectors

Multi-cut saddles encapsulate large-$N$ instanton sectors: configurations where a finite fraction $q<1$ of eigenvalues “tunnels” between different cuts, away from the dominant vacuum support. The gap-closing transition, defined by the merger of two real cuts (when $1/a_1=a_2$ in GWW), is marked by a critical line $q^*(\lambda)$ separating regimes of distinct saddle topology:

$$q^*(\lambda) = \frac{2}{\pi\lambda} \int_0^\pi \sqrt{\cos^2\theta + \lambda \cos\theta + \lambda - 1}\, d\theta$$

Instanton actions can be derived directly. For weak-coupling GWW,

$$S_{I}^{(w)} = \int_a^{-1} y(z)\, dz = \frac{1}{\lambda} \int_a^{-1} (z+1) \sqrt{(z-a)(z-\bar{a})} \frac{dz}{z^2}$$

while at strong coupling,

$$S_{I}^{(s)} \approx 2\left[ \frac{\lambda}{2} - \sqrt{1 - \frac{4}{\lambda^2}} \right]$$

Analogous instanton actions and eigenvalue transfer mechanisms underlie the multi-cut sector contributions in CS and ABJM models, identified both in spectral curve formalism and in the language of D2-brane instanton condensation (Álvarez et al., 2016, Morita et al., 2017).

5. Multi-Cut Solutions in Pure CS and ABJM Matrix Models

Pure CS matrix models on $S^3$ admit a countably infinite set of multi-cut solutions, stemming from the structure of their saddle-point equations. The prominent two-cut solution for $U(N)$ CS at finite $\lambda$ is characterized by explicit branch points $A$ and $B$ in the $Z$-plane ($Z = e^u$) and the exact resolvent $v(Z)$,

$$v(Z) = \frac{2n}{\pi\lambda} \arctan \left[ \sqrt{ \frac{ Z-A }{ Z-B } \frac{ B-1 }{ 1-A } } \right ] - \frac{n}{\lambda} + \frac{1}{\pi i \lambda} \log\left[ \frac{ Z + \sqrt{AB} - \sqrt{ (Z-A)(Z-B) } }{ \sqrt{A} + \sqrt{B} } \right]$$

with $n$ an integer specifying the cut configuration. In the ABJM matrix model, multi-cut solutions arise by integer shifts in eigenvalues, leading to a tower of instanton-like sectors. These sectors map precisely to D2-brane instanton condensates; for example, a single-eigenvalue shift yields an action

$$\Delta V_{\mathrm{eff}} = n N \pi \sqrt{2 / \lambda} + \dots, \quad |\lambda| \gg 1$$

which matches the D2-instanton action. Thus, multi-cut solutions in ABJM reconstruct the full nonperturbative transseries, including worldsheet and membrane instantons (Morita et al., 2017).

6. Physical Interpretation and Dominance

One-cut (vacuum) solutions in GWW and CS/ABJM models dominate the free energy in the perturbative regime: e.g., the weak-coupling GWW and CS one-cut densities have lowest energy, with multi-cut configurations exponentially suppressed by factors like $e^{-N/\lambda}$. However, multi-cut saddles are indispensable for the complete nonperturbative (transseries) description, particularly in strong-coupling regimes where instanton condensation and brane effects become relevant. In ABJM theory, multi-cut configurations reconstruct known membrane and D2-instanton corrections to the AdS$_4 \times$ CP$^3$ dual background, and their precise roles are encoded within the Fermi-gas approach (Álvarez et al., 2016, Morita et al., 2017).

7. Summary and Outlook

Multi-cut saddles represent a unified framework for understanding nonperturbative phenomena in large-$N$ matrix models. Through resolvent methods and spectral curve analysis, their explicit classification and instanton actions become accessible, with direct physical manifestations in phase transitions, gap-closings, and instanton expansions. In CS and ABJM theories, the infinite family of multi-cut solutions is crucial for capturing D2-brane and worldsheet instanton physics, completing the exact large-$N$ structure. A plausible implication is that any matrix model admitting multi-cut saddles possesses a rich transseries expansion, reconstructible only via full consideration of these sectors (Álvarez et al., 2016, Morita et al., 2017).