One-Eigenvalue Instanton Insights

• One-eigenvalue instanton is a nonperturbative solution characterized by a single distinguished eigenvalue or negative fluctuation mode, influencing tunneling and vacuum decay.
• It plays a pivotal role in matrix models and gauge theories by providing leading corrections through eigenvalue tunneling and controlling moduli in deformation theory.
• Its precise mathematical formulation unifies spectral analysis, resurgent transseries, and physical applications, impacting fields from quantum mechanics to string theory.

A one-eigenvalue instanton refers to a specialized class of instanton solutions in field theory, gauge theory, random matrix models, and mathematical physics, characterized by either the involvement of a single distinguished eigenvalue (as in matrix models and quantum mechanics) or possessing a single negative mode in its spectrum of fluctuations (as in vacuum decay and tunneling phenomena). Such instantons are pivotal in various nonperturbative phenomena, including vacuum transitions, spectral flow, dualities, and the full nonperturbative content of matrix models and string theory. The concept appears in several guises: as pointlike or localized instantons in gauge theories; as leading tunneling saddle points in matrix models; as singular objects controlling deformation theory in geometry; and as bounce solutions related to resonant states in complexified quantum mechanics.

1. Definition and Characterization

A one-eigenvalue instanton is typically defined with reference to either matrix models or differential operators:

• Matrix Model Context: A one-eigenvalue instanton is an eigenvalue tunneling event, in which a single eigenvalue is extracted (or tunnels) from the main support of the eigenvalue distribution (often the physical cut) and is placed at a nontrivial saddle point of the associated effective potential (Okuyama, 2018, Marino et al., 2022, Eniceicu et al., 2023, Chen et al., 11 Jul 2024).
• Field Theory and Tunneling: In vacuum decay or instanton operator interpretations, the one-eigenvalue instanton is the unique tunneling solution with a single unstable direction (single negative mode), controlling the decay rate between vacua (Brown et al., 2011, Bergman et al., 2016).
• Geometric/Analytic Setting: In singular instanton deformation theory, it refers to model instantons where the spectral problem reduces to a linear operator possessing one or a few distinguished eigenvalues in its discrete spectrum, often associated with moduli of the singularity (Wang, 2019).
• Quantum Mechanics: In Schrödinger-type eigenproblems with complex boundary conditions, the one-eigenvalue instanton reflects the isolated resonant eigenvalue whose imaginary part governs the decay rate, linking instanton methods to resonant state theory (Garbrecht et al., 30 Jul 2025).

2. Spectral Properties and Fluctuation Modes

The distinguishing property of a one-eigenvalue instanton in field theory is its fluctuation spectrum:

• For dynamical tunneling solutions, the spectrum of linearized fluctuations about the instanton includes exactly one negative mode. This negative mode typically corresponds to instability in the modulus controlling the physical transition (e.g., bubble radius in vacuum decay) (Brown et al., 2011).
• In gauge theory contexts, pointlike instantons becoming supersymmetric BPS objects are characterized by localization of their flux and the restoration of half the supersymmetry (Bergman et al., 2016).
• In matrix models, the action for extracting a single eigenvalue from the cut yields the leading nonperturbative correction, with contributions governed by the instanton action $A(t)$ and suppressed as $e^{-NA(t)}$ (Okuyama, 2018, Eniceicu et al., 2023).

3. Mathematical Formalism and Representative Equations

The mathematical structure of one-eigenvalue instantons varies by domain:

• Random Matrix Theory:
  • The one-point function capturing the tunneling of one eigenvalue:

  $$\mathcal{Z}(\tau) = \int dx \, \exp\left(-N V_{\mathrm{eff}}(x) + 2iN\tau x\right)$$ - Instanton action for the saddle at $x^*$:

  $S_{\mathrm{inst}}(\tau) = 2[\tau \sqrt{\tau^2-1} - \arccosh(\tau)]$

  with corrections scaling as $e^{-NS_{\mathrm{inst}}}$ (Okuyama, 2018).

• Instanton Operators:
  • In five dimensions, disorder operators $\mathcal{I}_n(x)$ insert $n$ units of instanton charge:

  $$\frac{1}{8\pi^2} \int_{S^4_x} \operatorname{Tr} (F \wedge F) = n$$

  The pointlike limit localizes the flux to a single point, leading to a one-eigenvalue instanton object (Bergman et al., 2016, Lambert et al., 2014).

• Spectral Flow and Deformation Theory:
  • The spectrum $Spec P$ of the deformation operator for singular instantons contains finitely many integer values (e.g., $-1$, $-2$), arising from sheaf cohomology, and an analytic continuum from the rough Laplacian:

  $$Spec(P) = S_{analytic} \cup S_{coh}, \quad S_{coh} = \{-1, -2\}$$

  (Wang, 2019).

• Path Integral for Resonant States:
  • The propagator along complex contours is related via path integrals and spectral representation:

  $$K_{\gamma, \theta}(s_i, s_f; T) = \sum_\ell e^{-i e^{-i\theta} E_\ell T/\hbar} \Psi_\ell[\gamma(s_i)] \Psi_\ell[\gamma(s_f)] \left(\int_\Gamma \Psi_\ell(z)^2 dz\right)^{-1}$$

  (Garbrecht et al., 30 Jul 2025).

4. Physical and Geometric Implications

One-eigenvalue instantons are central in both physical transitions and the structure of moduli spaces:

• Vacuum Transitions: The abrupt disappearance of a one-eigenvalue instanton means the decay channel closes while the tunneling rate remains finite, leading to non-smooth changes in the landscape of accessible vacua (Brown et al., 2011). This scenario underpins catastrophic changes in cosmological and string theory landscapes.
• Supersymmetric Index Contributions: In 5D $\mathcal{N}=1$ theories, only pointlike (one-eigenvalue) instantons contribute to the index and the symmetry enhancement (Bergman et al., 2016).
• Matrix Models and Resurgence: The exponential suppression from one-eigenvalue tunneling events is essential in nonperturbative corrections (e.g., plateau in spectral form factor), with anti-eigenvalue instantons contributing via nonphysical sheets in the spectral curve. The full resurgent transseries includes mixed sectors, all interpretable in terms of eigenvalue tunneling (Marino et al., 2022, Eniceicu et al., 2023).
• Giants Graviton Expansion: In finite-$N$ theories, each $m$-eigenvalue instanton event underpins an $e^{-mN}$ correction in the partition function and index (Chen et al., 11 Jul 2024).

5. Applications Across Domains

One-eigenvalue instantons manifest in several contexts:

• Gauge Theory: Moduli spaces for $k=1$ instantons (e.g., ADHM construction) are built on single instanton solutions, with the Hilbert series counting BPS operators by degree (Benvenuti et al., 2010).
• String Theory and Superstrings: Tracy–Widom distributions and Fredholm determinants encode the nonperturbative instanton sum, leading to smooth crossovers (rather than sharp phase transitions) in supersymmetry breaking (Nishigaki et al., 2014).
• Spectral Theory and Geometry: Deformation theory for singular G$_2$ instantons or Hermitian Yang–Mills connections is governed by the structure of discrete eigenvalues in operators associated with the singular locus (Wang, 2019).
• Quantum Mechanics: Path integral methods based on complex contours allow a unified treatment of metastable decay and PT-symmetric systems, with decay rates given by isolated resonant eigenvalues corresponding to instanton contributions (Garbrecht et al., 30 Jul 2025).

6. Abrupt vs. Smooth Disappearance and Catastrophe Theory

In vacuum decay dynamics, the fate of the one-eigenvalue instanton can be classified:

• Smooth Disappearance: The instanton dissolves as its action diverges (radius goes to infinity), the decay rate smoothly drops to zero (e.g., gravitational blocking) (Brown et al., 2011).
• Abrupt Disappearance: The instanton's action remains finite at the critical point until it annihilates with a double-bubble solution of higher index, leading to a discontinuous change in accessible vacua (Brown et al., 2011).

This bifurcation structure parallels catastrophe theory, where the merging and annihilation of saddles correspond to fold and cusp catastrophes.

7. Synthesis and Broader Significance

The paper of one-eigenvalue instantons illuminates several nonperturbative mechanisms:

• They supply the minimal action contributions to expansions of partition functions and indices in both gauge theory and matrix models, systematically accounting for nontrivial corrections at finite $N$ and strong coupling.
• Their abrupt disappearance or spectral restructuring determines the boundary between smoothly modulated and catastrophically reorganized landscapes in field theory, cosmology, and string theory.
• Their analytic properties in deformation theory offer precise control over moduli and singularities in geometric contexts.
• The equivalence between decay rates from instanton bounces and complex resonant states bridges semiclassical physics with spectral theory in non-Hermitian or PT-symmetric systems.
• In supersymmetric gauge theories, their localization properties dictate symmetry enhancement and multiplicity structures for current multiplets.

The one-eigenvalue instanton thus serves as a conceptual and computational nexus for the interplay between topology, spectral theory, nonperturbative physics, and algebraic geometry, underlining its foundational role in modern mathematical and theoretical physics.