Gapped Flux Vacua in Gauge & String Theories

• Gapped flux vacua are defined as discrete, stable ground states in flux compactifications and gauge theories that exhibit a strictly positive spectral gap, ensuring moduli stabilization and duality tests.
• They are constructed in frameworks like 2D (8,8) SU(N) SYM and 4D gauged supergravity, where quantized flux and center charges partition the Hilbert space into superselection sectors.
• Practical implications include precise non-perturbative tests of string dualities, controlled moduli lifting, and scalable models for anisotropic scale separation in AdS compactifications.

Gapped flux vacua are discrete, stable ground states in flux compactifications or gauge theories, characterized by a strictly positive spectral gap separating the vacuum from the tower of excitations. These configurations play central roles in non-perturbative string theory, gauge theory dualities, and the microscopic analysis of moduli stabilization and scale separation in higher-dimensional supergravity. Rigorous constructions of gapped flux vacua have been achieved in two principal frameworks: the flux sectors of $(8,8)$ $SU(N)$ super-Yang–Mills theory in two dimensions (matrix string theory context) and in four-dimensional gauged supergravity arising from geometric and non-geometric flux backgrounds in type II string compactification. The following sections provide a systematic account of these settings, the conditions for gapped vacua, classification of sectors, illustrative examples, and their implications for dualities and moduli stabilization.

1. Definition and Core Characteristics

A gapped flux vacuum is a unique ground state in a flux sector of a gauge or supergravity theory, defined by the presence of conserved discrete charges (e.g., center 1-form charges or quantized flux integers), for which the fluctuation spectrum exhibits a strictly positive mass gap above the vacuum (excluding decoupled massless modes). The gap persists in the strict large-$N$ or large-flux limit, and reflects the absence of moduli or unstable flat directions in the effective potential. In Yang–Mills theory, gapped flux vacua arise as supersymmetric ground states in superselection sectors labeled by (generalized) electric or magnetic flux, whereas in string compactifications, they correspond to critical points of the scalar potential with all moduli stabilized and the mass matrix possessing only non-negative eigenvalues above the cosmological constant.

2. Gapped Flux Vacua in 2D $(8,8)$ $SU(N)$ SYM and Matrix String Theory

In the framework of 2D ${\cal N}=(8,8)$ $SU(N)$ super-Yang–Mills theory compactified on a spatial circle of radius $R$, the system exhibits a conserved $\mathbb Z_N$ electric 1-form center charge, resulting in a decomposition of the Hilbert space into superselection sectors, ${\cal H}_k$, each labeled by $k\in\mathbb{Z}_N$ units of electric flux winding the circle (Cho et al., 6 Jan 2026).

The essential features are as follows:

• Classification: Each sector ${\cal H}_k$ contains a unique supersymmetric ground state—termed the flux vacuum of sector $k$—and the full Hilbert space is ${\cal H} = \bigoplus_{k=0}^{N-1} {\cal H}_k$.
• Vacuum Solutions: In temporal gauge $A_0=0$, the $k$-flux vacuum is realized by a constant non-Abelian electric field, with the gauge field holonomy $\oint_{S^1}\!A_1\,dx = 2\pi k/N$. The classical field configuration and energy are

$$A_1 = \frac{k}{N R}\,t\,\mathbb{I}_N,\quad F_{01} = \frac{g_{\rm YM}^2}{N} k\,\mathbb{I}_N,\quad E_k = \frac{g_{\rm YM}^2}{2}\frac{k^2}{N}(2\pi R).$$

• Spectral Gap: The mass gap above the $k$-flux vacuum is given by $\Delta_k = \frac{k g_{\rm YM}}{N} \sqrt{2\pi}$, arising from the quadratic fluctuations of the $SU(N)$ off-diagonal modes. This gap is positive for any $k\neq 0$ sector after removing the $U(1)$ photon which remains gapless.
• Stability: The flux vacua are BPS and preserve half of the supersymmetry; all non-Abelian modes are gapped, ensuring perturbative and nonperturbative stability.

This structure enables precise matching to the D0-brane sectors in dual matrix string theory. The $k$-flux sector corresponds to a state with $k$ coincident D0-branes, with a spectrum of massive excitations mirroring open string oscillator levels, $$m_\ell^{(k)} = \frac{k g_{\rm YM}}{N} \sqrt{2\pi\ell}$$, lying on a Regge trajectory (Cho et al., 6 Jan 2026).

3. Classification and Stability Conditions in Supergravity Flux Vacua

The existence of a gapped flux vacuum in string compactifications requires that all moduli be stabilized—i.e., the Hessian of the scalar potential at the vacuum has only positive (or at worst, Breitenlohner–Freedman-allowed negative) eigenvalues above the constant vacuum energy. This has been systematically explored in maximal $D=4$ supergravity obtained from type IIA and IIB flux compactifications (Dibitetto et al., 2012), as well as in semi-realistic models with scale separation in AdS$_4$ (Tringas, 2023).

Key points include:

• Potential Structure: In these supergravity models, the scalar potential is induced by the background fluxes and gauging, and depends on the embedding tensor; in explicit $SU(8)$ notation

$$V = g^2 \left[ -\tfrac{3}{4}|A_1|^2 + \tfrac{1}{24}|A_2|^2 \right],$$

where $A_1$ and $A_2$ are the gravitino and dilatino mass tensors, respectively.

• Critical Points: Critical points ($\partial V/\partial\phi=0$) correspond to possible vacua, which can be AdS$_4$ or Minkowski. The spectrum is computed from the second derivative of $V$ (mass matrix).
• Spectral Gap and Stability: A genuine gapped vacuum possesses a mass matrix whose smallest nonzero eigenvalue is strictly positive (in units of $|V_0|$ for AdS), and all scalars respect the Breitenlohner–Freedman bound, $m^2 \geq -\frac{3}{4}|V_0|$ for AdS$_4$. For example, a geometric IIA solution associated with $SO(4)\ltimes \mathrm{Nil}_{22}$ gauging features a non-supersymmetric, perturbatively stable AdS vacuum with normalized mass eigenvalues $m^2/|V_0| \geq 2/3$, above 11 flat Goldstone modes (Dibitetto et al., 2012).
• Moduli Lifting: A necessary condition for a finite gap is that all would-be moduli are lifted via gauge interactions; otherwise, residual flat directions preclude a positive gap.

4. AdS$_4$ Gapped Flux Vacua: Anisotropic Scale Separation and Parametric Control

In massive IIA compactifications on $T^6/(\mathbb{Z}_3 \times \mathbb{Z}_3)$ orbifolds, a rich structure of gapped AdS$_4$ flux vacua emerges, notably with parametrically tunable scale separation and anisotropic internal geometry (Tringas, 2023). The critical parameters are the quantized fluxes (e.g., $m_0, p, e_i$), whose scaling exponents ($f_i$) control both the mass gap and the degree of internal anisotropy.

Principal features:

• Flux Scaling and Moduli Stabilization: Solutions to the 4d effective potential yield

$$\nu_i \sim N^{(-f_i + f_j + f_k)/2},\quad e^\phi\sim N^{-(f_1+f_2+f_3)/4},\quad \mathrm{vol} \sim N^{(f_1+f_2+f_3)/2},$$

where $N\gg 1$ is a scaling parameter.

• Mass Gap Realization: The lightest Kaluza–Klein mass in the $i$-th direction is

$m_{KK,i} \sim N^{-(f_i + 3(f_j + f_k))/4},$

and the ratio $L_{KK,i}^2 / L_{AdS}^2 \sim N^{-f_i}$, admitting parametric scale separation for $f_i>0$.

• Scaling Regimes: Four regimes are classified depending on the $f_i$, including cases where all internal directions are large (anisotropic scale separation), one or more are constant or shrinking, or scale separation is broken.
• Physical Probes and Distance Conjecture: The interpolation between vacua by a probe anti-D4-brane enables calculation of geodesic distance in field space and the Swampland "distance conjecture" parameter $\gamma \simeq 0.88$, directly relating flux jumps to changes in light spectrum (Tringas, 2023).

5. Exceptional Flux Compactifications and Conditions for Spectral Gaps

The embedding of both geometric and non-geometric flux vacua into $D=4$ maximal supergravity provides a grand unified framework for classifying and realizing gapped flux vacua, as detailed in the work of Dibitetto, Guarino, and Roest (Dibitetto et al., 2012). The structure is summarized as follows:

• Flux Data and Gauging: All relevant fluxes—R–R, NS–NS, and T-dual non-geometric—are encoded in the SL(2)$\times$SO(6,6) embedding tensor and organize the possible gaugings of the theory.
• Vacuum Structure: Solutions are grouped into families with distinct symmetry and supersymmetry properties; some geometric IIA solutions correspond to $SO(4)\ltimes \mathrm{Nil}_{22}$ gaugings and yield AdS vacua with a positive mass gap and full stability.
• Spectral Gap Quantification: For an explicit geometric IIA gapped vacuum, the smallest nonzero scalar mass is normalized as $m^2/|V_0| = 2/3$ (besides Goldstones); no instabilities are present.
• Lifting of Flat Directions: Achieving a gapped spectrum requires that gauge interactions break all residual global symmetries (beyond AdS isometries), removing massless moduli.

6. Dualities, Matrix String Correspondence, and Physical Interpretation

Gapped flux vacua provide a precise non-perturbative testing ground for string dualities:

• Matrix String Duality: The mapping between gapped flux sectors in 2D SYM and D0-brane sectors in matrix string theory enables detailed matching of the gauge resonance spectrum to open string oscillator levels; the Regge behavior of masses is manifest in both pictures (Cho et al., 6 Jan 2026).
• Open/Closed String Sectors: The $k=0$ sector in SYM theory corresponds to the standard closed string vacuum, while $k>0$ flux vacua map to bound states of $k$ D0-branes with an infinite tower of massive open string excitations.
• Finite-N Effects and Metastability: At finite $N$ or non-infinite flux, corrections induce resonance widths and possible metastabilities, linking to physical processes such as closed-string emission.
• Swampland Considerations: The structure of geodesic distances and spectrum changes in interpolating between gapped vacua has direct implications for Swampland criteria and the distance conjecture (Tringas, 2023).

7. Open Questions and Future Directions

Several unresolved challenges and avenues for progress are identified:

• Deriving, from first principles, the conformal boundary conditions in symmetric-product orbifolds that correspond to specific flux sectors in gauge theory (Cho et al., 6 Jan 2026).
• Non-perturbative verification of the mass spectra and gap predictions, for example using Hamiltonian truncation, Discrete Light-Cone Quantization, or modern bootstrap methods for quantum mechanics.
• Systematic study of finite-$N$ and finite-radius corrections, particularly the properties of metastable resonances and decay channels.
• Exploration of strong-coupling dualities and the S-matrix structure in sectors with high flux and high oscillator levels.

A plausible implication is that the interplay between flux quantization, gap formation, and dual string theory structures provides robust, calculable laboratories for non-perturbative quantum gravity and for constraints arising in the Swampland program.