Supersymmetric Chiral Gauge Theories

Updated 2 September 2025

Supersymmetric chiral gauge theories are quantum field theories that merge non-Abelian gauge groups with chiral matter and supersymmetry, enforcing strict anomaly cancellation.
They display a range of IR phases, including superconformal fixed points, s-confinement, and dynamical symmetry breaking via nonperturbative effects and intricate dualities.
Analytical tools such as a-maximization, holomorphy, and deconfinement techniques enable precise characterization of the IR spectrum and deep insights into gauge dynamics.

Supersymmetric chiral gauge theories are quantum field theories in which a non-Abelian gauge symmetry coexists with chiral matter content—matter representations for which no gauge-invariant mass terms can be written—and supersymmetry. These theories are distinguished by the interplay between gauge anomalies, supersymmetric vacua, nonperturbative effects, and intricate dualities, leading to a spectrum of possible infrared (IR) phases with dynamical symmetry breaking, exotic fixed points, and a range of massless composite spectra. The synergy between supersymmetry and chirality has provided a laboratory where exact results can be established for nonperturbative strong coupling dynamics that are otherwise unapproachable in purely non-supersymmetric chiral gauge theories.

1. Structure of Supersymmetric Chiral Gauge Theories

Supersymmetric chiral gauge theories generally consist of a non-Abelian gauge group (G), chiral superfields transforming in (potentially reducible) complex representations of G, and a set of allowed gauge-invariant superpotential interactions. The archetype theories studied in the modern literature frequently involve:

Gauge group: $\mathrm{SU}(N)$ , $\mathrm{SO}(N)$ , $E_6$ , or other simple Lie groups.
Chiral superfields: Antisymmetric (A), symmetric (S), and (anti)fundamental (Q, $\widetilde{Q}$ ) fields, in patterns such that mass terms $\widetilde{Q} Q$ may not be gauge-invariant.
Superpotential deformations: Terms such as $W = Q A Q$ , or other cubic/marginal interactions.

Anomaly cancellation is essential; typically, the net gauge anomaly (the cubic index sum over all chiral fields) vanishes precisely, sometimes requiring an intricate balance between matter content and representation structure. For example, models with $SU(N)$ gauge group, an antisymmetric tensor, $N_F$ fundamentals, and $N_C+N_F-4$ antifundamentals (with $N_C = N$ ), are anomaly-free and exhibit highly nontrivial IR phenomena (Leedom et al., 11 Mar 2025).

Theories may also be constructed via orbifold projections, dimensional reduction, or deformations of higher-supersymmetry gauge theories, giving rise to product groups with chiral bifundamental matter (as in $SU(N)^3$ or $SU(3)^3$ “trinification” models) (Chatzistavrakidis et al., 2010).

2. Dynamical Phases and Symmetry Breaking

The IR behavior of supersymmetric chiral gauge theories displays a spectrum of phases, depending on the gauge group, matter content, and superpotential:

Conformal Window: For appropriately chosen flavor numbers ( $F$ ), marginal or “dangerously irrelevant” superpotentials can drive the theory to an interacting superconformal fixed point (SCFT). For example, $SU(N)$ with an antisymmetric plus $F=N+3$ fundamental/antifundamental pairs and $W=Q A Q$ flows to a nontrivial SCFT; these points admit a-maximization analysis and can exhibit (self-)duality (Craig et al., 2011, Craig et al., 2011).
s-Confinement and Mixed Phases: For certain flavor numbers, the theory confines without breaking chiral symmetry (“s-confinement”), or splits into product sectors where one sector is IR-free, and the other remains interacting (“mixed” phase) (Craig et al., 2011).
Dynamical Symmetry Breaking Patterns: Nonperturbative effects, often captured by dynamically generated superpotentials or soft supersymmetry breaking (such as anomaly mediation with Weyl compensator $\Phi = 1 + \theta^2 m$ $Φ = 1 + θ^{2} m$ ), can result in spontaneous flavor symmetry breaking:
- For $SU(N_C)$ with antisymmetric, $N_F$ fundamental, and $N_C + N_F -4$ antifundamentals, the IR flavor symmetry depends on whether $N_C$ is odd or even. For odd $N_C$ , the IR symmetry is anomalous (e.g., $Sp(N_C-3)\times U(1)$ ) and massless composite fermions appear to match anomalies. For even $N_C$ , the IR symmetry is nonanomalous and the spectrum is gapped (Leedom et al., 11 Mar 2025).
- For $SO(10)$ with $N_f$ spinors, $N_f=1,2$ produces a fully gapped spectrum, whereas $N_f\geq3$ triggers flavor symmetry breaking $SU(N_f)\to SO(N_f)$ (Kondo et al., 2022).
Higgs and Abelianization Phases: Bifermion condensates, either in color–flavor–locked patterns (dynamical Higgs phase) or diagonal Cartan patterns (dynamical Abelianization), break gauge symmetry down to subgroups such as $SU(N)\to U(1)^{N-1}$ . In supersymmetric examples, these are often realized and precisely solvable in $\mathcal{N}=2$ gauge theories and are mirrored in the non-supersymmetric context (Bolognesi et al., 2023, Konishi et al., 23 Mar 2024).

3. Dualities, Superconformal Fixed Points, and a-Maximization

Duality structures in these theories are both extensive and subtle:

Seiberg-like Duality: Electric–magnetic dualities exist for chiral theories, generalizing Seiberg duality for vectorlike SQCD. For $SU(N)$ chiral theories with antisymmetrics and fundamentals, there exists an infinite family of magnetic duals with arbitrarily large gauge groups and additional global symmetries (which are truncated nonperturbatively) (Craig et al., 2011, Craig et al., 2011).
Self-Dual Theories: For odd $N$ , some chiral theories are self-dual—the electric and magnetic theories share the same gauge group but distinct matter content and superpotential (Craig et al., 2011).
Chiral/Nonchiral Dualities: Certain chiral theories with antisymmetric representations can be dual to nonchiral theories with only (anti)fundamentals and singlets, enabling analysis of IR dynamics without two-index tensors (Craig et al., 2011).
a-Maximization: The IR R-charge assignments, superconformal central charge ( $a$ ), and operator dimensions are determined precisely via a-maximization [Intriligator–Wecht]. This method is critical in establishing that all duals flow to the same SCFT with identical anomalies, regardless of UV matter content (Craig et al., 2011, Craig et al., 2011).
Chiral Ring Stability: Operators violating the unitarity bound are decoupled by introducing flipping singlets, ensuring all physical computations (e.g., anomaly coefficients, partition functions, duality maps in 3d reduction) remain consistent and stable under RG flow (Benvenuti et al., 2017).

4. Methods for Nonperturbative Dynamics and Anomaly Matching

Supersymmetric chiral gauge theories permit several nonperturbative tools:

Holomorphy and Dynamically Generated Superpotentials: The form of the nonperturbative superpotential is tightly constrained by holomorphy, symmetry, and anomaly matching. For instance, in $E_6$ with $N_F$ fundamentals, the exact superpotential is built from $S$ and $T$ gauge invariants, e.g., $W_{N_F=3} = \left(\Lambda^{27}(T^3 + 6\sqrt{3}TS^4 - \frac{28\sqrt{3}}{5}S^6)\right)^{1/3}$ (Goh et al., 12 May 2025).
Anomaly-Mediated Supersymmetry Breaking (AMSB): Adding an infinitesimal soft mass by AMSB generates a scalar potential selecting vacua among D-flat directions, revealing IR symmetry structure and spectrum. Massless composite fermions that saturate ’t Hooft anomaly matching are produced only when required by the residual anomalous global symmetry in the vacuum (Csáki et al., 2021, Leedom et al., 11 Mar 2025, Goh et al., 12 May 2025).
Generalized Anomaly Constraints: By including anomalies of higher-form symmetries (e.g., one-form center symmetry), certain confining phases with unbroken flavor symmetry and purely massless composite fermions are ruled out, forcing dynamical Higgsing or Abelianization (Konishi et al., 23 Mar 2024).
Deconfinement, Seiberg Duality, and Deformation Techniques: Dualities are derived via deconfinement methods (replacing two-index tensors by bifundamentals and auxiliary gauge groups), by RG flows upon giving masses to matter fields, and by considering possible deformations within the SCFT window (Craig et al., 2011).
Lattice Formulations: While exact numerical studies are still at an early stage, lattice implementations for supersymmetric chiral gauge theories have been advanced by tuning mass terms and quartic scalar operators to maintain both supersymmetry and chirality, with promising preliminary results for $\mathcal{N}=1$ $SU(3)$ SQCD (Wellegehausen et al., 2018, Steinhauser et al., 2018).

5. Applications and Phenomenological Implications

Supersymmetric chiral gauge theories furnish a variety of mechanisms for model building and the realization of Standard Model extensions:

Grand Unification: The trinification model based on $SU(3)_C \times SU(3)_L \times SU(3)_R$ admits a natural sequence of spontaneous symmetry breaking to the Minimal Supersymmetric Standard Model (MSSM) and then to $SU(3)_C\times U(1)_{em}$ , with dynamical generation of "twisted fuzzy spheres" providing both the Higgs mechanism and a finite Kaluza–Klein spectrum (Chatzistavrakidis et al., 2010).
Composite Axion Models: Non-perturbative dynamics in supersymmetric chiral gauge theories can spontaneously break PQ symmetry, yielding natural QCD axion models. The PQ scale is determined by the composite operator VEV (set by the scale of strong dynamics and soft masses), and model-building can be tailored to avoid a Landau pole below the Planck scale when embedded in a GUT framework (Sato et al., 29 Aug 2025). This structure ensures an "accidental" PQ symmetry of high quality, protected from explicit breaking among dimension-5 operators, and links axion decay constant constraints to GUT unification.
Non-Kähler Heterotic Compactifications: Two-dimensional $(0,2)$ chiral gauge theories, central to string compactifications, encode non-Kähler target space geometry and fluxes, with effective action corrections (including anomalies and localized NS–brane sources) controlled via supersymmetric supergraph computations (Melnikov et al., 2012).
Classification of IR Phases: Exact results for the IR universality classes in strongly-coupled chiral gauge theories demonstrate that traditional tumbling hypotheses—stepwise condensation down to vectorlike or confined phases—are often invalid; non-tumbling “symplectic lock” vacua, unbroken flavor symmetries with massless composite fermions, or fully gapped phases are more typical in the presence of supersymmetry (and persist in the non-supersymmetric m→∞ limit) (Leedom et al., 11 Mar 2025, Goh et al., 12 May 2025, Csáki et al., 2021, Kondo et al., 2022).
Confinement as Deformation from Conformal Fixed Points: In softly broken supersymmetric theories, some confining gauge theories can be explicitly characterized as small RG deformations away from strongly coupled nonlocal conformal fixed points—establishing connections to the physics of mass gap generation and quark confinement (Konishi et al., 23 Mar 2024).

6. Open Problems and Future Directions

The paper of supersymmetric chiral gauge theories continues to confront several active questions:

Non-supersymmetric Continuity: It is conjectured (but not rigorously proven) that the dynamical symmetry breaking patterns and spectrum identified with small soft supersymmetry breaking can be continued to the full non-supersymmetric theory, provided no intervening phase transition occurs as the soft scale is increased (Leedom et al., 11 Mar 2025, Csáki et al., 2021, Goh et al., 12 May 2025). Testing the smoothness of this limit remains a crucial direction, especially in lattice studies.
Classification of Dynamical Phases: New criteria based on counting colored Nambu–Goldstone bosons have been proposed to distinguish confining, Higgs, and Abelian phases in both vectorlike and chiral gauge theories, but complete classification across the full space of chiral gauge models is an open problem (Konishi et al., 23 Mar 2024).
High-Scale Model Building: The constraints arising from Landau poles, GUT threshold corrections, and axion decay constants in composite axion models built from chiral gauge dynamics motivate more systematic exploration of GUT-axion interplay and its cosmological signatures (Sato et al., 29 Aug 2025).
Operator Ring, Moduli Space, and Dualities in 3d/4d: The full structure of the chiral ring, including the spectrum of dressed monopole operators (especially in 3d $\mathcal{N}=2$ chiral theories), and their mapping under duality transformations, remains an area of ongoing development (Nii, 2018, Cremonesi, 2017). Techniques such as Hilbert series calculations and chiral ring stability are central analytical tools in these studies (Benvenuti et al., 2017).
String Compactifications and Non-Kähler Geometry: The relation between $(0,2)$ chiral gauge theory anomalies, target space metric corrections, and fluxes in non-Kähler string compactifications remains an active research area with implications for heterotic model-building and the swampland conjectures (Melnikov et al., 2012).

Supersymmetric chiral gauge theories thus serve as an arena for exploring IR strong coupling dynamics, anomaly matching, phase transitions, and exact dualities, and provide access to phenomena relevant for both formal quantum field theory and phenomenological extensions of the Standard Model. The mathematical structure, nonperturbative analytic control, and deep connections to geometry and topology all underscore their central role in modern high-energy theoretical physics.