Supersymmetry Preserving RG Flows

• Supersymmetry preserving RG flows are constructed via first-order BPS equations that interpolate between UV conformal regimes and gapped IR phases while maintaining a fraction of supercharges.
• Explicit supergravity backgrounds in IIB, IIA, and M-theory enable controlled deformations that illuminate mass gaps, confinement phenomena, and the effects of topological twists.
• Universal observables such as Wilson loops, central charges, and complexity factorize into UV and flow-dependent components, providing a unifying perspective on strongly coupled dynamics.

Supersymmetry Preserving RG Flows are renormalization group trajectories in quantum field theory and holographic duals in supergravity that interpolate between fixed points (generally conformal field theories) while maintaining a non-trivial fraction of the supersymmetry at each energy scale. These flows, constructed via first-order (BPS) equations, provide a controlled setting to study the interplay of relevant and marginal deformations, mass gaps, confinement phenomena, and universal observable structures in strongly coupled systems. Recent advances leverage explicit supergravity backgrounds—often in Type IIB, Type IIA, or eleven-dimensional M-theory—and sophisticated holographic techniques to extract precise field-theoretic information.

1. Geometric Structure of Supersymmetric RG Flows

Supersymmetric RG flows in holography are realized via domain-wall-type solutions in higher-dimensional supergravity. The universal gravitational ansatz involves a metric of the form: $$ds_5^2 = e^{2A(r)}(-dt^2 + dx_1^2 + dx_2^2) + e^{2B(r)} dr^2$$ where $r$ is the holographic RG scale interpolating from the UV region ($r \to \infty$) with $A(r) \sim \ln(r/L)$—recovering AdS$_5$ geometry—and ending smoothly in the IR ($r \to r_\star$) where the space is capped off, giving rise to a gapped three-dimensional theory (Chatzis et al., 11 Jun 2025, Chatzis et al., 22 Nov 2025).

A supersymmetry-preserving twist is implemented by fibering a compact U(1) (or higher torus) over internal R-symmetry directions, canceling the anti-periodic boundary conditions that would otherwise break supersymmetry on the circle, and leaving a specified number of Poincaré supercharges unbroken. This is achieved by introducing gauge fields $A_i(r)\,d\phi$ which realize a topological twist in the gravity sector.

2. Supergravity, BPS Flow Equations, and Twists

All classes of supersymmetric RG flow backgrounds in (Chatzis et al., 11 Jun 2025, Chatzis et al., 22 Nov 2025) descend from first-order BPS equations determined by a superpotential $W(\phi)$ for the relevant scalars: $$A'(r) = + e^{B(r)} W(\phi), \qquad \phi^I{}'(r) = -2 G^{IJ}(\phi) \partial_J W(\phi)$$ where $G^{IJ}$ is the scalar metric and $\phi^I$ are the active fields. This BPS system is derived from requiring the vanishing of supersymmetry variations of the gravitino and other fermionic fields. Solutions typically preserve four supercharges (3d $\mathcal{N}=2$) throughout the flow (Chatzis et al., 22 Nov 2025).

The universal closed-form solution for flows of interest is: $$\lambda(r) = (1 + \varepsilon \ell^2 / r^2)^{1/6}, \qquad F(r) = 1/L^2 - \varepsilon \ell^2 L^2 Q^2 / r^4, \qquad A(r) = \ln(r \lambda(r) / L)$$ with $\lambda$ and $F$ controlling internal warpings and circle behavior, and $Q$ parameterizing the twist strength.

3. Universal Factorization of Holographic Observables

Physical observables extracted holographically—such as rectangular Wilson loops, central charge flows, and complexity—exhibit a striking universal factorization structure: $$O_{\text{flow}} = O_{\text{CFT}} \times F_{\text{flow}}(\hat{\nu}, r_\star)$$ where $O_{\text{CFT}}$ is the value at the undeformed UV fixed point and $F_{\text{flow}}$ captures dynamical modifications due to deformation parameters $\hat{\nu}$ and IR cap $r_\star$. In all three supergravity classes (IIB, IIA, M-theory), the same 5d metric functions $(A(r), \lambda(r), F(r))$ appear in the formulae for Wilson-line energy, central charge, and complexity integrals, illustrating the unifying role of BPS flow geometry (Chatzis et al., 11 Jun 2025, Chatzis et al., 22 Nov 2025).

Representative expressions include:

• Wilson loop energy: $$E_{QQ}(L) = E_{QQ}^{\text{CFT}}(L) \times F_W(r_\star, \hat{\nu})$$
• Central charge: $$c_{\text{flow}}(\xi) = c_{\text{UV}} \times f(\xi; \hat{\nu})$$ where $f$ interpolates from $1$ in the UV to $0$ at the gap scale
• Complexity: $$\mathcal{C}_V = c_{\text{UV}} \times \int_{r_\star}^\infty dr\, r^2 \lambda(r)$$

This universality extends to other nonperturbative observables such as ’t Hooft loops and entanglement entropy, all factorizing into UV prefactors and identical radial integrals of flow-dependent functions.

4. Physical Interpretation: Gapped IR Phases and Parameter Regimes

Supersymmetry-preserving RG flows constructed in (Chatzis et al., 11 Jun 2025, Chatzis et al., 22 Nov 2025) model the compactification of four-dimensional SCFTs (e.g., $\mathcal{N}=4$ SYM, Gaiotto theories, linear quivers) on a circle with a supersymmetry-preserving twist and Coulomb-branch deformations.

• The twist involves a dimension-3 operator's vev, mixing the compactification circle with the R-symmetry, ensuring four preserved supercharges.
• The Coulomb branch deformation (a dimension-2 vev) by itself leads to a singular geometry, but the twist regularizes the solution, producing a smooth cap-off in the bulk at $r = r_\star$.

The IR geometry is confining and gapped: Wilson loops exhibit linear confinement at large separation, the central charge vanishes as $r\to r_\star$, and entanglement entropy undergoes a phase transition characteristic of confining vacua. The essential RG data is the parameter $\hat{\nu} = \varepsilon \ell^2 / r_\star^2$; for $\hat{\nu}\to \infty$ the deformation vanishes, while as $\hat{\nu} \to -1^+$ supergravity breaks down and higher-curvature corrections dominate (Chatzis et al., 11 Jun 2025).

5. Higher-Curvature and Quantum Corrections

For most $\hat{\nu}$, curvature invariants remain finite. However, near $\hat{\nu} \to -1$, key invariants such as $R_{ab} R^{ab}$ and $R_{abcd} R^{abcd}$ become large at the IR cap ($\theta=0$), indicating the need to incorporate $\alpha'$ (stringy) corrections or M-theory effects to maintain physical reliability. This is reflected in artifacts such as swallowtail phase transitions in Wilson loop energy (Chatzis et al., 11 Jun 2025).

The "dangerous window," $\hat{\nu} \in (-1, -0.9)$, demarcates regimes where supergravity alone is insufficient; for $\hat{\nu} \gtrsim -0.9$ the solutions are robust and match holographic expectations for universal observables.

6. Extensions: Defect Flows, Supersymmetry Enhancement, and Phenomenology

Supersymmetry-preserving RG flows have been generalized to defect/impurity settings (Castiglioni et al., 2023, Erdmenger et al., 2020), including marginally relevant interface operators, Kondo-like impurity flows, and the full network of BPS and non-BPS fixed points in ABJM-type theories. The beta functions for defect RG flows are computed exactly for leading order in the coupling and showcase a wealth of supersymmetric, non-supersymmetric, and mixed RG trajectories, classified by fixed-point stability, R-symmetry, and preserved supercharges (Castiglioni et al., 2023).

Furthermore, flows with enhancement of supersymmetry in the IR are precisely characterized by anomaly matching and operator decoupling criteria (Giacomelli, 2017, Giacomelli, 2018, Maruyoshi et al., 2016). The technique to determine enhancement relies on coupling an adjoint chiral to the global moment map, giving a nilpotent vev, and tracking the fate of Goldstone and Coulomb-branch multiplets. Explicit formulas relate UV and IR central charges, providing a practical algorithm for identifying supersymmetry enhancement along RG flows.

Phenomenologically, non-perturbative constraints on light sparticle spectra and emergent supersymmetry in models with R-symmetric flows have been established, with exact bounds on the suppression of SUSY-breaking operators and predictions for stop mass hierarchies in strongly coupled extensions of MSSM (Buican, 2012).

7. Summary Table: Universal Structure of SUSY-Preserving RG Flows

Flow Background	Preserved SUSY	Universal Observable Structure	IR Physics
IIB/IIA/M-theory soliton	4 supercharges	Factorization: $$O_{\text{flow}} = O_{\text{CFT}} \times F_{\text{flow}}$$	Gap, confinement
ABJM defect flows	Variable (0–8)	$\beta$-functions, g-theorem	Defect RG, mixing
N=2→N=1→N=2 flows	Enhancement	a-maximization, anomaly matching	Strong coupling
Stringy corrections	Preserved up to critical $\hat{\nu}$	Appearance near IR cap	$\alpha'$/M-theory regime

Supersymmetry-preserving RG flows thus offer a comprehensive and coherent framework for accessing nonperturbative and universal features of strong-coupling physics. They facilitate explicit calculability under supersymmetric protection, underpin advanced holographic techniques, and unravel connections between high-energy model building, quantum gravity, and gauge theory dynamics.