---
title: Supersymmetry Preserving RG Flows
url: https://www.emergentmind.com/topics/supersymmetry-preserving-rg-flows
type: topic
---

# Supersymmetry Preserving RG Flows

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 [2506.10062][2511.18128].

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 [2506.10062][2511.18128] 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 [2511.18128].

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 [2506.10062][2511.18128].

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 [2506.10062][2511.18128] 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 [2506.10062].

## 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 [2506.10062].

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 [2305.01647][2001.04991], 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 [2305.01647].

Furthermore, flows with enhancement of supersymmetry in the IR are precisely characterized by anomaly matching and operator decoupling criteria [1710.06469][1808.00592][1606.05632]. 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 [1206.3033].

## 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.

Source: https://www.emergentmind.com/topics/supersymmetry-preserving-rg-flows