Emergent Spacetime Supersymmetry

Updated 2 January 2026

Emergent spacetime supersymmetry is a phenomenon where SUSY arises dynamically at low energies or critical points from non-supersymmetric microscopic models.
Matrix models and lattice simulations demonstrate that quantum fluctuation cancellations allow for spontaneous symmetry breaking and a dynamic selection of a (3+1)D spacetime.
Field theory and condensed matter studies reveal that equal boson-fermion scaling dimensions, mass degeneracies, and protected indices serve as signatures of emergent SUSY.

Emergent spacetime supersymmetry refers to the dynamical appearance of supersymmetric (SUSY) field theory descriptions at low energy or large scales, starting from a microscopic model or quantum system that is not manifestly supersymmetric in its fundamental degrees of freedom or action. In this framework, supersymmetry and sometimes even the very notion of spacetime arise effectively in the infrared (IR) or at critical points, as a consequence of renormalization group (RG) flow, nonperturbative dynamics, or collective phenomena. This concept is central to both high energy theory, in the context of nonperturbative string and matrix models, and condensed matter physics, especially at quantum critical points and in topological systems.

1. Emergence in Lorentzian IIB Matrix Models

In the Lorentzian type IIB (IKKT) matrix model, spacetime and its dimensionality are not input but are conjectured to arise dynamically from matrix degrees of freedom. The model is defined by the action

$S = S_b + S_f + S_\gamma,$

where the bosonic part is

$S_b = -\frac{N}{4} \Tr\{ -2 [A_0, A_i]^2 + [A_i, A_j]^2 \},$

the fermionic part is

$S_f = -\frac{N}{2} \Tr\{ \Psi_\alpha (C \Gamma^\mu)_{\alpha\beta}[A_\mu, \Psi_\beta] \},$

and $S_\gamma$ is a Lorentz-invariant infrared mass regulator. The core observable for emergent spacetime geometry is the moment of inertia tensor of spatial matrices,

$T_{ij}(t) = \frac{1}{n} \tr\bigl(\bar A_i(t)\, \bar A_j(t)\bigr),$

where $\lambda_i(t)$ denote its eigenvalues. If $d$ directions expand, the top $d$ eigenvalues grow as $|t|$ increases.

Without the fermionic (SUSY) sector, apparent dimensional reduction is an artifact of residual Lorentz boosts; after frame-fixing, SO(9) symmetry remains unbroken and no genuine expansion is observed. By contrast, including the full supersymmetric sector and properly regulating the infrared leads to spontaneous symmetry breaking $SO(9) \to SO(3)$ : specifically, three eigenvalues $\lambda_{1,2,3}$ increase at late $|t|$ , signaling an emergent expanding (3+1)D Lorentzian spacetime. Quantum fluctuation cancellations in the effective potential $V_{\rm eff}(R_i)$ are crucial: only with SUSY does $V_{\rm eff}$ become nearly flat along three directions, enabling expansion and stabilizing compactification in the other six dimensions. Thus, SUSY fundamentally drives dynamical spacetime emergence in nonperturbative matrix models, providing a statistically robust mechanism for selecting (3+1)D geometry (Hirasawa et al., 2024).

2. Renormalization Group Flows and Emergent Superconformal Symmetry

A class of RG flows in 1+1D quantum field theory exhibit emergent spacetime supersymmetry at their IR fixed points, even when the UV theory is non-supersymmetric. An explicit realization starts with a fermionic minimal model at $c_{UV}=4/5$ and flows under a relevant scalar perturbation to the $c_{IR}=7/10$ (tricritical Ising) model with full $\mathcal{N}=1$ superconformal symmetry. The IR theory contains supercurrents $G(z,\bar z)$ furnishing the super-Virasoro algebra; emergent supercharges $Q$ organize the IR Hilbert space into supermultiplets. The new "spin constraint" on topological defect lines is used to analytically track symmetry evolution, verifying that only lines compatible with supersymmetry persist. Truncated Conformal Space Approach (TCSA) numerically confirms central charge flow, operator spectrum, and degeneracies consistent with supermultiplets. This RG-based paradigm generalizes to other lower-dimensional systems and provides a first-principles route for emergent spacetime supersymmetry from defect symmetries in the absence of microscopic SUSY (Kikuchi, 2022).

3. Quantum Criticality and Emergent SUSY in Condensed Matter

At quantum critical points (QCPs), low-energy theories in certain strongly correlated systems manifest spacetime SUSY and associated superconformal symmetry:

Edge SUSY at Topological Phase Transitions: In two-dimensional topological superconductors, sign-problem-free Majorana quantum Monte Carlo demonstrates that the edge quantum critical point exhibits emergent $\mathcal{N}=1$ SUSY. At the QCP, bosonic (Ising order parameter) and fermionic (Majorana mode) excitations acquire identical anomalous dimensions ( $\eta_\phi = \eta_\psi = 0.4$ ) and degenerate dynamically generated masses, as required by SUSY Ward identities. The bulk remains topologically nontrivial and gapped, while the edge criticality sharpens the distinction between edge and bulk universality (Li et al., 2016).
Superconducting QCP and 2+1D N=2 SUSY: An interacting lattice Dirac fermion model with attractive Hubbard interaction flows, at its SC QCP, to an infrared theory described by the $\mathcal{N}=2$ Wess–Zumino model in 2+1D. Precise QMC numerics extract equal anomalous dimensions for fermions and pairing bosons ( $\eta_\phi = \eta_\psi \approx 1/3$ ), conformal exponents $\nu \approx 0.87$ , and agreement with conformal bootstrap predictions. Experimental signatures, such as the scaling of local density of states and universal optical conductivity, can test these predictions in candidate platforms (e.g., topological-insulator surfaces) (Li et al., 2017).
Disorder and Robustness: RG analysis shows that emergent SUSY at these QCPs is robust against weak disorder: for $(2+1)$ D Dirac and $(3+1)$ D Weyl semimetals near the PDW transition, random scalar, mass, and vector potentials are irrelevant at the SUSY fixed point, preserving the dynamical SUSY structure (Yu et al., 2022).
Fractionalized QCPs and Topological Orders: In the SSH–Kitaev honeycomb model with Z $_2$ spin-liquid and dynamic phonons, the transition from a Dirac spin liquid to a VBS phase with topological order is governed by an emergent $\mathcal{N}=2$ SUSY fixed point in 2+1D. The emergent superfield structure is explicitly realized, and the protected dimension $[\phi]=2/3$ stabilizes the universality class against perturbations. This leads to universal predictions for thermal conductivity and dynamical viscosity governed by operator-product expansion coefficients in the N=2 superconformal theory (Wu et al., 8 Oct 2025).

4. Lattice Realizations and Experimental Proposals

Lattice models with precisely controllable microscopic SUSY provide concrete settings to observe emergent spacetime supersymmetry and study transitions between phases with and without SUSY:

Interacting Kitaev Chain: A one-dimensional interacting Kitaev chain with explicit $\mathcal{N}=1$ quantum-mechanical SUSY, constructed from bilinear and quartic Majorana couplings, shows a phase transition as the coupling is tuned. Below the critical point, SUSY is spontaneously broken and the IR behavior is described by the Ising CFT ( $c=1/2$ ); above, SUSY is restored and the ground state is gapped. Exactly at the transition, the universality class is governed by the tricritical Ising CFT ( $c=7/10$ ), with direct numerical evidence for superconformal operator content and universal gaps. This model provides a fully controlled platform for studying emergent spacetime superconformal symmetry in $1+1$D (Miura et al., 2024).
Quantum Simulators and Rydberg Arrays: Proposals exist for realizing emergent SUSY in programmable quantum simulators. Dual-species Rydberg atom ladders can realize the tricritical Ising model and hence emergent (1+1)D $\mathcal{N}=1$ SUSY. String correlators encoding Majorana fermion superpartners to local bosonic order can be measured using hybrid analog-digital protocols, exploiting high-fidelity quantum circuits to resolve the nonlocal structure of the SUSY correlators (Li et al., 2024). The hybrid analog-digital protocol enables direct extraction of scaling exponents ( $\eta_\psi - \eta_\varepsilon = 1$ ) characteristic of the supersymmetric spectrum.

5. Effective Potential, Spontaneous Symmetry Breaking, and Mechanisms

In matrix and field theory approaches, the emergence of spacetime supersymmetry can be traced to the interplay between quantum fluctuations and the algebraic structure of the effective potential:

In the Lorentzian IIB matrix model, bosonic quantum fluctuations alone favor minimal dimensionality and symmetric collapse in $V_{\rm eff}(R_i)$ , while SUSY-induced fermionic fluctuations cancel these effects along three directions, generating flat directions and thus enabling a (3+1)D universe (Hirasawa et al., 2024).
In supersymmetric constraints on vector (super)fields (Chkareuli, 2014), imposing nonlinear $\sigma$ -model–type length-fixing constraints and spontaneous SUSY breaking promote global internal symmetry $G$ to an emergent local gauge symmetry $G_{loc}$ . The resulting massless vector supermultiplets arise as Goldstone and pseudo-Goldstone modes, while the exact cancellation of Lorentz-violating couplings implies the symmetry breaking is "inactive"—identical with a gauge choice. In these models, spontaneous SUSY breaking via D-terms is the dynamical trigger for emergent gauge fields and local symmetries.

6. Holography, Emergent Geometry, and Protected Indices

In the AdS/CFT correspondence, emergent spacetime supersymmetry is identified via growth properties of protected indices such as the elliptic genus in 2D $\mathcal{N} = (2,2)$ superconformal field theories. The key condition for emergent (large-radius, weakly curved) AdS $_3$ spacetime is that the Fourier coefficients of the elliptic genus exhibit sub-Hagedorn growth, $c(n, \ell) \lesssim e^{2\pi n^p}$ with $p<1$ , corresponding to a sparse spectrum below the string scale and a separation of Planck, string, and AdS scales. This index-theoretic technique allows direct diagnosis of emergent spacetime gravity and supergravity phases purely from boundary CFT data (Benjamin et al., 2015).

7. Pure Spinor Construction and Spacetime SUSY from Gauge-Fixing

In the pure spinor formalism for the superstring, starting from a bosonic worldsheet action with twistor-like constraints, gauge fixing via BRST quantization introduces ghost degrees of freedom that organize into the full ten-dimensional super-Yang–Mills (or supergravity) multiplet. After an appropriate ghost number twist, the physical state cohomology matches the on-shell supersymmetric spectrum, and spacetime supersymmetry emerges as the remnant gauge symmetry of the twistor constraint—providing a conceptual bridge between worldsheet and spacetime notions of supersymmetry (Berkovits, 2011).

Emergent spacetime supersymmetry thus encompasses a broad set of phenomena: nonperturbative matrix models dynamically selecting spacetime dimension, RG flows generating superconformal fixed points, condensed matter criticality manifesting SUSY spectra, explicit lattice and quantum simulator realizations, and protected structural criteria for holographic emergence. In each setting, the detailed interplay of quantum fluctuations, RG structure, algebraic constraints, and anomaly matching underpins the spontaneous or emergent realization of spacetime supersymmetry. Empirical signatures—such as equal boson-fermion scaling dimensions, mass degeneracies, and specific operator content—provide experimentally and numerically accessible hallmarks of these dynamics. The collective body of results substantiates the claim that supersymmetry, while not always present in the UV, can robustly emerge as a universal organizing principle in the IR or at criticality across theoretical and physical systems.