Unconventional Supersymmetry

• Unconventional supersymmetry is a framework that realizes SUSY without standard boson–fermion supermultiplet pairing by embedding fields into a unified Lie-superalgebra connection.
• It spans diverse approaches—from gauge-superconnection and lattice models to phenomenological adaptations—each redefining the role of supersymmetry in field theory and condensed matter systems.
• These constructions offer actionable insights into SUSY-breaking mechanisms, modified gauge dynamics, and novel interpretations of torsion and boundary effects beyond the conventional super-Poincaré setup.

Searching arXiv for recent and foundational papers on unconventional supersymmetry to ground the article in the literature. {"^^^^^^^^4query4 supersymmetry\"^^^^^^^^4 OR ti:\4^^^^^^^^"unconventional supersymmetry\"^^^^^^^^4 OR abs:\4^^^^^^^^"unconventional supersymmetry\"","max_results":^^^^4all:\4query4^^^^,"sort_by":"relevance"} {"^^^^^^^^4query4 OR id:(&&&^^^^4all:\4^^^^&&&) OR id:(&&&^^^^4 OR ti:\4^^^^&&&) OR id:(&&&^^^^4 OR abs:\4^^^^&&&) OR id:(Alvarez et al., 2023) OR id:(Alvarez et al., 2020) OR id:(Andrianopoli et al., 2019) OR id:(Andrianopoli et al., 2018) OR id:(D'Adda et al., 2017) OR id:(Alvarez et al., 4 Jun 2026)","max_results":^^^^4all:\4query4^^^^,"sort_by":"relevance"} Unconventional supersymmetry is a family of non-equivalent constructions in which supersymmetry is realized without the standard requirement that observable states form boson–fermion supermultiplet pairs. In the best-developed line, bosonic gauge fields and fermionic matter are combined into a single Lie-superalgebra-valued connection, while the physical fermionic sector is carried by spin-PRESERVED_PLACEHOLDER_^^^^4query4^^^^ fields rather than propagating spin-PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^ gravitini; in this setting there need be no superpartners, no boson–fermion degree-of-freedom matching, and no fully off-shell invariance under the full supergroup (&&&^^^^4all:\4^^^^&&&, &&&^^^^4 OR ti:\4^^^^&&&). The same label is also used for exact-lattice constructions based on nonlocal products and modified translation generators, for energy-dependent supersymmetric quantum mechanics of spin systems, and for phenomenological proposals that retain selected boson–fermion cancellation properties while discarding the conventional super-Poincaré realization (D'Adda et al., 2017, &&&^^^^4all:\4 OR abs:\4^^^^&&&, &&&^^^^4all:\44^^^^&&&).

^^^^4all:\4^^^^. Terminological scope and defining features

The literature uses “unconventional supersymmetry” in several technically distinct senses. What they share is a departure from the conventional supermultiplet picture, but they differ on which part of the standard framework is abandoned and which part is retained.

Strand	Defining move	Representative papers
Gauge-superconnection uSUSY	Bosonic gauge fields and fermionic matter assembled in one superconnection	(&&&^^^^4 OR ti:\4^^^^&&&, &&&^^^^4all:\4^^^^&&&, &&&^^^^4 OR abs:\4^^^^&&&)
Boundary/condensed-matter uSUSY	Spin-PRESERVED_PLACEHOLDER_^^^^4 OR ti:\4^^^^ Dirac matter extracted from AdS supergravity or Dirac-material geometry with torsion	(Andrianopoli et al., 2018, Andrianopoli et al., 2019, &&&^^^^4query4^^^^&&&)
Conformal/gauge-geometric 4D uSUSY	Broken PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4^^^^ or $su(2,2|3)$ gauge-superalgebra with gravity, Yang–Mills, torsion, and induced NJL sectors	(Alvarez et al., 2020, Alvarez et al., 2023, Alvarez et al., 4 Jun 2026)
Lattice uSUSY	Exact SUSY through a periodic lattice derivative and nonlocal $\star$-product	(D'Adda et al., 2017)
Quantum-mechanical uSUSY	Energy-dependent supercharges for $2\times2$ spin Hamiltonians	(&&&^^^^4all:\4 OR abs:\4^^^^&&&)
Phenomenological alternatives	Boson–fermion balancing without conventional super-Poincaré multiplets	(&&&^^^^4all:\44^^^^&&&)

In the gauge-superconnection program, the departure from standard supersymmetry is structural rather than merely phenomenological. Supersymmetry is retained as a graded gauge principle, but the physical spectrum need not display paired sleptons, squarks, gauginos, or gravitini. In the lattice and quantum-mechanical uses, by contrast, “unconventional” refers mainly to altered kinematics or algebraic closure. In particle-phenomenology papers, the term can denote looser boson–fermion correspondences that preserve selected cancellations or signatures while abandoning the conventional algebraic setup.

^^^^4 OR ti:\4^^^^. Gauge-superconnection supersymmetry without superpartners

A central formulation places the fields in the adjoint representation of a superalgebra and organizes them into a superconnection rather than into ordinary supermultiplets. In the general review formulation, one writes

$$\mathbb{A} = A^K T_K + \omega^A J_A + \bar Q_\alpha \chi^\alpha + \bar\chi_\alpha Q^\alpha ,$$

with $J_A$ spacetime generators, $T_K$ internal generators, and PRESERVED_PLACEHOLDER_^^^^4all:\4query4^^^^ the fermionic one-form (&&&^^^^4all:\4^^^^&&&). In the earlier four-dimensional construction, the same idea appears schematically as

PRESERVED_PLACEHOLDER_^^^^4all:\4all:\4^^^^

so that a spin-PRESERVED_PLACEHOLDER_^^^^4all:\4 OR ti:\4^^^^ fermion enters the connection through the soldering form PRESERVED_PLACEHOLDER_^^^^4all:\4 OR abs:\4^^^^ (&&&^^^^4 OR ti:\4^^^^&&&).

The defining matter ansatz identifies the odd one-form with a composite built from the vielbein and an ordinary spinor. In the AVZ form used repeatedly in PRESERVED_PLACEHOLDER_^^^^4all:\44^^^^ dimensions,

PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^5

or, in the review notation,

PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^6

This is what removes the independent spin-PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^7 gravitino interpretation and leaves a propagating spin-PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^8 Dirac sector (&&&^^^^4all:\4^^^^&&&).

Several consequences distinguish this framework from standard supergravity. The metric may be taken to be supersymmetry-invariant, with

PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^9

so there is no need for a gravitino gauge field generated by PRESERVED_PLACEHOLDER_^^^^4 OR ti:\4query4^^^^. The spin-PRESERVED_PLACEHOLDER_^^^^4 OR ti:\4all:\4^^^^ sector is projected out by the condition

PRESERVED_PLACEHOLDER_^^^^4 OR ti:\4 OR ti:\4^^^^

and the theory can have neither boson–fermion degree-of-freedom matching nor equal-mass superpartners (&&&^^^^4 OR ti:\4^^^^&&&). The ^^^^4 OR ti:\4query4 OR ti:\4all:\4^^^^ review formulates this succinctly: bosonic gauge fields and fermionic matter are components of one superconnection, states do not come in SUSY pairs, and the action is generally not fully off-shell invariant under the full supersymmetry (&&&^^^^4all:\4^^^^&&&).

Odd and even dimensions separate sharply. In odd dimensions, the natural action is Chern–Simons, and the full supergroup can act quasi-invariantly. In even dimensions, the natural local form is of Yang–Mills or MacDowell–Mansouri type,

PRESERVED_PLACEHOLDER_^^^^4 OR ti:\4 OR abs:\4^^^^

and the symmetry is reduced to a bosonic subgroup such as PRESERVED_PLACEHOLDER_^^^^4 OR ti:\44^^^^ or PRESERVED_PLACEHOLDER_^^^^4 OR ti:\4^^^^5 (&&&^^^^4 OR ti:\4^^^^&&&, &&&^^^^4all:\4^^^^&&&).

A major conceptual clarification came from the Dressing Field Method. The AVZ matter ansatz was previously treated as an ad hoc projection or as a gauge fixing. The ^^^^4 OR ti:\4query4 OR ti:\44^^^^ analysis instead starts from the general PRESERVED_PLACEHOLDER_^^^^4 OR ti:\4^^^^6 odd one-form PRESERVED_PLACEHOLDER_^^^^4 OR ti:\4^^^^7, decomposes it as

PRESERVED_PLACEHOLDER_^^^^4 OR ti:\4^^^^8

and constructs a SUSY dressing field PRESERVED_PLACEHOLDER_^^^^4 OR ti:\4^^^^9 such that the dressed field satisfies

PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4query4^^^^

In this formulation, PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4all:\4^^^^ is a SUSY-invariant dressed spinor, the AVZ connection is the dressed superconnection, and the residual gauge symmetry is the bosonic PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4 OR ti:\4^^^^ subgroup (&&&^^^^4 OR abs:\4^^^^&&&). This suggests that the matter ansatz is not a model-specific trick but a systematic reduction of the odd gauge sector.

^^^^4 OR abs:\4^^^^. Boundary theories, graphene, torsion, and Dirac materials

One influential branch derives unconventional supersymmetry on a PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4 OR abs:\4^^^^-dimensional boundary of AdSPRESERVED_PLACEHOLDER_^^^^4 OR abs:\44^^^^ supergravity and relates it to graphene-like Dirac matter. In the PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4^^^^5 derivation, the boundary theory has PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4^^^^6 invariance and contains the AVZ model with a specific parameter prescription; the propagating fermion is a spin-PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4^^^^7 Dirac field originating from the radial components of the four-dimensional gravitini, while the spin-PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4^^^^8 boundary components are projected out by the ansatz

PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4^^^^9

The resulting Dirac equation takes the form

$su(2,2|3)$^^^^4query4^^^^

so the boundary fermion mass is fixed by the AdS radius (Andrianopoli et al., 2018).

The $su(2,2|3)$^^^^4all:\4^^^^-extended generalization produces a boundary $su(2,2|3)$^^^^4 OR ti:\4^^^^ Chern–Simons theory. After the AVZ ansatz, the two sectors yield spin-$su(2,2|3)$^^^^4 OR abs:\4^^^^ fermions $su(2,2|3)$4 satisfying

$su(2,2|3)$5

For $su(2,2|3)$6, parity exchanges the two sectors, and they are identified with the $su(2,2|3)$7 and $su(2,2|3)$8 valleys of graphene-like systems. The corresponding masses

$su(2,2|3)$9

are then matched to Semenoff- and Haldane-type masses via

$\star$^^^^4query4^^^^

In this top-down interpretation, torsion parameters of the substrate encode valley masses (Andrianopoli et al., 2019).

A complementary condensed-matter route starts directly from low-energy Dirac materials. For graphene, silicene, and germanene, the low-energy $\star$^^^^4all:\4^^^^-electrons are described by a $\star$^^^^4 OR ti:\4^^^^-dimensional Dirac action

$\star$^^^^4 OR abs:\4^^^^

and dislocations are described geometrically by torsion rather than curvature. The fermions couple only to the totally antisymmetric torsion component,

$\star$4

through

$\star$5

Because the two valley spinors couple with opposite signs to $\star$6, the theory suggests a net particle–hole transport effect if such torsion is present (&&&^^^^4query4^^^^&&&).

The geometric obstruction is that a fully antisymmetric torsion term in two spatial dimensions appears to require a missing third index. The proposal is to use time as that third direction. The Burgers vector relation

$\star$7

admits a nonzero contribution on a spacetime surface, and the preferred construction uses an edge dislocation probed by a loop in the $\star$8-plane. The resulting “time-loop” is not a literal closed timelike curve but an emergent particle–hole loop enabled by half filling and the particle/antiparticle reinterpretation of the low-energy Dirac theory. Within that framework, the paper distinguishes three levels of status: the Dirac–torsion coupling and the need for a spacetime torsion flux are established in the model; engineering particle–hole time-loops is heuristic but physically motivated; laboratory realization of unconventional supersymmetry is an outlook rather than an experimental result (&&&^^^^4query4^^^^&&&).

4. Four-dimensional conformal, gauge-geometric, and instanton realizations

In four dimensions, unconventional supersymmetry is often built from a broken $\star$9 or $2\times2$^^^^4query4^^^^ gauge structure. One explicit $2\times2$^^^^4all:\4^^^^ model starts from the superconnection

$2\times2$^^^^4 OR ti:\4^^^^

and a MacDowell–Mansouri-type action

$2\times2$^^^^4 OR abs:\4^^^^

The components $2\times2$4 and $2\times2$5, associated with AdS boosts and special conformal translations, have no kinetic terms and are auxiliary. Fixing them reduces the theory to two sectors. For finite $2\times2$6, one obtains a vector-like theory with Einstein gravity, $2\times2$7 Yang–Mills, a Dirac fermion, torsion couplings, and an NJL term. In the singular limit $2\times2$8, one gets a chiral $2\times2$9 theory in which gravity decouples. The model predicts bare couplings

$$\mathbb{A} = A^K T_K + \omega^A J_A + \bar Q_\alpha \chi^\alpha + \bar\chi_\alpha Q^\alpha ,$$^^^^4query4^^^^

and a Planckian cosmological constant

$$\mathbb{A} = A^K T_K + \omega^A J_A + \bar Q_\alpha \chi^\alpha + \bar\chi_\alpha Q^\alpha ,$$^^^^4all:\4^^^^

The NJL interaction is

$$\mathbb{A} = A^K T_K + \omega^A J_A + \bar Q_\alpha \chi^\alpha + \bar\chi_\alpha Q^\alpha ,$$^^^^4 OR ti:\4^^^^

This sector is presented as a chiral gauge theory and gravity derived from broken unconventional supersymmetry rather than from ordinary supermultiplets (Alvarez et al., 2020).

A related conformal construction studies self-duality in a four-dimensional $$\mathbb{A} = A^K T_K + \omega^A J_A + \bar Q_\alpha \chi^\alpha + \bar\chi_\alpha Q^\alpha ,$$^^^^4 OR abs:\4^^^^ theory written in Townsend–MacDowell–Mansouri form. The generalized self-duality condition is

$$\mathbb{A} = A^K T_K + \omega^A J_A + \bar Q_\alpha \chi^\alpha + \bar\chi_\alpha Q^\alpha ,$$4

and in the purely bosonic sector it reduces to

$$\mathbb{A} = A^K T_K + \omega^A J_A + \bar Q_\alpha \chi^\alpha + \bar\chi_\alpha Q^\alpha ,$$5

The solutions include torsionful generalizations of Taub-NUT-AdS, Taub-Bolt-AdS, and Eguchi–Hanson geometries. Their on-shell action is topological,

$$\mathbb{A} = A^K T_K + \omega^A J_A + \bar Q_\alpha \chi^\alpha + \bar\chi_\alpha Q^\alpha ,$$6

and the solutions saturate a BPS bound. Torsion is controlled by a nonconstant $$\mathbb{A} = A^K T_K + \omega^A J_A + \bar Q_\alpha \chi^\alpha + \bar\chi_\alpha Q^\alpha ,$$7 through

$$\mathbb{A} = A^K T_K + \omega^A J_A + \bar Q_\alpha \chi^\alpha + \bar\chi_\alpha Q^\alpha ,$$8

so the instantons are not ordinary Einstein gravitational instantons but genuinely torsionful configurations in an unconventional conformal supersymmetric geometry (Alvarez et al., 2023).

A more recent $$\mathbb{A} = A^K T_K + \omega^A J_A + \bar Q_\alpha \chi^\alpha + \bar\chi_\alpha Q^\alpha ,$$9 gauge-geometric model pushes the same logic into dynamical mass generation. Here gravity, $J_A$^^^^4query4^^^^, $J_A$^^^^4all:\4^^^^, and fermions are components of one superconnection

$J_A$^^^^4 OR ti:\4^^^^

and the fundamental action is

$J_A$^^^^4 OR abs:\4^^^^

The low-energy effective theory contains no elementary scalar fields and no ad hoc four-fermion interaction; instead, nonminimal couplings and torsion generate an NJL-type quartic sector. For $J_A$4, Fierz reduction removes the tensor channel and yields

$J_A$5

In an AdS background with vanishing torsion, the mean-field gap equation is

$J_A$6

with curvature-induced effective mass

$J_A$7

and a critical cutoff near

$J_A$8

The paper interprets this as chiral symmetry breaking and mass-gap formation emerging from the gauge-geometric structure itself, though with severe fine-tuning if one asks for GeV-scale masses (Alvarez et al., 4 Jun 2026).

5. Lattice, quantum-mechanical, and dynamical-breaking uses

A distinct use of the term appears in lattice field theory. There, exact SUSY is pursued by abandoning the ordinary local product and ordinary lattice momentum conservation. The conserved additive quantity is taken to be a periodic lattice derivative $J_A$9, not the lattice momentum $T_K$^^^^4query4^^^^, and the local product is replaced by a nonlocal $T_K$^^^^4all:\4^^^^-product satisfying

$T_K$^^^^4 OR ti:\4^^^^

Because $T_K$^^^^4 OR abs:\4^^^^-conservation is built in, the Leibniz rule holds exactly. Associativity is obtained only for special $T_K$4, notably the inverse Gudermannian derivative

$T_K$5

for which the lattice theory becomes exactly equivalent to the continuum theory via an invertible map. The same framework reinterprets doublers as same-chirality states that can either be identified or regarded as members of an extended supermultiplet. Its central limitation is that a gauge-invariant regularization is still lacking: the regularized non-associative theory breaks gauge invariance (D'Adda et al., 2017).

In supersymmetric quantum mechanics, “unconventional” denotes an energy-dependent superalgebra attached to the eigenproblem of a generic $T_K$6 matrix Hamiltonian

$T_K$7

For $T_K$8, one defines resolvents

$T_K$9

and reduced Hamiltonians

PRESERVED_PLACEHOLDER_^^^^4all:\4query4query4^^^^

The supercharges

PRESERVED_PLACEHOLDER_^^^^4all:\4query4all:\4^^^^

satisfy

PRESERVED_PLACEHOLDER_^^^^4all:\4query4 OR ti:\4^^^^

but the algebra depends explicitly on the eigenvalue PRESERVED_PLACEHOLDER_^^^^4all:\4query4 OR abs:\4^^^^. In this formulation, the two spinor components are superpartners only up to an PRESERVED_PLACEHOLDER_^^^^4all:\4query44^^^^ rotation, and each component carries the full information of the eigenspinor. The authors extend the formalism to a single spin-PRESERVED_PLACEHOLDER_^^^^4all:\4query4^^^^5, to spin-boson models such as Jaynes–Cummings and Rabi systems, and to many-spin systems where the recursive reduction exposes a matrix-product-state structure (&&&^^^^4all:\4 OR abs:\4^^^^&&&).

A third technical strand concerns dynamical SUSY breaking in a model with one chiral superfield and a strong quartic Kähler interaction,

PRESERVED_PLACEHOLDER_^^^^4all:\4query4^^^^6

The nonstandard ingredient is a composite real superfield

PRESERVED_PLACEHOLDER_^^^^4all:\4query4^^^^7

introduced through

PRESERVED_PLACEHOLDER_^^^^4all:\4query4^^^^8

A PRESERVED_PLACEHOLDER_^^^^4all:\4query4^^^^9-component condensate of PRESERVED_PLACEHOLDER_^^^^4all:\4all:\4query4^^^^ generates a soft scalar mass,

PRESERVED_PLACEHOLDER_^^^^4all:\4all:\4all:\4^^^^

without hidden sectors or mediation sectors. The preferred branch has PRESERVED_PLACEHOLDER_^^^^4all:\4all:\4 OR ti:\4^^^^, PRESERVED_PLACEHOLDER_^^^^4all:\4all:\4 OR abs:\4^^^^, and a Goldstino appears as a composite fermionic zero mode. The composite superfield is called unconventional because it contains a spin-one component but is not an ordinary gauge vector superfield (Cheng et al., 2016).

6. Phenomenological reinterpretations, adjacent proposals, and open problems

Some recent particle-phenomenology papers use “unconventional supersymmetry” in a looser sense, centered on retaining boson–fermion balancing while abandoning the conventional super-Poincaré framework. One explicit proposal begins with primitive bosonic spinor partners PRESERVED_PLACEHOLDER_^^^^4all:\4all:\44^^^^ for each Weyl fermion PRESERVED_PLACEHOLDER_^^^^4all:\4all:\4^^^^5,

PRESERVED_PLACEHOLDER_^^^^4all:\4all:\4^^^^6

but argues that vacuum stability forces a reinterpretation of the bosonic sector into three kinds of scalar-boson fields: ordinary complex scalars PRESERVED_PLACEHOLDER_^^^^4all:\4all:\4^^^^7, auxiliary fields PRESERVED_PLACEHOLDER_^^^^4all:\4all:\4^^^^8, and new real scalar fields PRESERVED_PLACEHOLDER_^^^^4all:\4all:\4^^^^9. In this framework, the primitive symmetry is broken precisely by imposing a stable Lorentz-invariant vacuum. The resulting “sfermions” have only quartic gauge couplings,

PRESERVED_PLACEHOLDER_^^^^4all:\4 OR ti:\4query4^^^^

so conventional first-order production and decay channels are absent. The paper states that all sfermion decay processes are absent, a top squark does not decay at all, and superpartners with masses around PRESERVED_PLACEHOLDER_^^^^4all:\4 OR ti:\4all:\4^^^^ TeV may therefore have escaped standard searches because of reduced cross-sections and modified detector signatures (&&&^^^^4all:\44^^^^&&&).

Two neighboring programs delimit the concept from outside. “Superworld without supersymmetry” keeps an MSSM-like extra spectrum and a stabilizing PRESERVED_PLACEHOLDER_^^^^4all:\4 OR ti:\4 OR ti:\4^^^^ symmetry but explicitly discards supersymmetry itself; it is best described as a supersymmetry-adjacent construction rather than unconventional supersymmetry proper (Chakdar et al., 2015). Conversely, the conformal program of a dynamical Higgs boson and renormalizable four-fermion interactions presents itself as a replacement for supersymmetry, not as a variant of it; it is included in the surrounding discourse because it claims to perform jobs often assigned to SUSY, but it introduces no supercharges, supermultiplets, or SUSY algebra (Mannheim, 2015).

Across all strands, several issues remain open. In the gauge-superconnection program, the Dressing Field Method has clarified the status of the AVZ matter ansatz, but a general higher-dimensional and phenomenologically complete implementation is still incomplete (&&&^^^^4 OR abs:\4^^^^&&&). In lattice formulations, exact SUSY has been achieved for non-gauge models, but a gauge-invariant regularization is still missing (D'Adda et al., 2017). In the Dirac-material line, the time-loop mechanism and laboratory realization of unconventional supersymmetry remain proposals rather than demonstrations (&&&^^^^4query4^^^^&&&). In four-dimensional conformal and gauge-geometric models, chirality can emerge at the price of losing dynamical gravity, and several constructions inherit large bare cosmological constants or severe fine-tuning problems (Alvarez et al., 2020, Alvarez et al., 4 Jun 2026).

Unconventional supersymmetry is therefore not a single theory but a technical umbrella for attempts to preserve a graded unification of bosonic and fermionic structures while giving up the standard phenomenology of superpartners. Its most characteristic forms replace supermultiplet doubling by geometric unification in a superconnection, reinterpret torsion and auxiliary gauge sectors as dynamical ingredients, and treat supersymmetry as a contingent or reduced symmetry rather than as a universally manifest off-shell principle (&&&^^^^4all:\4^^^^&&&).