Bosonic Spinning Particle Model

• Bosonic spinning particle model is a framework that encodes spin on the worldline via bosonic degrees of freedom, providing alternatives to conventional fermionic methods.
• It employs worldline supersymmetry, gauge-fixing conditions, and BRST quantization to derive conserved Poincaré charges and consistent equations of motion.
• Applications range from demonstrating integrability in curved backgrounds like Kerr–NUT–(A)dS to mapping bosonic sectors in supergravity and higher-spin extensions.

Searching arXiv for the specified topic and closely related worldline/spinning-particle papers. Bosonic spinning particle model denotes a family of first-quantized relativistic particle constructions in which spin is carried by worldline degrees of freedom, by bosonic auxiliary variables, or by the geometry of the trajectory itself, rather than being introduced only through a second-quantized fermion field. In the literature, this designation covers at least three closely related uses: the bosonic sector of the worldline supersymmetric spinning particle; commuting-spinor, tensor, or oscillator realizations of spin; and geometric or topological models whose conserved Poincaré charges reproduce those of a massive or massless spinning boson (Kubiznak et al., 2011, Markov et al., 2013, Haddad et al., 2024, Karataeva et al., 2024).

1. Minimal worldline formulation

A standard starting point is the classical spinning-particle theory with worldline coordinates $x^\mu(\tau)$, Grassmann-odd spin variables $\psi^a(\tau)$, vielbein $e^a{}_\mu(x)$, and metric $g_{\mu\nu}=e^a{}_\mu e^b{}_\nu\eta_{ab}$. A convenient worldline Lagrangian is

$$L=\tfrac12\,g_{\mu\nu}(x)\,\dot x^\mu\dot x^\nu+\tfrac{i}{2}\,\eta_{ab}\,\psi^a\,\frac{D\psi^b}{d\tau}, \qquad \frac{D\psi^a}{d\tau}=\dot x^\mu\,\omega_\mu{}^a{}_b\,\psi^b .$$

The conjugate momentum is

$$p_\mu=g_{\mu\nu}\dot x^\nu-\tfrac{i}{2}\,\omega_{\mu ab}\,\psi^a\psi^b,$$

and the covariant momentum

$$\Pi_\mu=p_\mu+\tfrac{i}{2}\,\omega_{\mu ab}\,\psi^a\psi^b$$

gives the minimal Hamiltonian

$H=\tfrac12\,g^{\mu\nu}\,\Pi_\mu\,\Pi_\nu .$

The graded Poisson brackets satisfy

$$\{x^\mu,p_\nu\}=\delta^\mu{}_\nu,\qquad \{\psi^a,\psi^b\}=-\,i\,\eta^{ab},$$

and the generic supercharge

$Q=\psi^a\,e_a{}^\mu\,\Pi_\mu$

obeys $\psi^a(\tau)$0 and $\psi^a(\tau)$1. The gauge-fixing conditions $\psi^a(\tau)$2 and $\psi^a(\tau)$3 fix $\psi^a(\tau)$4 to be proper time and remove unphysical spin components (Kubiznak et al., 2011).

An equivalent one-dimensional supergravity presentation uses the worldline coordinate $\psi^a(\tau)$5, its superpartner $\psi^a(\tau)$6, the einbein $\psi^a(\tau)$7, and the gravitino $\psi^a(\tau)$8, with Lagrangian

$\psi^a(\tau)$9

This makes the bosonic reparametrization symmetry and the fermionic gauge symmetry manifest at the level of the action (Kiriushcheva et al., 2013).

These formulations already exhibit a recurring feature of the subject: the phrase “bosonic” need not mean that every worldline variable is commuting. In part of the literature it refers instead to the bosonic observables, the bosonic gauge sector, or the bosonic conserved quantities extracted from a supersymmetric particle model.

2. Constraints, gauge symmetries, and quantization

In the one-dimensional supergravity formulation, the nontrivial bosonic gauge symmetry is worldline reparametrization with parameter $e^a{}_\mu(x)$0,

$e^a{}_\mu(x)$1

Its algebra closes as

$e^a{}_\mu(x)$2

so the bosonic gauge algebra is a true Lie algebra with field-independent structure (Kiriushcheva et al., 2013).

Higher-spin generalizations are obtained by extending the worldline supersymmetry. In the SO($e^a{}_\mu(x)$3) model one introduces bosonic variables $e^a{}_\mu(x)$4, $e^a{}_\mu(x)$5, fermions $e^a{}_\mu(x)$6, an einbein $e^a{}_\mu(x)$7, $e^a{}_\mu(x)$8 gravitini $e^a{}_\mu(x)$9, and an SO($g_{\mu\nu}=e^a{}_\mu e^b{}_\nu\eta_{ab}$0) gauge field $g_{\mu\nu}=e^a{}_\mu e^b{}_\nu\eta_{ab}$1. The first-order action is

$g_{\mu\nu}=e^a{}_\mu e^b{}_\nu\eta_{ab}$2

with constraints

$g_{\mu\nu}=e^a{}_\mu e^b{}_\nu\eta_{ab}$3

Their graded Poisson brackets define the first-class constraint algebra. Compactifying one direction and imposing $g_{\mu\nu}=e^a{}_\mu e^b{}_\nu\eta_{ab}$4 yields a massive spinning-particle action in odd $g_{\mu\nu}=e^a{}_\mu e^b{}_\nu\eta_{ab}$5, while Dirac quantization produces the Fierz–Pauli system

$g_{\mu\nu}=e^a{}_\mu e^b{}_\nu\eta_{ab}$6

In the massless limit the resulting geometric equations can be partially integrated to recover the Fronsdal–Labastida equations, and on $g_{\mu\nu}=e^a{}_\mu e^b{}_\nu\eta_{ab}$7 the deformed worldline gauge algebra becomes nonlinear but remains first class (Bastianelli et al., 2014).

A BRST formulation pushes the same logic into background-field reconstruction. For the $g_{\mu\nu}=e^a{}_\mu e^b{}_\nu\eta_{ab}$8 spinning particle, the phase-space action contains the constraints $g_{\mu\nu}=e^a{}_\mu e^b{}_\nu\eta_{ab}$9, $$L=\tfrac12\,g_{\mu\nu}(x)\,\dot x^\mu\dot x^\nu+\tfrac{i}{2}\,\eta_{ab}\,\psi^a\,\frac{D\psi^b}{d\tau}, \qquad \frac{D\psi^a}{d\tau}=\dot x^\mu\,\omega_\mu{}^a{}_b\,\psi^b .$$0, and $$L=\tfrac12\,g_{\mu\nu}(x)\,\dot x^\mu\dot x^\nu+\tfrac{i}{2}\,\eta_{ab}\,\psi^a\,\frac{D\psi^b}{d\tau}, \qquad \frac{D\psi^a}{d\tau}=\dot x^\mu\,\omega_\mu{}^a{}_b\,\psi^b .$$1, and the BFV BRST operator $$L=\tfrac12\,g_{\mu\nu}(x)\,\dot x^\mu\dot x^\nu+\tfrac{i}{2}\,\eta_{ab}\,\psi^a\,\frac{D\psi^b}{d\tau}, \qquad \frac{D\psi^a}{d\tau}=\dot x^\mu\,\omega_\mu{}^a{}_b\,\psi^b .$$2 is built from the corresponding ghosts, antighosts, and structure-constant terms so that $$L=\tfrac12\,g_{\mu\nu}(x)\,\dot x^\mu\dot x^\nu+\tfrac{i}{2}\,\eta_{ab}\,\psi^a\,\frac{D\psi^b}{d\tau}, \qquad \frac{D\psi^a}{d\tau}=\dot x^\mu\,\omega_\mu{}^a{}_b\,\psi^b .$$3. When the generators are deformed by couplings to $$L=\tfrac12\,g_{\mu\nu}(x)\,\dot x^\mu\dot x^\nu+\tfrac{i}{2}\,\eta_{ab}\,\psi^a\,\frac{D\psi^b}{d\tau}, \qquad \frac{D\psi^a}{d\tau}=\dot x^\mu\,\omega_\mu{}^a{}_b\,\psi^b .$$4, $$L=\tfrac12\,g_{\mu\nu}(x)\,\dot x^\mu\dot x^\nu+\tfrac{i}{2}\,\eta_{ab}\,\psi^a\,\frac{D\psi^b}{d\tau}, \qquad \frac{D\psi^a}{d\tau}=\dot x^\mu\,\omega_\mu{}^a{}_b\,\psi^b .$$5, and $$L=\tfrac12\,g_{\mu\nu}(x)\,\dot x^\mu\dot x^\nu+\tfrac{i}{2}\,\eta_{ab}\,\psi^a\,\frac{D\psi^b}{d\tau}, \qquad \frac{D\psi^a}{d\tau}=\dot x^\mu\,\omega_\mu{}^a{}_b\,\psi^b .$$6, the single consistency condition $$L=\tfrac12\,g_{\mu\nu}(x)\,\dot x^\mu\dot x^\nu+\tfrac{i}{2}\,\eta_{ab}\,\psi^a\,\frac{D\psi^b}{d\tau}, \qquad \frac{D\psi^a}{d\tau}=\dot x^\mu\,\omega_\mu{}^a{}_b\,\psi^b .$$7 reproduces the NS–NS field equations

$$L=\tfrac12\,g_{\mu\nu}(x)\,\dot x^\mu\dot x^\nu+\tfrac{i}{2}\,\eta_{ab}\,\psi^a\,\frac{D\psi^b}{d\tau}, \qquad \frac{D\psi^a}{d\tau}=\dot x^\mu\,\omega_\mu{}^a{}_b\,\psi^b .$$8

$$L=\tfrac12\,g_{\mu\nu}(x)\,\dot x^\mu\dot x^\nu+\tfrac{i}{2}\,\eta_{ab}\,\psi^a\,\frac{D\psi^b}{d\tau}, \qquad \frac{D\psi^a}{d\tau}=\dot x^\mu\,\omega_\mu{}^a{}_b\,\psi^b .$$9

$$p_\mu=g_{\mu\nu}\dot x^\nu-\tfrac{i}{2}\,\omega_{\mu ab}\,\psi^a\psi^b,$$0

together with the effective action

$$p_\mu=g_{\mu\nu}\dot x^\nu-\tfrac{i}{2}\,\omega_{\mu ab}\,\psi^a\psi^b,$$1

The bosonic sector of ten-dimensional supergravity therefore appears as a BRST-consistency condition of an $$p_\mu=g_{\mu\nu}\dot x^\nu-\tfrac{i}{2}\,\omega_{\mu ab}\,\psi^a\psi^b,$$2 spinning-particle model (Boffo, 2023).

3. Hidden symmetries and integrability in curved backgrounds

A particularly developed bosonic sector arises for spinning-particle motion in higher-dimensional Kerr–NUT–(A)dS spacetimes. In dimensions $$p_\mu=g_{\mu\nu}\dot x^\nu-\tfrac{i}{2}\,\omega_{\mu ab}\,\psi^a\psi^b,$$3, the geometry admits a principal conformal Killing–Yano tensor

$$p_\mu=g_{\mu\nu}\dot x^\nu-\tfrac{i}{2}\,\omega_{\mu ab}\,\psi^a\psi^b,$$4

satisfying

$$p_\mu=g_{\mu\nu}\dot x^\nu-\tfrac{i}{2}\,\omega_{\mu ab}\,\psi^a\psi^b,$$5

Its wedge powers generate a tower of Killing–Yano forms $$p_\mu=g_{\mu\nu}\dot x^\nu-\tfrac{i}{2}\,\omega_{\mu ab}\,\psi^a\psi^b,$$6, whose quadratic “squares” define rank-2 Killing tensors $$p_\mu=g_{\mu\nu}\dot x^\nu-\tfrac{i}{2}\,\omega_{\mu ab}\,\psi^a\psi^b,$$7 with $$p_\mu=g_{\mu\nu}\dot x^\nu-\tfrac{i}{2}\,\omega_{\mu ab}\,\psi^a\psi^b,$$8 (Kubiznak et al., 2011).

The corresponding bosonic superinvariants are quadratic in momentum. For each $$p_\mu=g_{\mu\nu}\dot x^\nu-\tfrac{i}{2}\,\omega_{\mu ab}\,\psi^a\psi^b,$$9,

$$\Pi_\mu=p_\mu+\tfrac{i}{2}\,\omega_{\mu ab}\,\psi^a\psi^b$$0

and in the purely bosonic limit $$\Pi_\mu=p_\mu+\tfrac{i}{2}\,\omega_{\mu ab}\,\psi^a\psi^b$$1,

$$\Pi_\mu=p_\mu+\tfrac{i}{2}\,\omega_{\mu ab}\,\psi^a\psi^b$$2

The Poisson brackets $$\Pi_\mu=p_\mu+\tfrac{i}{2}\,\omega_{\mu ab}\,\psi^a\psi^b$$3 decompose by degree in $$\Pi_\mu=p_\mu+\tfrac{i}{2}\,\omega_{\mu ab}\,\psi^a\psi^b$$4. At zeroth order one obtains the Schouten–Nijenhuis bracket of Killing tensors, and in Kerr–NUT–(A)dS this vanishes for all pairs. The mixed $$\Pi_\mu=p_\mu+\tfrac{i}{2}\,\omega_{\mu ab}\,\psi^a\psi^b$$5 and $$\Pi_\mu=p_\mu+\tfrac{i}{2}\,\omega_{\mu ab}\,\psi^a\psi^b$$6 terms lead to differential conditions on the $$\Pi_\mu=p_\mu+\tfrac{i}{2}\,\omega_{\mu ab}\,\psi^a\psi^b$$7 that are satisfied because of the PCKY equation and its integrability conditions. In $$\Pi_\mu=p_\mu+\tfrac{i}{2}\,\omega_{\mu ab}\,\psi^a\psi^b$$8-, $$\Pi_\mu=p_\mu+\tfrac{i}{2}\,\omega_{\mu ab}\,\psi^a\psi^b$$9-, $H=\tfrac12\,g^{\mu\nu}\,\Pi_\mu\,\Pi_\nu .$0-, and $H=\tfrac12\,g^{\mu\nu}\,\Pi_\mu\,\Pi_\nu .$1-dimensional black-hole spacetimes these cancellations can be verified directly, so the bosonic part of the spinning-particle motion is integrable in those cases (Kubiznak et al., 2011).

Together with the $H=\tfrac12\,g^{\mu\nu}\,\Pi_\mu\,\Pi_\nu .$2 linear integrals from the explicit Killing vectors $H=\tfrac12\,g^{\mu\nu}\,\Pi_\mu\,\Pi_\nu .$3, the $H=\tfrac12\,g^{\mu\nu}\,\Pi_\mu\,\Pi_\nu .$4 independent bosonic integrals $H=\tfrac12\,g^{\mu\nu}\,\Pi_\mu\,\Pi_\nu .$5 yield the full set of $H=\tfrac12\,g^{\mu\nu}\,\Pi_\mu\,\Pi_\nu .$6 integrals required for Liouville integrability of the bosonic sector. The result generalizes the integrability of geodesic motion established for the same backgrounds, and the same mutual commutation is conjectured to hold in all higher dimensions (Kubiznak et al., 2011).

4. Bosonic realizations of spin beyond Grassmann variables

A distinct line of work replaces Grassmann spin variables by commuting ones. One such formulation uses a commuting Dirac–Majorana spinor $H=\tfrac12\,g^{\mu\nu}\,\Pi_\mu\,\Pi_\nu .$7, an auxiliary anticommuting Majorana spinor $H=\tfrac12\,g^{\mu\nu}\,\Pi_\mu\,\Pi_\nu .$8, the einbein $H=\tfrac12\,g^{\mu\nu}\,\Pi_\mu\,\Pi_\nu .$9, and a color charge $$\{x^\mu,p_\nu\}=\delta^\mu{}_\nu,\qquad \{\psi^a,\psi^b\}=-\,i\,\eta^{ab},$$0. The Lagrangian contains the terms

$$\{x^\mu,p_\nu\}=\delta^\mu{}_\nu,\qquad \{\psi^a,\psi^b\}=-\,i\,\eta^{ab},$$1

From $$\{x^\mu,p_\nu\}=\delta^\mu{}_\nu,\qquad \{\psi^a,\psi^b\}=-\,i\,\eta^{ab},$$2 one forms the five real bilinears

$$\{x^\mu,p_\nu\}=\delta^\mu{}_\nu,\qquad \{\psi^a,\psi^b\}=-\,i\,\eta^{ab},$$3

which satisfy a complete system of bilinear Fierz identities. The model also has a local bosonic symmetry generated by a commuting Majorana spinor $$\{x^\mu,p_\nu\}=\delta^\mu{}_\nu,\qquad \{\psi^a,\psi^b\}=-\,i\,\eta^{ab},$$4; the commutator of two $$\{x^\mu,p_\nu\}=\delta^\mu{}_\nu,\qquad \{\psi^a,\psi^b\}=-\,i\,\eta^{ab},$$5-transformations closes onto a reparametrization and, in the gauge-field case, an infinitesimal color rotation. Under the map

$$\{x^\mu,p_\nu\}=\delta^\mu{}_\nu,\qquad \{\psi^a,\psi^b\}=-\,i\,\eta^{ab},$$6

the commuting-spinor model becomes equivalent to the usual pseudoclassical description with anticommuting pseudovector and pseudoscalar variables (Markov et al., 2013).

The same tensor aggregate $$\{x^\mu,p_\nu\}=\delta^\mu{}_\nu,\qquad \{\psi^a,\psi^b\}=-\,i\,\eta^{ab},$$7 can be analyzed directly as a bosonic description of a relativistic spin-$$\{x^\mu,p_\nu\}=\delta^\mu{}_\nu,\qquad \{\psi^a,\psi^b\}=-\,i\,\eta^{ab},$$8 particle. The bilinear identities admit an explicit solution in terms of an orthonormal tetrad $$\{x^\mu,p_\nu\}=\delta^\mu{}_\nu,\qquad \{\psi^a,\psi^b\}=-\,i\,\eta^{ab},$$9 and an antisymmetric tensor $Q=\psi^a\,e_a{}^\mu\,\Pi_\mu$0, for example

$Q=\psi^a\,e_a{}^\mu\,\Pi_\mu$1

After eliminating auxiliary variables and choosing proper-time gauge, one arrives at a purely bosonic higher-derivative Lagrangian of Polyakov type, and the resulting equations of motion take the form of a generalized Lorentz force together with spin precession of Mathisson–Papapetrou–Dixon type (Markov et al., 2016).

A more recent bosonic realization introduces complex tangent-space oscillators $Q=\psi^a\,e_a{}^\mu\,\Pi_\mu$2, $Q=\psi^a\,e_a{}^\mu\,\Pi_\mu$3 with Poisson brackets

$Q=\psi^a\,e_a{}^\mu\,\Pi_\mu$4

and spin tensor

$Q=\psi^a\,e_a{}^\mu\,\Pi_\mu$5

Coupling to gravity uses the covariant momentum

$Q=\psi^a\,e_a{}^\mu\,\Pi_\mu$6

and the covariant spin-supplementary condition

$Q=\psi^a\,e_a{}^\mu\,\Pi_\mu$7

A Hamiltonian $Q=\psi^a\,e_a{}^\mu\,\Pi_\mu$8 is then constructed so that the SSC is preserved under time evolution, and a Legendre transform yields a second-order Lagrangian with bosonic worldline oscillators. Varying this action reproduces the MPD equations at pole-dipole order, while the same framework supports $Q=\psi^a\,e_a{}^\mu\,\Pi_\mu$9PM calculations to all orders in spin and $\psi^a(\tau)$00PM calculations up to quartic spin order (Haddad et al., 2024).

5. Twistor, geometric, and topological realizations

For a free massless spinning bosonic particle in four dimensions, the gauged Shirafuji model uses twistor variables $\psi^a(\tau)$01, their duals, a real scalar density $\psi^a(\tau)$02, a $\psi^a(\tau)$03 gauge field $\psi^a(\tau)$04, and a real constant $\psi^a(\tau)$05 interpreted as helicity. In the gauge $\psi^a(\tau)$06,

$\psi^a(\tau)$07

Dirac analysis produces first-class spacetime and helicity constraints, and quantization gives the differential equations

$\psi^a(\tau)$08

The general spin-$\psi^a(\tau)$09 plane-wave solution is

$\psi^a(\tau)$10

and Fourier–Laplace transforms turn the corresponding wave functions into nonprojective or projective Penrose transforms (Deguchi et al., 2013).

A purely geometric realization of a massive spinning particle in four-dimensional Minkowski space is obtained by demanding that every classical worldline lie on a fixed two-dimensional cylinder. The cylinder is defined by

$\psi^a(\tau)$11

with $\psi^a(\tau)$12, $\psi^a(\tau)$13, $\psi^a(\tau)$14, and $\psi^a(\tau)$15. Its parameters are in one-to-one correspondence with the Poincaré charges

$\psi^a(\tau)$16

The resulting equations of motion are fourth-order, non-Lagrangian, and gauge-invariant under reparametrizations and under sliding of the worldline along the same cylinder; all curves on a given cylinder are gauge-equivalent (Kaparulin et al., 2019).

In three dimensions, a topological string with an action involving the Nambu–Goto term, Gauss curvature, mean curvature, and two Lagrange multipliers has an extra scalar gauge symmetry with second derivatives of the gauge parameter. Hamiltonian analysis yields seven first-class constraints and two second-class constraints on a $\psi^a(\tau)$17-dimensional phase space, so there are no local degrees of freedom. The world sheet is a right circular cylinder with time-like axis, and its global modes describe a single irreducible massive $\psi^a(\tau)$18 particle with spin: $\psi^a(\tau)$19 The mass and spin are fixed by the string action parameters $\psi^a(\tau)$20 (Karataeva et al., 2024).

At the massless end of the spectrum, Wigner’s continuous-spin particle provides a scalar-like bosonic first-quantized system with wavefunction $\psi^a(\tau)$21 obeying

$\psi^a(\tau)$22

Its coadjoint-orbit description imposes the first-class constraints $\psi^a(\tau)$23, $\psi^a(\tau)$24, and $\psi^a(\tau)$25, with Pauli–Lubański invariant $\psi^a(\tau)$26 (Gracia-Bondía et al., 2018).

6. Background dynamics, applications, and limitations

When the spinning particle is treated as a pole-dipole test body, the equations of motion are the MPD system with a supplementary condition. In a rotating boson-star background, equatorial motion under the Tulczyjew condition $\psi^a(\tau)$27 leads to conserved energy and angular momentum, closed-form expressions for $\psi^a(\tau)$28, $\psi^a(\tau)$29, and $\psi^a(\tau)$30, and an effective radial potential

$\psi^a(\tau)$31

As $\psi^a(\tau)$32, $\psi^a(\tau)$33 generically when $\psi^a(\tau)$34, but for the special tuning

$\psi^a(\tau)$35

the effective potential remains finite and the spinning particle can pass through the center of the boson star. Circular orbits satisfy $\psi^a(\tau)$36 and $\psi^a(\tau)$37, while stability is determined by the sign of $\psi^a(\tau)$38. The particle spin shifts the regions of no circular orbits, unstable circular orbits, and stable circular orbits, and it shifts $\psi^a(\tau)$39, $\psi^a(\tau)$40, and $\psi^a(\tau)$41, with consequences for gravitational-wave phasing and cutoff frequency (Zhang et al., 2022).

In five-dimensional smooth Randall–Sundrum backgrounds, the bosonic test particle with dilaton-dependent mass $\psi^a(\tau)$42 has effective potential

$\psi^a(\tau)$43

and for $\psi^a(\tau)$44 this potential can develop a strict minimum at $\psi^a(\tau)$45, trapping bosonic particles of any $\psi^a(\tau)$46. The spinning extension introduces Grassmann-odd vectors $\psi^a(\tau)$47, an auxiliary Grassmann scalar $\psi^a(\tau)$48, and a worldline gravitino $\psi^a(\tau)$49. The effective potential becomes

$\psi^a(\tau)$50

and the spin–curvature coupling shifts or removes the minimum. In particular, $\psi^a(\tau)$51 generically, and $\psi^a(\tau)$52, so spinning particles are not localized on the brane; they can localize away from it or escape to infinity (Souza et al., 2019).

The oscillator-based worldline theory provides a current computational application of bosonic spinning-particle methods. In Worldline Quantum Field Theory, expanding

$\psi^a(\tau)$53

one computes the momentum impulse and spin kick from the one-point functions of $\psi^a(\tau)$54 and $\psi^a(\tau)$55. At $\psi^a(\tau)$56PM order the formalism yields all-order-in-spin impulse and spin-kick expressions in closed form, and at $\psi^a(\tau)$57PM order up to quartic spin it reproduces known conservative results, including the aligned-spin quartic contribution to the scattering angle (Haddad et al., 2024).

Taken together, these constructions indicate that there is no single universal bosonic spinning-particle model. Some formulations are worldline-supersymmetric and use Grassmann spin variables but isolate a bosonic integrable sector; some replace the spin variables by commuting spinors, tensors, or bosonic oscillators; some are twistor-based; and some encode spin purely through cylinder geometry or topological world-sheet data. A plausible implication is that “bosonic” in this context names a method of encoding spin on the worldline, rather than one unique dynamical system.