Supersymmetric World Line Dynamics

• Supersymmetric World Line is a one-dimensional manifold endowed with local supersymmetry via superfields and graded transformations, forming the basis for spinning particle models.
• It is constructed using a nonlinear realization of vector super-Poincaré symmetry, blending Nambu–Goto action components with Wess–Zumino terms to capture geometric and topological effects.
• Quantization yields two decoupled Dirac sectors and BPS configurations preserving 1/5 of the supersymmetries, underscoring its significance in advanced quantum mechanical frameworks.

A supersymmetric world line is a one-dimensional manifold endowed with local (world-line) supersymmetry, often realized by formulating dynamics for point particles or spinning particles using superfields and graded symmetries, such that supersymmetry transformations act both globally and, for particular parameter choices, locally along the world line. In the context of massive spinning particles with vector supersymmetry, the world-line supersymmetry emerges as a local fermionic gauge invariance, crucially affecting the spectrum, equations of motion, and BPS sector, and shaping the quantum theory as detailed through explicit nonlinear realizations of vector-extended super-Poincaré symmetry.

1. Nonlinear Realization of Space-Time Vector Supersymmetry

The construction begins by developing a nonlinear realization of a vector-extended super-Poincaré group. The coset element is parametrized as

$$g_\ell = \exp(i x^\mu P_\mu) \exp(i \xi^5 G_5) \exp(i \xi^\mu G_\mu) \exp(i c Z) \exp(i \hat{c} \tilde{Z}),$$

where $P_\mu$ are translations, $G_5$ and $G_\mu$ are odd (vector and scalar) supersymmetry generators, and $Z$, $\tilde{Z}$ are central charges. Constructing the massive model involves

$g = g_\ell U,$

where $U$ is a finite Lorentz boost parameterizing transition to the rest frame, leaving an $O(3)$ stability subgroup.

The Maurer–Cartan form,

$\Omega = U^{-1} \Omega_\ell U - i U^{-1} dU,$

is projected onto a basis of invariant one-forms, and the invariant world-line action is expressed as a sum of pullbacks:

$$S[x(\tau), \xi(\tau), v(\tau)] = \int d\tau [ -\mu L_x^0 - \beta L_Z - \gamma L_{\tilde{Z}} ],$$

where $L_x^0$ is of the Nambu–Goto type and $L_Z$, $L_{\tilde{Z}}$ are Wess–Zumino terms. The real parameters $\mu$, $\beta$, and $\gamma$ control the mass and central charge structure.

2. Emergence of Gauge (World-Line) Supersymmetry

While the action is invariant under rigid (global) vector supersymmetry for arbitrary coefficients, it develops a local fermionic gauge symmetry for a particular parameter relation:

$-\beta \gamma = \mu \,.$

With this condition, the action admits a local supersymmetry—analogous to $\kappa$-symmetry—given by

$$\delta \xi^i = -\frac{\beta}{2} \cosh(v) \kappa(\tau), \qquad \delta \xi^5 = \kappa(\tau), \qquad \delta x^\mu = i \xi^\mu \kappa(\tau),$$

$\delta v = 0$, where $\kappa(\tau)$ is a Grassmann-valued world-line gauge parameter, and similar transformations occur for the central-charge coordinates. This local fermionic symmetry allows for the gauging away (or reduction) of redundant fermionic degrees of freedom and is a manifestation of "gauge world-line supersymmetry".

3. BPS Solutions and Fractional Supersymmetry Preservation

Within the regime of gauge world-line supersymmetry ($-\beta\gamma=\mu$), one can investigate BPS-type world-line configurations. By requiring the vanishing of the world-line supersymmetry variation of the fermions: $\delta \xi = 0, \quad \delta \xi^5 = 0,$ one finds constraints on the supersymmetry transformation parameters, leading to a BPS condition: $$\left( \frac{d}{d\tau} x^\mu \dot{x}_\mu \right)^{1/2} = \text{const}, \qquad \frac{d}{d\tau} p^\mu = 0.$$ This implies that the momentum is conserved and the bosonic configuration describes a free relativistic particle. The crucial feature is that these BPS solutions preserve $1/5$ of the (vector) supersymmetries, rather than the typical $1/2$ seen in spinor-based models. The fraction is dictated by the representation content of the underlying vector supersymmetry algebra.

4. Supersymmetric Quantization and Dirac Sector Structure

Quantization proceeds by gauge-fixing (e.g., $x^0 = \tau$, partial setting of fermions to zero), leading to reduced canonical pairs $(x^i, p_j)$ and anticommuting variables $\xi^\mu$ with brackets: $$\{ x^i, p_j \}^* = -\delta^i_j, \qquad \{ \xi^\mu, \xi^\nu \}^* = i \eta^{\mu\nu}.$$ Quantization promotes these brackets to commutators and anticommutators. Fermions are realized as

$$\xi^\mu = \left(-\frac{1}{2\beta}\right)^{1/2} \Gamma^\mu,$$

where $\Gamma^\mu$ are $8 \times 8$ gamma matrices.

State conditions arising from the (Dirac) first-class constraints yield a Schrödinger equation

$$\Big(\Gamma^\mu (-i \partial_\mu) + \mu \Big) \Psi(x) = 0,$$

which, after an inverse Foldy–Wouthuysen transformation, is equivalent to two decoupled Dirac equations: $$\left(i \gamma^\mu \partial_\mu - \mu\right) \Psi_+(x) = 0, \qquad \left(i \gamma^\mu \partial_\mu - \mu\right) \Psi_-(x) = 0,$$ where $\Psi_\pm$ are independent 4-component spinors. Global vector supersymmetry transformations mix these two Dirac sectors.

5. Physical Implications and Theoretical Significance

The nonlinear realization of the vector super–Poincaré algebra, together with the emergence of gauge world-line supersymmetry upon certain parameter tuning, leads to several key features:

• Action construction: The massive spinning particle model is built as a pullback of coset-invariant one-forms, capturing both geometric (Nambu–Goto) and topological (Wess–Zumino) aspects.
• Local supersymmetry: World-line local supersymmetry imposes constraints that are analogous to $\kappa$-symmetry, reducing the number of propagating fermions and providing consistency with target space supersymmetry.
• BPS and protected sectors: The model exhibits BPS configurations preserving a reduced ($1/5$) fraction of supersymmetry, delineating a sector protected from quantum corrections—a recurrent phenomenon in supersymmetric field theories but with a vector supersymmetry-dependent fraction.
• Spectrum and representation theory: Upon quantization, the presence of two decoupled 4d Dirac equations captures the pseudo-classical description of a massive spinning particle with vector supersymmetry, and the structure of global supersymmetry transformations reflects the underlying representation theory.

This framework offers a cohesive mathematical apparatus for analyzing massive spinning particles within supersymmetric contexts, emphasizing the role of world-line local supersymmetry in both classical and quantum regimes. The approach generalizes to other systems—such as spinning particles on more general cosets or with different supersymmetry algebra extensions—by analogous nonlinear realization procedures and coset constructions.