Tensionless Spinning String: Theory and Applications

• Tensionless spinning string is a limit of superstring theory where the worldsheet tension vanishes, leading to null dynamics and characteristic Carrollian symmetries.
• The limit is achieved by rescaling worldsheet coordinates, preserving the Majorana condition for spinors and contracting the super-Virasoro algebra to a super-BMS₃ structure.
• Duality with gauge theories emerges as the null string reproduces super-Wilson loop dynamics, linking higher-spin physics and enabling anomaly cancellation.

A tensionless spinning string is a limit of superstring (or spinning string) theory in which the worldsheet tension $T$ is sent to zero, fundamentally altering the dynamics, symmetry structure, and physical interpretation of the string. In this regime, the induced worldsheet metric degenerates, and the string's internal excitations collapse to a null or Carrollian surface where every point travels at the speed of light in target space. The theory is distinguished by nullification of the usual Regge slope, contraction of the worldsheet super-Virasoro algebra to a super-BMS$_3$ algebra, emergence of distinctive geometric features, nontrivial behavior of spinor degrees of freedom (notably the persistence of a well-defined Majorana condition), and connections to high-energy limits, gauge-theoretic dualities, and higher-spin physics (Delage, 2022, Zheltukhin, 17 Jan 2026, Yamauchi et al., 2014).

1. Construction: Tensionless Limit of the Super-Polyakov Action

The starting point for the tensionless spinning string is the standard super-Polyakov action:

$$S = -\frac{1}{4\pi \ell} \int d^2x\, \sqrt{-g}\, g^{\mu\nu} \left[ \partial_\mu X\, \partial_\nu X + i\, \bar\psi\, \gamma_\mu\, \partial_\nu \psi \right]$$

with $\ell$ the string length, $T=1/(2\pi\ell)$ the tension, $g^{\mu\nu}$ the worldsheet metric of signature $(+,-)$, and $\psi$ a two-component Majorana spinor.

The tensionless or "null string" limit is effected by rescaling the spatial coordinate $x\mapsto x/\lambda$ (with $\lambda\in(0,1]$), sending the inverse metric to $g^{\mu\nu}(\lambda) = (-1,\,\lambda^2)$ and ultimately taking $\lambda\to0$:

$$S(\lambda) = -\frac{1}{4\pi\ell}\int dt\,dx\,\left[ -\dot{X}^2 + \lambda^2 X'^2 + i\, \bar\psi\,\gamma^0\,\dot{\psi} + i\, \lambda^2 \bar\psi\,\gamma^1\,\psi' \right]$$

The limit $\lambda\to0$ removes all $X'^2$ and $\psi'$ dependence, rendering the spatial worldsheet direction non-dynamical. The bosonic part reduces to $$S_{T=0} = \frac{1}{4\pi\ell}\int dt\,dx\, \dot{X}^2$$ (Delage, 2022).

2. Spinor Field Deformation and Majorana Condition

To keep fermionic kinetic terms finite under the tensionless scaling, $\psi$ must be rescaled appropriately: in the "up/down" basis,

$$\psi(\lambda)= \begin{pmatrix} \psi_u / \sqrt{\lambda} \ \sqrt{\lambda}\, \psi_d \end{pmatrix}$$

So for $\lambda\to0$, only $\psi_u$ survives in spatial derivatives and $\psi_d$ in the time derivatives of the action. The Clifford algebra is similarly contracted and embedded into a higher-dimensional fixed algebra for consistency. The explicit matrix relation is $$P^{-1}(\lambda)\gamma^\mu P(\lambda) = \Gamma^\mu(\lambda)$$ (Delage, 2022).

The crucial physical consequence of this deformation is retention of the Majorana condition in the tensionless regime. The charge conjugation matrix $C$ (taken to be real and antisymmetric) ensures that both components $\psi_u$ and $\psi_d$ remain real:

$\psi_u^* = \psi_u,\quad \psi_d^* = \psi_d$

Previous approaches to tensionless strings lost the reality condition, resulting in loss of physical fermionic degrees of freedom or a proliferation of negative-norm states. Delage's group-theoretic construction preserves real (Majorana) spinors at $T=0$ (Delage, 2022).

3. Symmetry Structure: Super-BMS$_3$ Algebra

At $T=0$ the worldsheet symmetry is dramatically enhanced and contracted. The usual two copies of the super-Virasoro algebra collapse to the super-BMS$_3$ algebra generated by:

• $K(f)$: spatial reparameterizations ($x$-reparametrizations)
• $M(g)$: time translations and $t$-dependent shifts
• $G(\zeta), H(\rho)$: two real supercharges

The commutation relations for the contracted modes (schematically):

$$[K_n, K_m] = \frac{2\pi i}{T} (m-n) K_{n+m},\quad \{ G_r, G_s \} = 2i K_{r+s},\quad \{ G_r, H_s \} = 2i M_{r+s}$$

This algebra encodes a Carrollian or Galilean structure, mixing spatial and time translations with supersymmetry. The worldsheet energy-momentum tensor vanishes on shell, and its covariant divergence also vanishes under the mutual cancellation of bosonic and fermionic terms, enforced by supersymmetry and the specific vector density architecture of the action (Zheltukhin, 17 Jan 2026).

4. Worldsheet Geometry and Emergence of Torsion Trace

The degeneration of the worldsheet metric (i.e., $\det g \to 0$) necessitates a redefinition of worldsheet geometric quantities. Rather than a zweibein and metric, the theory employs a vector density $\rho^\mu$ (analogous to $\sqrt{-g}g^{\mu\nu}$) and a composite fermionic scalar density $\chi$ constructed as $\chi = \rho^\mu \chi_\mu$, where $\chi_\mu$ is a worldsheet gravitino analog. The affine connection survives only through its trace, and the torsion tensor $$S^\alpha{}_{\mu\nu} = \Gamma^\alpha{}_{\mu\nu} - \Gamma^\alpha{}_{\nu\mu}$$ enters only through its contracted trace $S_\mu = S^\alpha{}_{\mu\alpha}$.

The Noether identity associated to supersymmetry variations enforces the conservation law $\mathbf{D}_\mu \rho^\mu = 0$. This directly links the torsion trace to the structure of $\rho^\mu$ and $\chi$, constraining torsion on the tensionless worldsheet to a single component, in contrast to the multi-component, geometry-rich setting of tensile strings. This is interpreted as a "torsion-trace supergravity" (Zheltukhin, 17 Jan 2026).

5. Canonical Quantization, Ghost Sector, and Critical Dimension

Canonical quantization of tensionless spinning (super)strings requires gauge-fixing and introduction of intrinsic BMS $bc$ and $\beta\gamma$ ghost systems derived from Faddeev–Popov determinants in gauge fixing. The path integral over gauge-fixed worldsheet action introduces novel field-theoretic structures; for instance, inhomogeneous doublet tensionless superstrings require a BMS-algebraic $\beta\gamma$ ghost system with nontrivial central charges.

Anomaly cancellation stipulates the critical dimension for various models:

• Bosonic tensionless string: $D=26$
• Homogeneous $\mathcal N=2$ superstring: $D=10$
• Inhomogeneous doublet tensionless superstring (true ambitwistor string): $D=10$

The algebraic structure underlying BMS ghost systems is BMS-Kac-Moody, built from a three-dimensional non-semi-simple Lie algebra whose structure constants and central terms reflect the tensionless worldsheet's nontrivial conformal and scale properties (Chen et al., 2023).

6. Duality to Gauge Theories and Physical Interpretation

Tensionless spinning strings have a direct interpretation as lines of electric flux in Abelian gauge theories, most notably quantum electrodynamics (QED) with spinor matter. In the tensionless (null-string) limit, worldsheet contact interactions constructed from delta-functions localize all interactions to the boundaries, effectively collapsing the worldsheet interior. The partition function becomes that of a collection of super-Wilson loops on charged world-lines, and the sum over string worldsheets reproduces the path integral of spinor QED via super-Wilson loops:

$$W[A,\psi] = \mathcal{P} \exp \left( i \int_B dx\, [\dot w^\mu A_\mu(w) + \tfrac{1}{2} \psi^\mu F_{\mu\nu}(w) \psi^\nu ] \right)$$

Worldsheet supersymmetry guarantees cancellation of bulk divergences and protection against metric anomalies; world-line supersymmetry ensures proper coupling of boundary spinors. This establishes a direct string-theoretic duality for electric lines of force in Abelian gauge theory, emerging entirely from the tensionless spinning string formalism (Edwards et al., 2014, Edwards et al., 2014).

7. Connections to Higher-Dimensional Compactification and Geometric Invariants

In higher-dimensional backgrounds (e.g., $M_4 \times S^1$), the tensionless condition can be realized physically via winding and spinning in compact directions. The string becomes tensionless (static in large dimensions) precisely when half the squared length and half the squared velocity lie in compact space. Dimensionally reducing leads to an effective worldsheet current structure, equivalent to superconducting and chiral-current carrying cosmic strings (Yamauchi et al., 2014).

In curved backgrounds admitting Killing and Killing-Yano tensors, tensionless spinning strings inherit all point-particle invariants, admitting stringy generalizations of conserved charges (e.g., Carter-type invariants in Kerr–Newman backgrounds) with all constraints remaining first-class under canonical analysis (Lindström et al., 2022, Uvarov, 2017).

Summary Table: Tensionless Spinning String Key Features

Feature/Aspect	Tensionless Limit ($T\to 0$)	Physical Significance
Supersymmetry algebra	super-BMS$_3$	Carrollian/Galilean regime, extended symmetry
Metric structure	Degenerate, density $\rho^\mu$	Null worldsheet, causal degenerate geometry
Fermionic condition	Majorana preserved	No negative-norm ghost fermions
Critical dimension	$D=10$/$26$ (model-dependent)	Ambitwistor/heterotic constraints
Ghost sector	Intrinsic BMS $bc$, $\beta\gamma$	Novel 2D algebra, anomaly control
Gauge theory duality	QED via super-Wilson loops	Electric flux lines, boundary matter coupling
Higher-spin connection	Massless spectrum, integrability	AdS/CFT, higher-spin dualities

All content summarized above is directly derived from foundational works by Delage (Delage, 2022), Zheltukhin (Zheltukhin, 17 Jan 2026), Yamauchi & Lake (Yamauchi et al., 2014), Lindström & Sarıoğlu (Lindström et al., 2022), Uvarov (Uvarov, 2017), Edwards & Mansfield (Edwards et al., 2014, Edwards et al., 2014), and Cheng et al (Chen et al., 2023).