Torsion Parallel Spinors

• Torsion parallel spinors are spinor fields that remain parallel under metric connections with contorsion, ensuring holonomy reduction and precise differential constraints.
• The analysis employs Cartan–Kähler techniques to reformulate torsion effects as an exterior differential system acting on spinor bilinears and null coframes.
• This framework connects Lorentzian supersymmetry with gerbe curvatures and NS–NS flux flows, offering deep insights into gravitational wave and Kundt geometries.

Searching arXiv for recent and foundational papers on torsion parallel spinors, especially the cited dissertation and closely related works. Torsion parallel spinors are spinor fields that are parallel not with respect to the Levi–Civita connection alone, but with respect to a metric connection carrying contorsion or, in special cases, totally skew-symmetric torsion. In the Lorentzian four-dimensional setting, their structure is governed by spinorial polyforms, null bilinears, and a reduced holonomy preserving a parabolic null datum; in the supersymmetric NS–NS system, they furnish the gravitino sector and couple torsion to a gerbe-theoretic flux $H_b$ and to a dilatino-type constraint (Shahbazi, 8 Jul 2025). Closely related formulations appear in Lorentzian three dimensions, in signature $(4,3)$, and in Riemannian characteristic-connection geometry, where torsion parallelism is likewise reformulated as a differential system on spinor bilinears or stable forms (Shahbazi, 2024, Gil-García et al., 2024, Chrysikos, 2015).

1. Definition and connection-theoretic framework

On an oriented, time-oriented Lorentzian four-manifold $(M,g)$ of mostly-plus signature, the basic object is a metric connection with contorsion $A$,

$\nabla^{g,A}_w X=\nabla^g_w X+A(w,X),$

where $A\in \Gamma(T^*M\otimes \mathrm{End}_g(TM))$. In the “strongly spin” case, meaning that $(M,g)$ admits a $\mathrm{Spin}_o(3,1)$-structure, spinors live in an irreducible real bundle $(S,\Gamma)$ with admissible bilinear pairing. The lifted spin connection is fixed by

$$\nabla^{g,A}_w \varepsilon=\nabla^g_w \varepsilon-\frac12 A_w\cdot \varepsilon,$$

which determines the sign and coefficient conventions used in the Lorentzian four-dimensional theory (Shahbazi, 8 Jul 2025).

The contorsion decomposes as

$(4,3)$0

with $(4,3)$1, $(4,3)$2, and $(4,3)$3 the trace-free torsion component. Tensorially,

$(4,3)$4

This separates vectorial, trace-free, and totally skew contributions, and it is the contorsion rather than the torsion tensor that is primary in the dissertation’s formulation (Shahbazi, 8 Jul 2025).

Across the broader literature, equivalent geometric content is often written with different normalizations. In the Riemannian characteristic-connection setting one frequently uses

$(4,3)$5

with the characteristic connection at $(4,3)$6, while the cone correspondence with skew torsion uses

$(4,3)$7

These are convention changes rather than substantive disagreements about the geometry of torsion parallelism (Chrysikos, 2015, Agricola et al., 2013).

2. Spinor bilinears, null structure, and holonomy reduction

In four Lorentzian dimensions, the square of a nonzero irreducible real spinor is exceptionally rigid. The spinorial polyform attached to $(4,3)$8 is

$(4,3)$9

where $(M,g)$0 is the Dirac current and is isotropic, while $(M,g)$1 is a unit one-form orthogonal to $(M,g)$2, defined only modulo the gauge transformation $(M,g)$3. In standard bilinear language, the scalar, pseudoscalar, and axial-vector pieces vanish, the Dirac current $(M,g)$4 is null, and the two-form bilinear is $(M,g)$5 (Shahbazi, 8 Jul 2025).

A direct consequence is that any nowhere-vanishing irreducible real spinor on a Lorentzian four-manifold has null Dirac current. The underlying geometry is therefore forced into the wave sector, specifically Brinkmann or Kundt type. The fundamental invariant is a parabolic pair $(M,g)$6, where $(M,g)$7, $(M,g)$8, $(M,g)$9, $A$0, and $A$1 denotes the equivalence class modulo $A$2. Extending this to an isotropic parallelism $A$3, one reconstructs the metric by

$A$4

This null coframe formalism packages the entire local geometry determined by the spinor square (Shahbazi, 8 Jul 2025).

The stabilizer of $A$5 in $A$6 is the subgroup preserving the parabolic pair,

$A$7

and is isomorphic to $A$8; its lift in $A$9 is connected and again isomorphic to $\nabla^{g,A}_w X=\nabla^g_w X+A(w,X),$0. Hence a torsionful metric connection fixing $\nabla^{g,A}_w X=\nabla^g_w X+A(w,X),$1 has holonomy contained in the subgroup preserving $\nabla^{g,A}_w X=\nabla^g_w X+A(w,X),$2 and the associated null-frame torsor. A common misconception is that timelike irreducible real parallel spinors might occur in this dimension; the Lorentzian four-dimensional analysis excludes this, because the Dirac current is necessarily null (Shahbazi, 8 Jul 2025).

The same stabilization phenomenon reappears in other signatures with different algebraic representatives. In signature $\nabla^{g,A}_w X=\nabla^g_w X+A(w,X),$3, isotropic irreducible spinors are encoded by a degenerate three-form $\nabla^{g,A}_w X=\nabla^g_w X+A(w,X),$4, and the stabilizer becomes $\nabla^{g,A}_w X=\nabla^g_w X+A(w,X),$5 (Gil-García et al., 2024). This suggests a general pattern: torsion parallelism is best understood through the geometry of the spinor square rather than through the spinor field alone.

3. Torsion parallel spinors as an exterior differential system

A torsion parallel spinor is a section $\nabla^{g,A}_w X=\nabla^g_w X+A(w,X),$6 satisfying

$\nabla^{g,A}_w X=\nabla^g_w X+A(w,X),$7

In bilinear form this immediately constrains the associated parabolic data. The Dirac current satisfies

$\nabla^{g,A}_w X=\nabla^g_w X+A(w,X),$8

and the complementary one-form obeys

$\nabla^{g,A}_w X=\nabla^g_w X+A(w,X),$9

for a one-form $A\in \Gamma(T^*M\otimes \mathrm{End}_g(TM))$0. In an adapted null frame one further has

$A\in \Gamma(T^*M\otimes \mathrm{End}_g(TM))$1

which makes explicit that the torsionful connection preserves the null direction and transports the rest of the frame through a reduced set of coefficients (Shahbazi, 8 Jul 2025).

The decisive reformulation is the exterior differential system obtained by decomposing the contorsion as $A\in \Gamma(T^*M\otimes \mathrm{End}_g(TM))$2. A torsion parallel spinor exists if and only if there is a global coframe $A\in \Gamma(T^*M\otimes \mathrm{End}_g(TM))$3 and one-forms $A\in \Gamma(T^*M\otimes \mathrm{End}_g(TM))$4 such that

$A\in \Gamma(T^*M\otimes \mathrm{End}_g(TM))$5

$A\in \Gamma(T^*M\otimes \mathrm{End}_g(TM))$6

$A\in \Gamma(T^*M\otimes \mathrm{End}_g(TM))$7

$A\in \Gamma(T^*M\otimes \mathrm{End}_g(TM))$8

This system is independent of any prior metric choice: the metric is reconstructed afterward from the isotropic parallelism. The dissertation emphasizes that this opens the subject to Cartan–Kähler analysis, prolongation, and Spencer cohomology (Shahbazi, 8 Jul 2025).

The system is invariant under the torsor action on null coframes. If one representative $A\in \Gamma(T^*M\otimes \mathrm{End}_g(TM))$9 in the isotropic parallelism solves the connection or exterior equations, then any coframe in the same isotropic parallelism also solves them after the induced transformation of $(M,g)$0 and $(M,g)$1. That invariance is the differential expression of the $(M,g)$2 stabilizer (Shahbazi, 8 Jul 2025).

Related results in neighboring signatures sharpen the general picture. In Lorentzian three dimensions, a skew-torsion parallel spinor is equivalent to a null one-form $(M,g)$3 satisfying

$(M,g)$4

so the geometry is necessarily Kundt (Shahbazi, 2024). In signature $(M,g)$5, an isotropic torsion parallel spinor exists exactly when a coherent isotropic triple of one-forms is preserved by the torsionful connection (Gil-García et al., 2024). This suggests that exterior or bilinear reformulations are not an artifact of the four-dimensional case but a recurrent structural feature of torsion parallelism.

4. Curvature constraints and the supersymmetric NS–NS system

The Lorentzian four-dimensional theory is motivated by a supersymmetric NS–NS system with variables $(M,g)$6, where $(M,g)$7 is a gerbe curving, $(M,g)$8 is a function on the presenting fibration $(M,g)$9 satisfying an integrality condition, and $\mathrm{Spin}_o(3,1)$0 is an irreducible real spinor. Supersymmetry is expressed by the gravitino and dilatino-type equations

$\mathrm{Spin}_o(3,1)$1

with $\mathrm{Spin}_o(3,1)$2 the totally skew torsion three-form and $\mathrm{Spin}_o(3,1)$3 the curvature of $\mathrm{Spin}_o(3,1)$4. The bosonic system is

$\mathrm{Spin}_o(3,1)$5

The formulation uses directly the torsionful Ricci tensor $\mathrm{Spin}_o(3,1)$6 rather than expanding it into Levi–Civita curvature plus flux terms (Shahbazi, 8 Jul 2025).

Supersymmetry forces the torsion parallel spinor constraints with $\mathrm{Spin}_o(3,1)$7, $\mathrm{Spin}_o(3,1)$8, and $\mathrm{Spin}_o(3,1)$9. Equivalently, the torsionful holonomy reduces to the $(S,\Gamma)$0 stabilizer of the null spinor, and the four-dimensional supersymmetric geometry is encoded by the null coframe system with totally skew torsion only. The dilatino equation then constrains $(S,\Gamma)$1 relative to the null structure induced by $(S,\Gamma)$2; the dissertation states that this coupling is drastic when written along the $(S,\Gamma)$3 directions (Shahbazi, 8 Jul 2025).

From the physical perspective, torsion parallel spinors are the mathematical avatars of supersymmetry in the NS–NS sector. The spinor $(S,\Gamma)$4 is the supersymmetry generator, $(S,\Gamma)$5 is the gravitino condition, and $(S,\Gamma)$6 expresses a dilatino–flux balance. Because the spinor is null, the spacetime acquires a Kundt or Brinkmann structure, so the Lorentzian geometry is closely tied to exact gravitational-wave sectors with additional flux degrees of freedom carried by torsion (Shahbazi, 8 Jul 2025).

A broader supergravity context appears in the Riemannian instanton literature, where canonical torsionful connections on nearly Kähler, nearly parallel $(S,\Gamma)$7, Sasaki–Einstein, and 3–Sasakian manifolds admit torsion-parallel spinors and furnish supersymmetry equations of heterotic type. In that setting the instanton condition $(S,\Gamma)$8 implies the Yang–Mills equation even in the presence of torsion (Harland et al., 2011). The Lorentzian NS–NS system extends this interaction between torsion, spinor parallelism, and field equations into a null and globally hyperbolic regime.

5. Bundle gerbes, higher gauge symmetry, and evolution

A distinctive feature of the four-dimensional Lorentzian theory is its gauge-theoretic interpretation of torsion through abelian bundle gerbes. A bundle gerbe $(S,\Gamma)$9 with connective structure carries a curving $$\nabla^{g,A}_w \varepsilon=\nabla^g_w \varepsilon-\frac12 A_w\cdot \varepsilon,$$0 and a three-curvature

$$\nabla^{g,A}_w \varepsilon=\nabla^g_w \varepsilon-\frac12 A_w\cdot \varepsilon,$$1

which models $$\nabla^{g,A}_w \varepsilon=\nabla^g_w \varepsilon-\frac12 A_w\cdot \varepsilon,$$2. In this formulation, pseudo-Riemannian skew torsion is not treated as an ad hoc auxiliary field; it is the gerbe curvature itself, and the gerbe automorphism groupoid encodes the relevant higher gauge symmetry (Shahbazi, 8 Jul 2025).

This identification matters in the globally hyperbolic setting. By slicing $$\nabla^{g,A}_w \varepsilon=\nabla^g_w \varepsilon-\frac12 A_w\cdot \varepsilon,$$3, the curving $$\nabla^{g,A}_w \varepsilon=\nabla^g_w \varepsilon-\frac12 A_w\cdot \varepsilon,$$4 decomposes into initial data and evolution, and one obtains supersymmetric evolution flow equations for $$\nabla^{g,A}_w \varepsilon=\nabla^g_w \varepsilon-\frac12 A_w\cdot \varepsilon,$$5 coupled to the NS–NS evolution flow. The flow preserves the constraints inherited from the torsion parallel spinor system together with the abelian gerbe Bianchi identity

$$\nabla^{g,A}_w \varepsilon=\nabla^g_w \varepsilon-\frac12 A_w\cdot \varepsilon,$$6

The dissertation proves a compatibility theorem stating that the first-order supersymmetric flow is consistent with the second-order NS–NS flow provided the Hamiltonian constraint of the Levi–Civita parallel case holds; this is described there as a rigidity statement (Shahbazi, 8 Jul 2025).

The gerbe viewpoint also clarifies the global geometry. Since isotropic parallelisms trivialize $$\nabla^{g,A}_w \varepsilon=\nabla^g_w \varepsilon-\frac12 A_w\cdot \varepsilon,$$7, the existence of a torsion parallel spinor implies parallelizability of $$\nabla^{g,A}_w \varepsilon=\nabla^g_w \varepsilon-\frac12 A_w\cdot \varepsilon,$$8. In the globally hyperbolic case with spacelike Cauchy surfaces, which are 3-manifolds, this topological condition becomes automatic. A plausible implication is that the higher-gauge formulation is especially well adapted to Lorentzian evolution problems, because both the torsion variable and the required global frame data are naturally compatible with spacetime slicing (Shahbazi, 8 Jul 2025).

6. Examples, obstructions, and connections with adjacent literatures

The four-dimensional dissertation gives explicit solution families. One class consists of standard conformally Brinkmann metrics with real Killing spinors, defined on $$\nabla^{g,A}_w \varepsilon=\nabla^g_w \varepsilon-\frac12 A_w\cdot \varepsilon,$$9, where each slice $(4,3)$00 is an elementary hyperbolic surface of scalar curvature $(4,3)$01, locally expressible as

$(4,3)$02

The resulting spacetimes generalize the Siklos class of exact AdS gravitational waves, and every Siklos metric appears as a special case when $(4,3)$03 are $(4,3)$04-independent and $(4,3)$05 is exact (Shahbazi, 8 Jul 2025).

A second family consists of axionic parallel spinors, for which the contorsion is pure volume-form coupling,

$(4,3)$06

The isotropic parallelism then satisfies

$(4,3)$07

$(4,3)$08

Any such solution is Brinkmann because $(4,3)$09, and these geometries define axionic Brinkmann four-manifolds (Shahbazi, 8 Jul 2025).

The principal obstruction theory is likewise explicit. In Lorentzian four dimensions, irreducible real spinors are necessarily null, so timelike parallel spinors do not arise. The stabilizer is noncompact, isomorphic to $(4,3)$10, and a torsion parallel spinor implies parallelizability of the manifold. These obstructions are specific to the Lorentzian four-dimensional irreducible real setting and distinguish it sharply from positive-definite skew-torsion spin geometry (Shahbazi, 8 Jul 2025).

That contrast is substantial. Riemannian work of Friedrich–Ivanov and Agricola and collaborators classifies parallel spinors with skew torsion in positive-definite signature, including nearly Kähler, $(4,3)$11, and $(4,3)$12 geometries. Canonical or characteristic connections with totally skew torsion admit torsion-parallel spinors on nearly parallel $(4,3)$13, nearly Kähler, Sasaki–Einstein, and 3–Sasakian manifolds (Harland et al., 2011). The cone correspondence with torsion shows that Killing spinors with skew torsion on $(4,3)$14-manifolds induce spinors parallel with respect to the characteristic connection of the cone, with explicit almost contact $(4,3)$15 almost Hermitian and $(4,3)$16 correspondences (Agricola et al., 2013). Under the additional condition $(4,3)$17, $(4,3)$18-parallel $(4,3)$19-eigenspinors become simultaneously Riemannian Killing spinors, Killing spinors with torsion, and twistor spinors with torsion, forcing Einstein and $(4,3)$20-Einstein geometry and yielding one-parameter families on nearly Kähler manifolds, nearly parallel $(4,3)$21-manifolds, and $(4,3)$22 (Chrysikos, 2015).

A separate four-dimensional spinorial treatment, formulated in two-component language for torsionful affinities, writes torsion-parallel spinors as solutions of $(4,3)$23 and derives integrability conditions requiring curvature annihilation along $(4,3)$24 together with vanishing torsional field strength $(4,3)$25 (Cardoso, 2014). Taken together with the Lorentzian spinorial-polyform approach, these results show that torsion parallelism is simultaneously a holonomy-reduction problem, a bilinear or form-theoretic differential system, and, in supergravity applications, a flux constraint encoded by higher gauge geometry.