Torsion Condensation in Gravity & Cosmology

• Torsion Condensation (TorC) is a family of theories where torsion acts as a dynamic geometric field influencing gravity, cosmic expansion, and observable phenomena.
• The framework covers diverse constructions including Poincaré-gauge cosmology, inflationary relics, topologically quantized torsion, and condensate mechanisms in fermionic and supersymmetric sectors.
• TorC models provide practical insights by linking modified Friedmann equations, effective four-fermion interactions, and defects in condensed-matter analogues with measurable astrophysical and quantum effects.

Torsion Condensation (TorC) is used in the literature in several closely related senses. In its most explicit cosmological form, it denotes a Poincaré-gauge extension of gravity in which intrinsic torsion is a dynamical degree of freedom and a torsion scalar field modifies the cosmic expansion history, while the theory reduces to $\Lambda$CDM at specific parameter values (Legner et al., 12 Jul 2025). Closely related work uses the same label, or motivates it implicitly, for mechanisms in which torsion remains relevant as the extrinsic geometry of spacelike slices after inflation, appears as a topologically quantized torsional texture, or generates effective condensates and perfect-square structures in fermionic and supersymmetric sectors (McInnes, 2024, Randono et al., 2010, Poplawski, 2011, Kallosh, 2019). This suggests that TorC is best understood as a family of torsion-dominated mechanisms rather than as a single universally fixed formalism.

1. Semantic range and unifying motif

The literature attaches the TorC label to several distinct constructions. In one branch, TorC is a concrete cosmological model in Poincaré gauge theory. In another, it denotes or motivates situations in which torsion becomes dynamically important through inflationary relics, topological winding, or condensate-induced effective interactions.

Usage	Defining structure	Representative source
Cosmological TorC in PGT	dynamical intrinsic torsion, torsion scalar $\varpi$, bare dark-energy parameter $\Omega_L$	(Legner et al., 12 Jul 2025)
Inflationary relic of extrinsic torsion	antisymmetric part of the second fundamental form	(McInnes, 2024)
Torsion-induced fermion condensate	four-fermion interaction from spin-torsion coupling	(Poplawski, 2011, Castillo-Felisola et al., 2013)
Topological torsion state	torsional monopole with integer-valued charge $Q$	(Randono et al., 2010)
Superspace geometric condensation	perfect square $Y_{abc}Y^{abc}$ from superspace torsion	(Kallosh, 2019)

Several papers relevant to TorC do not explicitly introduce the term. Instead, they formulate mechanisms in which torsion becomes condensed, coherent, quantized, or vacuum-relevant. This suggests a common motif: torsion is treated not merely as an auxiliary bookkeeping variable, but as a geometrical, topological, or effective field-theoretic structure that can organize low-energy observables or vacuum properties.

2. Poincaré-gauge cosmology and the explicit TorC model

In (Legner et al., 12 Jul 2025), TorC is a cosmological extension of general relativity built from Poincaré gauge theory. The gauge fields are a tetrad $e^i{}_\mu$ and an independent spin connection $\omega^{ij}{}_\mu$, with torsion and curvature field strengths

$$T^k{}_{ij}\equiv 2 e_i{}^\mu e_j{}^\nu\left(\partial_{[\mu} e^k{}_{\nu]}+\omega^k{}_{m[\mu}e^m{}_{\nu]}\right),$$

$$\mathcal{R}^{kl}{}_{ij}\equiv 2 e_i{}^\mu e_j{}^\nu\left(\partial_{[\mu}\omega^{kl}{}_{\nu]}+\omega^k{}_{m[\mu}\omega^{ml}{}_{\nu]}\right).$$

The TorC Lagrangian is a quadratic torsion/curvature theory. Rewritten schematically in metric/affine-connection language, it contains $\mu R^2$, $\varpi$0, $\varpi$1, $\varpi$2, $\varpi$3, and $\varpi$4. The paper identifies the quadratic and quartic torsion terms as the reason the dynamics favor a nonzero torsion condensate $\varpi$5, and compares this to a Higgs-style “sombrero” potential.

For homogeneous and isotropic cosmology, the theory is mapped to a scalar-tensor equivalent with fields $\varpi$6 and $\varpi$7,

$\varpi$8

$\varpi$9

The cosmological torsion tensor is written as

$\Omega_L$0

with $\Omega_L$1. The corresponding modified Friedmann equation is

$\Omega_L$2

and the torsion scalar satisfies

$\Omega_L$3

The two new cosmological parameters beyond standard $\Omega_L$4CDM are the bare dark-energy density $\Omega_L$5 and the early-Universe torsion scalar value $\Omega_L$6. The theory reproduces standard cosmology when

$\Omega_L$7

because then $\Omega_L$8 and the modified Friedmann equations reduce to the standard ones. More broadly, $\Omega_L$9 is an attractor, so standard cosmology emerges dynamically at late times.

The mechanism by which TorC can raise the inferred Hubble constant is explicitly early-Universe. Smaller $Q$0 reduces the comoving Hubble horizon at early times, reduces the sound horizon at recombination, shifts acoustic peaks to higher multipoles, and enhances early ISW driving. The paper finds a negative correlation between $Q$1 and $Q$2, and a positive correlation between $Q$3 and $Q$4. Constraints are obtained with the PolyChord nested sampling algorithm, interfaced via Cobaya with a modified version of CAMB, using Planck 2018 high-$Q$5 and low-$Q$6 TT, TE, EE, and lensing likelihoods together with the SH0ES 2020 measurement $Q$7. The reported consistency statistic changes from $Q$8 in $Q$9CDM to $Y_{abc}Y^{abc}$0 in TorC, but the Bayesian evidence still does not decisively favor TorC over $Y_{abc}Y^{abc}$1CDM.

3. Intrinsic versus extrinsic torsion and inflationary relics

A geometrically distinct TorC-related construction is developed in (McInnes, 2024). There the central distinction is between intrinsic torsion, meaning the torsion of the induced spatial $Y_{abc}Y^{abc}$2-geometry, and extrinsic torsion, meaning the antisymmetric part of the second fundamental form of a spacelike hypersurface embedded in a torsional spacetime. For a spacelike hypersurface $Y_{abc}Y^{abc}$3, the torsional Gauss–Codazzi relation is

$Y_{abc}Y^{abc}$4

and the extrinsic torsion is defined by

$Y_{abc}Y^{abc}$5

so that

$Y_{abc}Y^{abc}$6

A key consequence stated in the paper is that, unlike curvature, torsion cannot be “cancelled away” by choosing a different foliation, because the intrinsic and extrinsic pieces live in orthogonal components.

The same work argues that torsion is part of the actual spacetime geometry and enters the geodesic deviation equation directly: $Y_{abc}Y^{abc}$7 or equivalently

$Y_{abc}Y^{abc}$8

This implies that torsion can produce relative acceleration even if curvature is negligible, and that a rapidly varying torsion can produce large deviations even when torsion itself is small, a phenomenon described there as a “torsion escarpment” effect.

The cosmological proposal is that inflation makes intrinsic curvature tiny while making extrinsic curvature large, and that the same may happen with torsion. Intrinsic torsion may be driven to negligibility during inflation, but extrinsic torsion may grow during the inflationary era and remain non-negligible at reheating and later times. The second fundamental form is the central object because, in the torsion-free FRW case,

$Y_{abc}Y^{abc}$9

where $e^i{}_\mu$0, whereas with torsion

$e^i{}_\mu$1

so schematically

$e^i{}_\mu$2

with $e^i{}_\mu$3 the antisymmetric torsion-induced part. The symmetric part of $e^i{}_\mu$4 is controlled by Hubble expansion, while the antisymmetric part is the extrinsic torsion.

In the Einstein–Cartan example studied there, the generalized Friedmann constraint becomes

$e^i{}_\mu$5

and in the isotropic torsion parametrization,

$e^i{}_\mu$6

If the parameter $e^i{}_\mu$7 is ignored, one infers the wrong $e^i{}_\mu$8; the paper therefore identifies extrinsic torsion as a natural source of Hubble-parameter anomalies. It also notes an important caveat: for a highly symmetric FRW torsion ansatz, both intrinsic and extrinsic torsion can vanish identically, so the mechanism requires that the initial spacetime not be assumed too symmetric.

4. Torsion-induced condensates in fermionic and supersymmetric sectors

In Einstein–Cartan–Sciama–Kibble gravity, torsion is algebraically determined by the spin density of matter rather than propagating independently. Eliminating torsion from the action generates local fermion self-interactions. In (Poplawski, 2011), this leads to an effective Lagrangian containing axial–axial, vector–axial, and vector–vector terms,

$e^i{}_\mu$9

with an effective vacuum-energy contribution

$\omega^{ij}{}_\mu$0

and effective cosmological constant

$\omega^{ij}{}_\mu$1

Under the Shifman–Vainshtein–Zakharov vacuum-state-dominance approximation, quark condensation during the quark–gluon/hadron phase transition gives a positive vacuum expectation value, with scale

$\omega^{ij}{}_\mu$2

The paper states that the earlier pure ECSK analysis yielded a value about a factor of $\omega^{ij}{}_\mu$3 above the observed cosmological constant, and that the Holst/nonminimal-coupling factor $\omega^{ij}{}_\mu$4 can modify this result, with $\omega^{ij}{}_\mu$5 matching observation in the $\omega^{ij}{}_\mu$6 case.

A different mass-generation mechanism is proposed in (Castillo-Felisola et al., 2013). There, spacetime torsion in Cartan–Einstein gravity generates a universal local four-fermion contact interaction. In four dimensions the interaction is Planck-suppressed, but in a five-dimensional warped or compactified setting the effective gravitational scale can be lowered to $\omega^{ij}{}_\mu$7. The paper then assumes a fourth family of quarks condenses, and that condensate feeds masses to the ordinary quarks. In this construction, only the Lorentz- and color-singlet scalar channel $\omega^{ij}{}_\mu$8 condenses.

Supersymmetric and string-theoretic work gives a further geometric reinterpretation. In (Kallosh, 2019), gaugino condensation in ten dimensions is organized by the combination

$\omega^{ij}{}_\mu$9

with $$T^k{}_{ij}\equiv 2 e_i{}^\mu e_j{}^\nu\left(\partial_{[\mu} e^k{}_{\nu]}+\omega^k{}_{m[\mu}e^m{}_{\nu]}\right),$$0. The action contains the perfect square $$T^k{}_{ij}\equiv 2 e_i{}^\mu e_j{}^\nu\left(\partial_{[\mu} e^k{}_{\nu]}+\omega^k{}_{m[\mu}e^m{}_{\nu]}\right),$$1, and the paper states that this perfect square is the square of a superspace torsion tensor component at $$T^k{}_{ij}\equiv 2 e_i{}^\mu e_j{}^\nu\left(\partial_{[\mu} e^k{}_{\nu]}+\omega^k{}_{m[\mu}e^m{}_{\nu]}\right),$$2. The supercovariant gaugino equation of motion depends on the same combination,

$$T^k{}_{ij}\equiv 2 e_i{}^\mu e_j{}^\nu\left(\partial_{[\mu} e^k{}_{\nu]}+\omega^k{}_{m[\mu}e^m{}_{\nu]}\right),$$3

The paper does not explicitly define TorC, but it strongly motivates a torsion-based geometric condensation mechanism in which the condensate is absorbed into supertorsion.

5. Topological torsion states and condensed-matter realizations

A topological realization of TorC is constructed in (Randono et al., 2010) through torsional monopoles in “torqued geometries.” The setting is kinematical rather than dynamical: the manifold is topologically $$T^k{}_{ij}\equiv 2 e_i{}^\mu e_j{}^\nu\left(\partial_{[\mu} e^k{}_{\nu]}+\omega^k{}_{m[\mu}e^m{}_{\nu]}\right),$$4 with flat Minkowski metric induced by the tetrad, the spin connection satisfies $$T^k{}_{ij}\equiv 2 e_i{}^\mu e_j{}^\nu\left(\partial_{[\mu} e^k{}_{\nu]}+\omega^k{}_{m[\mu}e^m{}_{\nu]}\right),$$5, and asymptotically $$T^k{}_{ij}\equiv 2 e_i{}^\mu e_j{}^\nu\left(\partial_{[\mu} e^k{}_{\nu]}+\omega^k{}_{m[\mu}e^m{}_{\nu]}\right),$$6 approaches the Levi-Civita connection $$T^k{}_{ij}\equiv 2 e_i{}^\mu e_j{}^\nu\left(\partial_{[\mu} e^k{}_{\nu]}+\omega^k{}_{m[\mu}e^m{}_{\nu]}\right),$$7. Writing

$$T^k{}_{ij}\equiv 2 e_i{}^\mu e_j{}^\nu\left(\partial_{[\mu} e^k{}_{\nu]}+\omega^k{}_{m[\mu}e^m{}_{\nu]}\right),$$8

the relative gauge relation

$$T^k{}_{ij}\equiv 2 e_i{}^\mu e_j{}^\nu\left(\partial_{[\mu} e^k{}_{\nu]}+\omega^k{}_{m[\mu}e^m{}_{\nu]}\right),$$9

makes torsion a gauge-relative mismatch between connection and tetrad.

Flatness implies a conserved torsional current. The associated topological charge is

$$\mathcal{R}^{kl}{}_{ij}\equiv 2 e_i{}^\mu e_j{}^\nu\left(\partial_{[\mu}\omega^{kl}{}_{\nu]}+\omega^k{}_{m[\mu}\omega^{ml}{}_{\nu]}\right).$$0

In a convenient gauge,

$$\mathcal{R}^{kl}{}_{ij}\equiv 2 e_i{}^\mu e_j{}^\nu\left(\partial_{[\mu}\omega^{kl}{}_{\nu]}+\omega^k{}_{m[\mu}\omega^{ml}{}_{\nu]}\right).$$1

which is the winding number of a map $$\mathcal{R}^{kl}{}_{ij}\equiv 2 e_i{}^\mu e_j{}^\nu\left(\partial_{[\mu}\omega^{kl}{}_{\nu]}+\omega^k{}_{m[\mu}\omega^{ml}{}_{\nu]}\right).$$2. Because $$\mathcal{R}^{kl}{}_{ij}\equiv 2 e_i{}^\mu e_j{}^\nu\left(\partial_{[\mu}\omega^{kl}{}_{\nu]}+\omega^k{}_{m[\mu}\omega^{ml}{}_{\nu]}\right).$$3, the charge is localized, conserved, integer-valued, and topological. The paper gives an explicit single-monopole ansatz with $$\mathcal{R}^{kl}{}_{ij}\equiv 2 e_i{}^\mu e_j{}^\nu\left(\partial_{[\mu}\omega^{kl}{}_{\nu]}+\omega^k{}_{m[\mu}\omega^{ml}{}_{\nu]}\right).$$4, shows that multiple monopoles add by

$$\mathcal{R}^{kl}{}_{ij}\equiv 2 e_i{}^\mu e_j{}^\nu\left(\partial_{[\mu}\omega^{kl}{}_{\nu]}+\omega^k{}_{m[\mu}\omega^{ml}{}_{\nu]}\right).$$5

and interprets the construction as a generalization of Cartan’s spiral staircase. In this usage, TorC means a quantized topological condensate of torsion.

The same paper emphasizes that such objects are not ordinary disclinations or dislocations and argues that their condensed-matter realization would require micropolar (Cosserat) elasticity, where local rotational degrees of freedom are fundamental. A separate condensed-matter analogue appears in (Lima et al., 2017), where a uniform distribution of parallel screw dislocations along the $$\mathcal{R}^{kl}{}_{ij}\equiv 2 e_i{}^\mu e_j{}^\nu\left(\partial_{[\mu}\omega^{kl}{}_{\nu]}+\omega^k{}_{m[\mu}\omega^{ml}{}_{\nu]}\right).$$6-axis produces the effective metric

$$\mathcal{R}^{kl}{}_{ij}\equiv 2 e_i{}^\mu e_j{}^\nu\left(\partial_{[\mu}\omega^{kl}{}_{\nu]}+\omega^k{}_{m[\mu}\omega^{ml}{}_{\nu]}\right).$$7

with homogeneous axial torsion $$\mathcal{R}^{kl}{}_{ij}\equiv 2 e_i{}^\mu e_j{}^\nu\left(\partial_{[\mu}\omega^{kl}{}_{\nu]}+\omega^k{}_{m[\mu}\omega^{ml}{}_{\nu]}\right).$$8. The resulting dynamics exhibit Landau-like quantization,

$$\mathcal{R}^{kl}{}_{ij}\equiv 2 e_i{}^\mu e_j{}^\nu\left(\partial_{[\mu}\omega^{kl}{}_{\nu]}+\omega^k{}_{m[\mu}\omega^{ml}{}_{\nu]}\right).$$9

together with zero modes for spinless carriers and a charge-independent Zeeman-like splitting $\mu R^2$0 for spinful carriers. The paper does not use the TorC label, but it gives a concrete effective-medium torsion field generated by a defect distribution.

6. Empirical status, phenomenology, and conceptual divergences

The empirical status of torsion remains open rather than settled. The essay (Bonder, 2016) argues that the vanishing-torsion hypothesis has no direct empirical support and that all successful experimental tests of general relativity are compatible with nonzero torsion. In the Einstein–Cartan/Palatini setting used there, torsion is sourced algebraically by the axial current,

$\mu R^2$1

Because ordinary macroscopic matter typically has negligible net spin polarization, standard tests are largely insensitive. The paper therefore proposes spin-polarized probes: polarized neutron transmission through a polarized Holmium target, neutron spin-rotation experiments with spin-polarized liquid helium, and interferometry inside a spin-polarized medium.

Neutrino phenomenology provides a more recent torsion-signature program. In (Antonio et al., 30 May 2026), torsion is treated as a classical external axial background in flat spacetime with vanishing curvature,

$\mu R^2$2

For constant torsion aligned along the third axis, the effective masses become spin dependent,

$\mu R^2$3

and the oscillation amplitudes and phases inherit explicit spin dependence. For linearly time-dependent torsion, the mode functions acquire a quadratic-in-time phase. The paper further identifies spin-dependent condensate densities in the flavor vacuum and suggests that low-energy experiments such as PTOLEMY could be sensitive, whereas high-energy experiments such as DUNE are expected to be largely insensitive.

A recurrent source of ambiguity is that the dynamical status of torsion changes from one TorC-related framework to another. In ECSK-based condensation mechanisms torsion is an auxiliary field that is integrated out; in Poincaré-gauge TorC it is a dynamical intrinsic degree of freedom; in the intrinsic/extrinsic analysis it is part of the actual spacetime geometry and enters geodesic deviation directly. This difference is not a minor technicality, because it determines whether TorC is interpreted as a vacuum expectation value, a topological sector, an inflationary relic, or a late-time cosmological field. Some highly symmetric FRW torsion ansätze force both intrinsic and extrinsic torsion to vanish identically, so not every torsional cosmology realizes the inflationary mechanism (McInnes, 2024). Conversely, the explicit cosmological TorC model improves the Planck–SH0ES consistency measure but is not yet decisively favored over $\mu R^2$4CDM by Bayesian evidence (Legner et al., 12 Jul 2025).

Taken together, these constructions delimit the present meaning of TorC in the literature. It can denote a specific Poincaré-gauge cosmological model, but it also names or motivates a broader class of theories in which torsion becomes organized into a nonzero condensate, survives inflation as extrinsic geometry, appears as an integer-valued topological texture, or is encoded in fermionic and supersymmetric vacuum structure.