Four-Derivative Heterotic Supergravity

• Four-derivative heterotic supergravity is a higher-derivative extension of the ten-dimensional theory that incorporates leading α′ corrections via curvature-squared terms and modified three-forms.
• It employs a torsionful formulation where curvature, Chern-Simons modifications, and deformed supersymmetry work in tandem to ensure anomaly cancellation and duality covariance.
• Applications span corrected black-hole physics, non-Ricci-flat compactifications, and consistent truncations, revealing deep interplay between geometry and supersymmetry.

Searching arXiv for recent and foundational papers on four-derivative heterotic supergravity to support a comprehensive article. Four-derivative heterotic supergravity is the first higher-derivative extension of the ten-dimensional heterotic low-energy effective theory, retaining the leading $\mathcal O(\alpha')$ corrections beyond the two-derivative NS-NS and Yang-Mills sector. In the standard formulation, these corrections are encoded by curvature-squared terms, Chern-Simons modifications of the Kalb-Ramond three-form, and a corresponding deformation of supersymmetry. This sector is central to anomaly cancellation, torsionful geometry, duality-covariant reformulations, compactification, and corrected black-hole physics. Across recent work, it appears in several closely related guises: the Bergshoeff–de Roo torsionful completion in ten dimensions (Jayaprakash et al., 2024), T-duality-covariant heterotic Double Field Theory constructions (Lescano et al., 2021, Lescano et al., 2021), compactification and consistent truncation analyses (Liu et al., 2023), explicit nonsupersymmetric Minkowski vacua with non-Ricci-flat internal spaces (Tsuyuki, 2021), six-dimensional higher-derivative descendants and dualizations (Chang et al., 2023, Chang et al., 2022), and four-derivative corrected rotating heterotic black holes (Hu et al., 25 Jun 2025).

1. Ten-dimensional structure and the Bergshoeff–de Roo completion

In the ten-dimensional heterotic theory, the four-derivative sector is the leading correction to the bosonic Lagrangian and supersymmetry rules. One formulation keeps the Yang-Mills sector explicitly and uses

$$\mathcal L = \sqrt{-g}\, e^{-2\phi} \left( R + 4(\partial\phi)^2 - \frac{1}{12}H_{MNP}H^{MNP} + \frac{\alpha'}{4} R_{MNPQ}R^{MNPQ} - \frac{\alpha'}{4}\operatorname{tr}(F_{MN}F^{MN}) \right),$$

together with the heterotic Bianchi identity

$$dH=\frac{\alpha'}{4}\left(\operatorname{tr}R\wedge R-\operatorname{tr}F\wedge F\right),$$

which is the anomaly-cancellation condition used in flux compactifications (Tsuyuki, 2021).

A second, standard torsionful formulation truncates away the ten-dimensional gauge fields and writes the action in terms of the connection

$$\Omega_+=\omega+\frac12\tilde{\mathcal H}, \qquad \tilde H=H-\frac{\alpha'}{4}\omega_{3L}(\Omega_+),\qquad H=dB,$$

with bosonic action

$$e^{-1}\mathcal L_{\mathrm{het}}=e^{-2\phi}\Bigl(R+4(\partial\phi)^2-\frac{1}{12}\tilde H^2+\frac{\alpha'}{8}\bigl(R_{MNAB}(\Omega_+)\bigr)^2\Bigr).$$

Supersymmetry then uses the opposite torsionful connection,

$$\delta_\epsilon\psi_M=\nabla_M(\Omega_-)\epsilon,\qquad \delta_\epsilon\lambda=\left(\Gamma^M\partial_M\phi-\frac{1}{12}\tilde H_{MNP}\Gamma^{MNP}\right)\epsilon,$$

with $\Omega_-=\omega-\frac12\tilde{\mathcal H}$; the use of $\Omega_+$ in the action and $\Omega_-$ in the fermion variations is essential for supersymmetry (Jayaprakash et al., 2024).

This torsionful BdR structure reappears in several later developments. The torus-reduction analysis of higher-derivative heterotic supergravity adopts

$$e^{-1}\mathcal L = e^{-2\phi}\left[ R+4(\partial_M\phi)^2-\frac{1}{12}\tilde H_{MNP}^2 +\frac{\alpha'}{8}\big(R_{MNAB}(\Omega_+)\big)^2 \right] +\mathcal O(\alpha'^3),$$

with

$$\mathcal L = \sqrt{-g}\, e^{-2\phi} \left( R + 4(\partial\phi)^2 - \frac{1}{12}H_{MNP}H^{MNP} + \frac{\alpha'}{4} R_{MNPQ}R^{MNPQ} - \frac{\alpha'}{4}\operatorname{tr}(F_{MN}F^{MN}) \right),$$0

and corresponding bosonic equations of motion for the dilaton, metric, and three-form (Liu et al., 2023). The same general structure underlies the four-dimensional black-hole analysis, where the four-derivative term is the torsionful curvature-squared invariant accompanied by the Lorentz Chern-Simons modification of $$\mathcal L = \sqrt{-g}\, e^{-2\phi} \left( R + 4(\partial\phi)^2 - \frac{1}{12}H_{MNP}H^{MNP} + \frac{\alpha'}{4} R_{MNPQ}R^{MNPQ} - \frac{\alpha'}{4}\operatorname{tr}(F_{MN}F^{MN}) \right),$$1 (Hu et al., 25 Jun 2025).

A recurrent misconception is that the four-derivative sector is exhausted by an ordinary $$\mathcal L = \sqrt{-g}\, e^{-2\phi} \left( R + 4(\partial\phi)^2 - \frac{1}{12}H_{MNP}H^{MNP} + \frac{\alpha'}{4} R_{MNPQ}R^{MNPQ} - \frac{\alpha'}{4}\operatorname{tr}(F_{MN}F^{MN}) \right),$$2 deformation. The ten-dimensional heterotic completion is instead organized by torsionful curvature, Chern-Simons modifications, and deformed supersymmetry, so the curvature-squared term is inseparable from the Green-Schwarz structure in the standard heterotic scheme (Jayaprakash et al., 2024, Lescano et al., 2021).

2. Green-Schwarz mechanism, torsionful geometry, and supersymmetry

The Green-Schwarz mechanism is the structural principle that ties anomaly cancellation to the higher-derivative completion. In ordinary supergravity variables, the corrected three-form is written as

$$\mathcal L = \sqrt{-g}\, e^{-2\phi} \left( R + 4(\partial\phi)^2 - \frac{1}{12}H_{MNP}H^{MNP} + \frac{\alpha'}{4} R_{MNPQ}R^{MNPQ} - \frac{\alpha'}{4}\operatorname{tr}(F_{MN}F^{MN}) \right),$$3

or, in the notation of the T-duality-covariant construction,

$$\mathcal L = \sqrt{-g}\, e^{-2\phi} \left( R + 4(\partial\phi)^2 - \frac{1}{12}H_{MNP}H^{MNP} + \frac{\alpha'}{4} R_{MNPQ}R^{MNPQ} - \frac{\alpha'}{4}\operatorname{tr}(F_{MN}F^{MN}) \right),$$4

The two-form therefore acquires Lorentz and gauge Green-Schwarz shifts, and its supersymmetry variation is correspondingly deformed (Lescano et al., 2021).

The T-duality-covariant construction based on exact ten-dimensional $$\mathcal L = \sqrt{-g}\, e^{-2\phi} \left( R + 4(\partial\phi)^2 - \frac{1}{12}H_{MNP}H^{MNP} + \frac{\alpha'}{4} R_{MNPQ}R^{MNPQ} - \frac{\alpha'}{4}\operatorname{tr}(F_{MN}F^{MN}) \right),$$5 supergravity coupled to Yang-Mills makes this mechanism manifest in a generalized-geometry framework with global duality group

$$\mathcal L = \sqrt{-g}\, e^{-2\phi} \left( R + 4(\partial\phi)^2 - \frac{1}{12}H_{MNP}H^{MNP} + \frac{\alpha'}{4} R_{MNPQ}R^{MNPQ} - \frac{\alpha'}{4}\operatorname{tr}(F_{MN}F^{MN}) \right),$$6

and local double Lorentz symmetry

$$\mathcal L = \sqrt{-g}\, e^{-2\phi} \left( R + 4(\partial\phi)^2 - \frac{1}{12}H_{MNP}H^{MNP} + \frac{\alpha'}{4} R_{MNPQ}R^{MNPQ} - \frac{\alpha'}{4}\operatorname{tr}(F_{MN}F^{MN}) \right),$$7

Its central result is a supersymmetric and T-duality-covariant generalization of the Green-Schwarz mechanism that fixes the first-order $$\mathcal L = \sqrt{-g}\, e^{-2\phi} \left( R + 4(\partial\phi)^2 - \frac{1}{12}H_{MNP}H^{MNP} + \frac{\alpha'}{4} R_{MNPQ}R^{MNPQ} - \frac{\alpha'}{4}\operatorname{tr}(F_{MN}F^{MN}) \right),$$8 corrections to both the transformation rules and the invariant action (Lescano et al., 2021).

In heterotic Double Field Theory, the same logic appears as a generalized Green-Schwarz deformation of the frame,

$$\mathcal L = \sqrt{-g}\, e^{-2\phi} \left( R + 4(\partial\phi)^2 - \frac{1}{12}H_{MNP}H^{MNP} + \frac{\alpha'}{4} R_{MNPQ}R^{MNPQ} - \frac{\alpha'}{4}\operatorname{tr}(F_{MN}F^{MN}) \right),$$9

which the GKSA analysis identifies as the DFT counterpart of the heterotic anomalous $$dH=\frac{\alpha'}{4}\left(\operatorname{tr}R\wedge R-\operatorname{tr}F\wedge F\right),$$0-field transformation (Lescano et al., 2021). This DFT presentation is not merely a reformulation; it organizes the four-derivative sector directly in duality-covariant variables and shows how higher-curvature data arise geometrically from generalized fluxes (Lescano et al., 2021, Lescano et al., 2021).

Supersymmetry algebra closure is likewise deformed. The commutator of two supersymmetries still closes into generalized diffeomorphisms, local Lorentz transformations, and supersymmetry, but with field-dependent $$dH=\frac{\alpha'}{4}\left(\operatorname{tr}R\wedge R-\operatorname{tr}F\wedge F\right),$$1 corrections to the parameters,

$$dH=\frac{\alpha'}{4}\left(\operatorname{tr}R\wedge R-\operatorname{tr}F\wedge F\right),$$2

where $$dH=\frac{\alpha'}{4}\left(\operatorname{tr}R\wedge R-\operatorname{tr}F\wedge F\right),$$3, $$dH=\frac{\alpha'}{4}\left(\operatorname{tr}R\wedge R-\operatorname{tr}F\wedge F\right),$$4, and $$dH=\frac{\alpha'}{4}\left(\operatorname{tr}R\wedge R-\operatorname{tr}F\wedge F\right),$$5 each receive higher-derivative corrections built from generalized fluxes and fermionic curvatures (Lescano et al., 2021). This suggests that four-derivative heterotic supergravity is not merely a corrected action but a deformed local symmetry system.

3. Duality-covariant reduction and torus compactification

Compactification on a $$dH=\frac{\alpha'}{4}\left(\operatorname{tr}R\wedge R-\operatorname{tr}F\wedge F\right),$$6-torus turns ten-dimensional heterotic supergravity without gauge fields into half-maximal supergravity in $$dH=\frac{\alpha'}{4}\left(\operatorname{tr}R\wedge R-\operatorname{tr}F\wedge F\right),$$7 dimensions coupled to $$dH=\frac{\alpha'}{4}\left(\operatorname{tr}R\wedge R-\operatorname{tr}F\wedge F\right),$$8 vector multiplets (Jayaprakash et al., 2024). At two derivatives, the standard Kaluza-Klein ansatz is

$$dH=\frac{\alpha'}{4}\left(\operatorname{tr}R\wedge R-\operatorname{tr}F\wedge F\right),$$9

with $$\Omega_+=\omega+\frac12\tilde{\mathcal H}, \qquad \tilde H=H-\frac{\alpha'}{4}\omega_{3L}(\Omega_+),\qquad H=dB,$$0. The internal moduli reorganize into the $$\Omega_+=\omega+\frac12\tilde{\mathcal H}, \qquad \tilde H=H-\frac{\alpha'}{4}\omega_{3L}(\Omega_+),\qquad H=dB,$$1 scalar matrix

$$\Omega_+=\omega+\frac12\tilde{\mathcal H}, \qquad \tilde H=H-\frac{\alpha'}{4}\omega_{3L}(\Omega_+),\qquad H=dB,$$2

while the reduced scalar manifold is

$$\Omega_+=\omega+\frac12\tilde{\mathcal H}, \qquad \tilde H=H-\frac{\alpha'}{4}\omega_{3L}(\Omega_+),\qquad H=dB,$$3

A key result is that the continuous $$\Omega_+=\omega+\frac12\tilde{\mathcal H}, \qquad \tilde H=H-\frac{\alpha'}{4}\omega_{3L}(\Omega_+),\qquad H=dB,$$4 symmetry persists to all perturbative orders in the string $$\Omega_+=\omega+\frac12\tilde{\mathcal H}, \qquad \tilde H=H-\frac{\alpha'}{4}\omega_{3L}(\Omega_+),\qquad H=dB,$$5 expansion, and after suitable field redefinitions the reduced four-derivative action and supersymmetry variations become manifestly $$\Omega_+=\omega+\frac12\tilde{\mathcal H}, \qquad \tilde H=H-\frac{\alpha'}{4}\omega_{3L}(\Omega_+),\qquad H=dB,$$6 invariant (Jayaprakash et al., 2024).

In the generalized-vielbein basis, vector fields split into $$\Omega_+=\omega+\frac12\tilde{\mathcal H}, \qquad \tilde H=H-\frac{\alpha'}{4}\omega_{3L}(\Omega_+),\qquad H=dB,$$7 and $$\Omega_+=\omega+\frac12\tilde{\mathcal H}, \qquad \tilde H=H-\frac{\alpha'}{4}\omega_{3L}(\Omega_+),\qquad H=dB,$$8 components, and the Maurer-Cartan form decomposes into a composite connection $$\Omega_+=\omega+\frac12\tilde{\mathcal H}, \qquad \tilde H=H-\frac{\alpha'}{4}\omega_{3L}(\Omega_+),\qquad H=dB,$$9 and coset vielbein $$e^{-1}\mathcal L_{\mathrm{het}}=e^{-2\phi}\Bigl(R+4(\partial\phi)^2-\frac{1}{12}\tilde H^2+\frac{\alpha'}{8}\bigl(R_{MNAB}(\Omega_+)\bigr)^2\Bigr).$$0,

$$e^{-1}\mathcal L_{\mathrm{het}}=e^{-2\phi}\Bigl(R+4(\partial\phi)^2-\frac{1}{12}\tilde H^2+\frac{\alpha'}{8}\bigl(R_{MNAB}(\Omega_+)\bigr)^2\Bigr).$$1

In the conventions of the reduction, $$e^{-1}\mathcal L_{\mathrm{het}}=e^{-2\phi}\Bigl(R+4(\partial\phi)^2-\frac{1}{12}\tilde H^2+\frac{\alpha'}{8}\bigl(R_{MNAB}(\Omega_+)\bigr)^2\Bigr).$$2 is the supergravity $$e^{-1}\mathcal L_{\mathrm{het}}=e^{-2\phi}\Bigl(R+4(\partial\phi)^2-\frac{1}{12}\tilde H^2+\frac{\alpha'}{8}\bigl(R_{MNAB}(\Omega_+)\bigr)^2\Bigr).$$3-symmetry and $$e^{-1}\mathcal L_{\mathrm{het}}=e^{-2\phi}\Bigl(R+4(\partial\phi)^2-\frac{1}{12}\tilde H^2+\frac{\alpha'}{8}\bigl(R_{MNAB}(\Omega_+)\bigr)^2\Bigr).$$4 is a flavor symmetry of the $$e^{-1}\mathcal L_{\mathrm{het}}=e^{-2\phi}\Bigl(R+4(\partial\phi)^2-\frac{1}{12}\tilde H^2+\frac{\alpha'}{8}\bigl(R_{MNAB}(\Omega_+)\bigr)^2\Bigr).$$5 vector multiplets (Jayaprakash et al., 2024). This asymmetry is specific to the heterotic theory and is tied to the Lorentz Chern-Simons term; a bosonic-string analogue would not distinguish the two $$e^{-1}\mathcal L_{\mathrm{het}}=e^{-2\phi}\Bigl(R+4(\partial\phi)^2-\frac{1}{12}\tilde H^2+\frac{\alpha'}{8}\bigl(R_{MNAB}(\Omega_+)\bigr)^2\Bigr).$$6 factors in the same way (Jayaprakash et al., 2024).

The first pass of the reduction produces many terms not manifestly covariant under $$e^{-1}\mathcal L_{\mathrm{het}}=e^{-2\phi}\Bigl(R+4(\partial\phi)^2-\frac{1}{12}\tilde H^2+\frac{\alpha'}{8}\bigl(R_{MNAB}(\Omega_+)\bigr)^2\Bigr).$$7, including cross terms such as

$$e^{-1}\mathcal L_{\mathrm{het}}=e^{-2\phi}\Bigl(R+4(\partial\phi)^2-\frac{1}{12}\tilde H^2+\frac{\alpha'}{8}\bigl(R_{MNAB}(\Omega_+)\bigr)^2\Bigr).$$8

The crucial step is then a field redefinition of the internal metric/vielbein,

$$e^{-1}\mathcal L_{\mathrm{het}}=e^{-2\phi}\Bigl(R+4(\partial\phi)^2-\frac{1}{12}\tilde H^2+\frac{\alpha'}{8}\bigl(R_{MNAB}(\Omega_+)\bigr)^2\Bigr).$$9

with

$$\delta_\epsilon\psi_M=\nabla_M(\Omega_-)\epsilon,\qquad \delta_\epsilon\lambda=\left(\Gamma^M\partial_M\phi-\frac{1}{12}\tilde H_{MNP}\Gamma^{MNP}\right)\epsilon,$$0

which removes the non-invariant pieces and makes the action and supersymmetry rules manifestly covariant (Jayaprakash et al., 2024).

After redefinition, the four-derivative action is organized as

$$\delta_\epsilon\psi_M=\nabla_M(\Omega_-)\epsilon,\qquad \delta_\epsilon\lambda=\left(\Gamma^M\partial_M\phi-\frac{1}{12}\tilde H_{MNP}\Gamma^{MNP}\right)\epsilon,$$1

with $$\delta_\epsilon\psi_M=\nabla_M(\Omega_-)\epsilon,\qquad \delta_\epsilon\lambda=\left(\Gamma^M\partial_M\phi-\frac{1}{12}\tilde H_{MNP}\Gamma^{MNP}\right)\epsilon,$$2 pure gravity-multiplet terms, $$\delta_\epsilon\psi_M=\nabla_M(\Omega_-)\epsilon,\qquad \delta_\epsilon\lambda=\left(\Gamma^M\partial_M\phi-\frac{1}{12}\tilde H_{MNP}\Gamma^{MNP}\right)\epsilon,$$3 pure vector-multiplet terms, and $$\delta_\epsilon\psi_M=\nabla_M(\Omega_-)\epsilon,\qquad \delta_\epsilon\lambda=\left(\Gamma^M\partial_M\phi-\frac{1}{12}\tilde H_{MNP}\Gamma^{MNP}\right)\epsilon,$$4 mixed couplings (Jayaprakash et al., 2024). This decomposition is structurally important: it shows that higher derivatives do not destroy the lower-dimensional multiplet splitting, but they do induce controlled gravity/vector-multiplet mixing fixed by the coset geometry and composite connection.

4. Consistent truncations and lower-dimensional effective theories

Torus reduction generically produces extra vector multiplets, and a central question is whether they can be consistently removed once four-derivative corrections are included. The consistent truncation analysis shows that they can (Liu et al., 2023).

At leading order, the bosonic supersymmetric truncation is

$$\delta_\epsilon\psi_M=\nabla_M(\Omega_-)\epsilon,\qquad \delta_\epsilon\lambda=\left(\Gamma^M\partial_M\phi-\frac{1}{12}\tilde H_{MNP}\Gamma^{MNP}\right)\epsilon,$$5

equivalently

$$\delta_\epsilon\psi_M=\nabla_M(\Omega_-)\epsilon,\qquad \delta_\epsilon\lambda=\left(\Gamma^M\partial_M\phi-\frac{1}{12}\tilde H_{MNP}\Gamma^{MNP}\right)\epsilon,$$6

The resulting two-derivative Lagrangian is

$$\delta_\epsilon\psi_M=\nabla_M(\Omega_-)\epsilon,\qquad \delta_\epsilon\lambda=\left(\Gamma^M\partial_M\phi-\frac{1}{12}\tilde H_{MNP}\Gamma^{MNP}\right)\epsilon,$$7

with

$$\delta_\epsilon\psi_M=\nabla_M(\Omega_-)\epsilon,\qquad \delta_\epsilon\lambda=\left(\Gamma^M\partial_M\phi-\frac{1}{12}\tilde H_{MNP}\Gamma^{MNP}\right)\epsilon,$$8

This retains the lower-dimensional gravity multiplet plus graviphotons (Liu et al., 2023).

At $$\delta_\epsilon\psi_M=\nabla_M(\Omega_-)\epsilon,\qquad \delta_\epsilon\lambda=\left(\Gamma^M\partial_M\phi-\frac{1}{12}\tilde H_{MNP}\Gamma^{MNP}\right)\epsilon,$$9, the truncation requires corrections: $\Omega_-=\omega-\frac12\tilde{\mathcal H}$0 with the key choice

$\Omega_-=\omega-\frac12\tilde{\mathcal H}$1

together with

$\Omega_-=\omega-\frac12\tilde{\mathcal H}$2

These shifts cancel the four-derivative sources in the vector-multiplet equations of motion (Liu et al., 2023). The reduced truncated action then acquires curvature, $\Omega_-=\omega-\frac12\tilde{\mathcal H}$3, $\Omega_-=\omega-\frac12\tilde{\mathcal H}$4, and quartic $\Omega_-=\omega-\frac12\tilde{\mathcal H}$5 terms, while the three-form becomes

$\Omega_-=\omega-\frac12\tilde{\mathcal H}$6

with

$\Omega_-=\omega-\frac12\tilde{\mathcal H}$7

The fermionic consistency check is equally nontrivial: the gaugini require the redefinition

$\Omega_-=\omega-\frac12\tilde{\mathcal H}$8

after which

$\Omega_-=\omega-\frac12\tilde{\mathcal H}$9

so the truncation is compatible with the Killing spinor equations (Liu et al., 2023).

A plausible implication is that four-derivative consistent truncation is highly scheme-sensitive: the lower-dimensional field content survives only after $\Omega_+$0-dependent redefinitions of both bosons and fermions.

5. Compactification beyond Ricci-flatness

One of the most concrete applications of the four-derivative heterotic action is the construction of four-dimensional Minkowski vacua with curved internal manifolds that are not Ricci-flat (Tsuyuki, 2021). The starting point is ten-dimensional heterotic supergravity with constant dilaton and $\Omega_+$1, for which the remaining equations reduce to

$\Omega_+$2

$\Omega_+$3

$\Omega_+$4

An important point is that the trace of the Einstein equation implies

$\Omega_+$5

but this does not imply Ricci flatness $\Omega_+$6 (Tsuyuki, 2021). Positive and negative curvature components can cancel in the scalar curvature, opening a route to Minkowski compactifications with curved internal spaces.

The compactification ansatz takes

$\Omega_+$7

where $\Omega_+$8 is four-dimensional Minkowski spacetime and each $\Omega_+$9 is a two-dimensional space of constant sectional curvature,

$\Omega_-$0

Depending on the sign of $\Omega_-$1, the factor is $\Omega_-$2, $\Omega_-$3, or a compact hyperbolic surface $\Omega_-$4 (Tsuyuki, 2021). Gauge flux is chosen block-diagonally with Freund-Rubin-like form

$\Omega_-$5

which automatically solves the gauge equation and permits anomaly cancellation without the standard embedding (Tsuyuki, 2021).

The equations reduce to algebraic conditions,

$\Omega_-$6

Flux quantization and the Gauss-Bonnet theorem enforce discreteness,

$\Omega_-$7

implying

$\Omega_-$8

The allowed solutions fall into exactly three classes (Tsuyuki, 2021).

Internal space	Curvature balance	Noted feature
$\Omega_-$9	$$e^{-1}\mathcal L = e^{-2\phi}\left[ R+4(\partial_M\phi)^2-\frac{1}{12}\tilde H_{MNP}^2 +\frac{\alpha'}{8}\big(R_{MNAB}(\Omega_+)\big)^2 \right] +\mathcal O(\alpha'^3),$$0	minimal nonzero flux uses two $$e^{-1}\mathcal L = e^{-2\phi}\left[ R+4(\partial_M\phi)^2-\frac{1}{12}\tilde H_{MNP}^2 +\frac{\alpha'}{8}\big(R_{MNAB}(\Omega_+)\big)^2 \right] +\mathcal O(\alpha'^3),$$1 fluxes
$$e^{-1}\mathcal L = e^{-2\phi}\left[ R+4(\partial_M\phi)^2-\frac{1}{12}\tilde H_{MNP}^2 +\frac{\alpha'}{8}\big(R_{MNAB}(\Omega_+)\big)^2 \right] +\mathcal O(\alpha'^3),$$2	$$e^{-1}\mathcal L = e^{-2\phi}\left[ R+4(\partial_M\phi)^2-\frac{1}{12}\tilde H_{MNP}^2 +\frac{\alpha'}{8}\big(R_{MNAB}(\Omega_+)\big)^2 \right] +\mathcal O(\alpha'^3),$$3	no flat torus is required
$$e^{-1}\mathcal L = e^{-2\phi}\left[ R+4(\partial_M\phi)^2-\frac{1}{12}\tilde H_{MNP}^2 +\frac{\alpha'}{8}\big(R_{MNAB}(\Omega_+)\big)^2 \right] +\mathcal O(\alpha'^3),$$4	$$e^{-1}\mathcal L = e^{-2\phi}\left[ R+4(\partial_M\phi)^2-\frac{1}{12}\tilde H_{MNP}^2 +\frac{\alpha'}{8}\big(R_{MNAB}(\Omega_+)\big)^2 \right] +\mathcal O(\alpha'^3),$$5	at least three flux components are needed

Representative examples include $$e^{-1}\mathcal L = e^{-2\phi}\left[ R+4(\partial_M\phi)^2-\frac{1}{12}\tilde H_{MNP}^2 +\frac{\alpha'}{8}\big(R_{MNAB}(\Omega_+)\big)^2 \right] +\mathcal O(\alpha'^3),$$6 for $$e^{-1}\mathcal L = e^{-2\phi}\left[ R+4(\partial_M\phi)^2-\frac{1}{12}\tilde H_{MNP}^2 +\frac{\alpha'}{8}\big(R_{MNAB}(\Omega_+)\big)^2 \right] +\mathcal O(\alpha'^3),$$7, and $$e^{-1}\mathcal L = e^{-2\phi}\left[ R+4(\partial_M\phi)^2-\frac{1}{12}\tilde H_{MNP}^2 +\frac{\alpha'}{8}\big(R_{MNAB}(\Omega_+)\big)^2 \right] +\mathcal O(\alpha'^3),$$8 or $$e^{-1}\mathcal L = e^{-2\phi}\left[ R+4(\partial_M\phi)^2-\frac{1}{12}\tilde H_{MNP}^2 +\frac{\alpha'}{8}\big(R_{MNAB}(\Omega_+)\big)^2 \right] +\mathcal O(\alpha'^3),$$9 for $$\mathcal L = \sqrt{-g}\, e^{-2\phi} \left( R + 4(\partial\phi)^2 - \frac{1}{12}H_{MNP}H^{MNP} + \frac{\alpha'}{4} R_{MNPQ}R^{MNPQ} - \frac{\alpha'}{4}\operatorname{tr}(F_{MN}F^{MN}) \right),$$00 (Tsuyuki, 2021). These are nonsupersymmetric, non-Ricci-flat Minkowski vacua whose existence relies essentially on the curvature-squared term. Without the $$\mathcal L = \sqrt{-g}\, e^{-2\phi} \left( R + 4(\partial\phi)^2 - \frac{1}{12}H_{MNP}H^{MNP} + \frac{\alpha'}{4} R_{MNPQ}R^{MNPQ} - \frac{\alpha'}{4}\operatorname{tr}(F_{MN}F^{MN}) \right),$$01 contribution, the sign structure preventing such positive/negative curvature balance would remain (Tsuyuki, 2021).

6. Six-dimensional descendants, dualization, and independent invariants

In six dimensions, the four-derivative heterotic sector exhibits a richer invariant structure than in ten dimensions. For $$\mathcal L = \sqrt{-g}\, e^{-2\phi} \left( R + 4(\partial\phi)^2 - \frac{1}{12}H_{MNP}H^{MNP} + \frac{\alpha'}{4} R_{MNPQ}R^{MNPQ} - \frac{\alpha'}{4}\operatorname{tr}(F_{MN}F^{MN}) \right),$$02 supergravity coupled to a tensor multiplet, there are two known off-shell supersymmetric invariants at four derivatives: a Riemann-squared invariant $$\mathcal L = \sqrt{-g}\, e^{-2\phi} \left( R + 4(\partial\phi)^2 - \frac{1}{12}H_{MNP}H^{MNP} + \frac{\alpha'}{4} R_{MNPQ}R^{MNPQ} - \frac{\alpha'}{4}\operatorname{tr}(F_{MN}F^{MN}) \right),$$03 and a Gauss-Bonnet invariant $$\mathcal L = \sqrt{-g}\, e^{-2\phi} \left( R + 4(\partial\phi)^2 - \frac{1}{12}H_{MNP}H^{MNP} + \frac{\alpha'}{4} R_{MNPQ}R^{MNPQ} - \frac{\alpha'}{4}\operatorname{tr}(F_{MN}F^{MN}) \right),$$04 (Chang et al., 2022). The general two-parameter theory is

$$\mathcal L = \sqrt{-g}\, e^{-2\phi} \left( R + 4(\partial\phi)^2 - \frac{1}{12}H_{MNP}H^{MNP} + \frac{\alpha'}{4} R_{MNPQ}R^{MNPQ} - \frac{\alpha'}{4}\operatorname{tr}(F_{MN}F^{MN}) \right),$$05

A key result is that these two invariants are not related by field redefinitions. Although some $$\mathcal L = \sqrt{-g}\, e^{-2\phi} \left( R + 4(\partial\phi)^2 - \frac{1}{12}H_{MNP}H^{MNP} + \frac{\alpha'}{4} R_{MNPQ}R^{MNPQ} - \frac{\alpha'}{4}\operatorname{tr}(F_{MN}F^{MN}) \right),$$06 and $$\mathcal L = \sqrt{-g}\, e^{-2\phi} \left( R + 4(\partial\phi)^2 - \frac{1}{12}H_{MNP}H^{MNP} + \frac{\alpha'}{4} R_{MNPQ}R^{MNPQ} - \frac{\alpha'}{4}\operatorname{tr}(F_{MN}F^{MN}) \right),$$07 terms can be shifted away, the difference still contains structures such as

$$\mathcal L = \sqrt{-g}\, e^{-2\phi} \left( R + 4(\partial\phi)^2 - \frac{1}{12}H_{MNP}H^{MNP} + \frac{\alpha'}{4} R_{MNPQ}R^{MNPQ} - \frac{\alpha'}{4}\operatorname{tr}(F_{MN}F^{MN}) \right),$$08

so the invariants remain genuinely distinct (Chang et al., 2022).

The same paper develops direct action-level dualization with a Lagrange multiplier,

$$\mathcal L = \sqrt{-g}\, e^{-2\phi} \left( R + 4(\partial\phi)^2 - \frac{1}{12}H_{MNP}H^{MNP} + \frac{\alpha'}{4} R_{MNPQ}R^{MNPQ} - \frac{\alpha'}{4}\operatorname{tr}(F_{MN}F^{MN}) \right),$$09

treating $$\mathcal L = \sqrt{-g}\, e^{-2\phi} \left( R + 4(\partial\phi)^2 - \frac{1}{12}H_{MNP}H^{MNP} + \frac{\alpha'}{4} R_{MNPQ}R^{MNPQ} - \frac{\alpha'}{4}\operatorname{tr}(F_{MN}F^{MN}) \right),$$10 as independent and obtaining a dual two-form theory with field strength

$$\mathcal L = \sqrt{-g}\, e^{-2\phi} \left( R + 4(\partial\phi)^2 - \frac{1}{12}H_{MNP}H^{MNP} + \frac{\alpha'}{4} R_{MNPQ}R^{MNPQ} - \frac{\alpha'}{4}\operatorname{tr}(F_{MN}F^{MN}) \right),$$11

The special case $$\mathcal L = \sqrt{-g}\, e^{-2\phi} \left( R + 4(\partial\phi)^2 - \frac{1}{12}H_{MNP}H^{MNP} + \frac{\alpha'}{4} R_{MNPQ}R^{MNPQ} - \frac{\alpha'}{4}\operatorname{tr}(F_{MN}F^{MN}) \right),$$12 gives the dual of the BdR-type combination; $$\mathcal L = \sqrt{-g}\, e^{-2\phi} \left( R + 4(\partial\phi)^2 - \frac{1}{12}H_{MNP}H^{MNP} + \frac{\alpha'}{4} R_{MNPQ}R^{MNPQ} - \frac{\alpha'}{4}\operatorname{tr}(F_{MN}F^{MN}) \right),$$13 yields a different higher-derivative extension whose curvature-squared sector can be removed by field redefinitions at the cost of more complicated supersymmetry rules (Chang et al., 2022).

In ten dimensions, by contrast, dualizing the BdR action does not produce an analogous clean split into two independent invariants. The dualized result contains terms such as

$$\mathcal L = \sqrt{-g}\, e^{-2\phi} \left( R + 4(\partial\phi)^2 - \frac{1}{12}H_{MNP}H^{MNP} + \frac{\alpha'}{4} R_{MNPQ}R^{MNPQ} - \frac{\alpha'}{4}\operatorname{tr}(F_{MN}F^{MN}) \right),$$14

and the paper identifies an obstacle to separating the result into a sum of two independent invariants because some terms do not lift from $$\mathcal L = \sqrt{-g}\, e^{-2\phi} \left( R + 4(\partial\phi)^2 - \frac{1}{12}H_{MNP}H^{MNP} + \frac{\alpha'}{4} R_{MNPQ}R^{MNPQ} - \frac{\alpha'}{4}\operatorname{tr}(F_{MN}F^{MN}) \right),$$15 to $$\mathcal L = \sqrt{-g}\, e^{-2\phi} \left( R + 4(\partial\phi)^2 - \frac{1}{12}H_{MNP}H^{MNP} + \frac{\alpha'}{4} R_{MNPQ}R^{MNPQ} - \frac{\alpha'}{4}\operatorname{tr}(F_{MN}F^{MN}) \right),$$16 (Chang et al., 2022). This underpins the view that the ten-dimensional BdR completion is much closer to unique than its six-dimensional descendants.

A related six-dimensional development constructs the four-derivative supersymmetric extension of $$\mathcal L = \sqrt{-g}\, e^{-2\phi} \left( R + 4(\partial\phi)^2 - \frac{1}{12}H_{MNP}H^{MNP} + \frac{\alpha'}{4} R_{MNPQ}R^{MNPQ} - \frac{\alpha'}{4}\operatorname{tr}(F_{MN}F^{MN}) \right),$$17 supergravity coupled to Yang-Mills and hypermultiplets on

$$\mathcal L = \sqrt{-g}\, e^{-2\phi} \left( R + 4(\partial\phi)^2 - \frac{1}{12}H_{MNP}H^{MNP} + \frac{\alpha'}{4} R_{MNPQ}R^{MNPQ} - \frac{\alpha'}{4}\operatorname{tr}(F_{MN}F^{MN}) \right),$$18

with a new independent coupling $$\mathcal L = \sqrt{-g}\, e^{-2\phi} \left( R + 4(\partial\phi)^2 - \frac{1}{12}H_{MNP}H^{MNP} + \frac{\alpha'}{4} R_{MNPQ}R^{MNPQ} - \frac{\alpha'}{4}\operatorname{tr}(F_{MN}F^{MN}) \right),$$19 governing higher-derivative hypermultiplet interactions (Chang et al., 2023). The hyperscalar kinetic term

$$\mathcal L = \sqrt{-g}\, e^{-2\phi} \left( R + 4(\partial\phi)^2 - \frac{1}{12}H_{MNP}H^{MNP} + \frac{\alpha'}{4} R_{MNPQ}R^{MNPQ} - \frac{\alpha'}{4}\operatorname{tr}(F_{MN}F^{MN}) \right),$$20

is not deformed, so the quaternionic Kähler geometry is preserved (Chang et al., 2023). In the special $$\mathcal L = \sqrt{-g}\, e^{-2\phi} \left( R + 4(\partial\phi)^2 - \frac{1}{12}H_{MNP}H^{MNP} + \frac{\alpha'}{4} R_{MNPQ}R^{MNPQ} - \frac{\alpha'}{4}\operatorname{tr}(F_{MN}F^{MN}) \right),$$21 case, comparison with heterotic supergravity compactified on $$\mathcal L = \sqrt{-g}\, e^{-2\phi} \left( R + 4(\partial\phi)^2 - \frac{1}{12}H_{MNP}H^{MNP} + \frac{\alpha'}{4} R_{MNPQ}R^{MNPQ} - \frac{\alpha'}{4}\operatorname{tr}(F_{MN}F^{MN}) \right),$$22 shows agreement for $$\mathcal L = \sqrt{-g}\, e^{-2\phi} \left( R + 4(\partial\phi)^2 - \frac{1}{12}H_{MNP}H^{MNP} + \frac{\alpha'}{4} R_{MNPQ}R^{MNPQ} - \frac{\alpha'}{4}\operatorname{tr}(F_{MN}F^{MN}) \right),$$23 or $$\mathcal L = \sqrt{-g}\, e^{-2\phi} \left( R + 4(\partial\phi)^2 - \frac{1}{12}H_{MNP}H^{MNP} + \frac{\alpha'}{4} R_{MNPQ}R^{MNPQ} - \frac{\alpha'}{4}\operatorname{tr}(F_{MN}F^{MN}) \right),$$24, depending on how $$\mathcal L = \sqrt{-g}\, e^{-2\phi} \left( R + 4(\partial\phi)^2 - \frac{1}{12}H_{MNP}H^{MNP} + \frac{\alpha'}{4} R_{MNPQ}R^{MNPQ} - \frac{\alpha'}{4}\operatorname{tr}(F_{MN}F^{MN}) \right),$$25 is embedded (Chang et al., 2023). This suggests that four-derivative heterotic descendants can preserve target-space geometry while still admitting new independent couplings not fixed solely by the BdR invariant.

7. Black holes, observables, and broader extensions

Four-derivative heterotic supergravity has recently been used to obtain explicit corrections to the Kerr-Sen black hole (Hu et al., 25 Jun 2025). The starting point is the four-dimensional string-frame BdR action

$$\mathcal L = \sqrt{-g}\, e^{-2\phi} \left( R + 4(\partial\phi)^2 - \frac{1}{12}H_{MNP}H^{MNP} + \frac{\alpha'}{4} R_{MNPQ}R^{MNPQ} - \frac{\alpha'}{4}\operatorname{tr}(F_{MN}F^{MN}) \right),$$26

with

$$\mathcal L = \sqrt{-g}\, e^{-2\phi} \left( R + 4(\partial\phi)^2 - \frac{1}{12}H_{MNP}H^{MNP} + \frac{\alpha'}{4} R_{MNPQ}R^{MNPQ} - \frac{\alpha'}{4}\operatorname{tr}(F_{MN}F^{MN}) \right),$$27

After dualizing $$\mathcal L = \sqrt{-g}\, e^{-2\phi} \left( R + 4(\partial\phi)^2 - \frac{1}{12}H_{MNP}H^{MNP} + \frac{\alpha'}{4} R_{MNPQ}R^{MNPQ} - \frac{\alpha'}{4}\operatorname{tr}(F_{MN}F^{MN}) \right),$$28 to an axion $$\mathcal L = \sqrt{-g}\, e^{-2\phi} \left( R + 4(\partial\phi)^2 - \frac{1}{12}H_{MNP}H^{MNP} + \frac{\alpha'}{4} R_{MNPQ}R^{MNPQ} - \frac{\alpha'}{4}\operatorname{tr}(F_{MN}F^{MN}) \right),$$29, the Einstein-frame two-derivative sector becomes

$$\mathcal L = \sqrt{-g}\, e^{-2\phi} \left( R + 4(\partial\phi)^2 - \frac{1}{12}H_{MNP}H^{MNP} + \frac{\alpha'}{4} R_{MNPQ}R^{MNPQ} - \frac{\alpha'}{4}\operatorname{tr}(F_{MN}F^{MN}) \right),$$30

while the $$\mathcal L = \sqrt{-g}\, e^{-2\phi} \left( R + 4(\partial\phi)^2 - \frac{1}{12}H_{MNP}H^{MNP} + \frac{\alpha'}{4} R_{MNPQ}R^{MNPQ} - \frac{\alpha'}{4}\operatorname{tr}(F_{MN}F^{MN}) \right),$$31 correction is an explicit functional of $$\mathcal L = \sqrt{-g}\, e^{-2\phi} \left( R + 4(\partial\phi)^2 - \frac{1}{12}H_{MNP}H^{MNP} + \frac{\alpha'}{4} R_{MNPQ}R^{MNPQ} - \frac{\alpha'}{4}\operatorname{tr}(F_{MN}F^{MN}) \right),$$32, $$\mathcal L = \sqrt{-g}\, e^{-2\phi} \left( R + 4(\partial\phi)^2 - \frac{1}{12}H_{MNP}H^{MNP} + \frac{\alpha'}{4} R_{MNPQ}R^{MNPQ} - \frac{\alpha'}{4}\operatorname{tr}(F_{MN}F^{MN}) \right),$$33, and the axion current $$\mathcal L = \sqrt{-g}\, e^{-2\phi} \left( R + 4(\partial\phi)^2 - \frac{1}{12}H_{MNP}H^{MNP} + \frac{\alpha'}{4} R_{MNPQ}R^{MNPQ} - \frac{\alpha'}{4}\operatorname{tr}(F_{MN}F^{MN}) \right),$$34 (Hu et al., 25 Jun 2025).

The Kerr solution can be embedded with $$\mathcal L = \sqrt{-g}\, e^{-2\phi} \left( R + 4(\partial\phi)^2 - \frac{1}{12}H_{MNP}H^{MNP} + \frac{\alpha'}{4} R_{MNPQ}R^{MNPQ} - \frac{\alpha'}{4}\operatorname{tr}(F_{MN}F^{MN}) \right),$$35 at zeroth order, and an important structural feature is that in Einstein frame the Kerr metric itself is uncorrected at this order; only the scalars acquire hair (Hu et al., 25 Jun 2025). The Kerr-Sen solution is then generated by an $$\mathcal L = \sqrt{-g}\, e^{-2\phi} \left( R + 4(\partial\phi)^2 - \frac{1}{12}H_{MNP}H^{MNP} + \frac{\alpha'}{4} R_{MNPQ}R^{MNPQ} - \frac{\alpha'}{4}\operatorname{tr}(F_{MN}F^{MN}) \right),$$36 boost, but only after field redefinitions that restore manifest higher-derivative hidden symmetry in the reduced action (Hu et al., 25 Jun 2025). This requirement reflects the same general theme seen in torus compactification: duality symmetry survives at four derivatives, but only in an appropriate field basis.

The corrected multipole moments are a particularly sharp result. At two derivatives, Kerr-Sen, Kerr, and Kerr-Newman have coincident gravitational multipoles, but at four derivatives the heterotic solution develops distinct corrections such as

$$\mathcal L = \sqrt{-g}\, e^{-2\phi} \left( R + 4(\partial\phi)^2 - \frac{1}{12}H_{MNP}H^{MNP} + \frac{\alpha'}{4} R_{MNPQ}R^{MNPQ} - \frac{\alpha'}{4}\operatorname{tr}(F_{MN}F^{MN}) \right),$$37

$$\mathcal L = \sqrt{-g}\, e^{-2\phi} \left( R + 4(\partial\phi)^2 - \frac{1}{12}H_{MNP}H^{MNP} + \frac{\alpha'}{4} R_{MNPQ}R^{MNPQ} - \frac{\alpha'}{4}\operatorname{tr}(F_{MN}F^{MN}) \right),$$38

$$\mathcal L = \sqrt{-g}\, e^{-2\phi} \left( R + 4(\partial\phi)^2 - \frac{1}{12}H_{MNP}H^{MNP} + \frac{\alpha'}{4} R_{MNPQ}R^{MNPQ} - \frac{\alpha'}{4}\operatorname{tr}(F_{MN}F^{MN}) \right),$$39

$$\mathcal L = \sqrt{-g}\, e^{-2\phi} \left( R + 4(\partial\phi)^2 - \frac{1}{12}H_{MNP}H^{MNP} + \frac{\alpha'}{4} R_{MNPQ}R^{MNPQ} - \frac{\alpha'}{4}\operatorname{tr}(F_{MN}F^{MN}) \right),$$40

showing that the four-derivative heterotic corrections are distinguishable from both Kerr and Kerr-Newman, even allowing the most general choice of four-derivative Einstein-Maxwell corrections (Hu et al., 25 Jun 2025).

Beyond relativistic backgrounds, the four-derivative gravitational corrections have also been extended to the non-relativistic heterotic limit. There, the BdR identification is adapted to Newton-Cartan/string-Newton-Cartan geometry, and the gauge connection is identified with a torsionful $$\mathcal L = \sqrt{-g}\, e^{-2\phi} \left( R + 4(\partial\phi)^2 - \frac{1}{12}H_{MNP}H^{MNP} + \frac{\alpha'}{4} R_{MNPQ}R^{MNPQ} - \frac{\alpha'}{4}\operatorname{tr}(F_{MN}F^{MN}) \right),$$41 spin connection,

$$\mathcal L = \sqrt{-g}\, e^{-2\phi} \left( R + 4(\partial\phi)^2 - \frac{1}{12}H_{MNP}H^{MNP} + \frac{\alpha'}{4} R_{MNPQ}R^{MNPQ} - \frac{\alpha'}{4}\operatorname{tr}(F_{MN}F^{MN}) \right),$$42

A field redefinition

$$\mathcal L = \sqrt{-g}\, e^{-2\phi} \left( R + 4(\partial\phi)^2 - \frac{1}{12}H_{MNP}H^{MNP} + \frac{\alpha'}{4} R_{MNPQ}R^{MNPQ} - \frac{\alpha'}{4}\operatorname{tr}(F_{MN}F^{MN}) \right),$$43

then exposes an $$\mathcal L = \sqrt{-g}\, e^{-2\phi} \left( R + 4(\partial\phi)^2 - \frac{1}{12}H_{MNP}H^{MNP} + \frac{\alpha'}{4} R_{MNPQ}R^{MNPQ} - \frac{\alpha'}{4}\operatorname{tr}(F_{MN}F^{MN}) \right),$$44 Green-Schwarz mechanism in the non-relativistic theory (Lescano, 12 Aug 2025). This mechanism is trivializable by field redefinitions, unlike the relativistic Lorentz Green-Schwarz shift, and the resulting higher-curvature completion is finite and symmetry-consistent (Lescano, 12 Aug 2025). This suggests that the organizing role of BdR geometry and Green-Schwarz structure extends beyond relativistic supergravity.

Another direction is exact four-derivative couplings in lower-dimensional heterotic CHL orbifolds, where a three-dimensional scalar coupling

$$\mathcal L = \sqrt{-g}\, e^{-2\phi} \left( R + 4(\partial\phi)^2 - \frac{1}{12}H_{MNP}H^{MNP} + \frac{\alpha'}{4} R_{MNPQ}R^{MNPQ} - \frac{\alpha'}{4}\operatorname{tr}(F_{MN}F^{MN}) \right),$$45

is proposed as an exact U-duality invariant modular integral over $$\mathcal L = \sqrt{-g}\, e^{-2\phi} \left( R + 4(\partial\phi)^2 - \frac{1}{12}H_{MNP}H^{MNP} + \frac{\alpha'}{4} R_{MNPQ}R^{MNPQ} - \frac{\alpha'}{4}\operatorname{tr}(F_{MN}F^{MN}) \right),$$46, reproducing tree-level, one-loop, and non-perturbative effects as well as exact four-dimensional $$\mathcal L = \sqrt{-g}\, e^{-2\phi} \left( R + 4(\partial\phi)^2 - \frac{1}{12}H_{MNP}H^{MNP} + \frac{\alpha'}{4} R_{MNPQ}R^{MNPQ} - \frac{\alpha'}{4}\operatorname{tr}(F_{MN}F^{MN}) \right),$$47 couplings (Bossard et al., 2017). Although this is not a ten-dimensional supergravity construction, it shows how the four-derivative sector of heterotic theories is tightly constrained by supersymmetric Ward identities, automorphic structure, and BPS spectra.

Taken together, these developments define four-derivative heterotic supergravity as a broad but tightly structured framework. Its characteristic ingredients are torsionful curvature-squared invariants, Green-Schwarz-modified three-form geometry, deformed supersymmetry, and nontrivial field-redefinition dependence. Its applications range from explicit compactifications and consistent truncations to exact duality-covariant formulations and corrected black-hole observables. The current literature indicates that the ten-dimensional BdR completion remains the central organizing principle, while reduction, dualization, and symmetry-covariant reformulation reveal both its rigidity and its diverse lower-dimensional consequences (Jayaprakash et al., 2024, Lescano et al., 2021, Liu et al., 2023, Chang et al., 2022, Hu et al., 25 Jun 2025).