Compactification of Heterotic Supergravity

• Compactification of heterotic supergravity is the process of reducing 10D N=1 supergravity with Yang-Mills to effective lower-dimensional theories via internal manifolds with nontrivial H-flux and gauge bundles.
• Effective theories are constructed under strict anomaly cancellation, supersymmetry, and higher-derivative corrections, ensuring moduli stabilization and consistent chiral spectra.
• Concrete models on nearly Kähler, half-flat, G2, and Spin(7) backgrounds illustrate the rich geometric, algebraic, and physical phenomena emerging from the compactification procedure.

The compactification of heterotic supergravity systematically constructs lower-dimensional effective theories by reducing ten-dimensional $N=1$ supergravity coupled to Yang-Mills on suitable internal manifolds, generically equipped with nontrivial $H$-flux and gauge bundles. This procedure, highly constrained by anomaly cancellation, supersymmetry, and the structure of higher-derivative corrections (notably at $\mathcal{O}(\alpha')$), produces a wealth of geometric, algebraic, and physical phenomena. Key insights have emerged from studies on nearly Kähler, half-flat, $G_2$, and Spin(7) backgrounds, as well as through the analysis of both moduli spaces and worldsheet aspects. Compactification on non-Ricci-flat spaces and explicit models with moduli stabilization or chiral matter have also demonstrated the broader scope of heterotic backgrounds.

1. Ten-Dimensional Setup and Anomaly Cancellation

The ten-dimensional heterotic supergravity action to leading order in $\alpha'$ includes the metric $g_{MN}$, dilaton $\Phi$, $B$-field with 3-form field strength $H$, and either $E_8\times E_8$ or $SO(32)$ gauge multiplets. The critical correction at $\mathcal{O}(\alpha')$ is the Green–Schwarz mechanism, encoded by the Bianchi identity: $$dH = \frac{\alpha'}{4}\left(\mathrm{Tr}\,R\wedge R - \mathrm{Tr}\,F\wedge F\right)$$ where $R$ is the curvature 2-form of a specified connection on $TX$, and $F$ the field strength of the gauge bundle. Precise connection choice is crucial: only the "minus" (Hull) connection $\nabla^- = \nabla^{LC} + \frac{1}{2} H$ is compatible with the full set of supersymmetry and field equations at this order, as confirmed by the analysis of compactifications on nearly Kähler coset spaces and the intrinsic constraints found by Ivanov (Lechtenfeld et al., 2010, Ossa et al., 2014).

Supersymmetry transformations for the gravitino, dilatino, and gaugino depend on $H$ and $F$, and their simultaneous vanishing imposes further restrictions on background geometry and fluxes. Field redefinitions correspond to changes in the tangent bundle connection by $(0,1)$-Dolbeault classes in $\operatorname{End}(TX)$; these deformations are nonphysical at leading order but suggest a symmetry between gauge and tangent bundle sectors to all orders in $\alpha'$ (Ossa et al., 2014).

2. Internal Geometries and Structure Groups

The class of admissible internal manifolds is tightly constrained. Key classes are:

• Nearly Kähler (NK) Manifolds: Six-dimensional homogeneous spaces such as $SU(3)/U(1)^2$ admit an $SU(3)$-structure where the only nontrivial torsion class is $W_1$ (the "nearly Kähler" class). The intrinsic torsion exactly matches the required $H$-flux for supersymmetry. Compactification yields AdS$_4$ backgrounds with both supersymmetric and explicit non-supersymmetric vacua, with all moduli fixed at tree level by combined geometric and gauge data (Lechtenfeld et al., 2010, Matti, 2012).
• Half-Flat and Complex Non-Kähler Spaces: The heterotic Strominger system admits solutions on half-flat or complex non-Kähler six-manifolds with specific patterns of $SU(3)$ torsion classes. Nearly Kähler cosets generalize Strominger's original construction and are systematically realizable via homogeneous coset spaces (Matti, 2012).
• $G_2$ and $\operatorname{Spin}(7)$ Manifolds: Compactification on seven-manifolds with integrable $G_2$ structure or eight-manifolds with Spin(7) structure provides vacua with $\mathcal{N}=1$ supersymmetry in $d=3$ or $1/4$-BPS structure in $d=4$. The $G_2$ torsion classes correspond to modulated fluxes, and the effective superpotentials encapsulate all necessary BPS conditions (Ossa et al., 2019, McOrist et al., 22 Feb 2025, Angus et al., 2015).
• Product Spaces and Non-Ricci-Flat Examples: Non-supersymmetric, non-Ricci-flat compactifications can be realized on products of constant-curvature two-manifolds such as $S^2$, $T^2$, and $H^2/\Gamma$, with gauge field strengths determined by flux quantization and index constraints. These backgrounds fix all metric moduli and admit explicit chiral spectra (Tsuyuki, 2021, Takeuchi et al., 2023).

3. Moduli Space, Supersymmetry Conditions, and Stabilization

Supersymmetric heterotic compactifications are determined by vanishing fermion variations and the Bianchi identity, implemented as the Strominger system for $SU(3)$-structure backgrounds, or its $G_2$/$\operatorname{Spin}(7)$ analogues:

• The holomorphicity and instanton conditions for gauge and tangent bundle connections: $F^{(0,2)}=R^{(0,2)}=0$, primitivity $F\wedge\Omega=R\wedge\Omega=0$.
• The conformally balanced condition $d(e^{-2\phi}\omega\wedge\omega)=0$.
• Hermitian Yang–Mills and balanced metric conditions appear as D-term constraints in the effective theory (Ossa et al., 2015, Ossa et al., 2019).

Moduli spaces are described by the cohomology of an explicit holomorphic extension bundle $Q$ and a double-extension differential $\bar{D}$, itself required by anomaly cancellation. Infinitesimal deformations appear in $H^{0,1}(Q)$, generalizing the familiar Dolbeault cohomologies of Calabi–Yau backgrounds to include bundle and Hermitian moduli, jointly constrained by the Bianchi identity and anomaly extensions (Ashmore et al., 8 Jul 2025, Ossa et al., 2015). $\alpha'$-corrections deform the Weil–Petersson metric on the moduli space by explicit curvature and bundle-dependent terms (McOrist et al., 22 Feb 2025).

Notably, in certain NK and product vacua, both AdS radius and internal volume are fixed at leading order—no geometric moduli remain, and any residual moduli are discrete (Lechtenfeld et al., 2010, Tsuyuki, 2021).

4. Explicit Models and Phenomenological Features

Several classes of constructive compactifications have distinct physical signatures:

• Nearly Kähler Coset Models: On $SU(3)/U(1)^2$, all supersymmetry and Bianchi equations are satisfied for specific choices of metric scale, flux normalization, and gauge bundle—a discrete set of vacua remains, with moduli stabilization and absence of scale moduli (Lechtenfeld et al., 2010, Matti, 2012).
• Three-Generation Product Geometries: By placing $U(1)$ fluxes on products like $S^2 \times H^2/\Gamma \times H^2/\Gamma$ and solving Einstein and Bianchi equations, explicit three-generation spectra can be realized. The generation number and allowed fluxes are set by Euler characteristics, curvature-quantization, and systematized via index theorems (Takeuchi et al., 2023).
• Local Non-Kähler and Resolved Singularities: Local models using resolved Eguchi–Hanson or resolved conifold spaces, when combined with torus fibrations and Abelian gauge extensions, admit exact worldsheet CFT descriptions in certain double-scaling limits. These backgrounds are noncompact but highlight the role of flux, warping, and resolution, including worldsheet instanton corrections and charge quantization (Carlevaro et al., 2011).
• Half-BPS Domain Wall and Standard Embedding Schemes: Kähler-with-torsion (HKT) constructions and their T-duals produce smooth domain-wall solutions and demonstrate the impact of standard embedding on the gauge sector and bundle topology, e.g., $E_8 \rightarrow SO(10)$ via duality-induced instantons, with geometric moduli linked to harmonic function parameters (Hinoue et al., 2014).
• Consistent Group Coset Reductions: Compactifying on coset spaces with isometry-generated gauge bosons, such as $\mathbb{R} \times T^{1,1}$, allows exact truncations to four-dimensional $\mathcal{N}=1$ SUGRA with explicit $SU(2)\times SU(2)$ gauge groups, revealing the interplay between geometric automorphisms and Yang-Mills symmetry (Pope, 28 Dec 2025).

5. Higher-Derivative Corrections, Anomaly Structures, and Holomorphic Theories

At $\mathcal{O}(\alpha'^2)$ and beyond, the low-energy effective theory receives additional corrections affecting anomaly cancellation and the structure of moduli spaces. The ten-dimensional holomorphic Kodaira–Spencer/BV theory approach encodes the moduli problem via a classically nilpotent BV differential $\bar D$; its refined cohomology computes physical deformations including leading-order $\alpha'$ effects. The one-loop anomaly polynomial factorizes as required by both $SO(32)$ and $E_8\times E_8$ heterotic supergravity, with a Green–Schwarz-like counterterm ensuring gauge invariance and deforming the extension complex to its "double" structure (Ashmore et al., 8 Jul 2025).

The interplay of worldsheet and spacetime anomalies is manifest in orbifold compactifications with Wilson lines and anomalous $U(1)$ sectors, such as the $Z_3$ orbifold with two Wilson lines. There, combined Pauli–Villars and Green–Schwarz regularizations achieve full cancellation of chiral and conformal anomalies, reflected in explicit effective supergravity Lagrangians (1908.10470).

Holographic analysis of compactifications (e.g., on AdS$_4 \times M_6$) demonstrates that higher-derivative terms in the 4D scalar effective action are controlled by compactification data (volume, curvature, flux). These modify the Breitenlohner-Freedman bound and associated operator dimensions in the dual CFT, clarifying the impact of curvature and flux on the infrared and ultraviolet dynamics (Acosta et al., 28 Apr 2025).

6. Mathematical and Physical Generalizations

The moduli problem in heterotic compactification has evolved into a rich interface of derived algebraic geometry, supersymmetric gauge theory, and bundle stability:

• The infinitesimal coupled moduli problem is formulated via a differential acting on an explicit extension bundle (the generalized Atiyah/hull-strominger complex), with obstruction classes and higher products reflecting Yukawa couplings and obstruction theory (Ossa et al., 2015, Ossa et al., 2019).
• Extension to $G_2$ and Spin(7) backgrounds leads to the appropriate versions of the Atiyah and anomaly complexes, with generalized superpotentials governed by intrinsic torsion and NS flux (Ossa et al., 2019, Angus et al., 2015).
• In the large volume limit, or upon reduction to Calabi–Yau settings, all expressions reduce to familiar Hodge-theoretic moduli, but non-Kähler torsional backgrounds remain a distinct class allowing, for instance, the stabilization of moduli and three-generation GUT constructions (Matti, 2012, McOrist et al., 22 Feb 2025).

7. Summary Table: Notable Backgrounds and Features

Type / Geometry	Supersymmetry	Moduli Stabilization	Gauge / Bundle Structure	Distinctive Features
$SU(3)/U(1)^2$ (nearly Kähler)	AdS$_4$ $\mathcal{N}=1$ (or none)	All geometric moduli fixed	$SU(3)$-instanton, Hull conn.	$\mathcal{O}(\alpha')$ complete, Ivanov’s theorem, discrete spectra (Lechtenfeld et al., 2010)
Product $S^2 \times H^2/\Gamma$	Broken	All metric moduli fixed	Multiple $U(1)$ fluxes	Explicit three-generation models, no Ricci-flatness (Tsuyuki, 2021, Takeuchi et al., 2023)
Torsional $SU(3)$, Half-flat	Domain wall (BPS)	Moduli by extension complex	Line/higher-rank bundles	Cohomological moduli, explicit bundle constructions (Matti, 2012, Ossa et al., 2015)
$G_2$ manifolds	3D $\mathcal{N}=1$	Moduli by $G_2$–extension	Instanton bundles	$\alpha'$-corrected moduli metric, limit to $SU(3)$ case (McOrist et al., 22 Feb 2025, Ossa et al., 2019)
$T^{1,1}$ coset reduction	4D $\mathcal{N}=1$	Truncated chiral+vector sector	$SU(2)\times SU(2)$	Fully consistent fermionic and bosonic reduction (Pope, 28 Dec 2025)
Resolved conifold, Eguchi–Hanson	Local models	N/A	Abelian bundles, fibred	Smooth CFT backgrounds in double-scaling limit (Carlevaro et al., 2011)

Comprehensive analyses now incorporate holomorphic field-theoretic techniques, higher-order anomalies, and exact solutions, confirming the intricate interplay between differential geometry, bundle theory, quantum consistency, and phenomenological engineering in heterotic compactification.