10D Non-Tachyonic String Models

• Ten-dimensional non-tachyonic string models are quantum gravity theories that eliminate tachyonic states using orbifold/orientifold projections and generalized GSO phases.
• They achieve modular invariance and anomaly cancellation via Green–Schwarz mechanisms and cobordism analyses, ensuring consistent, non-supersymmetric spectra.
• These models support realistic phenomenological constructions, such as Standard-like and Pati–Salam vacua, with controlled effective potentials and duality connections.

Ten-dimensional non-tachyonic string models are quantum theories of gravity formulated in ten dimensions that avoid spacetime tachyons in their spectra, do not require spacetime supersymmetry, and satisfy anomaly-consistency requirements. These models provide an enlarged landscape of consistent string vacua beyond the five classic supersymmetric theories, permit non-trivial constructions of phenomenological models, and display rich worldsheet and cobordism structures amenable to rigorous analysis.

1. Catalog of Ten-Dimensional Non-Tachyonic String Models

The exhaustive set of tachyon-free, non-supersymmetric string models in ten dimensions comprises three families, each defined by a distinct worldsheet construction, gauge group, and anomaly profile:

Model	Origin/Construction	Gauge Algebra
SO(16)×SO(16) heterotic	$\mathbb{Z}_2$ orbifold of $E_8 \times E_8$ (Faraggi, 2019, Faraggi et al., 2020)	$\mathfrak{so}(16)\oplus\mathfrak{so}(16)$
Sagnotti 0′B Model	Orientifold of Type 0B string (Dudas et al., 6 Nov 2025, Basile et al., 2023)	$\mathfrak{u}(32)$ (broken to SU(32))
Sugimoto Model	Orientifold of Type IIB with anti-D9s, O9$^+$ (Basile et al., 2023)	$\mathfrak{sp}(16)$

These models are constructed through orbifold or orientifold projections that eliminate spacetime fermions and tachyons, and are characterized by their absence of unbroken spacetime supersymmetry. The underlying worldsheet models are consistent with modular invariance and implement generalized GSO projections designed to expunge tachyonic states (Faraggi, 2019, Dudas et al., 6 Nov 2025).

2. Worldsheet Consistency and Elimination of Tachyons

Modular invariance and generalized GSO projections are central for consistency and for removing tachyonic excitations:

• Modular invariance ensures invariance of the one-loop torus amplitude under $\tau\mapsto\tau+1$ and $\tau\mapsto -1/\tau$. This constrains the allowed combinations of SO($2n$) fermion characters and boundary conditions in the partition function (Faraggi, 2019).
• Generalized GSO (GGSO) projections: In the free-fermion formalism, vectors $v_i$ define boundary conditions, and physical state survival is dictated by GGSO phases $C[v_i,v_j]$. Only combinations satisfying all GGSO conditions survive, systematically projecting out tachyonic sectors (Faraggi et al., 2020).
• The one-loop partition function for, e.g., the SO(16)$^2$ heterotic string is

$$Z_{SO(16)\times SO(16)} = \frac{1}{\tau_2^{4}\,\eta^{8}\bar\eta^{24}} \Bigl\{ V_8(O_{16}O_{16}+S_{16}S_{16}) - S_8(O_{16}S_{16}+S_{16}O_{16}) + O_8(C_{16}V_{16}+V_{16}C_{16}) - C_8(C_{16}C_{16}+V_{16}V_{16}) \Bigr\}.$$

This structure automatically projects out the potentially tachyonic $O_8 \bar{O}_{16} \bar{O}_{16}$ sector (Faraggi, 2019).

• The mass formula in light-cone gauge,

$M^2 = \frac{4}{\alpha'}(N_L-a_L+N_R-a_R),$

together with selected GGSO phases, restricts $M^2 \geq 0$ across the spectrum.

Construction details differ for orientifold-based models; here, GSO projections and orientifold involutions (e.g., $\Omega'$ with Klein bottle amplitude removing NS–NS tachyons at the closed-string level (Dudas et al., 6 Nov 2025)) are used to eliminate tachyons. For open-string sectors, requirement of vanishing NS–NS tadpole fixes Chan–Paton degrees (e.g., $N=32$ in O$'$B) and removes open-string tachyons.

3. Anomaly Cancellation and Global Consistency

All non-tachyonic models feature local anomaly cancellation via generalizations of the Green–Schwarz mechanism (Adams et al., 2010, Basile et al., 2023):

• The anomaly polynomial factorizes as $P_{12} = X_4 \wedge X_8$ (or $P_{12} = X_2 X_{10} + X_4 X_8 + X_6^2$ in 0′B), where $X_4$ contains gravitational and gauge terms and $X_8$ encodes higher-degree contributions (Basile et al., 2023, Adams et al., 2010).
• The counterterm $S_{GS} = -\int B_2 \wedge X_8$ cancels local anomalies provided the Bianchi identity $dH_3 = X_4$ is maintained.
• For SO(16)$^2$ heterotic: $X_4 = \frac{1}{2} p_1(T) + c_2(\text{gauge})$.
• For Sagnotti 0′B: $X_2, X_4, X_6$ are constructed from Chern classes and Pontryagin numbers.
• Computation of twisted String bordism groups (using Adams spectral sequence) reveals full vanishing in degrees $\leq$11 for SO(16)$^2$ and Sugimoto models, and partial vanishing for Sagnotti 0′B; thus, no global anomalies exist in these models (Basile et al., 2023).

The Cobordism Conjecture further relates non-trivial lower-dimensional bordism groups to the existence and classification of non-BPS branes; for SO(16)$^2$, $$\Omega_6^{\text{String–Spin}(16)^2} \cong \mathbb{Z}_2$$ yields a 6-brane charge, and $\Omega_8 \cong \mathbb{Z}^6$ includes the NS5-brane as an S$^8$ kink in $H_3$.

4. Model Building: Phenomenological Examples and Fertility

Non-tachyonic vacua support systematic construction of phenomenologically viable compactifications, including realizations of Standard-like Models (SLM):

• Starting from SO(16)×E$_8$ or SO(16)$^2$ heterotic in ten dimensions, four-dimensional compactifications use $\mathbb{Z}_2 \times \mathbb{Z}_2$ orbifolds and a basis of free-fermion shift and twist vectors (e.g., $b_1$, $b_2$, $b_3$, $\alpha$, $\beta$, $\gamma$, $a$, $b$) (Faraggi, 2019, Faraggi et al., 2020).
• GGSO phase assignments $C[v_i,v_j]$ are exhaustively scanned using fertility preselection to avoid tachyonic sectors and ensure the presence of complete spinorial families and Higgs multiplets (Faraggi et al., 2020).
• Concrete results include:
  • A six-generation SLM (from (Faraggi, 2019)): Gauge group SU(3)$_C \times$ SU(2)$_L \times$ U(1)$_Y \times$ U(1)$^6$ in the observable sector, SU(5)$_H \times$ SU(3)$_H \times$ U(1)$^2$ in the hidden sector; six chiral families from $b_i$ twisted sectors, three candidate Higgs pairs, and full tachyon projection.
  • In Pati–Salam heterotic models, O$(10^5)$ distinct three-generation models can be constructed with boson–fermion degeneracy ($a_{00}=0$) and realistic Yukawa couplings.
• The probability to obtain fully tachyon-free compactifications in model scans is $\sim$0.01 for SO(16)$^2$ “S-models” (Faraggi et al., 2020).

5. Effective Potentials, Warped Vacua, and Cosmological Behavior

The ten-dimensional effective dynamics are governed by string-frame actions with dilaton tadpole potentials and, in some models, form-field fluxes (Mourad et al., 2021, Dudas et al., 6 Nov 2025):

• The generic Einstein-frame action takes

$$S_E = \frac{1}{2(\alpha')^4} \int d^{10}x \sqrt{-g} \left\{ R - \tfrac{4}{8} (\partial\phi)^2 - V(\phi) - \tfrac{1}{2(p+2)!} e^{-2\beta_p\phi} \mathcal{H}_{p+2}^2 \right\},$$

where $V(\phi) = T e^{\gamma\phi}$ and parameters $(\gamma, \beta_p)$ distinguish the models.

• For SO(16)$^2$ heterotic, $\gamma = 5/2$.
• Vacuum solutions include warped geometries $M_{p+1} \times \mathbb{R} \times T^{8-p}$ admitting two families in orientifold models and four in heterotic, with explicit ODEs for metric functions and dilaton (Mourad et al., 2021).
• Inclusion of symmetric fluxes (e.g., $H_7$ field) produces three exact one-parameter families of flux-tadpole vacua with distinct boundary and asymptotic behaviors.
• Cosmologies derived by analytic continuation in the “radial” coordinate exhibit “climbing” and “descending” dilaton regimes controlled by $\gamma$ and the potential parameters; climbing occurs for large $\gamma$ ($\gamma \geq 3/2$), while descending requires $\gamma < 2\sqrt{2}$ (Mourad et al., 2021).

Runaway tadpole potentials of the form $V_{\rm tad}(\Phi, R) \sim e^{-\Phi} R^{-9}$ (Type 0′B) result in decompactification unless stabilized by additional fluxes or potential terms; these can influence early universe cosmology by triggering episodes of accelerated expansion, though they do not generically yield slow-roll inflation unless supplemented with extra moduli dynamics (Dudas et al., 6 Nov 2025).

6. Connectivity, Dualities, and Swampland Criteria

A salient feature of non-tachyonic ten-dimensional string models is their connectivity to supersymmetric vacua via orbifolds or moduli-space interpolation:

• In heterotic compactifications, $(2,0)$ worldsheet supersymmetry models (e.g., SO(10)$\times$U(1) gauge symmetry) can be connected to generic $(2,2)$ vacua with unbroken E$_6$ via discrete GGSO phase changes or orbifold/Wilson-line actions (Faraggi, 2019).
• The number of massless states is preserved in spinor–vector duality transitions.
• Conjecture: any four-dimensional EFT not connected by interpolation or orbifold to a $(2,2)$ point belongs to the swampland and cannot be UV-completed in string theory (Faraggi, 2019).
• Non-supersymmetric vacua are unified with supersymmetric ones in the enlarged N=4 toroidal moduli space, suggesting effective field theory properties can be interpreted as phases of a more fundamental worldsheet conformal field theory.

7. Topological, K-theoretic, and Matrix Model Perspectives

K-theory and cobordism classifications elucidate the flux, brane, and anomaly structure of ten-dimensional non-tachyonic string models:

• K-theory calculations reveal that, for Type I, RR fluxes are classified by $\tilde{KO}(S^1)\cong \mathbb{Z}_2$, identifying two discrete choices of RR axion backgrounds; Sethi’s “new string” background distinguishes itself by orientifolding at $C_0=\frac{1}{2}$ (Sethi, 2013).
• Matrix model decoupling limits suggest that both conventional Type I and the “new string” admit a Lightcone DLCQ description via a 2+1D membrane theory, potentially capturing the full nonperturbative spectrum, though explicit Lagrangians remain undeveloped (Sethi, 2013).
• Cobordism groups classify branes and worldvolume anomaly inflow; nonzero classes correspond to end-of-the-world branes, and anomaly cancellation matches predicted worldvolume spectra to inflow from bulk Green–Schwarz-type counterterms (Basile et al., 2023).

Tables of Main Model Properties

Model	Gauge Group	Partition Function Structure	Local/Global Anomaly Status
SO(16)$^2$ heterotic	$\mathfrak{so}(16)\oplus\mathfrak{so}(16)$	GGSO-projected, modular-invariant, tachyon-free (Faraggi, 2019)	Both cancelled, bordism group vanishes (Basile et al., 2023)
Sagnotti 0′B	$\mathfrak{u}(32)\to \mathfrak{su}(32)$	Type 0B orientifold with $\Omega'$ involution (Dudas et al., 6 Nov 2025)	Local GS, global $\mathbb{Z}_2$ ambiguity resolved (Basile et al., 2023)
Sugimoto	$\mathfrak{sp}(16)$	Type IIB orientifold with O9$^+$, anti-D9s (Basile et al., 2023)	Both cancelled, bordism group vanishes (Basile et al., 2023)

Summary and Implications

Ten-dimensional non-tachyonic string models are rigorously constructed worldsheet CFTs and ten-dimensional quantum gravities devoid of tachyons and consistent at both perturbative and non-perturbative levels. Systematic orbifold and orientifold projections, together with careful GGSO-phase engineering, generate modular-invariant spectra and eliminate pathological sectors. These models provide the foundation for phenomenological vacua—such as multi-generation Standard-like and Pati–Salam models—with manifest gauge anomaly cancellation and controlled boson–fermion spectra. The global anomaly analysis via cobordism, and connectivity to supersymmetric vacua, suggest that ten-dimensional non-tachyonic models are robust candidates for quantum gravity compactifications, and reinforce swampland constraints on low-energy EFTs. Their rich cosmological and topological structure continues to yield insights into string landscape dynamics, non-supersymmetric model building, and new avenues for investigating quantum gravity beyond the supersymmetric paradigm.