O(16)$\times$O(16) heterotic theory on $AdS_3\times S^3\times T^4$ (2510.20915v1)

Abstract: In this paper, we study non-supersymmetric $O(16)\times O(16)$ heterotic theory on an $AdS_{3}\times S^{3}\times T^{4}$ background, finding a family of vacua parameterized by a pair of flux integers. Adding the one-loop scalar potential to the effective theory contributes positively to the cosmological constant, but we find that there is no uplift to de Sitter for any values of the fluxes. We study the fluctuations around these vacua and show that all scalar and tensor modes from the six-dimensional effective theory lie above the Breitenlohner-Freedman bound. The moduli coming from the torus compactification will also be above the bound, at least for a large range of fluxes.

• The paper demonstrates that one-loop corrections keep the cosmological constant negative and do not uplift AdS to dS.
• The analysis uses quantized electric and magnetic fluxes to derive analytic expressions for moduli and the string coupling.
• The fluctuation spectrum remains above the BF bound, indicating perturbative stability, though potential moduli instabilities arise in the quantum regime.

Analysis of $O(16)\times O(16)$ Heterotic Theory on $AdS_3\times S^3\times T^4$

Introduction and Motivation

This work investigates the non-supersymmetric $O(16)\times O(16)$ heterotic string theory compactified on the background $AdS_3\times S^3\times T^4$. The primary focus is on the existence and stability of AdS vacua, the impact of one-loop corrections on the cosmological constant, and the spectrum of fluctuations, particularly in relation to the Breitenlohner-Freedman (BF) bound. The paper is motivated by the ongoing challenge of constructing stable de Sitter (dS) vacua in string theory, especially in non-supersymmetric settings, and by the need to understand the perturbative and non-perturbative stability of such vacua.

Construction of the Background and Effective Potential

The compactification is characterized by two quantized $H_3$ fluxes: $n_1$ (electric, on $AdS_3$) and $n_5$ (magnetic, on $S^3$). The tree-level string coupling is fixed by the fluxes and the torus volume $v$ as

$g_s^2 = \frac{v|n_5|}{(2\pi)^4|n_1|}.$

At tree level, the effective three-dimensional cosmological constant is negative, corresponding to an $AdS_3$ vacuum. The potential is extremized at a square torus at the self-dual radius, and the explicit dependence of the AdS and $S^3$ radii, as well as $g_s$, on the fluxes is derived.

The one-loop correction to the potential is computed by integrating the genus-one partition function over the fundamental domain, following the standard approach for heterotic compactifications. The correction is positive and parameterized by a dimensionless constant $\lambda$, numerically evaluated to be $\sim 1.565$ for the square torus.

One-Loop Effects and the Cosmological Constant

The inclusion of the one-loop term modifies the extremization conditions for the moduli. The resulting equations are solved exactly, yielding analytic expressions for the corrected radii and string coupling. The key result is that, for all values of the fluxes, the one-loop correction does not uplift the AdS vacuum to de Sitter; the cosmological constant remains negative. This is robust both in the regime of large $n_1$ (small $g_s$) and in the "intrinsically quantum" regime $n_1\to 0$, where the one-loop term regularizes the otherwise divergent string coupling.

Figure 1: Plots of $L_o$, $g_o$, and $V_{\text{min}}$ against $\log n_1$ for various $n_5$, showing the smooth behavior of the moduli and the persistent negativity of the cosmological constant.

Fluctuation Spectrum and Stability

A detailed analysis of the spectrum of fluctuations is performed by expanding the six-dimensional effective action around the vacuum. The analysis distinguishes between "free" scalars (from torus moduli and Wilson lines) and "coupled" scalars (from the metric, dilaton, and $B$-field). The equations of motion are expanded in spherical harmonics on $S^3$, and the mass matrices for the various modes are computed.

For all values of the fluxes, the scalar and tensor modes from the six-dimensional supergravity sector are found to lie above the BF bound. The eigenvalues of the mass matrices are computed both analytically (in small and large $s = \lambda n_5^2/|n_1|$ expansions) and numerically for representative flux values.

Figure 2: The four eigenvalues for the $\ell=2$ mass matrix plotted against $\log n_5$ for $n_1=10$, showing that all eigenvalues remain above the BF bound.

Figure 3: The three eigenvalues for the $\ell=1$ mass matrix plotted against $\log n_5$ for $n_1=10$, with all eigenvalues above the BF bound.

Figure 4: The two eigenvalues for the $\ell=0$ mass matrix plotted against $\log n_5$ for $n_1=10$, confirming positivity throughout the flux range.

The analysis of the torus moduli is more subtle. While the square torus at the self-dual radius is shown to be a critical point of the potential, the stability of these moduli depends on the eigenvalues of the Hessian of the one-loop potential. For a large range of fluxes, the moduli masses are above the BF bound, but in the $n_1\to 0$ limit, some moduli may drop below the bound, indicating a potential instability in the intrinsically quantum regime.

Implications and Open Questions

The results confirm that, in this non-supersymmetric heterotic compactification, the one-loop correction is insufficient to uplift the AdS vacuum to de Sitter, in agreement with previous studies on related backgrounds. The absence of tachyons and the satisfaction of the BF bound for the supergravity sector indicate perturbative stability for a wide range of fluxes. However, the possible instability of torus moduli in the quantum regime remains an open issue.

The work highlights the technical challenges in constructing stable dS vacua in non-supersymmetric string theory and underscores the importance of understanding both perturbative and non-perturbative instabilities, including bubble nucleation and flux transitions. The analysis also suggests that further paper of the moduli space, including other critical points and worldsheet approaches, is warranted.

Conclusion

This paper provides a comprehensive analysis of the $O(16)\times O(16)$ heterotic string on $AdS_3\times S^3\times T^4$, demonstrating that the one-loop potential does not yield de Sitter vacua and that the spectrum of fluctuations is perturbatively stable for a broad range of fluxes. The results reinforce the difficulty of achieving dS uplift in non-supersymmetric string theory and point to several directions for future research, including a more detailed exploration of moduli stability and non-perturbative decay channels.