Gauge Unification in Highly Anisotropic String Compactifications

TL;DR

This work tackles the long‑standing problem of achieving precise gauge coupling unification in perturbative heterotic string theory. By defining the weak‑coupling boundary through duality and EFT arguments, it shows that isotropic compactifications struggle to reproduce MSSM unification, while highly anisotropic geometries with one or two large radii open two viable routes: non‑local GUT breaking via freely acting orbifolds and local GUT breaking at singularities with bulk domination. The authors construct explicit orbifold models and discuss their gauge embeddings, dualities, and mass scales, demonstrating that non‑local breaking can yield exact unification at the GUT scale with MSSM spectra, whereas local breaking requires careful handling of threshold corrections and UV completions. The findings suggest a promising string‑theory realization of GUTs through orbifold constructions with large extra dimensions, guiding future model building in heterotic string theory. Overall, the paper clarifies how anisotropic compactifications and non‑local breaking mechanisms can reconcile string theory with low‑energy gauge coupling precision, with implications for realistic string GUTs and beyond.

Abstract

It is well-known that heterotic string compactifications have, in spite of their conceptual simplicity and aesthetic appeal, a serious problem with precision gauge coupling unification in the perturbative regime of string theory. Using both a duality-based and a field-theoretic definition of the boundary of the perturbative regime, we reevaluate the situation in a quantitative manner. We conclude that the simplest and most promising situations are those where some of the compactification radii are exceptionally large, corresponding to highly anisotropic orbifold models. Thus, one is led to consider constructions which are known to the effective field-theorist as higher-dimensional or orbifold grand unified theories (orbifold GUTs). In particular, if the discrete symmetry used to break the GUT group acts freely, a non-local breaking in the larger compact dimensions can be realized, leading to a precise gauge coupling unification as expected on the basis of the MSSM particle spectrum. Furthermore, a somewhat more model dependent but nevertheless very promising scenario arises if the GUT breaking is restricted to certain singular points within the manifold spanned by the larger compactification radii.

Figures (4)

• Figure 1: This figure shows $m_H$ and $m_I$ as functions of the coupling $u$ as defined by Eqs. (\ref{['mh']}) and (\ref{['mi']}). The critical point $u_c$ separates heterotic (left) and type I (right) weak coupling regimes.
• Figure 2: This figure shows $m_H$ and $m_{KK}$ as functions of the coupling $u$ as defined by Eqs. (\ref{['mh']}) and (\ref{['mthu']}). The critical point $u_c$ separates heterotic-string (left) and 11d-supergravity (right) weak coupling regimes.
• Figure 3: The structure of the internal space $(T^2\times T^2\times T^2)/(Z_2\times Z^\prime_2)$. Note that the two larger dimensions are shared between the first and the second torus. The action of $Z^\prime_2$ is a rotation in the first two tori with fixed points shown as black dots with labels $A$ and $B$. The action of $Z_2$ is instead a translation along $x_1$ (arrow) and a rotation in the second and third torus with would-be fixed points shown as crosses with labels $C$ and $D$.
• Figure 4: Visualisation of the 6d internal space after orbifolding. The two larger extra dimensions $S_{x_1}$ and $S_{x_2}$ parameterize a 'half-pillow' with crosscap identification on the left. Away from the two conical singularities on the right, every point of this 2d space supports 4 smaller extra dimensions with $T^4$ geometry.