A localisation bound for Standard-Model representations in heterotic orbifolds

Abstract

In heterotic string compactifications on toroidal orbifolds, the massless states at a twisted fixed point are constrained by the evenness of the sixteen-dimensional gauge lattice. We show that two massless left-chiral states in the same Standard-Model representation, built on the same constructing space-group element and distinct modulo the Weyl group of $SU(3{)}_c\times SU(2{)}_L$, must satisfy $\sqrt{R}+\sqrt{R^{\prime }}\ge \sqrt{2}$, where $R$ is the squared length of the part of a state's gauge momentum orthogonal to the Standard-Model Cartan directions. Without left-moving oscillator excitations this reads $R\ge 1/2$. When $R<1$, $n$ such states, pairwise distinct in the same sense, carry the weights of the fundamental representation of a local $SU(n)$ commuting with the Standard Model, and $n\le 1/(1-R)$. Since the representation fixes the projected length (at the canonical hypercharge normalisation, $6/5$ for the 10-like $Q$, $u^c$, $e^c$ and $4/5$ for the 5bar-like $d^c$ and lepton doublet), the bound becomes a condition on the sector: a repeated 10-like representation requires a massless budget of at least $17/20$, a repeated 5bar-like one at least $13/20$. In the complete massless spectra of 16,138 models from two orbifold catalogues the multiplicity ceiling is reached at four values of $R$. We exhibit an explicit model on $T^6/(Z_3\times Z_6)$ whose budget clears the higher threshold and carries a repeated 10-like representation at the multiplicity the bound permits. For the Standard Model embedded through $SU(5)$ in one $E_8$ factor we show that, for that geometry and budget, exactly two inequivalent order-six shift classes admit such a doubling, and among the embeddings over them with the Standard Model intact we find none with three net generations of $Q$, $u^c$ and $e^c$ at once.

arXiv comment26 pages. Reproduction package: https://doi.org/10.5281/zenodo.23575403