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Niemeier lattices in the free fermionic heterotic-string formulation. (English) Zbl 1400.81158
Summary: The spinor-vector duality was discovered in free fermionic constructions of the heterotic string in four dimensions. It played a key role in the construction of heterotic-string models with an anomaly-free extra $$Z^{\prime}$$ symmetry that may remain unbroken down to low energy scales. A generic signature of the low scale string derived $$Z^{\prime}$$ model is via diphoton excess that may be within reach of the LHC. A fascinating possibility is that the spinor-vector duality symmetry is rooted in the structure of the heterotic-string compactifications to two dimensions. The two-dimensional heterotic-string theories are in turn related to the so-called moonshine symmetries that underlie the two-dimensional compactifications. In this paper, we embark on exploration of this connection by the free fermionic formulation to classify the symmetries of the two-dimensional heterotic-string theories. We use two complementary approaches in our classification. The first utilises a construction which is akin to the one used in the spinor-vector duality. Underlying this method is the triality property of $$\mathrm{SO}(8)$$ representations. In the second approach, we use the free fermionic tools to classify the twenty-four-dimensional Niemeier lattices.
##### MSC:
 81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory 81T25 Quantum field theory on lattices 81R25 Spinor and twistor methods applied to problems in quantum theory 81S05 Commutation relations and statistics as related to quantum mechanics (general) 81R05 Finite-dimensional groups and algebras motivated by physics and their representations 81V80 Quantum optics 22E70 Applications of Lie groups to the sciences; explicit representations
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