Niemeier Lattices in the Free Fermionic Heterotic–String Formulation

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′ symmetry that may remain unbroken down to low energy scales. A generic signature of the low scale string derived Z′ 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 SO(8) representations. In the second approach, we use the free fermionic tools to classify the twenty-four-dimensional Niemeier lattices.

(0,2) CALABI–YAU SIGMA MODELS AND THEIR DUALITY

N=(0,2) heterotic string theories on Calabi–Yau spaces are considered based on linear sigma approaches. We construct (0,2) sigma models on complete intersection Calabi–Yau spaces, ℳ, realized in a product of weighted projective spaces. The definition of the associating vector bundle V over ℳ is given. These models are formulated as the IR limits of (0,2) U (1)N gauged linear sigma models. Examining their phase structures, we observe that there are hybrid phases where one cannot intrinsically distinguish between the defining polynomials of the Calabi–Yau space ℳ and those of the gauge bundle V. Thus we construct dual pairs of two (0,2) Calabi–Yau sigma models which are isomorphic in their hybrid phases. In addition, generalizing the gauge bundle into the direct sum of bundles, we study the duality of (0,2) string vacua.

WEYL ORBIFOLD MODELS

Superstring models on Weyl orbifolds are investigated in [Formula: see text] heterotic string theories. Some of the Weyl orbifold models are shown to be consistent with worldsheet supersymmetry, N=1 spacetime supersymmetry and modular invariance. Two ways of embedding in [Formula: see text] are studied and residual gauge groups are classified.

T-DUALITY OF p-BRANES

We investigate possible existence of duality symmetries which exchange the Kaluza–Klein modes with the wrapping modes of a BPS saturated p-brane on a torus. Assuming the validity of the conjectured U-duality symmetries of type-II and heterotic string theories and M-theory, we show that for a BPS saturated p-brane there is an SL (2, Z) symmetry that mixes the Kaluza–Klein modes on a (p+1)-dimensional torus T(p+1) with the wrapping modes of the p-brane on T(p+1). The field that transforms as a modular parameter under this SL (2, Z) transformation has as its real part the component of the (p+1)-form gauge field on T(p+1), and as its imaginary part the volume of T(p+1), measured in the metric that couples naturally to the p-brane.