On the Moduli Space of the $T^6/Z_3$ Orbifold and Its Modular Group

We discuss geometric aspects of orbifold conformal field theories in the moduli space of N=(4,4) superconformal field theories with central charge c=6. Part of this note consists of a summary of our earlier results on the location of these theories within the moduli space [NW01,Wen] and the action of a specific version of mirror symmetry on them [NW]. We argue that these results allow for a direct translation from geometric to conformal field theoretic data. Additionally, thi...

A geometrisation scheme internal to the category of Lie supergroups is discussed for the supersymmetric de Rham cocycles on the super-Minkowski group $\,\mathbb{T}\,$ which determine the standard super-$p$-brane dynamics with that target, and interpreted within Cartan's approach to the modelling of orbispaces of group actions by homotopy quotients. The ensuing higher geometric objects are shown to carry a canonical equivariant structure for the action of a discrete subgroup o...

We study the gauging of a discrete $\mathbb{Z}_3$ symmetry in the five-dimensional superconformal $T_N$ theories. We argue that this leads to an infinite sequence of five-dimensional superconformal theories with either $E_6 \times SU(N)$ or $SU(3)\times SU(N)$ global symmetry group. In the $M$-theory realisation of $T_N$ theories as residing at the origin in the Calabi-Yau orbifolds ${\mathbb{C}^3 \over {\mathbb{Z}_N \times \mathbb{Z}_N}}$ we identify the $\mathbb{Z}_3$ symme...

We present a 1-loop toroidal membrane winding sum reproducing the conjectured $M$-theory, four-graviton, eight derivative, $R^4$ amplitude. The $U$-duality and toroidal membrane world-volume modular groups appear as a Howe dual pair in a larger, exceptional, group. A detailed analysis is carried out for $M$-theory compactified on a 3-torus, where the target-space $Sl(3,\Zint)\times Sl(2,\Zint)$ $U$-duality and $Sl(3,\Zint)$ world-volume modular groups are embedded in $E_{6(6)...

In this paper, we study moduli spaces of low dimensional complex Lie superalgebras. We discover a similar pattern for the structure of these moduli spaces as we observed for ordinary Lie algebras, namely, that there is a stratification of the moduli space by projective orbifolds. The moduli spaces consist of some families as well as some singleton elements. The different strata are linked by jump deformations, which gives a uniques manner of decomposing the moduli space which...

We provide a complete classification of six-dimensional symmetric toroidal orbifolds which yield N>=1 supersymmetry in 4D for the heterotic string. Our strategy is based on a classification of crystallographic space groups in six dimensions. We find in total 520 inequivalent toroidal orbifolds, 162 of them with Abelian point groups such as Z_3, Z_4, Z_6-I etc. and 358 with non-Abelian point groups such as S_3, D_4, A_4 etc. We also briefly explore the properties of some orbif...

${\bf Z}_2\times {\bf Z}_2$ Coxeter orbifolds are constructed with the property that some twisted sectors have fixed planes for which the six-torus can not be decomposed into a direct sum ${\bf T}^2\bigoplus{\bf T}^4 $ with the fixed plane lying in ${\bf T}^2$. The string loop threshold corrections to the gauge coupling constants are derived, and display symmetry groups for the $T$ and $U$ moduli that are subgroups of the full modular group $PSL(2,Z)$. The effective potential...

We propose a candidate for the dual (in the weak/strong coupling sense) of the six-dimensional heterotic string compactification constructed recently by Chaudhuri, Hockney and Lykken. It is a type IIA string theory compactified on an orbifold $K3/Z_2$, where the $Z_2$ action involves an involution of $K3$ with fixed points, and also has an embedding in the U(1) gauge group associated with the Ramond-Ramond sector of the type IIA string theory. This introduces flux of the U(1)...

We classify orbifold geometries which can be interpreted as moduli spaces of four-dimensional $\mathcal{N}\geq 3$ superconformal field theories up to rank 2 (complex dimension 6). The large majority of the geometries we find correspond to moduli spaces of known theories or discretely gauged version of them. Remarkably, we find 6 geometries which are not realized by any known theory, of which 3 have an $\mathcal{N}=2$ Coulomb branch slice with a non-freely generated coordinate...

After pointing out the role of the compactification lattice for spectrum calculations in orbifold models, I discuss modular discrete symmetry groups for $Z_N$ or\-bi\-folds. I consider the $Z_7$ orbifold as a nontrivial example of a (2,2) model and give the generators of the modular group for this case, which does not contain $[SL(2,{\bf Z})]^3$ as had been speculated. I also discuss how to treat cases where quantized Wilson lines are present. I consider in detail an example,...