ar X iv : 0 70 9 . 01 76 v 2 [ m at h . D G ] 1 1 O ct 2 00 7 REPRESENTATIONS OF ORBIFOLD GROUPOIDS

ar X iv : 0 70 9 . 01 76 v 1 [ m at h . D G ] 3 S ep 2 00 7 REPRESENTATIONS OF ORBIFOLD GROUPOIDS

Orbifold groupoids have been recently widely used to represent both effective and ineffective orbifolds. We show that every orbifold group-oid can be faithfully represented on a continuous family of finite dimensional Hilbert spaces. As a consequence we obtain the result that every orbifold groupoid is Morita equivalent to the translation groupoid of an action of a bundle of compact topological...

ar X iv : 0 81 0 . 16 21 v 1 [ m at h . Q A ] 9 O ct 2 00 8 DRINFEL ’ D DOUBLES AND SHAPOVALOV DETERMINANTS

The Shapovalov determinant for a class of pointed Hopf algebras is calculated, including quantized enveloping algebras, Lusztig’s small quantum groups, and quantized Lie superalgebras. Our main tools are root systems, Weyl groupoids, and Lusztig type isomorphisms. We elaborate powerful novel techniques for the algebras at roots of unity, and pass to the general case using a density argument.

ar X iv : d g - ga / 9 51 00 06 v 1 2 3 O ct 1 99 5 GROUP SYSTEMS , GROUPOIDS , AND MODULI SPACES OF PARABOLIC BUNDLES

ar X iv : 0 70 4 . 35 30 v 2 [ m at h . D G ] 2 5 O ct 2 00 7 Invariant forms , associated bundles and Calabi - Yau metrics

We develop a method, initially due to Salamon, to compute the space of " invariant " forms on an associated bundle X = P ×G V , with a suitable notion of invariance. We determine sufficient conditions for this space to be d-closed. We apply our method to the construction of Calabi-Yau metrics on T CP 1 and T CP 2 .

ar X iv : h ep - l at / 0 01 00 76 v 1 3 0 O ct 2 00 0 1 Vortex structure of the vacuum and confinement

We discuss the recently published numerical computations of the vortex free energy. They dramatically demonstrate that the excitation probability for a sufficiently thick vortex approaches unity at large beta, i.e. that thick vortices 'condense' in the vacuum. This is known to imply confinement. An analytical approach for exploring this phenomenon is also outlined.