Orbifold equivalence: structure and new examples

Orbifold equivalence is a notion of symmetry that does not rely on group actions. Among other applications, it leads to surprising connections between hitherto unrelated singularities. While the concept can be defined in a very general category-theoretic language, we focus on the most explicit setting in terms of matrix factorisations, where orbifold equivalences arise from defects with special properties. Examples are relatively difficult to construct, but we uncover some structural features that distinguish orbifold equivalences -- most notably a finite perturbation expansion. We use those properties to devise a search algorithm, then present some new examples including Arnold singularities.Comment: 34 pages, web-link to Singular code provide

Supersymmetric quantum theory and (non-commutative) differential geometry

We reconsider differential geometry from the point of view of the quantum theory of non-relativistic spinning particles, which provides examples of supersymmetric quantum mechanics. This enables us to encode geometrical structure in algebraic data consisting of an algebra of functions on a manifold and a family of supersymmetry generators represented on a Hilbert space. We show that known types of differential geometry can be classified in terms of the supersymmetries they exhibit. Replacing commutative algebras of functions by non-commutative *-algebras of operators, while retaining supersymmetry, we arrive at a formulation of non-commutative geometry encompassing and extending Connes' original approach. We explore different types of non-commutative geometry and introduce notions of non-commutative manifolds and non-commutative phase spaces. One of the main motivations underlying our work is to construct mathematical tools for novel formulations of quantum gravity, in particular for the investigation of superstring vacua.Comment: 125 pages, Plain TeX fil

Dilogarithm Identities in Conformal Field Theory

Dilogarithm identities for the central charges and conformal dimensions exist for at least large classes of rational conformally invariant quantum field theories in two dimensions. In many cases, proofs are not yet known but the numerical and structural evidence is convincing. In particular, close relations exist to fusion rules and partition identities. We describe some examples and ideas, and present some conjectures useful for the classification of conformal theories. The mathematical structures seem to be dual to Thurston's program for the classification of 3-manifolds.Comment: 14 pages, BONN-preprint. (a few minor changes, two major corrections in chapter 3, namely: (3.10) only holds in the case of the A series, Goncharovs conjecture is not an equivalence but rather an implication and a theorem

Supersymmetric quantum theory and non-commutative geometry

Classical differential geometry can be encoded in spectral data, such as Connes' spectral triples, involving supersymmetry algebras. In this paper, we formulate non-commutative geometry in terms of supersymmetric spectral data. This leads to generalizations of Connes' non-commutative spin geometry encompassing non-commutative Riemannian, symplectic, complex-Hermitian and (Hyper-)Kaehler geometry. A general framework for non-commutative geometry is developed from the point of view of supersymmetry and illustrated in terms of examples. In particular, the non-commutative torus and the non-commutative 3-sphere are studied in some detail.Comment: 77 pages, PlainTeX, no figures; present paper is a significantly extended version of the second half of hep-th/9612205. Assumptions in Sect. 2.2.5 clarified; final version to appear in Commun.Math.Phy

The conformal boundary states for SU(2) at level 1

For the case of the SU(2) WZW model at level one, the boundary states that only preserve the conformal symmetry are analysed. Under the assumption that the usual Cardy boundary states as well as their marginal deformations are consistent, the most general conformal boundary states are determined. They are found to be parametrised by group elements in SL(2,C).Comment: 22 pages, harvmac (b), 5 figure

Non-commutative World-volume Geometries: Branes on SU(2) and Fuzzy Spheres

The geometry of D-branes can be probed by open string scattering. If the background carries a non-vanishing B-field, the world-volume becomes non-commutative. Here we explore the quantization of world-volume geometries in a curved background with non-zero Neveu-Schwarz 3-form field strength H = dB. Using exact and generally applicable methods from boundary conformal field theory, we study the example of open strings in the SU(2) Wess-Zumino-Witten model, and establish a relation with fuzzy spheres or certain (non-associative) deformations thereof. These findings could be of direct relevance for D-branes in the presence of Neveu-Schwarz 5-branes; more importantly, they provide insight into a completely new class of world-volume geometries.Comment: 19 pages, LaTeX, 1 figure; some explanations improved, references adde