The concept of a noncommutative Riemann surface (original) (raw)
Related papers
Noncommutative riemann surfaces
Arxiv preprint hep-th/0003131, 2000
We compactify M(atrix) theory on Riemann surfaces Σ with genus g > 1. Following [1], we construct a projective unitary representation of π 1 (Σ) realized on L 2 (H), with H the upper half-plane. As a first step we introduce a suitably gauged sl 2 (R) algebra. Then a uniquely determined gauge connection provides the central extension which is a 2-cocycle of the 2nd Hochschild cohomology group. Our construction is the double-scaling limit N → ∞, k → −∞ of the representation considered in the Narasimhan-Seshadri theorem, which represents the highergenus analog of 't Hooft's clock and shift matrices of QCD. The concept of a noncommutative Riemann surface Σ θ is introduced as a certain C ⋆ -algebra. Finally we investigate the Morita equivalence.
A Krichever-Novikov formulation of classical W algebras on Riemann surfaces
Physics Letters B, 1994
It is shown how the theory of classical W -algebras can be formulated on a higher genus Riemann surface in the spirit of Krichever and Novikov. An intriguing relation between the theory of A 1 embeddings into simple Lie algebras and the holomorphic geometry of Riemann surfaces is exihibited.
W Algebras on Riemann Surfaces
1994
It is shown how the theory of classical W –algebras can be formulated on a higher genus Riemann surface in the spirit of Krichever and Novikov. An intriguing relation between the theory of A 1 embeddings into simple Lie algebras and the holomorphic geometry of Riemann surfaces is exihibited.
The Drinfeld-Sokolov holomorphic bundle and classical W algebras on Riemann surfaces
Journal of Geometry and Physics, 1995
Developing upon the ideas of ref. [6], it is shown how the theory of classical W algebras can be formulated on a higher genus Riemann surface in the spirit of Krichever and Novikov. The basic geometric object is the Drinfeld-Sokolov principal bundle L associated to a simple complex Lie group G equipped with an SL(2, C) subgroup S, whose properties are studied in detail. On a multipunctured Riemann surface, the Drinfeld-Sokolov-Krichever-Novikov spaces are defined, as a generalization of the customary Krichever-Novikov spaces, their properties are analyzed and standard bases are written down. Finally, a WZWN chiral phase space based on the principal bundle L with a KM type Poisson structure is introduced and, by the usual procedure of imposing first class constraints and gauge fixing, a classical W algebra is produced. The compatibility of the construction with the global geometric data is highlighted.
Noncommutative geometry in M-theory and conformal field theory
1999
In the first part of the thesis I will investigate in the Matrix theory framework, the subgroup of dualities of the Discrete Light Cone Quantization of M-theory compactified on tori, which corresponds to T-duality in the auxiliary Type II string theory. After a review of matrix theory compactification leading to noncommutative supersymmetric Yang-Mills gauge theory, I will present solutions for the fundamental and adjoint sections on a two-dimensional twisted quantum torus and generalize to three-dimensional twisted quantum tori. After showing how M-theory T- duality is realized in supersymmetric Yang-Mills gauge theories on dual noncommutative tori I will relate this to the mathematical concept of Morita equivalence of C*-algebras. As a further generalization, I consider arbitrary Ramond-Ramond backgrounds. I will also discuss the spectrum of the toroidally compactified Matrix theory corresponding to quantized electric fluxes on two and three tori. In the second part of the thesis ...
2003
This paper continues the same-named article, Part I (math.QA/9812083). We give a global operator approach to the WZWN theory for compact Riemann surfaces of an arbitrary genus g with marked points. Globality means here that we use Krichever-Novikov algebras of gauge and conformal symmetries (i.e. algebras of global symmetries) instead of loop and Virasoro algebras (which are local in this context). The elements of this global approach are described in Part I. In the present paper we give the construction of conformal blocks and the projective flat connection on the bundle constituted by them.
Introduction to M(atrix) theory and noncommutative geometry
Physics Reports, 2002
Noncommutative geometry is based on an idea that an associative algebra can be regarded as "an algebra of functions on a noncommutative space". The major contribution to noncommutative geometry was made by A. Connes, who, in particular, analyzed Yang-Mills theories on noncommutative spaces, using important notions that were introduced in his papers (connection, Chern character, etc). It was found recently that Yang-Mills theories on noncommutative spaces appear naturally in string/M-theory; the notions and results of noncommutative geometry were applied very successfully to the problems of physics.
Quantum gauge symmetries in noncommutative geometry
Journal of Noncommutative Geometry, 2014
We discuss generalizations of the notion of i) the group of unitary elements of a (real or complex) finite dimensional C * -algebra, ii) gauge transformations and iii) (real) automorphisms, in the framework of compact quantum group theory and spectral triples. The quantum analogue of these groups are defined as universal (initial) objects in some natural categories. After proving the existence of the universal objects, we discuss several examples that are of interest to physics, as they appear in the noncommutative geometry approach to particle physics: in particular, the C * -algebras M n (R), M n (C) and M n (H), describing the finite noncommutative space of the Einstein-Yang-Mills systems, and the algebras A F = C⊕H⊕M 3 (C) and A ev = H ⊕ H ⊕ M 4 (C), that appear in Chamseddine-Connes derivation of the Standard Model of particle physics minimally coupled to gravity. As a byproduct, we identify a "free" version of the symplectic group Sp(n) (quaternionic unitary group).
1993
This talk reviews results on the structure of algebras consisting of meromorphic differential operators which are holomorphic outside a finite set of points on compact Riemann surfaces. For each partition into two disjoint subsets of the set of points where poles are allowed, a grading of the algebra and of the modules of λ−forms is introduced. With respect to this grading the Lie structure of the algebra and of the modules are almost graded ones. Central extensions and semi-infinite wedge representations are studied. If one considers only differential operators of degree 1 then these algebras are generalizations of the Virasoro algebra in genus zero, resp. of Krichever Novikov algebras in higher genus. 1 invited talk at the International Symposium on Generalized Symmetries in Physics at the Arnold-Sommerfeld-Institute, Clausthal, Germany, July 26 – July 29, 1993