Superstring theoryをめぐって(放談室)

Quantum field theory on noncommutative spaces

This paper introduces five notions, including π-algebras, π-ideals, Hopf π-algebras, π-modules and Hopf π-modules, verifies the fundamental isomorphism theorem of π-algebras and studies some algebraic properties of Hopf π-algebras as well.

Hopf π- Algebras

By Vyjayanthi Chari and Andrew Pressley: 651 pp., £22.95 (US$34.95), isbn 0 521 55884 0 (Cambridge University Press, 1994).

A guide to quantum groups

Quantum fields, noncommutative spaces, and motives The Riemann zeta function and noncommutative geometry Quantum statistical mechanics and Galois symmetries Endomotives, thermodynamics, and the Weil explicit formula Appendix Bibliography Index.

Noncommutative Geometry, Quantum Fields and Motives

Noncommutative Spaces and Algebras of Functions.- Projective Systems of Noncommutative Lattices.- Modules as Bundles.- A Few Elements of K-Theory.- The Spectral Calculus.- Noncommutative Differential Forms.- Connections on Modules.- Field Theories on Modules.- Gravity Models.- Quantum Mechanical Models on Noncommutative Lattices.

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An Introduction to Noncommutative Spaces and Their Geometries

These lectures notes are an intoduction for physicists to several ideas and applications of noncommutative geometry. The necessary mathematical tools are presented in a way which we feel should be accessible to physicists. We illustrate applications to Yang-Mills, fermionic and gravity models, notably we describe the spectral action recently introduced by Chamseddine and Connes. We also present an introduction to recent work on noncommutative lattices. The latter have been used to construct topologically nontrivial quantum mechanical and field theory models, in particular alternative models of lattice gauge theory. 
Here is the list of sections: 
1. Introduction. 2. Noncommutative Spaces and Algebras of Functions. 3. Noncommutative Lattices. 4. Modules as Bundles. 5. The Spectral Calculus. 6. Noncommutative Differential Forms. 7. Connections on Modules. 8. Field Theories on Modules. 9. Gravity Models. 10. Quantum Mechanical Models on Noncommutative Lattices. 
Appendices: Basic Notions of Topology. The Gel'fand-Naimark-Segal Construction. Hilbert Modules. Strong Morita Equivalence. Partially Ordered Sets. Pseudodifferential Operators

An Introduction to Noncommutative Spaces and their Geometry

We give new examples of noncommutative manifolds that are less standard than the NC-torus or Moyal deformations of ℝn. They arise naturally from basic considerations of noncommutative differential topology and have non-trivial global features.

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Noncommutative Manifolds, the Instanton Algebra and Isospectral Deformations

We give new examples of noncommutative manifolds that are less standard than the NC-torus or Moyal deformations of $\Rb^n$. They arise naturally from basic considerations of noncommutative differential topology and have non-trivial global features. The new examples include the instanton algebra and the NC-4-spheres $S^4_{\theta}$. The noncommutative algebras $\Ac=C^{\ify} (S^{4}_{\theta})$ of functions on NC-spheres are solutions to the vanishing, $ {\rm ch}_j (e) = 0, j = \g_5$, where $<$ is the projection on the commutant of $4 \ts 4$ matrices. Finally, we show that any compact Riemannian spin manifold whose isometry group has rank $r \geq 2$ admits isospectral deformations to noncommutative geometries.

Noncommutative Manifolds the Instanton Algebra and Isospectral Deformations

We construct a 3+-summable spectral triple Open image in new window over the quantum group SUq(2) which is equivariant with respect to a left and a right action of Open image in new window The geometry is isospectral to the classical case since the spectrum of the operator D is the same as that of the usual Dirac operator on the 3-dimensional round sphere. The presence of an equivariant real structure J demands a modification in the axiomatic framework of spectral geometry, whereby the commutant and first-order properties need be satisfied only modulo infinitesimals of arbitrary high order.

Giovanni Landi

Papers

An Introduction to Noncommutative Spaces and Their Geometries

An Introduction to Noncommutative Spaces and their Geometry

Noncommutative Manifolds, the Instanton Algebra and Isospectral Deformations

Noncommutative Manifolds the Instanton Algebra and Isospectral Deformations

The Dirac Operator on SU q (2)