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Zbl 1097.58004
Várilly, Joseph C.
An introduction to noncommutative geometry.
(English)
[B] EMS Series of Lectures in Mathematics. Zürich: European Mathematical Society Publishing House. viii, 113~p. EUR~28.00 (2006). ISBN 3-03719-024-8/pbk

This book is based on the author's course on Noncommutative Geometry and Applications at the European Mathematical Society Summer School, Monsaraz and Lisboa, Portugal, 1997. It focuses mainly on spectral triple (noncommutative vision of spin geometry) and its applications. Progress after 1997 is supplied in Chapter 9, the last chapter. Audience of the course were students and young researchers in mathematics and physics. It aims to give a rapid understanding of noncommutative geometry and its applications. So detailed proofs and calculations are sometimes omitted. But it presents rich examples and remarks on physical applications, which serve to illustrate ideas of noncommutative geometry for the readers. The book consists of the following 9 chapters: 1. Commutative geometry from the noncommutative point of view. 2. Spectral triples on the Riemann sphere. 3. Real spectral triples: the axiomatic foundation. 4. Geometries on the noncommutative torus. 5. The noncommutative integral. 6. Quantization and the tangent groupoid. 7. Equivalence of geometries. 8. Action functional. 9. Epilogue: new directions. Chapter 1 reviews ordinary geometry from the viewpoint of noncommutative geometry. First, the author remarks that even in the classical study of elliptic curves, essential study was done through the analysis of elliptic function fields. Then the Gelfand-Naimark functor forming the category of locally compact spaces to the category of (commutative) $C^*$-algebra is explained. Noncommutative geometry extends this correspondence to the correspondence between noncommutative $C^*$-algebras and virtual geometric objects. Algebraic theory of vector bundles, Hermitian metrics, $\text{spin}^C$-structures, the Dirac operator and the distance formula $$d(p,q)= \sup\{|(\widehat p-\widehat q)(a)|: a\in C(M),\Vert[D,a]\Vert\le 1\},$$ are also exposed in Chapter 1. Chapter 2 deals with the spinor bundle, the Dirac operator and its spectrum, and twisted spinor modules on $S^2$ concretely. So the readers obtain the concrete image of the spectral triple explained in Chapter 3. Formal explanation of noncommutative geometry begins with Chapter 3. In this chapter, after giving the definition of the real spectral triple $({\Cal A},{\Cal H}, D; \Gamma, J)$, $\Gamma$ is the $\bbfZ_2$-grading and $J$ is the real structure (Def. 6), the axiomatic scheme for noncommutative geometries is set forth. The metric dimension of the spectral triple is defined by using the singular values of $ds= |D|^{-1}$, whose finiteness is assumed as Axiom 1. Further assumptions on $D$ (the first-order condition and the regularity) are given as Axioms 2 and 3. Then by using Hochschild cocycle, orientability of the triple is assumed as Axiom 4. Then assuming the possibility of Hermitian pairing on ${\Cal H}= \bigcap^\infty_{k=1} \text{Dom}(D^k)$ (Axiom 5), existence of Poincaré duality of the $K$-theory ring of ${\Cal A}\otimes{\Cal A}^0$ is assumed as Axiom 6. Assumptions on reality, which provides the 8 steps Bott periodicity, is presented as Axiom 7. The author remarks Riemannian spin manifolds provide the commutative examples of the spectral triple satifying these axioms. In Chapter 4, definitions and axioms of spectral triple are examined for the noncommutative torus. Then the skelton of the noncommutative torus, the noncommutative version of cell deecomposition, is defined by using cyclic cocycles (Def. 9). This was used in the theory of the integer quantum Hall effect [{\it J. Bellisard}, {\it A. van Elst} and {\it H. Schulz-Baldes}, J. Math. Phys. 35, No. 10, 5373--5451 (1994; Zbl 0824.46086)]. The metric of the noncommutative torus is given by $D_\tau$, $\tau\in \bbfC$, $D_\tau$ is constructed by using $\partial_\tau= \delta_1+ \tau\delta_2$, $\delta_i$ corresponds to $\partial/\partial\phi_i$ on the algebra of Fourier series. The volume of the noncommutative torus by this metric is computed to be $1/{\germ I}_\tau$, which is inversely proportional to the area of the periodic parallelogram of the elliptic curve $E_\tau$ with period $(1,\tau)$ (calculation is only sketched). Then the $K$-theory and Poincaré duality of the noncommutative torus are described. Calculation of the volume of the noncommutative torus already uses the integral in noncommutative geometry, which is defined by using trace. But to define the integral only for trace class operators is to restrictive. So in Chapter 4, first Dixmier trace, which can be applied for operators having logarithmic divergent traces, is explained. As an example, on the torus $\bbfT^n$, $\Delta^{-w_2}$ is shown to have Dixmier trace and it gives the volume of the sphere. After dealing with pseudodifferential operators, the Wodzicki residue and the zeta residue are explained and remarks their coincidence is remarked [{\it J. C. Várilly} and {\it J. M. Gracia-Bond\'ia}, J. Geom. Phys. 12, No. 4, 223--301 (1993; Zbl 0806.46075)]. Since the Connes-Lott model only produces classical Lagrangean of the Standard Model, the author says the phrase noncommutative = quantum'' must be disregarded. But quantization must proceed from noncommutative theory as the starting point. In Chapter 6, Moyal quantization and the tangent groupoid, which acomplishes to the deformation of an algebra of functions on phase spaces to an algebra of operator kernels, are explained [cf. {\it J. F. Cariñena}, {\it J. Clemente-Gallardo}, {\it E. Follana}, {\it J. M. Garcia-Bond\'ia} {\it A. Rivero} and {\it J. C. Várilly}, J. Geom. Phys. 32, 79--96 (1999; Zbl 0961.53047)]. Remarks on the relation between quantization and the index theorem are also given [cf. {\it N. Berline}, {\it E. Getzler} and {\it M. Vergne}, Heat kernels and Dirac operators, Springer (1992; Zbl 0744.58001)]. Chapter 7 deals with classification of spin geometry (spectral triple). It provides the notion of connections and gives a Morita equivalent condition of two spin geometries in terms of connections. Then taking noncommutative torus as an example, these results are examined. This chapter is concluded to describe to noncommutative gauge potential. Spin geometry is not unique for a given Riemannian metric. But in the commutative case, selection of spin geometry is possible by minimization of an action functional. In the noncommutative case, such selections may not be possible. It leads to the phenomenon of spontaneous symmetry breaking [{\it M. Dubois-Violette}, {\it R. Kerner} and {\it J. Madore}, Gauge bosons in a noncommutative geometry, Phys. Lett. B 217, 485--488 (1989)]. Chapter 8 deals with action functional in noncommutative geometry, and the Chamseddine-Connes bosonic action $B_\phi(D)= \text{Tr\,}\phi(D^2/\Lambda^2)$, $\Lambda$ is the cutoff, is presented. For the Standard Model plus gravity, the expansion at $\Lambda\to\infty$ of this action is obtained by using Wodzicki residue [{\it A. H. Chamseddine} and {\it A. Connes}, Commun. Math. Phys. 186, No. 3, 731--750 (1997; Zbl 0894.58007)]. This chapter is concluded sketching this calculation. Developements of noncommutative geometry mainly motivate its application to string theory, and M-theory, reviewed in Chapter 9.
MSC 2000:
*58B34 Noncommutative geometry (a la Connes)
46L87 Noncommutative differential geometry
81T75 Noncommutative geometry methods
58-01 Textbooks (global analysis)
58-02 Research monographs (global analysis)
53C27 Spin and Spin$^c$ geometry

Keywords: spectral triple; noncommutative torus; Dixmier trace; Moyal quantization; action functional; spin geometry

Cited in: Zbl 1210.58006

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