Noncommutative Geometry. Nigel Higson Penn State University

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1 Noncommutative Geometry Nigel Higson Penn State University

2 Noncommutative Geometry Alain Connes 1

3 What is Noncommutative Geometry? Geometric spaces approached through their algebras of functions. The spaces are often very singular (defined by equivalence relations, or even groupoids). The function algebras are typically noncommutative. The algebras/spaces are analyzed using Hilbert space tools. In particular, spectral properties of algebras, viewed as algebras of operators on Hilbert space, are crucial. One might call the subject spectral geometry. 2

4 What are its Origins? Werner Heisenberg What Heisenberg understood... is that [the] Ritz-Rydberg combination principle actually dictates an algebraic formula for the product of any two observable physical quantities... 3

5 Heisenberg wrote down the formula for the product of two observables; and he noticed of course that this algebra is no longer commutative,... The right way to think about this new phenomenon is to think in terms of a new kind of space in which the coordinates do not commute. The starting point of noncommutative geometry is to take this new notion of space seriously. Alain Connes Noncommutative geometry, Year

6 Commentary of Riemann... it seems that the empirical notions on which the metric determinations of space are based... lose their validity in the infinitely small; it is therefore quite definitely conceivable that the metric relations of space in the infinitely small do not conform to the hypotheses of geometry; and in fact one ought to assume this as soon as it permits a simpler way of explaining phenomena. Bernhard Riemann On the Hypotheses which lie at the Foundations of Geometry 5

7 What Has Noncommutative Geometry Accomplished? Manifold topology (progress on the Novikov conjecture, Gromov-Lawson conjecture, etc). Harmonic analysis, especially of discrete groups. Models in physics (notably of the quantum Hall effect). Foliation theory and Atiyah-Singer index theory, on singular spaces, or parametrized by singular spaces. In addition, NCG may offer the prospect for progress in fundamental physics, arithmetic,... 6

Spectral Theory and Hilbert Space David Hilbert in 1900 In the winter of 1900-1901 the Swedish mathematician Holmgren reported in Hilbert s seminar on

8 Spectral Theory and Hilbert Space David Hilbert in 1900 In the winter of the Swedish mathematician Holmgren reported in Hilbert s seminar on Fredholm s first publications on integral equations, and it seems that Hilbert caught fire at once... Hermann Weyl David Hilbert and his mathematical work 7

9 ..., the equation! % & '(% $ Helmholtz Equation Hilbert saw two things: (1) after having constructed Green s function for a given region and for the potential equation!#" $ for the oscillating membrane changes into a homogeneous integral equation % *)+ & ' *)-,/.0 % $ with the symmetric,.-,4)+ *)-,/.+... ; (2) the problem of ascertaining the eigen values ' and eigen functions % *)+ of this integral equation is the analogue for integrals of the transformation of a quadratic form of 5 variables onto principal axes. Hermann Weyl David Hilbert and his mathematical work 8

10 Problem of H.A. Lorentz... there is a mathematical problem which will perhaps arouse the interest of mathematicians... In an enclosure with a perfectly reflecting surface there can form standing electromagnetic waves analogous to tones of an organ pipe... there arises the mathematical problem to prove that the number of sufficiently high overtones which lie between6 and is independent of the shape of the enclosure and is simply proportional to its area. H.A. Lorentz Wolfskehl Lecture,

11 B ' C # eigenvalues of! less than or equal to'. Reformulation!#"98 ':8+";8 " 8=<?>A@ $ DFEHG lim B ' ' Area constant This is equivalent to the asymptotic relation 8 EHG ' 8 lim 5 constant Area 10

The idea was one of many, as they probably come to every young person preoccupied with science but while others soon burst like soap bubbles, this one soon led, as a short inspection showed, to the

12 The idea was one of many, as they probably come to every young person preoccupied with science but while others soon burst like soap bubbles, this one soon led, as a short inspection showed, to the goal. I was myself rather taken aback by it as I had not believed myself capable of anything like it. Added to that was the fact that the result, although conjectured by physicists some time ago, appeared to most mathematicians as something whose proof was still far in the future. Hermann Weyl Gibbs Lecture,

13 Compact Operators Definition. A bounded linear operatori JK L K on a Hilbert space is compact if it maps the closed unit ball of Hilbert space to a (pre)compact set. Example. IfI is a norm-limit of finite-rank operators theni is compact. Elementary calculus M the maximum value of the function N POQ R S I OTSVU on the closed unit ball of K is an eigenvalue for IXWYI. Theorem (Hilbert et al). If I JZK L K is a compact and selfadjoint operator then there is an orthonormal basis for K comprised of eigenvectors for I. Thus I [ \ \ '^] ' U `` ':_... Theorem (Rellich Lemma).!ba ] is a compact operator. 12

14 Spectral Theory for the Laplacian c d! e W e The Laplace operator. Theorem. There is an orthonormal basis for f U c consisting of functions gf8 for which!hgv8 ':8igV8 in the distributional sense. The eigenvalues 'j8 are positive and converge to infinity. Spec! Remark. In fact one can show that gf8 k l G c. This follows from elliptic regularity. 13

15 `` Singular Values Definition. The singular values m ] I n, m U I n,opq of a bounded operatori are the scalars mi8 I r dim srut inf8 a ] vnw s sup S I OTS SYOTS m^] I x m U I yx qz Observe that and that I { is compact 8 EHG m 8 I r $2 lim I Now let I be O}, O=~ compact, x $ self-adjoint, and positive ' 8 I (meaning ). List the eigenvalues in decreasing order, and with multiplicity. Theorem. If I is compact, self-adjoint, and positive then m8 I r ':8 I. Proof. I \ \ ' ] ' U ' _... 14

16 Trace Class Operators Lemma. mi8 I ] 7 I U mi8 I ] 7 mi8 I U m U 8 I ] 7 I U V mi8 ƒ I n, mi8 I Q y S TS mi8 I o Definition. The trace ideal in K is ] K C I < mi8 I Qˆ Š= Definition. If I k ] K then Tr I r G Pt ] O, I O ~ O ],qzœ,*oj Ž is an orthonormal set then < O 8 t 8, I O 8 ~ < ] 8 t m 8 I V ] The sum is over an orthonormal basis. Note: if 15

17 As with the usual trace, k K F, I k ] K M Tr ƒ I Tr I Q V Example. If is smooth on c c and if 9g } C, H g P 1, then is a trace-class operator, and Tr y y, } 1 } 16

18 Dixmier Trace Definition. ] G K C I š š š š sup8 5 im 8 I rˆ Observe that ] K œ ] G K œ K. Definition. If I k ] G K is positive and LIMž is a Banach limit then define Trž I ƒ LIMž LIMž Ÿ B 8 mi8 I log Ÿ B 8: ' 8 I V log Theorem (Dixmier). If LIMž has the property LIMž Y ], U, _,pqz C LIMž / ], ], U, U, zqp then Trž I ] 7 I U y Trž I ] 7 Trž I U. 17

19 Integration Now back to Weyl s Theorem... ':8! [ vol c 5 c d Weyl s Theorem shows that Trž! a C vol c o More generally, given g3j c L we get Trž g Œ! a C g 1 vol This suggests that! a is some sort of volume element for the manifold c... 18

20 Spectral Triples Definition. A spectral triple is a triple, K,V consisting of a separable Hilbert space K, an algebra of bounded operators on K, and a (typically unbounded) selfadjoint operator on K, for which: the operator Y 7 U a ] is compact, and if ª k then the commutator «, ªi ª & ª extends to a bounded operator on K. According to Connes, spectral triples constitute an extension of the notion of Riemannian geometric space which is broadly applicable to problems in fundamental physics, number theory,.... Remark. If has no unit, replace A 7 U a ] with ª A 7 U a ] in the above. 19

21 Basic ideas: The Standard Example Regard as a square root of!. Think of «, ªi as a gradient of ª k. The simplest case is l G F, K f U F, & Ÿ Ÿ 1 1i The theory of Dirac-type operators in geometry provides further commutative examples in the context of Riemannian manifolds W Dirac Operator U e W e U e on forms W e on spinors 20

22 , The Operator F Write! U! ± and so that W U [ 4 and In the simplest case this is the Hilbert transform: g } C ²r³ Ÿ g P & 1 H The operator is important! Roughly! ± speaking the distinction between and corresponds to the distinction between densities and differential forms (on manifolds). 21

23 ¹ ± ¹ µ Groupoids and Quotients Let µ be a smooth étale groupoid: range Obj µ source (source and range are local diffeomorphisms). Let l G µ and define g ]i g U h C t ¹ g ] ] g U U V We obtain the convolution algebra of µ. 22

24 Examples Heisenberg example,º» ¼ ½ ¾ÁÀÃÂ Kronecker foliation,º» ÄoÅHÆÈÇ ÉÅCÇ» Ê ÌËÎÍÐÏ Ç^ÅZÑ 23

25 Infinitesimals and Differentials Definition. An infinitesimal of order is a compact operatori m8 I Ò 5 a for which. Definition. For ª k define 1 ª «Ó, ªi. Example. In the basic case, 1 g operator with integral kernel g } & g P &, give or take a factor of²r³. If we grade differentials ª=ÔÕ«Ó, ª ] qp «Ó, ªiÖ according to degree and use the graded commutator then 1 U Ø $=, as in de Rham theory. 24

26 Connes dictionary Geometric space Spectral triple Ì, K,V Complex variable Operator in Real variable Selfadjoint operator in Infinitesimal Compact operator 1 ª «Ó, ªi Differential Commutator Ù ª Integral Dixmier trace Trž ª

27 Zeta Functions Theorem. If ) Ú d U then! a:û is a trace-class operator. ÜÞÝ Proof. Follows from Rellich Lemma. Theorem (Minakshisundaram ß and Pleijel). The zeta function È)à C! a(á Tr is meromorphic on with only simple poles. Residues Vanish Actual Poles Singularities of âz¾äãåâ for a closed surface. 26

28 8 Weyl s Theorem Residues Vanish Actual Poles I Abelian-Tauberian Theorem. Let be a positive, invertible operator ) Ú I a:û and assume that is traceclass for all Ÿ. Then B æ DFEHG ' lim ' l { *) lim Ûƒç ] & Ÿ I a:û l Tr See Hardy, Divergent Series. The theorem says DFEHG Ÿ lim ' D/è D Ÿ l { lim Ûƒç ] È) & Ÿ ' 8a:Û l 27

29 Ÿ Ÿ è Û The proof of meromorphicity uses pseudodifferential operators [ Ié! a ápê, I differential of order5 Lemma (Guillemin). Suppose that for every holomorphic family Û ë ] there are pseudodifferential, í Û î and Ûa ] such that Y1 7 )+ ï «ðë ì ], í Û 27 î Ûa ] ì Then Tr ƒ Û is meromorphic, with simple poles. Proof. If Re È)à Tñ $ then Tr Û C 1 7 ) ò Tr «ðë ì ], í Û 7 Tr î Ûa ] ó 1 7 ) Tr î Ûa ] Hence Tr ƒ Û y /1 7 )à a ] Tr î Ûa ]. 28

30 The poles of Tr ƒ Û are located at& 1ô, Ÿ & 1ô,/õ & 1ô,qz. & 1 Ÿ & 1 Domain of Tr Û Domain of Tr î Ûa ] 29

31 Lemma. If is (pseudo)differential of order5 then d «,Î t ö 5 & î, ] ö where î has order5 & Ÿ, and hence /1 7 5 ï d t ],ï ö Èø & ö d t ], ö ø 7 î ö Proof. This follows from the Heisenberg relations ö,ï ø ù F ö As a result, Weyl s Theorem follows from Guillemin s Lemma. 30

32 Cyclic Cocycles Let, K,4 be a spectral triple. Proposition. The formula N ª Ô, ª ],qpœ, ª 8 C Tr /ú ª Ô «Ó, ª ] û z q Ó«Ó, ª 8 defines a multlinear functional on following properties: N ª Ô, ª ], zqõ, ª 8 C & Ÿ 8 N ª ],qzœ, ª 8, ª Ô with the ü N ªjÔ, zqõ, ª 80ý ] C $, where ü N ª Ô,qpŒ, ª 80ý ] C N ª Ô ª ],qpœ, ª 80ý ] & N ª Ô, ª ] ª U,qzŒ, ª 80ý ] 7 pq 7 & 8+ý ] Ÿ N ª 80ý ] ª Ô, pqþ, ª 8 V 31

33 Cyclic Cohomology Lemma. Let N be a cyclic5 -linear functional. Then ü N is a cyclic 5 7 Ÿ -linear functional, and ü U N $. Definition. Let be an algebra. cohomology group of is The 5 th cyclic K l 8 Ì y cyclic5 -cocycles modulo cyclic coboundaries. N The cocycle is a, K sort,v of fundamental class for the spectral triple. It reflects information from index theory: ÿt zÿ# T & è Index Ÿ N äÿ, ÿh,ppõ,èÿ# o 32

34 Continuation of the Dictionary. 1. ª «Ó, ªi Differential Commutator de Rham theory Cyclic cohomology.. 33

35 Ö Ô ± Local Index Formula Theorem (Connes and Moscovici). Let, K,V be a spectral triple with simple dimension spectrum. The local formula ª Ô,qzŒ, ª Ö C t Ö Res Û Ô ò ú ª Ô «, ª ] Tr ð q p à«, ª Ö ð! a a a:û ó, where Ö & Ÿ qp ] Ö ] 7 Ÿ < < 7 ÖU U 7 õ q p Ö 7 ÿx defines a cocycle which is cohomologous to the fundamental cocycle N. Notation. I ð U, U,qp «U, I qp. 34

36 Comments The formula is a starting point for Atiyah-Singer index theory in noncommutative geometry. In the classical case (Riemmanian manifolds) the residues are computable from the coefficients of (Seeley, Wodzicki, et al). Small (smoothing operator) changes in leave the index formula invariant. In the noncommutative world, local means concentrated at in momentum space (c.f. Fourier theory). 35

The Local Index Formula in Noncommutative Geometry

The Local Index Formula in Noncommutative Geometry Nigel Higson Pennsylvania State University, University Park, Pennsylvania, USA (Dedicated to H. Bass on the occasion of his 70th birthday) Lectures given