Noncommutative Geometry and Physics: Renormalisation, Motives, Index Theory (ESI Lectures in Mathematics and Physics)

ESI Lectures in Mathematics and Physics Editors Joachim Schwermer (Institute for Mathematics, University of Vienna) Jakob Yngvason (Institute for Theoretical Physics, University of Vienna) Erwin Schrödinger International Institute for Mathematical Physics Boltzmanngasse 9 A-1090 Wien Austria The Erwin Schrödinger International Institute for Mathematical Phyiscs is a meeting place for leading experts in mathematical physics and mathematics, nurturing the development and exchange of ideas in the international community, particularly stimulating intellectual exchange between scientists from Eastern Europe and the rest of the world. The purpose of the series ESI Lectures in Mathematics and Physics is to make selected texts arising from its research programme better known to a wider community and easily available to a worldwide audience. It publishes lecture notes on courses given by internationally renowned experts on highly active research topics. In order to make the series attractive to graduate students as well as researchers, special emphasis is given to concise and lively presentations with a level and focus appropriate to a student's background and at prices commensurate with a student's means. Previously published in this series: Arkady L. Onishchik, Lectures on Real Semisimple Lie Algebras and Their Representations Werner Ballmann, Lectures on Kähler Manifolds Christian Bär, Nicolas Ginoux, Frank Pfäffle, Wave Equations on Lorentzian Manifolds and Quantization Recent Developments in Pseudo-Riemannian Geometry, Dmitri V. Alekseevsky and Helga Baum (Eds.) Boltzmann's Legacy, Giovanni Gallavotti, Wolfgang L. Reiter and Jakob Yngvason (Eds.) Hans Ringström, The Cauchy Problem in General Relativity Emil J. Straube, Lectures on the
2-Sobolev Theory of the ∂⎯ -Neumann Problem

Noncommutative Geometry and Physics: Renormalisation, Motives, Index Theory Alan Carey Editor with the assistance of Harald Grosse and Steve Rosenberg

Editor: Alan Carey Mathematical Sciences Institute Australian National University Canberra, ACT, 0200 Australia E-mail: [email protected]

2010 Mathematics Subject Classification (primary; secondary): 58B34, 11M55, 11G09 11M06, 11M32, 47G30, 81Q30, 81T15, 17A30; 16T30, 18D50, 46L80, 46L87, 19K33, 19K56, 58J42, 58J20, 81Q30 Key words: Arithmetic geometry, motives, noncommutative geometry, index theory, K-theory, pre-Lie algebras, Feynman integrals, zeta functions, Birkhoff–Hopf factorisation, renormalisation

ISBN 978-3-03719-008-1 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained.

© 2011 European Mathematical Society Contact address: European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum FLI C4 CH-8092 Zürich Switzerland Phone: +41 (0)44 632 34 36 Email: [email protected] Homepage: www.ems-ph.org Typeset using the author's TEX files: I. Zimmermann, Freiburg Printed in Germany 987654321

Introduction This volume is a collection of expository articles on mathematical topics of current interest that have been partly stimulated or influenced by interactions with physical theory. Some of these articles are based on lectures given at various venues over the last few years, revised and written especially for this volume. They cover a diverse range of mathematical topics stemming from different parts of physics. The major underlying theme of the volume involves the interactions between physics and number theory. This theme is manifested in two ways, first through the study of Feynman integrals and renormalisation theory, and second, through the application of methods from quantum statistical mechanics. In the former, the work of Bogoliubov– Parasuik–Hepp–Zimmermann (BPHZ) on renormalisation theory gave a method for step by step control of divergences and of their regularisation in Feynman’s approach to perturbative quantum field theory. Already, in the evaluation of Feynman integrals, the occurrence of multizeta values hinted at deeper mathematical connections. This deeper underlying structure was found by Alain Connes and Dirk Kreimer in the form of associating to each renormalisable quantum field theory a Hopf algebra that provided a systematic understanding of the BPHZ procedure in terms of a Birkhoff factorisation in a Lie group associated to the Hopf algebra. A contemporary view of this fundamental work is provided here by Christoph Bergbauer. The latter strand, the relationship between statistical mechanics and number theory, began much earlier through the work of Jean-Benoît Bost and Alain Connes. Bost–Connes introduced a quantum statistical mechanical dynamical system that captures information on the primes and on the Riemann zeta function. They determined the equilibrium states of their model (the socalled Kubo–Martin–Schwinger (KMS) states) and its phase transitions. Subsequent extensions of this basic idea to number theoretic questions resulted in the theory of ‘endomotives’. Motives also arise in the study of Feynman integrals and hence we have provided here an introduction to these ideas in the Ramdorai–Plazas–Marcolli article. We now give a brief overview of the contents of this volume. Bergbauer’s article discusses Feynman integrals, regularization and renormalization following the algebraic approach to the Feynman rules developed by Bloch, Connes, Esnault, Kreimer, and others. It reviews several renormalization methods found in the literature from a single point of view using resolution of singularities, and includes a discussion of the motivic nature of Feynman integrals. Motives are explained in much greater detail in the article of Sujatha Ramdorai and Jorge Plazas. The construction of the category of pure motives is explained here starting from the category of smooth projective varieties. They also survey the theory of endomotives developed by D. C. Cisinski and G. Tabuada, which links the theory of motives to the quantum statistical mechanical techniques that connect number theory and noncommutative geometry. The appendix to this article, contributed by Matilde Marcolli, elaborates these latter ideas providing a useful introduction to, and summary

vi

Introduction

of, the role of KMS states. Also described is the view of motives that arises from noncommutative geometry along with a detailed account of the interweaving of number theory with statistical mechanics, noncommutative geometry and endomotives. Hopf algebras associated to rooted trees are known to systemetise the combinatorics of Feynman graphs through the work of Connes–Kreimer. Dominique Manchon describes another algebraic object associated to rooted trees, namely pre-Lie algebras. His article reviews the basic theory of pre-Lie algebras and also describes how they arise from operads. Their relation to other algebraic structures and their application to numerical analysis are described as well. Multiple zeta values arise from the evaluation of Feynman integrals. Sylvie Paycha’s contribution discusses generalisations of renormalised multiple zeta values at nonpositive integers. As with some of the other articles, the exposition is partly inspired by renormalised Feynman integrals in physics using pseudodifferential symbols. Zeta residues and pseudodifferential analysis, both of which play a role in earlier articles, also arise in index theory. In turn, index theory is well known to play a role in gauge field theories via the study of anomalies. These are described mathematically by the families index theorem. Under the influence of Alain Connes and others, a noncommutative approach to index theory, partly inspired by quantum theory, has emerged over the last two decades. This noncommutative index theory is described in the contribution of Alan Carey, John Phillips and Adam Rennie, beginning with a discussion of classical index theorems from a noncommutative point of view. This is followed by a review of K-theory and cyclic cohomology that culminates in a description of the local index formula in noncommutative geometry. The article concludes with an example that underlies recent applications of noncommutative geometry to Mumford curves. All of the articles in this volume were partly inspired by a program in Number Theory and Physics held at the Erwin Schrödinger Institute from March 2 to April 18, 2009. The editor would like to thank ESI for its hospitality and support for the Number Theory and Physics program, Arthur Greenspoon for his enthusiastic assistance with proof-reading and Steve Rosenberg and Irene Zimmermann for their very valuable contributions to the preparation of this volume.

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Notes on Feynman integrals and renormalization by Christoph Bergbauer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Introduction to motives. With an appendix by Matilde Marcolli by Sujatha Ramdorai, Jorge Plazas, and Matilde Marcolli . . . . . . . . . . . . . . . . . . . . . . . 41 A short survey on pre-Lie algebras by Dominique Manchon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 Divergent multiple sums and integrals with constraints: a comparative study by Sylvie Paycha . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Spectral triples: examples and index theory by Alan Carey, John Phillips, and Adam Rennie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 List of Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

Notes on Feynman integrals and renormalization Christoph Bergbauer

Contents 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

2

Feynman graphs and Feynman integrals . . . . . . . . . . . . . . . . . . .

2

3

Regularization and renormalization . . . . . . . . . . . . . . . . . . . . . .

14

4

Motives and residues of Feynman graphs . . . . . . . . . . . . . . . . . . .

28

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

1 Introduction In recent years there has been a growing interest in Feynman graphs and their integrals. Physicists use Feynman graphs and the associated integrals in order to compute certain experimentally measurable quantities out of quantum field theories. The problem is that there are conceptual difficulties in the definition of interacting quantum field theories in four dimensions. The good thing is that nonetheless the Feynman graph formalism is very successful in the sense that the quantities obtained from it match with the quantities obtained in experiment extremely well. Feynman graphs are interpreted as elements of a perturbation theory, i.e., as an expansion of an (interesting) interacting quantum field theory in the neighborhood of a (simple) free quantum field theory. One therefore hopes that a better understanding of Feynman graphs and their integrals could eventually lead to a better understanding of the true nature of quantum field theories, and contribute to some of the longstanding open questions in the field. A Feynman graph is simply a finite graph, to which one associates a certain integral: the integrand depends on the quantum field theory in question, but in the simplest case it is just the inverse of a direct product of rank 4 quadratic forms, one for each edge of the graph, restricted to a real linear subspace determined by the topology of the graph. For a general graph, there is currently no canonical way of solving this integral analytically. However, in this simple case where the integrand is algebraic, one can be convinced to regard the integral as a period of a mixed motive, another notion which is not rigorously defined as of today. All the Feynman periods that have been computed so far, are rational linear combinations of multiple zeta values, which are known to be periods of mixed Tate motives, a simpler and better understood kind of motives. A stunning theorem of Belkale and Brosnan however indicates that this is possibly a coincidence due to the relatively small number of Feynman periods known today: They showed that in fact any algebraic variety defined over Z is related to a Feynman graph

2

C. Bergbauer

hypersurface (the Feynman period is one period of the motive of this hypersurface) in a quite obscure way. The purpose of this article is to review selected aspects of Feynman graphs, Feynman integrals and renormalization in order to discuss some of the recent work by Bloch, Esnault, Kreimer and others on the motivic nature of these integrals. It is based on public lectures given at the ESI in March 2009, at the DESY and IHES in April and June 2009, and several informal lectures in a local seminar in Mainz in fall and winter 2009. I would like to thank the other participants for their lectures and discussions. Much of my approach is centered around the notion of renormalization, which seems crucial for a deeper understanding of Quantum Field Theory. No claim of originality is made except for Section 3.2 and parts of the surrounding sections, which is a review of my own research with R. Brunetti and D. Kreimer [10], and Section 3.6 which contains new results. This article is not meant to be a complete and up to date survey by any means. In particular, several recent developments in the area, for example the work of Brown [24], [26], Aluffi and Marcolli [3], [2], [1], Doryn and Schnetz [35], [76], and the theory of Connes and Marcolli [32], [67] are not covered here. Acknowledgements. I thank S. Müller-Stach, R. Brunetti, S. Bloch, M. Kontsevich, P. Brosnan, E. Vogt, C. Lange, A. Usnich, T. Ledwig, F. Brown and especially D. Kreimer for discussions on the subject of this article. I would like to thank the ESI and the organizers of the spring 2009 program on number theory and physics for hospitality during the month of March 2009, and the IHES for hospitality in January and February 2010. My research is funded by the SFB 45 of the Deutsche Forschungsgemeinschaft.

2 Feynman graphs and Feynman integrals For the purpose of this article, a Feynman graph is simply a finite connected multigraph where “multi” means that there may be several, parallel edges between vertices. Loops, i.e., edges connecting to the same vertex at both ends, are not allowed in this article. Roughly, physicists think of edges as virtual particles and of vertices as interactions between the virtual particles corresponding to the adjacent edges. If one has to consider several types of particles, one has several types (colors, shapes etc.) of edges. Here is an example of a Feynman graph: (1) This Feynman graph describes a theoretical process within a scattering experiment: a pair of particles annihilates into a third, intermediate, particle, and this third particle then decays into the two outgoing particles on the right.

Notes on Feynman integrals and renormalization

3

This Feynman graph (and the probability amplitude assigned to it) make sense only as a single term in a first order approximation. In order to compute the scattering cross section, one will have to sum over arbitrarily complicated Feynman graphs with four fixed external edges, and in this sum an infinity of graphs with cycles will occur, for example:

In this article we will be concerned only with Feynman graphs containing cycles, and I will simply omit the external edges that correspond to the (asymptotic) incoming and outgoing physical particles of a scattering experiment. I will come back to the physical interpretation in greater detail in Section 2.3. 2.1 Feynman rules. Feynman graphs are not only a nice tool for drawing complex interactions of virtual particles; they also provide a recipe to compute the probability that certain scattering processes occur. The theoretical reason for this will be explained later, but to state it very briefly, a Feynman graph is regarded as a label for a term in a perturbative expansion of this probability amplitude. This term in this expansion is called a Feynman integral, but at this point one must be careful with the word integral because of reasons of convergence. n Definition 2.1. An integral is a pair .A; Su/ where A is an open subset of some R or n n R0 , and u a distribution in A \ .R n Hi /, where Hi are affine subspaces.

A distribution in X is a continuous linear functional on the space of compactly supported test functions C01 .X / with the usual topology. Locally integrable functions (that is, functions integrable on compact subsets) define distributions in an obvious way. Let us denote by 1A the characteristic function of A in Rn . It is certainly not a test function unless A is compact, but for suitable u (decays rapidly enough at 1) we may evaluate u against 1A . We write uŒf  for the distribution applied toR the test function f . If u is given by a locally integrable function, we may also write u.x/f .x/dx. If u is given by a function which is integrable over all of A, then .A; u/ can be R associated with the usual integral A u.x/dx D uŒ1RA . Feynman integrals however are very often divergent: this means by definition that A u.x/dx is divergent, and this can either result from problems with local integrability at the Hi or lack of integrability at 1 away from the Hi (if A is unbounded), or both. (A more unified point of view would be to start with a P n instead of Rn in order to have the divergence at 1 as a divergence at the hyperplane H1 at 1, but I will not exploit this here.) A basic example of such a divergent integral is the pair A D R n f0g and u.x/ D jxj1 . The function u is locally integrable inside A, hence a distribution in A. But it is neither integrable as jxj ! 1, nor locally integrable at f0g. We will see in a

4

C. Bergbauer

moment that the divergent Feynman integrals to be defined are higher-dimensional generalizations of this example, with an interesting arrangement of the Hi . In this section I will introduce three kinds of Feynman integrals associated to a given graph: a position space, a momentum space and a parametric integral. The first two are related by a Fourier transform, and the second and the third by a change of variables. It may be useful to emphasize at this point that all three amount to the same “value” once they are properly and consistently renormalized (a notion that I will introduce in the next section). The following approach, which I learned from S. Bloch [15], [14], is quite powerful when one wants to understand the various Feynman rules from a common point of view. It is based on the idea that a Feynman graph first defines a point configuration in some Rn , and it is only this point configuration which determines the Feynman integral via the Feynman rules. Let  be a Feynman graph with set of edges E./ and set of vertices V ./. A subgraph  has by definition the same vertex set V . / D V ./ but E. /  E./. Impose temporarily an orientation of the edges, such that every edge has an incoming ve;in and an outgoing vertex ve;out . Since we do not allow loops, the two are different. Set .v W e/ D 1 if v is the outgoing vertex of e, .v W e/ D 1 if v is the incoming vertex and e, and .v W e/ D 0 otherwise. Let M D Rd , where d 2 2 C 2N, called space-time, with Euclidean metric j  j. We will mostly consider the case where d D 4, but it is useful to see the explicit dependence on d in the formulas. All the information of  is encoded in the map @

ZE./ !  ZV ./ P sending an edge e 2 E./ to @.e/ D v2V ./ .v W e/v D ve;out  ve;in . This is nothing but the chain complex for the oriented simplicial homology of the 1-dimensional simplicial complex , and it is a standard construction to build from this map @ an exact sequence @

0 ! H1 .I Z/ ! ZE./ !  ZV ./ ! H0 .I Z/ ! 0:

(2)

From this one obtains two inclusions of free abelian groups into ZE./ : i W H1 .I Z/ ,! ZE./ I the second one is obtained by dualizing @_

j W ZV ./_ =H 0 .I Z/ ,! ZE./_ : Here, and generally whenever a basis is fixed, we can canonically identify free abelian groups with their duals. All this can be tensored with R, and we get inclusions i , j of vector spaces into another vector space with a fixed basis. If one then replaces any Rn by Mn and denotes

Notes on Feynman integrals and renormalization

5

i˚d D .i ; : : : ; i /, j˚d D .j ; : : : ; j /, then two types of Feynman integrals .A; u/ are defined as follows: AM D H1 .I R/d ;

˚d  ˝jE./j ; uM  D .i / u0;M

AP D MV ./_ =H 0 .I R/d ;

˚d  ˝jE./j uP :  D .j / u0;P

The distributions u0;M ; u0;P 2 D 0 .M/ therein are called momentum space, resp. position space propagators. Several examples of propagators and how they are related will be discussed in the next section, but for a first reading u0;M .p/ D

1 ; jpj2

u0;P .x/ D

1 ; jxjd 2

inverse powers of a rank d quadratic form. As announced earlier, the pullbacks .i˚d / u˝jE./j and .j˚d / u˝jE./j are only defined as distributions outside certain 0;M 0;M affine spaces Hi , that is, for test functions supported on compact subsets which do not meet these Hi . The map  7! .AM ; uM  / is called momentum space Feynman rules, and the map  7! .AP ; uP / is called position space Feynman rules. Usually, in the physics literature, the restriction to the subspace is imposed by multiplying the direct product of propagators with several delta distributions which are interpreted as “momentum conservation” at each vertex in the momentum space picture, and dually “translation invariance” in the position space case. In position space, it is immediately seen that Q ˚d  ˝jE./j uP D  u0;P .xe;out  xe;in /;  D .j / u0;P e2E./

where  means pushforward along the projection  W M V ./_ ! M V ./_ =H 0 ./d , see [10]. In momentum space, things are a bit more complicated. Definition 2.2. A connected graph  is called core if rk H1 . n feg/ < rk H1 ./ for all e 2 E./. By Euler’s formula (which follows from the exactness of (2)) rk H1 ./  jE./j C jV ./j  rk H0 ./ D 0; it is equivalent for a connected graph  to be core and to be one-particle-irreducible (1PI), a physicists’ notion:  is one-particle-irreducible if removing an edge does not disconnect .

6

C. Bergbauer

Let now  be connected and core; then  P  Q Q ˚d  ˝jE./j uM D u0;M .pe / ı0 .v W e/pe :  D .i / u0;M e2E./

v2V ./

e2E./

This is simply because im i D ker @, and because for   P P P pe .v W e/v D 0 @ pe e D e2E./

v

e2E./

it is necessary that P

.v W e/pe D 0

for all v 2 V ./:

e2E./

(The requirement that  be core is really needed here because otherwise certain e 2 E./ would never show up in a cycle, and hence would be missing inside the delta function.) Moreover, one can define a version of uM  which depends additionally on external momenta PvP2 M, one for each v 2 V ./, up to momentum conservation for each component v2C Pv D 0: UM .fPv gv2V ./ / D

Q e2E./

u0;M .pe /

Q

ı0 .Pv C

v2V ./

P

.v W e/pe /:

e2E./

By a slight abuse of notation I keep the Pv , v 2 V ./, as coordinate vectors for MjV ./j =H 0 .; R/d D AP Q and identify P distributions on AP with distributions on MjV ./j that are multiples of C ı0 . v2C Pv /. UM is now a distribution on a subset of AP  AM , and UM jPv D0;v2V ./ D uM  : The vectors Pv 2 AP determine a shift of the linear subspace AM D H1 .I R/˚d ,! MjE./j to an affine one. Usually all but a few of the Pv are set to zero, namely all but those which correspond to the incoming or outgoing particles of an experiment (see Section 2.3). The relation between the momentum space and position space distributions is then a Fourier duality. I denote by F the Fourier transform. Proposition 2.1. If the basic propagators are Fourier-dual (F u0;P D u0;M ), as is the case for u0;M .p/ D jpj1 2 and u0;P .x/ D jxjd12 , then .UM Œ1AM /.fPv g/ D F uP  where only the (internal) momenta of AM are integrated out; and this holds up to convergence issues only, i.e., in the sense of Definition 2.1.

7

Notes on Feynman integrals and renormalization

For example, the graph

gives rise to 2 uM 3 D u0;M .p1 /u0;M .p2 /u0;M .p1 C p2 /u0;M .p3 /u0;M .p2 C p3 /; 2 uP 3 D u0;P .x1  x2 /u0;P .x1  x3 /u0;P .x2  x3 /u0;P .x2  x4 /u0;P .x3  x4 /;

where p1i ; : : : ; p3i , i D 0; : : : ; d  1 is a basis of coordinates for AM and x1i ; : : : ; x4i , i D 0; : : : ; d  1 is a basis of coordinates for MV .3 /_ (If  is connected, dividing by H 0 .I R/d takes care of the joint (diagonal) translations by M and, as previously, instead of writing distributions on M V ./_ =H 0 .I R/d , I take the liberty of writing translation-invariant distributions on M V ./_ /. Finally, the case of external momenta: UM3 .P1 ; P2 ; 0; P4 / D u0;M .p1 /u0;M .p1 C P1 /u0;M .p2 /  u0;M .p1 C p2 C P1 C P2 /u0;M .p3 /  u0;M .p2 C p3 C P4 /ı0 .P1 C P2 C P4 /:

(3)

I set one of the external momenta, P3 , to zero in order to have a constant number of 4 adjacent (internal and external) momenta at each vertex: P1 is the sum of two external momenta at the vertex 1 (see Section 2.3 for the reason). P We will come back to the question of the affine subspaces Hi where uM  , resp. u is not defined in the section about renormalization. In general, following [15], Section 2, a configuration is just an inclusion of a vector space W into another vector space RE with fixed basis E: The dual basis vectors e _ , e 2 E determine linear forms on W , and those linear forms (or dually the linear hyperplanes annihilated by them) are the “points” of the configuration in the usual sense. By the above construction, any such configuration, plus the choice of a propagator, defines an integral. If the configuration comes from a Feynman graph, the integral is called a Feynman integral. 2.2 Parametric representation. Integrals can be rewritten in many ways, using linearity of the integrand, of the domain, change of variables and Stokes’ theorem, and possibly a number of other tricks. For many purposes it will be useful to have a version of the Feynman rules with a domain A which is much lower-dimensional than in the previous section but has boundaries and corners. The first part of the basic trick here is to rewrite the propagator Z 1 u0 D exp.ae u1 0 /dae 0

8

C. Bergbauer

(whenever the choice of propagator allows this inversion; u0 .p/ D jpj1 2 certainly does), introducing a new coordinate ae 2 R0 for each edge e 2 E./. From this one has a distribution   N P exp.ae u1 ae u1 (4) 0 .pe // D exp  0 .pe / e2E./

e2E./

in .M  R0 /jE./j . From now on I assume u0 .p/ D

1 . jpj2

Suppose that i W W ,!

RjE./j is an inclusion. Once a basis of W is fixed, the linear form e _ i is a row vector in W and its transpose .e _ i /t a column vector in W . The product .e _ i /t .e _ i / is then a dim W -square matrix. Pulling back (4) along an inclusion i ˚d W W ,! MjE./j (such as i ˚d D i˚d or i ˚d D j˚d / means imposing linear relations on the pe . These relations can be transposed onto the ae : after integrating Gaussian integrals over W (this is the second part of the trick) and a change of variables, one is left with the distribution  d=2 P uS .fae g/ D det ae .e _ i /t .e _ i / e2E./

on AS D RjE./j except for certain intersections Hi of coordinate hyperplanes fae D 0 0g. I discarded a multiplicative constant C D .2/d dim W=2 which does not depend on the topology of the graph. Suppose that d D 4. Depending on whether i D i or j there is a momentum space and a position space version of this trick. The two are dual to each other in the following sense:  Q  P P det ae .e _ i /t .e _ i / D ae det ae1 .e _ j /t .e _ j / e2E./

e2E./

e2E./

See [15], Proposition 1.6, for a proof. In this article, we will only consider the momentum space version, where i D i . The map  7! .AS ; uS / with i D i is called the Schwinger or parametric Feynman rules. Just as in the previous section, there is also a version with external momenta, which I just quote from [47], [16], [14], US .fae g; fPv g/ D where N D

exp..N 1 P /t P / P .det e2E./ ae .e _ i /t .e _ i //2 P e2E./

ae1 .e _ j /t .e _ j /;

a d.jV ./j  dim H0 .I R//-square matrix.

9

Notes on Feynman integrals and renormalization

The determinant P

‰ .ae / D det

ae .e _ i /t .e _ i /

e2E./

is a very special polynomial in the ae . It is called the first graph polynomial, Kirchhoff polynomial or Symanzik polynomial. It can be rewritten ‰ .ae / D

Q

P

T sf of  e62E.T /

ae

(5)

as a sum over spanning forests (sf) T of : A spanning forest is a subgraph E.T /  E./ such that the map @jRE.T / W RE.T / ! RV ./ =H0 .I R/ is an isomorphism; in other words, a subgraph without cycles that has exactly the same components as . (In the special case where  is connected, a spanning forest is called a spanning tree (st) and is characterized by being connected as well and having no cycles.) For the second graph polynomial ˆ , which is a polynomial in the ae and a quadratic form in the Pv , let us assume for simplicity that  is connected. Then ˆ .ae ; Pv / D ‰  .N 1 P /t P D

P

P

T st of  e0 2E.T /

P1t P2 ae0

Q e62E.T /

ae

P where PA D P v2CA Pv is the sum of momenta in the first connected component CA and PB D v2CB Pv the sum of momenta in the second connected component CB of the graph E.T / n fe0 g (which has exactly two components since T is a spanning tree). See [15], [16], [47] for proofs. Here is a simple example: If

2 D

then ‰2 D a1 C a2 ; and US D All this holds if u0;M D

1 . jpj2

ˆ2 D P12 a1 a2

  a2 exp  P12 aa11Ca 2 .a1 C a2 /2

If u0;M D

 US D exp  m2

1 jpj2 Cm2

P e2E./

:

then  ae US jmD0 :

10

C. Bergbauer

2.3 The origin of Feynman graphs in physics. Before we continue with a closer analysis of the divergence locus of these Feynman integrals, it will be useful to have at least a basic understanding of why they were introduced in physics. See [88], [28], [54], [42], [45], [73], [87], [33], for a general exposition, and I follow in particular [73], [42], [45] in this section. Quantum Field Theory is a theory of particles which obey the basic principles of quantum mechanics and special relativity at the same time. Special relativity is essentially the study of mechanics covariant under the Poincaré group P D R1;3 Ì SL.2; C/ (where SL.2; C/ ! O.1; 3/C is the universal double cover of the identity component O.1; 3/C of O.1; 3//. In other words, P is the double cover of the group of (spaceand time-) orientation-preserving isometries of Minkowski space-time R1;3 (I assume d D 4 in this section). On the other hand, quantum mechanics always comes with a Hilbert space, a vacuum vector, and operators on the Hilbert space. By definition, a single particle is then an irreducible unitary representation of P on some Hilbert space H1 . Those have been classified by Wigner according to the joint spectrum of P D .P0 ; : : : ; P3 /, the vector of infinitesimal generators of the translations. Its joint spectrum (as a subset of R1;3 ) is either one of the following SL.2; C/-orbits: the hyperboloids (mass shells) S˙ .m/ D f.p 0 /2  .p 1 /2  .p 2 /2  .p 3 /2 D m2 ; p 0 ? 0g  R1;3

.m > 0/;

and the forward and backward light cones S˙ .0/  R1;3 .m D 0/. (There are two more degenerate cases, for example m < 0 which I do not consider further.) This gives a basic distinction between massive .m > 0/ and massless particles .m D 0/. For a finer classification, one looks at the stabilizer subgroups Gp at p 2 S˙ .m/. If m > 0, then Gp Š SU.2; C/, if m D 0, then Gp is the double cover of the group of isometries of the Euclidean plane. In any case, the Gp are pairwise conjugate in SL.2; C/ and Z H p d m .p/; H1 D ˚

p

where the H are pairwise isomorphic and carry an irreducible representation of Gp . By d m I denote the unique SL.2; C/-invariant measure on S˙ . The second classifying parameter is then an invariant of the representation of Gp on H p : In the case where m > 0 and Gp Š SU.2; C/, one can take the dimension: H p Š C2sC1 , and s 2 N=2 is called the spin. If m D 0, Gp acts on C by mapping a rotation by the angle  around the origin to e i n 2 C , and n=2 is called the helicity (again I dismiss a few cases that are of no physical interest). In summary, one identifies a single particle of mass m and spin s or helicity n with the Hilbert space H1 Š L2 .S˙ .m/; d m / ˝ C2sC1

resp. L2 .S˙ .0/; d 0 /;

Notes on Feynman integrals and renormalization

11

and a state of the given particle is an element of the projectivized Hilbert space P H1 . Quantum field theories describe many-particle systems, and particles can be generated and annihilated. A general result in quantum field theory, the Spin-Statistics theorem [62, 55], tells us that systems of particles with integer spin obey Bose (symmetric) statistics while those with half-integer spin obey Fermi (antisymmetric) statistics. We stick to the case of s D 0, and most of the time even m D 0, n D 0, (which can be considered as the limit m ! 0 of the massive case) in this article. The Hilbert space of infinitely many non-interacting particles of the same type, called Fock space, is then the symmetric tensor algebra H D Sym H1 D

1 L nD0

Symn H1

of H1 (for fermions, one would use the exterior algebra instead). P acts on H in the obvious way; denote the representation by U , and  D 1 2 C D Sym0 H1  H is called vacuum vector. Particles are created and annihilated as follows: If f 2 D.R1;3 / is a test function, then fO D F f jS˙ .m/ 2 H1 (the Fourier transform is taken with respect to the Minkowski metric) and a Œf  W Symn1 H1 ! Symn H1 W n P ˆ.p1 ; : : : ; pn1 / 7! fO.pi /ˆ.p1 ; : : : ; p bi ; : : : ; pn /; iD1

aŒf  W Sym

H1 ! Symn H1 W Z ˆ.p1 ; : : : ; pnC1 / 7! fO.p/ˆ.p; p1 ; : : : ; pn /d m .p/ nC1

S˙ .m/

define operator-on-H -valued distributions f 7! a Œf , f 7! aŒf  on R1;3 . The operator a Œf  creates a particle in the state fO (i.e., with smeared momentum fO), and aŒf  annihilates one. The sum  D a C a is called the field. It is the quantized version of the classical field, which is a C 1 function on Minkowski space. The field  on the other hand is an operator-valued distribution on Minkowski space. It satisfies the Klein–Gordon equation . C m2 / D 0

(6)

. is the Laplacian of R1;3 / which is the Euler-Lagrange equation for the classical Lagrangian 1 1 L0 D .@ /2  m2  2 : (7) 2 2

12

C. Bergbauer

The tuple .H; U; ; / and one extra datum which I omit here for simplicity is what is usually referred to as a quantum field theory satisfying the Wightman axioms [78]. The axioms require certain P -equivariance, continuity and locality conditions. The tuple I have constructed (called the free scalar field theory) is a very well understood one because (6), resp. the Lagrangian (7), are very simple indeed. As soon as one attempts to construct a quantum field theory .HI ; UI ; I ; I / for an interacting Lagrangian (which looks more like a piece of the Lagrangian of the Standard Model) such as 1 1 (8) L0 C LI D .@ I /2  m2 I2 C In 2 2 (n  3,  2 R is called the coupling constant) one runs into serious trouble. In this rigorous framework the existence and construction of non-trivial interacting quantum field theories in four dimensions is as of today an unsolved problem, although there is an enormous number of important partial results; see for example [75]. However, one can expand quantities of the interacting quantum field theory as a formal power series in  with coefficients quantities of the free field theory, and hope that the series has a positive radius of convergence. This is called the perturbative expansion. In general the power series has radius of convergence 0, but due to some non-analytic effects which I do not discuss further, the first terms in the expansion do give a very good approximation to the experimentally observed quantities for many important interacting theories (this is the reason why quantum field theories have played such a prominent role in the physics of the last 50 years). I will devote the remainder of this section to a sketch of this perturbative expansion, and how the Feynman integrals introduced in the previous section arise there. By Wightman’s reconstruction theorem [78], a quantum field theory .HI ; UI ; I ; I / is uniquely determined by and can be reconstructed from the Wightman functions (distributions) wnI D hI ; I .x1 / : : : I .xn /I i. Similar quantities are the timeordered Wightman functions tnI D hI ; T .I .x1 / : : : I .xn //I i; which appear directly in scattering theory. If one knows all the tnI , one can compute all scattering cross-sections. The symbol T denotes time-ordering: T.

1 .x1 /

2 .x2 //

D

1 .x1 /

2 .x2 /

if x10  x20 ;

D

2 .x2 /

1 .x1 /

if x20 > x10

for operator-valued distributions 1 , 2 . For the free field theory, all the wn and tn are well understood, in particular t2 .x1 ; x2 / D h; T ..x1 /.x2 //i D F 1

i .p 0 /2



.p 1 /2



.p 2 /2

 .p 3 /2  m2 C i

;

Notes on Feynman integrals and renormalization

13

where the Fourier transform is taken with respect to the difference coordinates x1  x2 (the tn are translation-invariant). t2 is a particular fundamental solution of equation (6) called the propagator. By a technique called Wick rotation, one can go back and forth between Minkowski space R1;3 and Euclidean space R4 [71], [48], turning Lorentz squares .p 0 /2  .p 1 /2  .p 2 /2  .p 3 /2 into Euclidean squares jpj2 , and the Minkowski space propagator t2 into the distribution u0;P D F 1 jpj21Cm2 introduced in the previous sections. In the massless case m D 0, we have u0;P D u0;M D jxj1 2 if d D 4. From the usual physics axioms for scattering theory and on a purely symbolic level, Gell-Mann and Low’s formula relates the interacting tnI with vacuum expectation values h; T .: : : /i of time-ordered products of powers of the free fields,

tnI .x1 ; : : : ; xn / Z 1 X (9) ik D h; T ..x1 / : : : .xn /LI0 .y1 / : : : LI0 .yk //id 4 y1 : : : d 4 yk ; kŠ kD0

as a formal power series in . I denote LI0 D LI jI ! D  n . There is a subtle point here in defining powers of  as operator-valued distributions. The solution is called Wick (or normal ordered) powers: in the expansion of  n D .a C a /n , all a are moved to the left of the a such that no monomials containing aa in this order appear. Consequently h; W .: : : /i D 0 for any normal ordered operator W .: : : /. Time- and normal ordered products are related by what is called Wick’s Theorem: T ..x1 /.x2 // D W ..x1 /.x2 // C t2 .x1 ; x2 /; T ..x1 / : : : .x3 // D W ..x1 /.x2 /.x3 // C W ..x1 //t2 .x2 ; x3 / C W ..x2 //t2 .x1 ; x3 / C W ..x3 //t2 .x1 ; x2 /; T ..x1 / : : : .x4 // D W ..x1 / : : : .x4 // C W ..x1 /.x2 //t2 .x3 ; x4 / C       C t2 .x1 ; x2 /t2 .x3 ; x4 / C    :: : Now within the free field theory, the h; T .: : : /i are well understood. It follows from the arguments above that h; T .: : : /i is a polynomial in the t2 ; more precisely, P (10) h; T . n1 : : :  nk /i D c   uP ; 

where the sum is over all Feynman graphs  with k vertices such that the ith vertex has degree ni , and where uP  is defined as in the previous sections, c is a combinatorial symmetry factor, and u0;P .x/ D t2 .x; 0/ up to a Wick rotation. For example, the graph (1) arises in h; T ..x1 /.x2 /.x3 /.x4 / 3 .y1 / 3 .y2 //i;

14

C. Bergbauer

a second-order (in ) contribution to t4I of the theory where LI D I3 . The points x1 ; : : : ; x4 are external and y1 , y2 the internal ones. Generally, if one uses (10) for (9) then one gets Feynman graphs with n external vertices of degree 1. The external edges, i.e., edges leading to those n vertices, appear simply as tensor factors, and can be omitted (amputated) in a first discussion. In this way we are left with the graphs considered in the previous section. It follows in particular that only Feynman graphs with vertices of degree n appear from the Lagrangian (8). Note that whereas external physical particles are always on-shell (i.e., their momentum supported on S˙ /, the internal virtual particles are integrated over all of momentum space in the Gell-Mann–Low formula. In summary, the perturbative expansion of an interacting quantum field theory (whose existence let alone construction in the sense of the Wightman axioms is an unsolved problem) provides a power series approximation in the coupling constant to the bona fide interacting functions tnI . The coefficients are sums of Feynman integrals which are composed of elements of the free theory alone.

3 Regularization and renormalization The Feynman integrals introduced so far are generally divergent integrals. At first sight it seems to be a disturbing feature of a quantum field theory that it produces divergent integrals in the course of calculations, but a closer look reveals that this impression is wrong: it is only a naive misinterpretation of perturbation theory that makes us think this way. Key to this is the insight that single Feynman graphs are really about virtual particles, and their parameters, for example their masses, have no real physical meaning. They have to be renormalized. In this way the divergences are compensated by so-called counterterms in the Lagrangian of the theory which provide some kind of dynamical contribution to these parameters [28]. I will not make further use of this physical interpretation but only consider the mathematical aspects. If the divergences can be compensated by adjusting only a finite number of parameters in the Lagrangian (i.e., by leaving the form of the Lagrangian invariant and not adding an infinity of new terms to it) the theory is called renormalizable. An important and somewhat nontrivial, but fortunately solved [19], [46], [90], [38], [53], [29], [30], problem is to find a way to organize this correspondence between removing divergences and compensating counterterms in the Lagrangian for arbitrarily complicated graphs. Since the terms in the Lagrangian are local terms, that is, polynomials in the field and its derivatives, a necessary criterion for this is the so-called locality of counterterms: if one has a way of removing divergences such that the correction terms are local ones, then this is a good indication that they fit into the Lagrangian in the first place. Regularization on the other hand is the physics term used for a variety of methods of writing the divergent integral or integrand as the limit of a holomorphic family of

Notes on Feynman integrals and renormalization

15

convergent integrals or integrands, say over a punctured disk. Sometimes also the integrand is fixed, and the domain of integration varies holomorphically, say over the punctured disk. We will see a number of such regularizations in the remainder of this article. 3.1 Position space. In position space, the renormalization problem has been known for a long time to be an extension problem of distributions [19], [38]. This follows already from our description in Section 2, but it will be useful to have a closer look at the problem. Recall the position space Feynman distribution ˚d  ˝jE./j uP  D .j / u0;P

is defined only as a distribution on AP D MjV ./j_ =H 0 .I R/˚d minus certain affine (in this case even linear) subspaces. Suppose, for example, that

2 D

1 with uP 2 D jxj2d 4 . If f is a non-negative test function supported in a ball N D fjxj  g around 0.

Z uP 2 Œf

D N

Z f .x/uP 2 .x/dx

 minx2N f .x/

Z



d 0

dr d 1 : r 2d 4

If d  1  .2d  4/  1, that is, d  4, the integral will be divergent at 0 and uP 2 not defined on test functions supported at 0. This is the very nature of ultraviolet (i.e., short-distance) divergences. On the other hand, divergences as some positionspace coordinates go to 1 are called infrared (long-distance) divergences. We will be concerned with ultraviolet divergences in this article. For simplicity we restrict ourselves to graphs with at most logarithmic divergences throughout the rest of the article, that is, d rk H1 . /  2jE. /j for all subgraphs E./  E./. A subgraph  where equality holds is called divergent. A detailed power-counting analysis, carried out in [10], shows that uP  is only defined as a distribution inside T S APı D AP n De ; (11) E./E./ d rk H1 . /D2jE. /j

e2E./

where De D fxe;out  xe;in D 0g. The singular support (the locus where uP  is not smooth) is S ı sing supp uP De :  D AP \ e2E./

16

C. Bergbauer

ı An extension of uP  from AP to AP is called a renormalization provided it satisfies certain consistency conditions to be discussed later. In the traditional literature, which dates back to a central paper of Epstein and ı Glaser [38], an extension of uP  from AP to all of AP was obtained inductively, by starting with the case of two vertices, and embedding the solution (extension) for this case into the three, four, etc. vertex case using a partition of unity. In this way, in each step only one extension onto a single point, say 0, is necessary, a well-understood problem with a finite-dimensional space of degrees of freedom: two extensions differ by a distribution supported atPthis point 0, and the difference is therefore, by elementary considerations, of the form j˛jn c˛ @˛ ı0 with c˛ 2 C. Some of these parameters c˛ are fixed by physical requirements such as probability conservation, Lorentz and gauge invariance, and more generally the requirement that certain differential equations be satisfied by the extended distributions. But even after these constants are fixed, there are degrees of freedom left, and various groups act on the space of possible extensions, which are collectively called the renormalization group. For the at most logarithmic graphs considered in this article, n D 0 and only one constant c0 needs to be fixed in each step.

3.2 Resolution of singularities. The singularities, divergences and extensions (renormalizations) of the Feynman distribution uP  are best understood using a resolution of singularities [10]. The Fulton–MacPherson compactification [43] introduced in a quantum field theory context by Kontsevich [51], [49] and Axelrod and Singer [6] serves as a universal smooth model where all position space Feynman distributions can be renormalized. In [10], a graph-specific De Concini–Procesi wonderful model [34] was used, in order to elaborate the striking match between De Concini and Procesi’s notions of building set, nested set and notions found in Quantum Field Theory. No matter which smooth model is chosen, one is led to a smooth manifold Y and a proper surjective map, in fact a composition of blowups, ˇ W Y ! AP ; which is a diffeomorphism on ˇ 1 .APı / but where ˇ 1 .AP n APı / is (the real locus of) a divisor with normal crossings. Instead of the nonorientable smooth manifold Y one can also find an orientable manifold with corners Y 0 and ˇ a composition of real spherical blowups as in [6]. In my figures, the blowups are spherical because they are easier to draw, but in the text they are projective. Here is an example: If 3 is again the graph

3 D

(12)

Notes on Feynman integrals and renormalization

17

and d D 4, then by (11) the locus where there are non-integrable singularities is D1234  D234  D34 ; where D1234 D D12 \ D13 \ D14 , D234 D D23 \ D24 . In AP , D1234 is a point, D234 is 4-dimensional and D24 is 8-dimensional. Blowing up something means replacing it by its projectivized normal bundle. The map ˇ is composed of three maps ˇ3

ˇ2

ˇ1

Y D Y34 ! Y234 ! Y1234 ! AP where ˇ1 blows up D1234 , ˇ2 blows up the strict transform of D234 , and ˇ3 blows up the strict transform of D34 .

ˇ

!

Now uP 3 can be pulled back along ˇ (because of lack of orientability of Y , it will become a distribution density). In a clever choice of local coordinates, for example y10 D x10  x20 ; y20 D .x20  x30 /=.x10  x20 /; y30 D .x30  x40 /=.x20  x30 /; y1i D .x1i  x2i /=.x10  x20 /; y2i D .x2i  x3i /=.x20  x30 /; y3i D .x3i  x4i /=.x30  x40 /; one has wP3 D ˇ  uP 3 D

fP3 jy10 y20 y30 j

;

(13)

where fP3 is a locally integrable density which is even C 1 in the coordinates y10 , y20 , y30 . The divergence is therefore isolated in the denominator, and only in three directions: y10 , y20 and y30 . The first is the local coordinate transversal to the exceptional divisor E1234 of the blowup of D1234 , the second transversal to the exceptional divisor E234 of the blowup of D234 , and the third transversal to the exceptional divisor E34 of the blowup of D34 (the difference between E34 and D34 is not seen in the figure for of dimensional reasons).

18

C. Bergbauer

For a general graph , the total exceptional divisor E D ˇ 1 .AP n APı / has normal crossings and the irreducible components E are indexed by connected divergent (consequently core) irreducible subgraphs  . Moreover, E1 \    \ Ek ¤ ; () the i are nested; where nested means each pair is either disjoint or one contained in the other. See [10] for the general result and more details. s Inspired by old papers of Atiyah [5], Bernstein and Gelfand [12] we used .uP / , where s in a complex number in a punctured neighborhood of 1, as a regularization [10]. Similarly, since the propagator u0;P .x/ D jxjd12 depends on the dimension, one can

also consider uP  with d in a punctured complex neighborhood of 4 as a regularization but I will not pursue this here. Definition 3.1. A connected graph  is called primitive if d rk H1 . / D 2jE. /j () E. / D E./ for all subgraphs E. /  E./.

For a primitive graph p , only the single point 0 2 AP needs to be blown up, and the pullback along ˇ yields in suitable local coordinates (y10 D x10  x20 , yij D j .xij  xiC1 /=.x10  x20 / otherwise) ˇ  uP p D

fp ; jy10 j

where fp is a locally integrable distribution density constant in the y10 -direction. Let d D d.jV .p /j  1/. Consequently s ˇ  .uP p / D

fsp jy10 jdp s.dp 1/

:

It is well known that the distribution-valued function jxj1 s can be analytically continued in a punctured neighborhood of s D 1, with a simple pole at s D 1. The residue of this pole is ı0 :

Z jxjsfin Œf

D

1

1

ı0 1 D C jxjsfin ; jxjs s1 Z s jxj .f .x/  f .0//dx C

jxjs f .x/dx:

RnŒ1;1

s This implies that the residue at s D 1 of ˇ  .uP p / is a density supported at the exceptional divisor (which is given in these coordinates by y10 D 0), and integrating

19

Notes on Feynman integrals and renormalization

this density against the constant function 1Y gives what is called the residue of the graph p , Z 2 s resP p D ressD1 ˇ  .uP / Œ1  D  f : Y p dp E p (The exceptional divisor can actually be oriented in such a way that fp is a degree .dp  1/ differential form.) Let us now come back to the case of 3 which is not primitive but has a nested set of three divergent subgraphs. Raising (13) to a power s results in a pole at s D 1 of order 3. The Laurent coefficient a3 of .s  1/3 is supported on E1234 \ E234 \ E34 ; for this is the set given in local coordinates by y10 D y20 D y30 D 0. Similarly, the coefficient of .s  1/2 is supported on .E1234 \ E234 / [ .E1234 \ E34 / [ .E234 \ E34 / and the coefficient of .s  1/1 on E1234 [ E234 [ E34 : (The non-negative part of the Laurent series is supported everywhere on Y .) Write jdyj D jdy10 : : : dy33 j. In order to compute the coefficient a3 , one needs to integrate f3 , restricted to the subspace y10 D y20 D y30 D 0: f3 D

.1 C 

y12 /.1

jdyj C y22 /.1 C y32 / 1

..1 C

y20 /2

C .y1 C

y20 y2 /2 /..1

C y30 /2 C .y2 C y30 y3 /2 //

;

where yi denotes the 3-vector .yi1 ; yi2 ; yi3 /. Consequently f3 jy 0 Dy 0 Dy 0 D0 D 1

2

3

.1 C

y12 /2 ; .1

jdyj D f˝3 1 C y22 /2 .1 C y32 /2

where 1 is the primitive graph with two vertices and two parallel edges joining them:

1 D

:

(14)

The chart where (13) holds covers actually all of YP up to a set of measure zero where there are no additional divergences. It suffices therefore to integrate in these coordinates

20

C. Bergbauer

only. Several charts must be taken into account however when there is more than one maximal nested set. In conclusion, a3 Œ1Y  D .resP 1 /3 ;

(15)

s which is a special case of a theorem in [10] relating pole coefficients of ˇ  .uP  / to residues of graphs obtained from  by contraction of divergent subgraphs. But the ultimate reason to introduce the resolution of singularities in the first place is as follows: In order to obtain an extension (renormalization) of uP  , one can now simply remove the simple pole at s D 1 along each component of the exceptional divisor: f f3 : wP3 D 0 03 0 ; .wP3 /R D 0 jy1 y2 y3 j jy1 jfin jy20 jfin jy30 jfin

The second distribution .wP3 /R is defined on all of Y , and consequently ˇ .wP3 /R on

ı all of AP . It agrees with uP 3 on test functions having support in AP and is therefore an extension. The difference between wP3 and .wP3 /R is a distribution supported on the exceptional divisor which gives rise to a candidate for a counterterm in the Lagrangian. I call this renormalization scheme local minimal subtraction, because locally, along each component of the exceptional divisor, the simple pole is removed in a “minimal way”, changing only the principal part of the Laurent series. See [10] for a proof that this results in local counterterms, a necessary condition for the extension to be a physically consistent one.

3.3 Momentum space. In momentum space, T the bad definition of the position space Feynman distribution at certain diagonals De is translated by a Fourier transform into ill-defined (divergent) integrals with divergences at certain strata at infinity. For 2 example, the position space integral .M; uP 1 D u0;P / in d D 4 dimensions for the graph 1 (see (14)) has a divergence at 0 (which is the image D12 of the diagonal). A formal Fourier transform would turn the pointwise product u20;P into a convolution product Z .F u20;P /.P / D

u0;M .p/u0;M .p  P /d 4 p:

In fact the right hand side is exactly UM1 .P /Œ1A1 , in agreement with Proposition 2.1. It does not converge at 1. (In order to see this we actually only need UM1 jP D0 D uM 1 , not the dependence upon external momenta.) On the other hand, the infrared singularities are to be found at affine subspaces in momentum space. Of course the program sketched in the previous section can be applied to the momentum space Feynman distribution as well. A resolution of singularities for the relevant strata at infinity can be found, and the pullback of the momentum space Feynman distribution can be extended onto all the irreducible components of the exceptional divisor. But I want to use this section in order to sketch another, algebraic,

Notes on Feynman integrals and renormalization

21

approach to the momentum space renormalization problem, which is due to Connes and Kreimer [53], [29], [30]. Assume UM Œ1AM  varies holomorphically with d in a punctured disk around d D 4. Physicists call this dimensional regularization [39], [32]: any integral of the form R 4 R d pu.p/dp is replaced by a d -dimensional integral d d pu.p/dp. In this way we can consider UM as a distribution on all of AP  AM with values in R D CŒŒ.d  4/1 ; .d  4/, the field of Laurent series in d  4. If UM Œf  is not convergent in d D 4 dimensions, then there will be a pole at d D 4. Now let  2 D 0 .AP / be a distribution with compact support. Since the distribution M U is smooth in the Pv , we can actually integrate it against the distribution  . (For example, if  D ı0 .jPv1 j2  E1 / ˝    ˝ ı0 .jPv2n j  En / then this amounts simply to evaluating UM on the subspaces jPv1 j2 D E1 ; : : : ; jPv2 j2 D En .) In any case we have a map  W .;  / 7! UM Œ1AM ˝   2 R sending pairs to Laurent series. Now let H be the polynomial algebra over C generated by isomorphism classes of connected core divergent graphs  of a given renormalizable quantum field theory. Define a coproduct  by P

./ D 1 ˝  C  ˝ 1 C

1 ttk ¨ conn. core div.

1 : : : k ˝ ==.1 t    t k /:

The notation == means that any connected component of  inside  is contracted to a (separate) vertex. By standard constructions [29], H becomes a Hopf algebra, called the Connes–Kreimer Hopf algebra. Denote the antipode by S . Now let H be the corresponding Hopf algebra of pairs .;  /. (In order to define this Hopf algebra of pairs, one needs the extra condition that  vanishes on all vertices that have no external edges, a standard assumption if one considers only graphs of a fixed renormalizable theory.) The map  W H ! R is a homomorphism of unital C-algebras. The space of these maps H ! R is a group with the convolution product 1 ? 2 D m.1 ˝ 2 /: On R; there is the linear projection ´ R W .d  4/n 7! onto the principal part.

0 .d  4/n

if n  0; if n < 0

(16)

22

C. Bergbauer

Theorem 3.1 (Connes, Kreimer). The renormalized Feynman integral R .;  /jd D4 and the counterterm SR .;  / are given as follows. I write  for the pair .;  /: SR ./ D R../ C

P

D1 ttk ¨ conn. core div.

R ./ D .1  R/../ C

SR . /.== //;

P

D1 ttk ¨ conn. core div.

SR . /.== //:

These expressions are assembled from the formula for the antipode and the convolution product. Combinatorially, the Hopf algebra encodes the BPHZ recursion [46] and Zimmermann’s forest formula [90]. The theorem can be interpreted as a Birkhoff decomposition of the character  into  D SR and C D R [30]. The renormalization scheme described here is what I call global minimal subtraction, because in the target field R, when all local information has been integrated out, the map 1  R removes only the entire principal part at d D 4. This coincides with the renormalization scheme described in [28]. In the case of m D 0 and zero-momentum transfer (all but two external momenta set to 0) one knows that at d D 4, R ./ D

N P nD0

pn ./.log jP j2 = 2 /n ;

pn ./ 2 R;

(17)

where is an energy scale, and the  can be dropped for convenience. Let us now do our standard example

3 D

using the Hopf algebra. We interpret 3 as a graph in  4 theory, so we think of two external edges at the first vertex, one at the second, and one at the fourth. Recall the momentum space Feynman rules (3) for 3 . Let P2 D 0 and write P D P1 D P4 such that P1 is the sum of the two external momenta entering at the first vertex. Then Z d d p1 d d p2 d d p3 .3 / D 2 R: p12 .p1 C P /2 p22 .p1 C p2 C P /2 p32 .p2 C p3  P /2 This integral can be evaluated as a Laurent series in d D 4 using standard techniques [28]. It has a pole of order 3 at d D 4, and one might think of simply taking .1R/.3 / as a renormalized value, for this kills the principal part, and the limit at d D 4 may be taken. But the resulting counterterms would not be local ones, and the renormalization would be physically inconsistent. The benefit of the Hopf algebra approach is that the necessary correction terms are provided right away.

Notes on Feynman integrals and renormalization

23

Let again 1 be the full subgraph with vertices 3 and 4, and 2 the full subgraph with vertices 2, 3 and 4. Then R .3 / D .1R/..3 /.R.2 //.3 ==2 /CR..R.1 //.2 ==1 //.3 ==2 //: Observe that, as a coincidental property of our example, 3 ==2 Š 2 ==1 Š 1 (compare this with (15), (21)). The Hopf algebra approach to renormalization has brought up a number of surprising connections to other fields; see for example [30], [31], [37], [68], [41], [79], [64], [80], [59]. Other developments starting from the Connes–Kreimer theory can be found in [32]. Kreimer and van Suijlekom have shown that gauge and other symmetries are compatible with the Hopf algebra structure [55], [85], [86], [84], [61]. A sketch of how the combinatorics of the Hopf algebra relates to the resolution of singularities in the previous section and to position space renormalization can be found in [10]; see also Section 3.6. 3.4 Parametric representation. In the parametric representation introduced in Section 2.2, the divergences can be found at certain intersections of the coordinate hyperplanes Ae D fae D 0g. This is in fact one of the very reasons why the parametric representation was introduced: Consider for example the divergent integral .R4 ; u20;M /, with u0;M D jpj1 2 , Z

d 4p D jpj4

ZZ

1Z 1

0

0

exp.a1 jpj2  a2 jpj2 / da1 da2 d 4 p

in R the sense of Definition 2.1. (In this section, instead of .A; u/ I will simply write A u.x/dx.) The integral on the left-hand side is divergent both at 0 and at 1. But splitting it into the two parts at the right, and interchanging the d 4 p with the da1 da2 integrations leaves a Gaussian integral Z exp. 2c jpj2 /d d p D .2=c/d=2 ; which is convergent but at the expense of getting .a1 C a2 /2 in the denominator: The integral Z 1Z 1 da1 da2 .a1 C a2 /2 0 0 has a logarithmic singularity at 0 and at 1. This can be seen by blowing up the origin in R20 , and pulling back: Z 1Z 1 db1 db2 : 2 b 1 .1 C b2 / 0 0 In other words, the trick with the parametric parameterization (called the Schwinger trick in [15]), does not get rid of any divergences. It just moves them into another, lower-dimensional space.

24

C. Bergbauer

Again, it is useful to have a resolution of singularities in order to separate the various singularities and divergences of a graph along irreducible components of a divisor with normal crossings. The most obvious and efficient such resolution is given in [15], [16]. Let  be core. For a subgraph E. /  E./, let T Ae D fae D 0; e 2 E. /g; L D e2E./

a linear subspace. Set Lcore D fL W  is a core subgraph of g, and L0 D fminimal element of Lcore g D f0g; F LnC1 D fminimal elements of Lcore n niD0 Li g: This partition of Lcore is made in such a way that (see [16], Proposition 3.1) a sequence of blowups  W Z S !    ! AS (18) is possible which starts by blowing up L0 and then successively the strict transforms of the elements of L1 , L2 , : : : . This ends up with ZS a manifold with corners. The map  is of course defined not only as a map onto AS D RjE./j but as a birational map 0 jE./j  W ZS ! C , with ZS a smooth complex variety. The total exceptional divisor E has normal crossings, and one component EL for each L 2 Lcore . (In the language of Section 3.2, Lcore is the “building set”.) Moreover, EL1 \    \ ELk ¤ 0 () the Li are totally ordered by inclusion. Since the coordinate divisor fae D 0 for some e 2 E./g already has normal crossings by definition, the purpose of these blowups is really only to pull out into codimension 1 all the intersections where there are possibly singularities or divergences, and to separate the integrable singularities of the integrand from this set as much as possible. Note that in the parametric situation where the domain of integration is the manifold with corners RjE./j , the blowups do not introduce an orientation issue on the real locus. 0 For the example graph 3 of the previous sections (see (12)), uS3 D

da1 : : : da6 ..a1 C a2 /..a3 C a4 /.a5 C a6 / C a5 a6 / C a3 a4 a5 C a3 a4 a6 C a3 a5 a6 /d=2

we examine the pullback of uS3 onto ZS . There are various core subgraphs to consider, but it is easily seen, in complete analogy with (11), that the divergences are located only at L3 , L2 and L1 , where 1 is the full subgraph with vertices 3 and 4, and 2 the full subgraph with vertices 2, 3 and 4. In order to see the divergences in ZS , it therefore suffices to look in a chart where EL3 , EL2 and EL1 intersect. In such a chart, given by coordinates b1 D a1 , b2 D a2 =a1 , b3 D a3 =a1 , b4 D a4 =a1 , b5 D a5 =a3 , b6 D a6 =a5 , we have   uS3 D

db1 : : : db6 b1 b3 b5 ..1 C b2 /..1 C b6 /.1 C b4 / C b5 b6 / C b3 .b5 b6 C b4 b6 C b4 //d=2 (19)

25

Notes on Feynman integrals and renormalization

Now we are in a very similar position as that in the previous section. If p is a primitive graph, then there is only the origin 0 2 AS which needs to be blown up in order to isolate the divergence. Since uS3 depends explicitly on d in the exponent, let us use d as an analytic regulator. One finds, using for example coordinates b1 D a1 , bi D ai =a1 , i ¤ 1, in a neighborhood of d D 4,   uSp .d /



D



ı0 .b1 / C finite gp d 4

with gp 2 L1loc . (If one wants moreover a regular gp one needs to perform the remaining blowups in (18).) Then we define Z Z   S gp D ; (20) resS p D .resd D4  up .d //Œ1 D 2 b1 D0;bi 0 ‰p

b

P n where D fai  0g  P jE./j1 .R/ and  D jE./j nD1 .1/ an da1 ^  ^ d an ^  ^ dajE./j . The last integral on the right is a projective integral, meaning that the ai are interpreted as homogeneous coordinates of P jE./j1 . By choosing affine coordinates bi , one finds that it is identical with the integral of gp over the exceptional divisor intersected with the total inverse image of AS . Coming back to the non-primitive graph 3 (see (19)) we find, in complete analogy with Section 3.2, that uS3 .d / D

1 P n3

cn .d  4/n

in a neighborhood of d D 4, and c3 Œ1AS  D .resS 1 /3 ;

(21)

which is easily seen by sending b1 , b3 , b5 to 0 in (19): g3 jb1 Db3 Db5 D0 D g˝3 . 1 Similarly, one can translate the results of Section 3.2 and [10] into this setting and obtain a renormalization (extension of uS ) by removing the simple pole along each component of the irreducible divisor. In Section 4.5 a different, motivic renormalization scheme for the parametric representation will be studied, following [16]. 3.5 Dyson–Schwinger equations. Up to now we have only considered single Feynman graphs, with internal edges interpreted as virtual particles, and parameters such as the mass subjected to renormalization. Another approach is to start with the full physical particles from the beginning, that is, with the non-perturbative objects. Implicit equations satisfied by the physical particles (full propagators) and the physical interactions (full vertices) are called Dyson–Schwinger equations. The equations can be imposed in a Hopf algebra of Feynman graphs [23], [58], [11], [57], [89] and turn into systems of integral equations when Feynman rules are applied.

26

C. Bergbauer

For general configurations of external momenta, Dyson–Schwinger equations are extremely hard to solve. But if one sets all but two external momenta to 0, a situation called zero-momentum transfer (see (17)), then the problem simplifies considerably. In [60], an example of a linear Dyson–Schwinger equation is given which can be solved nonperturbatively by a very simple Ansatz. More difficult non-linear Dyson– Schwinger equations, and finally systems of Dyson–Schwinger equations as above, are studied in [62], [63], [82], [83]; see also [89], [56], [40]. 3.6 Remarks on minimal subtraction. I come back at this point to the difference between what I call local (Section 3.2) and global (Section 3.3) minimal subtraction, which, I think, is an important one. I tried to emphasize in the exposition of the previous sections that the key concepts of renormalization are largely independent of whether momentum space, position space, or parameter space Feynman rules are used. This is immediately seen in the Connes– Kreimer Hopf algebra framework where a graph  and some external information  are sent directly to a Laurent series in d  4. For this we do not get to see and do not need to know if the integral has been computed in momentum, position, or parameter space. They all produce the same number (or rather Laurent series), provided the same regularization is chosen for all three of them. In position space, where people traditionally like to work with distributions as long as possible and integrate them against a test function only at the very end (or even against the constant function 1, the adiabatic limit), one is tempted to define the Feynman rules as a map into a space of distribution-valued Laurent series, as we have done in [10]. But one has to be aware that this space of distribution-valued Laurent series does not necessarily qualify as a replacement for the ring R in Section 3.3 if one looks for a new Birkhoff decomposition. In general, many questions and misconceptions that I have encountered in this area can be traced back to deciding at which moment one integrates, and minimal subtraction seems to be a good example of this. Let me now give a detailed comparison of what happens in local and global minimal subtraction, respectively. Assume for example the massless graph in 4 dimensions,

D

:

Clearly  itself and the full subgraph  on the vertices 2 and 3 are logarithmically divergent. No matter which kind of Feynman rules we use, assume there is a regularized Feynman distribution u . / varying holomorphically in a punctured disk around D 0, with a finite order pole at D 0. Assume after resolution of singularities that the regularized Feynman distribution, pulled back onto the smooth model, has a simple pole supported on the component E of the total exceptional divisor (for the superficial divergence), and another on the component E (for the subdivergence).

Notes on Feynman integrals and renormalization

27

Let E D fy D 0g and E D fy D 0g in local coordinates y ; y ; y3 ; : : : ; yn . We have    ı0 .y / ı0 .y / (22) C jy jfin . / C jy jfin . / f . /; u . / D where f is locally integrable and smooth in y and y , such that in particular f . / is holomorphic in . There is accordingly a second order pole supported at E \ E . We know from [10], as was also sketched in Section 3.2, that the leading coefficient of this second order pole is a product of delta functions restricting it to E \ E times the residue of  times the residue of ==. Consequently, integrating u . / against a fixed function (for a first reading take

D 1, but in the massless case one has to worry about infared divergences) provides a Laurent series u . /Π D a2 2 C a1 1 C a0 0 C    : Since  and == are primitive, u . /Π D b1 1 C b0 0 C b1 1 C    ; u== . /Π D c1 1 C c0 0 C b1 1 C    : We know from the previous remarks that a2 D res. / res.== / D b1 c1 and similarly a1 D b1 c0 C g, where I do not want to specify g. Let me now compare local and global minimal subtraction in this example. Local minimal subtraction is defined on distribution-valued Laurent series, but global minimal subtraction only on C-valued Laurent series. Therefore we need to integrate everything out before comparing. I start with local minimal subtraction (LMS). In order to get from (22) to .u /R;LMS . / D jy jfin . /jy jfin . /f . / one has to subtract three terms from (22): 



ı0 .y / ı0 .y / C jy jfin . / f . /;   ı0 .y / ı0 .y / ;== C jy jfin . / f . /; RLMS u . / D ı0 .y / ı0 .y / ;== u . / D  f . /: RRLMS  RLMS u . / D

 u has only a The first term eliminates the pole supported on E , such that u  RLMS ;== simple pole supported on E left. On the other hand, u  RLMS u has only a simple pole supported on E left, and the third term is a correction term supported on E \ E accounting for what has been subtracted twice. In summary, ;== ;==  u . /  RLMS u . / C RRLMS u . / .u /R;LMS . / D u . /  RLMS

(23)

28

C. Bergbauer

is the result of local minimal subtraction. Let us now integrate out (23). u . /Π D a2 2 C a1 1 C a0 0 C    ;  RLMS u . /Π D a2 2 C g 1 C h 0 C    ; ;== RLMS u . /Π D a2 2 C b1 c0 1 C b1 c1 0 C    ; ;== RRLMS u . /Π D a2 2 ;

These equations follow from (22), and I do not want to specify h. Consequently .u /R;LMS . /Π D a0  b1 c1  h as ! 0: In global minimal subtraction (GMS), where RGMS D R as in (16), something different happens. RGMS .u . /Π/ D a2 2 C a1 1 ; .RGMS u . /Π/u== . /Π D b1 c1 2 C b1 c0 1 C b1 c1 0 C    ; RGMS .RGMS u . /Π/u== . /Π/ D b1 c1 2 C b1 c0 1 : The first subtraction u Π  RGMS .u Π/ removes the poles everywhere, also the one supported on E which has nothing to do with the superficial divergence. The third and fourth term restore the locality of counterterms. We have .u /R;GMS . /Π D a0  b1 c1

as ! 0:

In summary: Unless h D 0, local and global minimal subtraction differ by a finite renormalization. Moreover, although there is a one-to-one-correspondence between terms to be subtracted in LMS and GMS, the values of those single terms do not agree. It seems to me that GMS is a quite clever but somehow special trick of defining the subtraction operator R on C-valued Laurent series where all the geometric information (i.e., where the pole is supported) has been forgotten. In [10] it is shown how to relate, for a general graph , the combinatorics of the total exceptional divisor of the resolution of singularities to the Connes–Kreimer Hopf algebra of Feynman graphs, such that the example presented here is a special case of a more general result. A similar analysis applies to other local renormalization prescriptions, called subtraction at fixed conditions in [10], as well.

4 Motives and residues of Feynman graphs 4.1 Motives, Hodge realization and periods. Much of the present interest in Feynman integrals is due to the more or less obvious fact that there is something motivic about them. In order to understand and appreciate this, one obviously needs to have an

Notes on Feynman integrals and renormalization

29

idea of what a motive is. I am not an expert in this area and will not even attempt to provide much background to the notion of motive. See [4] for an often cited introduction to the subject, which I follow closely in the beginning of this section. The theory of motives is a means to unify the various cohomology theories known for algebraic varieties X over a number field k. Such cohomology theories include the algebraic de Rham and the Betti cohomology, but there are many others. The algebraic  de Rham cohomology HdR .X / is defined over the ground field k, and Betti cohomology  HB .XI Q/ is the singular cohomology of X.C/ with rational coefficients. A motive of a variety is supposed to be a piece of a universal cohomology, such that all the usual cohomology theories (functors from varieties to graded vector spaces) factor through the category of motives. A particular cohomology theory is then called a realization. For example, the combination of de Rham and Betti cohomology, giving rise to a Hodge structure, is called the Hodge realization. The theory of motives is not yet complete. Only for the simplest kind of algebraic varieties, smooth projective ones, has a category of motives with the desired properties been constructed. These motives are called pure. For general, i.e., singular or non-projective varieties, the theory is conjectural in the sense that only a triangulated category as a candidate for the derived category of the category of these motives, called mixed motives, exists.  Let X be a smooth variety over Q. Let HdR .X / denote the algebraic de Rham cohomology of X, a graded Q-vector space, and HB .X I Q/ the rational Betti cohomology (singular cohomology of the complex manifold X.C/ with rational coefficients), a graded Q-vector space. A period of X is by definition a matrix element of the comparison isomorphism (integration) 



HdR .X / ˝Q C Š HB .X I Q/ ˝Q C for a suitable choice of basis. A period is therefore in particular an integral of an algebraic differential form over a topological cycle on X.C/. A standard example is the case of an elliptic curve X defined by the equation y 2 D x.x  1/.x  /, 1 .X / is the 1-form ! D dx and a basis of  2 Q n f0; 1g. A basis element of HdR 2y 1 the singular cohomology HB .X / is given by the duals of two circles around the cut between 0 and 1, resp. the cut between 1 and 1. Integrating ! against these cycles gives the generators of the period lattice of X . Similarly, matrix elements of a comparison isomorphism between relative cohomologies of pairs .X; A/ are called relative periods. Many examples considered below will be relative periods. 4.2 Multiple zeta values, mixed Tate motives and the work of Belkale and Brosnan. Let  be a primitive Feynman graph. I assume d D 4 and m D 0. Recall the graph polynomial P Q ‰ D ae 2 ZŒae W e 2 E./ T st of  e62E.T /

30

C. Bergbauer

from (5). The sum is over the spanning trees of . Following [15], we have a closer look at the parametric residue Z  resS  D 2 ‰ introduced in (20). Let X D f‰ D 0g  P jE./j1 and CX D f‰ D 0g  AjE./j its affine cone. X resp. CX are called the projective resp. affine graph hypersurface. The P chain of integration is D fae  0g  P jE./j1 .R/, and  D .1/n an da1 ^    ^ d an ^    ^ dajE./j . The residue resS  already looks likeSa relative period, since has its boundary contained in the coordinate divisor  D e2E./ fae D 0g, and the differential form

is algebraic (i.e., regular) in P jE./j1 n X . But in general X \  is quite big, ‰2

b



and

2 ‰

jE./j1 62 HdR .P jE./j1 n X ;  n .X \ //.

The solution is of course to work in the blowup ZS of Section 3.4 where things are separated. Let PS be the variety obtained from P jE./j1 by regarding all elements of the Ln (n  1) in Section 3.4 as subspaces of P jE./j1 and starting the blowup sequence at n D 1 instead of n D 0. In [15], [16] it is shown that PS has the desired properties: the strict transform of X does not meet the strict transform of . In this way resS  is a relative period of the pair .PS n Y ; B n .B \ Y //; where Y is the strict transform of X and B the total transform of the coordinate divisor . We call resS  a Feynman period of . An empirical observation due to Broadhurst and Kreimer [21], [22] was that all Feynman periods computed so far are rational linear combinations of multiple zeta values. A multiple zeta value of depth k and weight s D s1 C    C sk is a real number defined as X 1 .s1 ; : : : ; sk / D s ; s1 n1 : : : nkk 1n 0 and let l ¤ p be a prime number. The étale cohomology of a variety X in V .k/ is defined as the l-adic cohomology of N Étale cohomology is a Weil cohomology theory with coefficients X Spec.k/ Spec.k/. in Ql . Example 1.10. Let k be a field of characteristic p > 0. Crystalline cohomology was introduced by Grothendieck and developed by Berthelot as a substitute for l-adic étale cohomology in the l D p case (see [29]). Let W .k/ be the ring of Witt vectors with coefficients in k and let FWitt.k/ be its field of fractions. Crystalline cohomology is a Weil cohomology theory with coefficients in FWitt.k/ . Remark 1.11. It is possible to define cohomology theories in a more general setting where the functor H takes values on a linear tensor category C. Properties analogous to the aforementioned ones should then hold. In particular H should be a symmetric monoidal functor from V .k/ to C, where we view V .k/ as a symmetric monoidal category with product X  Y D X Spec k Y (this is just the Künneth formula). 1.3 Correspondences. Let  be a fixed adequate equivalence relation on algebraic cycles on smooth projective varieties over k. Definition 1.12. Let X and Y be two varieties in V .k/. An element f 2 A .X  Y / is called a correspondence between X and Y . Note that this definition depends on the choice of the adequate equivalence relation . Given varieties X1 ; X2 and X3 denote by pr i;j W X1  X2  X3 ! Xi  Xj ;

1  i < j  3;

the projection morphisms. Given two correspondences f 2 A .X1  X2 / and g 2 A .X2  X3 / we define their composition as the correspondence g B f D pr 1;3  .pr 1;2 .f / pr 2;3 .g// 2 A .X1  X3 /: It can be shown that composition of correspondences is associative for any adequate equivalence relation (cf. [42]). For a variety X in V .k/ we denote by X the class of the diagonal cycle X ,! X  X in A .X  X /. If X and Y are two varieties in V .k/ and f 2 A .X  Y /, g 2 A .Y  X / are correspondences, then f B X D f;

X B g D g:

48

R. Sujatha, J. Plazas, and M. Marcolli

Thus X behaves as the identity with respect to composition of correspondences. z' the correspondence in A .X  Y / For any morphism ' W X ! Y denote by  given by the class of the cycle ' ,! X  Y given by the graph of '.

2 From varieties to pure motives As in the previous section let k be a fixed base field and denote by V .k/ the category of smooth projective varieties over k. Throughout this section we fix an adequate equivalence relation  for algebraic cycles on varieties in V .k/. In this section we will use the formalism developed in Section 1.3 to “linearize” the category V .k/. 2.1 Linearization. Correspondences between varieties possess all the formal properties of morphisms, we can therefore construct a new category whose objects correspond to smooth projective varieties over k but whose morphisms are given by correspondences. Since cycles modulo an adequate equivalence relation form an abelian group, the category thus obtained will have the advantage of being an additive category. Given two varieties X and Y in V .k/ we set L dim X Corr .X; Y / D A i .X  Y / i

where Xi are the irreducible components of X . Definition 2.1. Let F be a field of characteristic 0. We define Corr .k; F /, the category of correspondences over k with coefficients in F , as the category whose objects are smooth projective varieties over k, Obj.Corr .k; F // D Obj.V .k//; and whose morphisms are given by HomCorr .k;F / .X; Y / D Corr .X; Y / ˝ F: Composition of morphisms is given by composition of correspondences. The identity morphism in HomCorr.k/ .X; X / is given by the correspondence X . It follows from the definitions that given any adequate equivalence relation  for algebraic cycles on varieties in V .k/ the category Corr .k; F / is an F -linear category. We denote Corr .k; Q/ by Corr .k/ and refer to it simply as the category of correspondences over k. The category V .k/ can be faithfully embedded into the category Corr .k; F / via the contravariant functor h W V .k/ ! Corr .k; F /;

Introduction to motives

49

which acts as the identity on objects and sends a morphism ' W Y ! X in V .k/ to the z' in HomCorr .k;F / .X; Y / (see 1.3). We denote therefore by h.X / correspondence  a smooth projective variety X when considered as an object in Corr .k; F /. The product of varieties in V.k/ induces a tensor structure in the category Corr .k; F / via h.X / ˝ h.Y / D h.X  Y / turning Corr .k; F / into a F -linear tensor category with identity object given by 1k ´ h.Spec k/: 2.2 Pseudo-abelianization. The category Corr .k; F / obtained in Section 2.1, although being F -linear, is still far from abelian. In particular not every idempotent morphism in Corr .k; F / corresponds to a direct sum decomposition of the underlying object. In this section, we will formally add the kernels of idempotent morphisms in Corr .k; F / in order to obtain a pseudo-abelian category. As this formal procedure can be carried out for any additive category, we start this section by describing it in this generality. Definition 2.2. An additive category A is called pseudo-abelian if for any object A in A and any idempotent endomorphism p D p 2 2 HomA .A; A/ there exist a kernel ker p and the canonical morphism: ker p ˚ ker.idA  p/ ! A is an isomorphism. Given any additive category D; it is possible to construct a pseudo-abelian category z into which D embeds fully faithfully via a functor D z ‰D W D ! D; which is universal in the sense that given any additive functor F W D ! A, where A z ! A such that the is a pseudo-abelian category, there exists an additive functor Fz W D z functors G and F ‰D are equivalent. z is obtained by formally adding kernels of idempotent endomorThe category D z are given by pairs .D; p/ where D is an object phisms in D. Objects in the category D 2 in D and p D p 2 HomD .D; D/ is an idempotent endomorphism: z D f.D; p/ j D 2 Obj.D/; p D p 2 2 HomD .D; D/g: Obj.D/ z we define Hom z ..D; p/; .D 0 ; p 0 // to be the If .D; p/ and .D 0 ; p 0 / are objects in D D quotient group ff 2 HomD .D; D 0 / such that fp D p 0 f g : ff 2 HomD .D; D 0 / such that fp D p 0 f D 0g

50

R. Sujatha, J. Plazas, and M. Marcolli

Composition of morphisms is induced from composition of morphisms in D. The category thus obtained is a pseudo-abelian category and the functor given on objects by ‰D W D 7! .D; idD / and sending a morphism f 2 HomD .D; D 0 / to its class in HomD .D; D 0 /=.0/ is fully faithful and satisfies the above universal property. z the pseudo-abelian envelope of D (also sometimes referred to as the We call D idempotent completion of D, or the Karoubi envelope D). If the category D is an z is an F -linear pseudo-abelian category. If the F -linear category for a field F then D category D has an internal tensor product ˝ then the product .D; p/ ˝ .D 0 ; p 0 / D .D ˝ D 0 ; p ˝ p 0 / z is an internal tensor product on D. Definition 2.3. Let F be a field of characteristic 0. The category of effective motives over k with coefficients in F , denoted by Moteff  .k; F /, is the pseudo-abelian envelope of the category Corr .k; F /. eff As above we denote Moteff  .k; Q/ by Mot  .k/ and refer to its objects simply as eff effective motives over k. The category Mot .k; F / is by construction a pseudo-abelian F -linear tensor category. We can extend the functor h from V .k/ to Corr .k; F / to a functor from V .k/ to Moteff  .k; F / by composing it with the canonical embedding ‰Corr .k;F / , we denote the functor thus obtained also by h. Spelling out the definition of the pseudo-abelian envelope in this particular case, we see that effective motives over k can be represented as pairs

.h.X /; p/ where X is a smooth projective variety over k and p 2 Corr.X; X / ˝ F is an idempotent correspondence. Since for any such idempotent we have ker p ˚ ker.X  p/ D h.X / in Moteff  .k; F /; we see that effective motives over k are essentially given by direct factors of smooth projective varieties over k. Consider the case of Pk1 . Let e be as in Lemma 1.6, then the correspondence .1  e/ 2 Corr.Pk1 ; Pk1 / is idempotent. We define the Lefschetz motive over k to be the effective motive given by Lk D .h.Pk1 /; .1  e//I in particular we get a decomposition of Pk1 in Moteff  .k; F / of the form h.Pk1 / D 1k ˚ L;

Introduction to motives

51

where as above we take 1k D h.Spec.k//. More generally, we obtain a decomposition of r-dimensional projective space over k as h.P r / D 1 ˚ Lk ˚    ˚ Lrk ; where Lik D Lk ˝    ˝ Lk ; i times: It can also be shown that an irreducible curve X in V .k/ admits a decomposition in Moteff  .k; F / of the form h.X / D 1 ˚ h1 .X / ˚ Lk : 2.3 Inversion. Tensoring with the Lefschetz motive induces a functor M ! M ˝ Lk ;

f 7! f ˝ idLk ;

from the category Moteff  .k; F / to itself. This functor is fully faithful. In particular, given two effective motives M and M 0 and integers n, m, N with N  n; m the F -vector space m n HomMoteff .M ˝ LN ; M 0 ˝ LN / k k  .k;F // is independent of the choice of N . This can be used to obtain the category of pure motives from the category of effective motives by formally inverting the element Lk . More precisely, define the category of pure motives Mot .k; F / as the category whose objects are given by pairs .M; n/, where M is an effective motive and m is an integer, Obj.Mot .k; F // D f.M; m/ j M 2 Obj.Moteff  .k; F //; m 2 Zg and whose morphisms are given by m n HomMot .k;F / ..M; m/; .M 0 ; n// D HomMoteff .M ˝ LN ; M 0 ˝ LN /; k k  .k;F //

where N  n; m. As above we let Mot .k; Q/ D Mot .k/. The category Mot .k; F / has a tensor product given by .M; m/ ˝ .M 0 ; n/ D .M ˝ M 0 ; m C n/: We can embed the category of effective motives Moteff  .k; F // as a subcategory of Mot .k; F / via the functor M 7! .M; 0/: As before we denote by h the functor from V.k/ to Mot .k; F / induced by the above embedding. Denote by Tk the object .1k ; 1/ in Mot .k; F / and write Tkn for .1k ; n/, n 2 Z. Then Tk0 D 1k and there is a canonical isomorphism Tk1 D Lk :

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The element Tk is called the Tate motive. Tk plays a role analogous to the Tate module in l-adic cohomology. We define M.n/ D M ˝ Tkn

(“Tate twisting”):

Any pure motive can be written as M.n/ for an effective motive M and an integer n. It can be shown (see [22]) that for any smooth projective variety X there are canonical isomorphisms Ar .X / ˝ F ' HomMot .k;F / .1k ; h.X /.r//: As mentioned in the Introduction some of the properties of the category Mot .k/ and the extent to which it depends on the choice of  remain largely conjectural. We end this section with an important result due to Jannsen in the case  is numerical equivalence on cycles (Example 1.4). Theorem 2.4 (Jannsen [30]). The category Motnum .k; F / is a semi-simple F -linear, rigid tensorial category.

3 Artin motives Artin motives are motives of zero-dimensional varieties, already at this level various facets of the theory make their appearance and some of the richness of the underlying structures manifest itself. In a sense which we will make more precise, the theory of Artin motives can be considered as a linearization of Galois theory. Let V 0 .k/ be the subcategory of V.k/ consisting of varieties of dimension 0 over k. Objects in V 0 .k/ are given by spectra of finite k-algebras: Obj.V 0 .k// D fX 2 Obj.V.k// j X D Spec.A/; dimk A < 1g: Fix a separable closure k sep of k. Then for any X D Spec.A/ 2 Obj.V 0 .k// the absolute Galois group Gk D Gal.k sep =k/ acts continuously on the set of algebraic points of X: X.k sep / D Homkalg .A; k sep /: The action of Gk commutes with morphisms in V 0 .k/ since these are given by rational maps. Taking algebraic points induces then a functor X 7! X.k sep / between V 0 .k/ and the category F 0 .Gk / consisting of finite sets endowed with a continuous Gk -action. The fact that this functor is an equivalence of categories is essentially a restatement of the main theorem of Galois theory. The inverse functor maps a finite Gk -set I 2 Obj.F 0 .Gk // to the spectrum of the ring of Gk -invariant functions from I to k sep . This equivalence of categories is usually referred as the Grothendieck–Galois correspondence.

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The category of Artin motives Mot0 .k; F /, is by definition the subcategory of Mot .k; F / spanned by objects of the form h.X / for X 2 Obj.V 0 .k//. It is important to note that since any adequate equivalence relation on algebraic cycles on varieties in V .k/ becomes trivial when restricted to V 0 .k/; this definition does not depend on the choice of . Also, since the Lefschetz motive corresponds to the decomposition of the one dimensional variety Pk1 there is no need to take into account the twisting by Tk . The category Mot0 .k; F / is therefore the pseudo-abelian envelope of the category of correspondences of zero-dimensional varieties Corr0 .k; F /, whose objects are given by varieties in V 0 .k/ and whose morphisms are given by HomCorr0 .k;F / .X; Y / D C 0 .X  Y / ˝ F: A correspondence between two varieties X and Y in V 0 .k/ (with coefficients in F ) is thus given by a formal linear combination of connected components of X  Y with coefficients in F . By taking characteristic functions we may identify such a correspondence with a Gk -invariant function from X.k sep /Y .k sep / to F . Composition of correspondences becomes matrix multiplication and passing to the pseudo–abelian envelope we get an equivalence of categories, Mot0 .k; F / ' Rep.Gk ; F /; where Rep.Gk ; F / is the category of finite dimensional F -representations of the group Gk . The functor of motivic cohomology restricted to dimension zero is then given by h W X 7! F X.k

sep /

; sep

where X 2 Obj.V 0 .k// and the F -vector space F X.k / is endowed with the natural Gk -action. The category Mot0 .k; F / has a rich structure coming from the fact that it can be identified with the category of representations of a group, the corresponding properties encoded thereby correspond to the fact that Mot0 .k; F / is a Tannakian category. When reference to an ambient category is relevant it is customary to view Mot0 .k; F / as a subcategory of Motnum .k; F /.

4 Vistas As mentioned in the introduction, the formal properties in the construction of Mot .k; F / imply the existence of realization functors ‡H W Mot .k; F / ! GrAlgF for various Weil cohomology theories H W V .k/op ! GrAlgF defined on V .k/. These functors enrich the structure of Mot .k; F / and lead to further important constructions.

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Realization functors play an important role in the definition of L-functions associated to motives the special values of which are of prime importance in number theory (cf. [21]). Realization functors also play a role in the definition of motivic Galois groups. In order to be able to modify the category Mot .k; F / to obtain a category equivalent to the category of representations of a group,it is necessary to have a fiber functor with values on F -vector spaces playing the role of forgetful functor. As mentioned in the previous section in the case of Artin motives the absolute Galois group of the base field is recovered from this formalism. However more general cases involve in a deep way the validity of the standard conjectures (see [50] for a review). Once a Tannakian category of motives has been constructed the corresponding motivic Galois group provides a rich higher dimensional analogue of Galois theory. From here on, the theory develops rapidly and branches in numerous directions, leading to a very rich landscape of results connecting and interrelating various areas of mathematics. The largely conjectural theory of mixed motives, that is motives of varieties which are not necessary smooth or projective, seems to underlay phenomena relevant to different fields. Areas like Hodge theory, K-theory and automorphic forms are enriched by the presence of structures of motivic origin leading in many cases to deep conjectures. For an account of various aspects of the theory the reader may consult the two volumes [32].

5 Endomotives Various interactions between noncommutative geometry and number theory have taken place recently. The tools and methods developed by noncommutative geometry are well suited for the study of structures relevant in number theory, and in many cases shed new light into old outstanding problems (cf. [44], [15]). Some aspects of the theory so developed make contact with the theory of motives in a natural way. In this section we describe the construction of a class of noncommutative spaces closely related to Artin motives and its relevance in the study of class field theory. The main reference for the material treated in this section is [11], which we follow closely (see also [15], [18]). 5.1 Adèles and idèles, basics in class field theory. Global class field theory describes the abelian extensions of a global field k in terms of analytic-arithmetic data coming from the field itself. We start this section by briefly recalling some basic notions from global class field theory (cf. [1]). A global field k is by definition a field of one of the following two kinds: • a number field, i.e., a finite extension of Q, the field of rational numbers; • a function field, i.e., a finite extension of Fq .t /, the field of rational functions in one variable over a finite field.

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An important part of the structure of a global field k is encoded by ideals in its ring of integers Ok which is given by the integral closure in k of Z in the number field case or that of Fq Œt in the function field case. A valuation on a global field k is by definition a nonnegative multiplicative function j  j from k to R, with j0j D 0 and non-vanishing on k  D k n f0g, satisfying the triangle inequality jx C yj  jxj C jyj for all x; y 2 k: The valuation j  j is called non-archimedean if it satisfies jx C yj  maxfjxj; jyjg

for all x; y 2 k;

otherwise we say that the valuation j  j is archimedean. Two valuations on a global field k are said to be equivalent if the corresponding metrics induce the same topology on k. A place  on a global field k is by definition an equivalence class of valuations on k. A place  on k is said to be archimedean (resp. non-archimedean) if it consist of archimedean (resp. non-archimedean) valuations. Given a place  on a global field k we denote by k the completion of k with respect to the metric induced by any of the valuations in . The space k is a locally compact field. If  is a non-archimedean place, the set Ok; D f 2 k j jj  1g is a subring of k where j  j is the norm in k induced by . We call Ok; the ring of integers of k . For a non-archimedean place  on k the ring Ok; is an open compact subring of k . If k is a number field, then non-archimedean places on k are in one-to-one correspondence with prime ideals in Ok while the archimedean places on k correspond to the finitely many different embeddings of k in C. This is essentially a consequence of Ostrowski’s theorem by which any valuation on Q is equivalent to the p-adic absolute value j  jp for some prime number p or to the ordinary absolute value induced by the embedding of Q in R. Given a collection f† g2ƒ of locally compact topological spaces and compact subspaces   † for all but finitely many  2 ƒ; there exists a unique topology on the set .res/ Q

Q † D f. / 2 † j  2  for all but finitely many  2 ƒg; Q Q for which  is a compact subspace. We call the set .res/ † together with this topology the restricted topological product of f† g2ƒ with respect to the subspaces  . Definition 5.1. Let k be a global field and let P be the collection of places on k. The ring of adèles of k, denoted by Ak , is the topological ring given by the restricted topological product of fk g2P with respect to the subspaces Ok; . The group of idèles of k, denoted by Ik , is the topological group given by the restricted topological product  . of fk g2P with respect to the subspaces Ok;

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Any global field k can be embedded as a discrete co-compact subfield of the locally compact ring Ak . Likewise k  embeds diagonally in Ik . The quotient topological group Ck D Ik =k  is called the idèle class group of k. One of the central results in class field theory associates to any abelian extension of a global field k a subgroup of the idèle class group. The main theorem of global class field theory can be stated as follows: Theorem 5.2. Let k be a number field and let k ab be its maximal abelian extension. Denote by Dk the connected component of the identity in Ck . Then there is a canonical isomorphism of topological groups: ‰ W Ck =Dk ! Gal.k ab =k/: In the case of a function field k an analogous result identifies in a canonical way the group Gal.k ab =k/ with the profinite completion of Ck . The isomorphism ‰ is usually referred as the global Artin map or reciprocity map. 5.2 Noncommutative spaces. One of the departure points of noncommutative geometry is the duality between various classes of spaces and algebras of functions canonically associated to these spaces. One classical instance of this duality is furnished by Gelfand’s theorem which provides a one-to-one correspondence between compact Hausdorff topological spaces and unital C*-algebras via the functor that sends a space X to its algebra of complex valued continuous functions. Other examples and of this type of duality and refinements thereof abound in the mathematical landscape. In these situations it might be possible to define structures relevant to the study of a space in terms of the corresponding algebras of functions and, in cases in which these definitions do not depend on the commutativity of the underlying algebra, extend them in order to study noncommutative algebras. Geometric information which is difficult to encode with traditional tools might be understood by enlarging the class of spaces in consideration in such a way as to allow the presence of “noncommutative coordinates”. In this way we view a noncommutative algebra as defining by duality a noncommutative space for which this algebra plays the role of algebra of coordinates. One may then for example view a noncommutative unital C*-algebra as defining by duality a “noncommutative compact Hausdorff topological space”. This process is far from being a mere translation of concepts to another framework even when such a translation might prove to be delicate. Many new phenomena arise in this context and in some situations the classical picture is also enriched (cf. [9]). 5.3 Quantum statistical mechanics. A particular area in which these ideas occur naturally is quantum statistical mechanics. A quantum statistical mechanical system is determined by a C*-algebra A (the algebra of observables of the system) and a

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one-parameter group of automorphisms t 2 Aut.A/, t 2 R (the time evolution of the system). A state on the C*-algebra A is by definition a norm-one linear functional ' W A ! C satisfying the positivity condition '.a a/  0 for all a 2 A. In a quantum statistical mechanical system .A; t / the time evolution t singles out a class of states of the algebra A, the equilibrium states of the system, these are families of states parametrized by a positive real number ˇ corresponding to a thermodynamical parameter (the inverse temperature of the system). The appropriate definition of equilibrium states in the context of quantum statistical mechanics was given by Haag, Hugenholtz and Winnink in [25]. This condition, known after Kubo, Martin and Schwinger as the KMS condition, is given as follows. Definition 5.3. Let .A; t / be a quantum statistical mechanical system. A state ' on A satisfies the KMS condition at inverse temperature 0 < ˇ < 1 if for every a; b 2 A there exists a bounded holomorphic function Fa;b on the strip fz 2 C j 0 < Im.z/ < ˇg, continuous on the closed strip, such that Fa;b .t/ D '.a t .b//;

Fa;b .t C iˇ/ D '. t .b/a/ for all t 2 R:

We call such state a KMSˇ state. A KMS1 state is by definition a weak limit of KMSˇ states as ˇ ! 1. For each 0 < ˇ  1 the set of KMSˇ states associated to the time evolution t is a compact convex space (cf. [6], Section 5:3). We denote by Eˇ the space of extremal points of the space of KMSˇ states. A group G  Aut.A/ such that t g D g t for all g 2 G and all t 2 R is called a symmetry group of the system .A; t /. If G is a symmetry group of the system .A; t / then G acts on the space of KMSˇ states for any ˇ and hence on Eˇ . Inner automorphisms coming from unitaries invariant under the time evolution act trivially on equilibrium states. Starting with the seminal work of Bost and Connes, [5], various quantum statistical mechanical systems associated to arithmetic data have been studied. Potential applications to the explicit class field theory problem make the understanding of such systems particularly valuable (see [44], [15] and references therein). We recall below the definition and properties of the quantum statistical mechanical system introduced in [5] in the form most adequate for our purposes, which in particular serves as a motivating example for the introduction of endomotives. 5.4 The Bost–Connes system. For a positive integer n consider the cyclic group of order n as a zero-dimensional variety over Q given by Xn D Spec.An /I

An D QŒZ=nZ:

We order N by divisibility. For njm the canonical morphism Xm ! Xn is a morphism in V 0 .Q/ and we can view fXn gn2N as a projective system of zero-dimensional algebraic varieties over Q. The profinite limit X D lim Xn  n

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corresponds to the direct limit of finite dimensional algebras: A D lim An D lim QŒZ=nZ ' QŒQ=Z: ! ! n

n

Denote by fe.r/ j r 2 Q=Zg the canonical basis of A. The multiplicative abelian semigroup N acts as a semigroup of endomorphisms of the algebra A by P e.l/:

n W A ! A; e.r/ 7! n1 l2Q=Z;lnDr

x coincides with The algebra of continuous functions on the profinite space X.Q/  C .Q=Z/, the group C*-algebra of Q=Z (i.e., the completion of CŒQ=Z in its regular representation). The semigroup action of N on A extends to an action of semigroup action of N on C  .Q=Z/ and the semigroup crossed product algebra AQ D A Ì N is a rational sub-algebra of the semigroup crossed product C*-algebra A D C  .Q=Z/ Ì N : The C*-algebra A coincides with the C*-algebra generated by elements e.r/, r 2 Q=Z (corresponding the generators of the group algebra of Q=Z) and n , n 2 N (corresponding the generators of the semigroup action) and satisfying the relations: e.0/ D 1;

e.r/e.s/ D e.r C s/; e.r/ D e.r/ for all r; s 2 Q=Z; n k D nk ; n n D 1 for all n; k 2 N ; n e.r/n D n .e.r// for all n 2 N ; r 2 Q=Z:

It is possible to define a time evolution t on the C*-algebra A by taking t .e.r// D e.r/;

t .n / D nit :

y  ' Hom.Q=Z; Q=Z/ defines a representation  of the C*Each element  2 Z algebra A as an algebra of operators on the Hilbert space l 2 .N / via  .e.r// k D e 2ır "n ;

 .n /"k D "nk ;

where f"k gk2N is the canonical basis for l 2 .N /. For any of these representations the time evolution t can be implemented via the Hamiltonian H."k / D .log k/ k as  . t .a// D e ıtH  .a/e ıtH : The partition function of the system, defined as Trace.e ˇH / is then given by the Riemann zeta function .ˇ/. The structure of equilibrium states for this system was studied in [5], where it is shown that for 0 < ˇ  1 there exist a unique KMSˇ state while for any 1 < ˇ  1 there are infinitely many states and the space of extremal states Eˇ O  . The group Z y  acts as a group of symmetries of the system is homeomorphic to Z .A; t / and the induced action on Eˇ for 1 < ˇ  1 is free and transitive. The arithmetic of abelian extensions of Q is encoded in this system in the following way:

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Theorem 5.4 (Bost, Connes). For every ' 2 E1 and every a 2 AQ the value '.a/ is algebraic over Q. Moreover Qab is generated by values of this form and for all ' 2 E1 ,  2 Gal.Qab jQ/ and a 2 AQ one has '.a/ D '.‰ 1 . /a/; y  ! Gal.Qab jQ/ is the class field theory isomorphism. where ‰ W CQ =DQ D Z 5.5 Algebraic and analytic endomotives. In this section we briefly survey the theory of endomotives as introduced in [11]. It provides a systematic way to construct associative algebras analogous to the rational sub-algebra of the Bost–Connes system in terms of arithmetic data. The resulting category provides an enlargement of the category of Artin motives over a number field to a category of arithmetic noncommutative spaces. Let k be a number field. Given a projective system fXi gi2I of varieties in V 0 .k/, I a countable partially ordered set, we can consider the direct limit of algebras A D lim Ai ; ! i

where Xi D Spec Ai . Assume that S is an abelian semigroup acting by algebra endomorphisms on A such that for any s 2 S the corresponding endomorphism s induces an isomorphism A ' ps A; where ps D ps2 D s .1/. Definition 5.5. An algebraic endomotive over k is given by an associative algebra of the form Ak D A Ì S corresponding a projective system fXi gi2I of varieties in V 0 .k/ and a semigroup S as above. Fix an algebraic closure kN of k. For an algebraic endomotive Ak with corresponding projective system fXi gi2I the set of algebraic points of the pro-variety X D lim Xi i is given by the profinite (compact Hausdorff) space: N D Homk-alg .A; k/: N X.k/ N Pure Given an embedding kN ,! C we can identify A with a sub-algebra of C.X.k//. N states in C.X.k// attain algebraic values when restricted to A. Moreover, the natural N N induces an action of Gal.k=k/ N action of the absolute Galois group Gal.k=k/ on X.k/ on the C*-algebra N Ì S; A D C.X.k// in which the algebraic endomotive Ak embeds as a rational sub-algebra. An analytic endomotive is by definition a C*-algebra of the above form. In the case of an endomotive given by abelian extensions of k the corresponding action factors through Gal.k ab =k/ and the following result holds:

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Theorem 5.6 (Connes, Consani, Marcolli). Let Ak be an algebraic endomotive over k with corresponding projective system of varieties fXi gi2I and semigroup S . Assume that for each i 2 I the algebra Ai is a finite product of normal abelian field extensions N of k. Then the algebras A and Ak are globally invariant under the action of Gal.k=k/ N N on A D C.X.k//ÌS . Moreover, any state ' on A induced by a pure state on C.X.k// satisfies '.a/ D '. 1 a/ for all a 2 Ak and all  2 Gal.k ab =k/. Example 5.7. Let .Y; y0 / be a pointed algebraic variety over k and let S be an abelian semigroup of finite morphisms from Y to itself. Assume moreover that any morphism in S fixes y0 and is unramified over y0 . For any s 2 S let Xs D fy 2 Y j s.y/ D y0 g: Order S by divisibility, then the system fXs gs2S defines an algebraic endomotive over k with S as semigroup of endomorphisms. Example 5.8. Let Gm denote the multiplicative group viewed as a variety over Q. The power maps z 7! z n for n 2 N define finite self-morphisms on Gm fixing the point 1 2 Gm and unramified over it. The algebraic endomotive associated to .Gm ; 1/ and S D N as in Example 5.7 is the arithmetic subalgebra AQ of the Bost–Connes system. Example 5.9. Let E be an elliptic curve with complex multiplication by an order O in an imaginary quadratic field K. Any endomorphism of E in O  fixes 0 and is unramified over it. The algebraic endomotive associated to .E; 0/ and S D O  as in Example 5.7 is the arithmetic subalgebra of the quantum statistical mechanical system considered in [16]. This system encodes the class field theory of the field K. An algebraic endomotive over k defines a groupoid in a natural way. By considering groupoid actions satisfying a suitable étale condition it is possible to extend morphisms in Corr0 .k; F / in order to define correspondences between algebraic endomotives. The pseudo-abelian envelope of the category so obtained is the category of algebraic endomotives EndMot0 .k; F /. By construction Mot0 .k; F / embeds as a full subcategory of EndMot0 .k; F /. The groupoid picture likewise allows to define a category of analytic endomotives C EndMot0 .k; F / as the pseudo-abelian envelope of the category of correspondences between analytic endomotives. The map that assigns to an algebraic endomotive over k N with projective system fXi gi2I and semigroup S the analytic endomotive C.X.k//ÌS, X D lim Xi , extends to a tensor functor i EndMot0 .k; F / ! C EndMot .k; F / 0

N on which the Galois group Gal.k=k/ acts by natural transformations.

Motivic ideas in noncommutative geometry An appendix by Matilde Marcolli There has been in recent years a very fruitful interplay between ideas originally developed in the context of Grothendieck’s theory of motives of algebraic varieties and techniques and notions arising in the context of noncommutative geometry. Two main directions have become prominent: one based on treating noncommutative spaces as algebras, and importing motivic ideas by extending the notion of morphisms of noncommutative spaces to include Morita equivalences through correspondences realized by bimodules and other types of morphisms in larger categories like cyclic modules. This allows for cohomological methods based on cyclic (co)homology to be employed in a setting that provides an analog of the motivic ideas underlying the Weil proof of the Riemann hypothesis for function fields. It is this approach, developed in [11], [12], [15], that we focus on mostly in this survey. It concentrates on a category of noncommutative motives that are built out of the simplest class of motives of algebraic varieties, the Artin motives, which are motives of zero-dimensional algebraic varieties. At the same time, there is a more general and very broad approach to motives in the noncommutative geometry setting, developed by Kaledin, Kontsevich, Tabuada, and others [33], [34], [51], based on the idea of representing noncommutative spaces as categories instead of algebras, and the related circle of ideas of derived algebraic geometry, [35], see also the short survey [41]. As we argue briefly in §4 below, one can expect that a merging of these two approaches will lead to some very interesting generalizations of some of the results that we review here.

1 Noncommutative motives and cyclic cohomology In noncommutative geometry one encounters a problem that is very familiar to the context of algebraic geometry. Namely, if one thinks of noncommutative spaces as being described by associative algebras, then the category of algebras over a field with algebra homomorphisms is not abelian or even additive. Moreover, it is well known that morphisms of algebras are too restrictive a notion of morphisms for noncommutative spaces, as they do not account for the well known phenomenon of Morita equivalence. Thus, one needs to embed the category of associative algebras with algebra homomorphisms inside an abelian (or at least pseudo-abelian) category with a larger collection of morphisms that include the correspondences given by tensoring with bimodules, as in the case of Morita equivalences. The objects of a category with these properties can be regarded as “noncommutative motives”, in the same sense as the motives of algebraic varieties are the objects of an abelian category (or pseudo-abelian, or triangulated in the mixed case) that contains the category of algebraic varieties.

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A first construction of such a category of noncommutative motives was obtained in [8], using the notion of cyclic modules. One first defines the cyclic category ƒ as the category that has an object Œn for each positive integer and has morphisms generated by the morphisms ıi W Œn  1 ! Œn, j W Œn C 1 ! Œn, and n W Œn ! Œn, with the relations ıj ıi D ıi ıj 1 for i < j; j i D i j C1 for i  j; 8 ˆ j C 1; n ıi D ıi1 n1 n ı0 D ın ;

for 1  i  n;

n i D i1 nC1

for 1  i  n;

n 0 D

2 n nC1

nnC1 D 1n : Given a category C , one defines as cyclic objects the covariant functors ƒ ! C. In particular, we consider the case where C D Vect K is the category of vector spaces over a field K and we refer to the cyclic objects as cyclic modules, or K.ƒ/-modules. In particular, consider a unital associative algebra A over a field K. One associates to A a K.ƒ/-module A\ , which is the covariant functor ƒ ! Vect K that assigns to objects in ƒ the vector spaces .nC1/

Œn ) A˝

D A ˝ A ˝  ˝ A

and to the generators of the morphisms of ƒ the linear maps ıi ) .a0 ˝    ˝ an / 7! .a0 ˝    ˝ ai aiC1 ˝    ˝ an /; j ) .a0 ˝    ˝ an / 7! .a0 ˝    ˝ ai ˝ 1 ˝ aiC1 ˝    ˝ an /; n ) .a0 ˝    ˝ an / 7! .an ˝ a0 ˝    ˝ an1 /: The category of cyclic modules is an abelian category, and the construction above shows that one can embed inside it a copy of the category of associative algebras, hence the cyclic modules can be regarded as noncommutative motives. Notice that there are many more objects in the category of cyclic modules than those that come from associative algebras. For example, being an abelian category, kernels and cokernels of morphisms of cyclic modules are still cyclic modules even when, for instance, one does not have cokernels in the category of algebras.

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Moreover, one sees that there are many more morphisms between cyclic modules than between algebras and that the type of morphisms one would like to have between noncommutative spaces, such as Morita equivalences, are included among the morphisms of cyclic modules. In particular, all the following types of morphisms exist in the category of cyclic modules. • Morphisms of algebras  W A ! B induce morphisms of cyclic modules  \ W A\ ! B \ . • Traces  W A ! K induce morphisms of cyclic modules  \ W A\ ! K\ by setting  \ .x 0 ˝    ˝ x n / D .x 0 : : : x n /. • A-B-bimodules E induce morphisms of cyclic modules, E \ D  \ B \ , by composing W A ! EndB .E/ and  W EndB .E/ ! B. Moreover, it was shown in [8] that in the abelian category of cyclic modules the Ext functors recover cyclic cohomology of algebras by HCn .A/ D Extn .A\ ; K\ /:

2 Artin motives and the category of endomotives The simplest category of motives of algebraic varieties is the category of Artin motives, which corresponds to zero-dimensional varieties over a field K (which we take here to be a number field), with correspondences that are given by formal linear combinations of subvarieties Z  X  Y in the product. Because everything is zero-dimensional, in this case one does not have to worry about the different equivalence relations on cycles. Artin motives over K not only form an abelian category, but in fact a Tannakian x category with motivic Galois group given by the absolute Galois group Gal.K=K/. The category of endomotives was introduced in [11] as a category of noncommutative spaces that are built out of towers of Artin motives with endomorphisms actions. At the algebraic level, one considers as objects crossed product algebras AK D AÌS , where A is a commutative algebra over K, obtained as a direct limit A D lim A˛ !˛ of finite dimensional reduced algebras over K, which correspond under X˛ D Spec.A˛ / to zero-dimensional algebraic varieties X˛ , Artin motives over K. The direct limit A of algebras corresponds to a pro-variety X D lim X˛ . ˛ The datum S is a unital abelian semigroup, which acts on A by endomorphisms ' with W A  ! eAe, where e D .1/ is an idempotent e 2 D e. Morphisms between these objects are also constructed out of morphisms (correspondences) in the category of Artin motives, via projective limits, compatibly with the semigroup actions. More precisely, if G .X˛ ; S / denotes the groupoid of the action of S on X, the morphisms of endomotives are given by étale correspondences, which are formal linear combinations of G .X˛ ; S/  G .X˛0 0 ; S 0 / spaces Z for which the right action of G .X˛0 0 ; S 0 / is étale. This means that, when representing Z D Spec.M /, with

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M an AK –A0K bimodule, M is finite projective as a right AK -module. Morphisms are 0 0 then given P by the Q-linear space M..X˛ ; S /; .X˛0 ; S // formal linear combinations U D i ai Zi of étale correspondences as above. The composition of morphisms is then given by the fibered product Z B W D Z G 0 W over the groupoid of the action of S 0 on X 0 . x with the action At the analytic level, one considers the topological space X D X.K/ x of the semigroup S . The topology is the one of the projective limit, which makes X.K/ into a totally disconnected (Cantor-like) compact Hausdorff space. One can therefore x Ì S D C  .G /. One imposes consider the crossed product C*-algebras A D C.X.K// a uniform condition, which provides a probability measure  D lim ˛ obtained using   x Integration the counting measures on the X˛ , with dd locally constant on X.K/. with respect to this measure gives a state ' on the C*-algebra A. One can also extend morphisms given by étale correspondences to this analytic setting. These are given by spaces Z with maps g W Z ! X with discrete fiber and such that 1 is a compact operator on the MZ over C.X/ from the Cc .G /P right module N valued inner product h; i.x; s/ ´ z2g 1 .x/ .z/.z B s/. For G -G 0 spaces defining morphisms of algebraic endomotives, one can consider x D Z and obtain a correspondence of analytic endomotives, with Cc .Z/ Z 7! Z.K/ a right module over Cc .G /. These morphisms induce morphisms in the KK category and in the category of cyclic modules. x The analytic endomotive A is endowed with a Galois action of G D Gal.K=K/, as x x an action on the characters X.K/ D Hom.A; K/ by 



g

x 7! A ! K x ! K: x A ! K The action is also compatible with the endomorphisms action of S since the latter acts on the characters by pre-composition. Thus, the Galois group acts by automorphisms of the crossed product algebra A D C.X/ Ì S . 2.1 The Bost–Connes endomotive. A prototype example of an endomotive is the noncommutative space associated to the quantum statistical mechanical system constructed by Bost and Connes in [5]. A more transparent geometric interpretation of this noncommutative space as the moduli space of 1-dimensional Q-lattices up to scale, modulo the equivalence relation of commensurability, was given in [14]. A Q-lattice in Rn is a pair .ƒ; / of a lattice ƒ  Rn and a (possibly degenerate) labeling of its torsion points via a group homomorphism  W Qn =Zn ! Qƒ=ƒ: In the special case where  is an isomorphism one says that the Q-lattice is invertible. The equivalence relation of commensurability is defined by setting .ƒ1 ; 1 /  .ƒ2 ; 2 / whenever Qƒ1 D Qƒ2 and 1 D 2 mod ƒ1 C ƒ2 .

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The quotient of the space of Q-lattices by the commensurability relation is a “bad quotient” in the sense of ordinary geometry, but it can be described by a noncommutative space whose algebra of functions is the convolution algebra of the equivalence relation. In the 1-dimensional case a Q-lattices is specified by the data .ƒ; / D .Z;  / y If we consider for some  > 0 and some 2 Hom.Q=Z; Q=Z/ D lim Z=nZ D Z. n the lattices up to scaling, we eliminate the factor  and we are left with a space whose y algebra of functions is C.Z/. The commensurability relation is then expressed by the action of the semigroup N D Z>0 which maps ˛n .f /. / D f .n1 / when one can divide by n and sets the result to zero otherwise. The quotient of the space of 1-dimensional Q-lattices up to scale by commensurability is then realized as a noncommutative space by the crossed product algebra y Ì N. This can also be written as a convolution algebra for a partially defined C.Z/ action of QC , with P f1 f2 .r; / D f1 .rs 1 ; s /f2 .s; / y s2Q C ;s2Z

with adjoint f  .r; / D f .r 1 ; r /. This is the algebra of the groupoid of the commensurability relation. It is isomorphic to the Bost–Connes (BC) algebra of [5]. As an algebra over Q, it is given by AQ;BC D QŒQ=Z Ì N, and it has an explicit presentation by generators and relations of the form n m D nm ; n m D m n when .n; m/ D 1; n n D 1; e.r C s/ D e.r/e.s/; e.0/ D 1; P e.s/:

n .e.r// D n e.r/n D n1 nsDr

y Ì N, where one uses the The C*-algebra is then given by C  .Q=Z/ Ì N D C.Z/ y identification, via Pontrjagin duality, between C.Z/ and C  .Q=Z/. The time evolution of the BC quantum statistical mechanical system is given in terms of generators and relations by t .e.r// D e.r/;

t .n / D nit n

d and it is generated by a Hamiltonian H D dt t j tD0 with partition function Tr.e ˇH / D .ˇ/, in the representations on the Hilbert space `2 .N/ parameterized by the invertible y  , are given on y  . These representation  on `2 .N /, for 2 Z Q-lattices 2 Z generators by n m D nm ;  .e.r// m D rm m ;

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where r D .e.r// is a root of unit. Given a C*-algebra with time evolution, one can consider states, that is, linear functionals ' W A ! C, with '.1/ D 1 and '.a a/  0, that are equilibrium states for the time evolution. As a function of a thermodynamic parameter (an inverse temperature ˇ), these are specified by the KMS condition: ' 2 KMSˇ for some 0 < ˇ < 1 if for all a; b 2 A there exists a holomorphic function Fa;b .z/ on the strip Iˇ D fz 2 C j 0 < Im.z/ < ˇg, continuous on the boundary @Iˇ , and such that, for all t 2 R, Fa;b .t / D '.a t .b//

and

Fa;b .t C iˇ/ D '. t .b/a/:

In the case of the BC system the KMS states are classified in [5]: the low temperature extremal KMS states, for ˇ > 1 are of the form 'ˇ; .a/ D

Tr.  .a/e ˇH / ; Tr.e ˇH /

y ;

2Z

while at higher temperatures there is a unique KMS state. At zero temperature the evaluations '1; .e.r// D r , which come from the projection on the kernel of the Hamiltonian, '1; .a/ D h 1 ;  .a/ 1 i; exhibit an intertwining of Galois action on the values of states on the arithmetic subalgebra and symmetries of the quantum statistical mechanical system: for a 2 AQ;BC y  , one has and  2 Z '1; .a/ D  .'1; .a//; where

' y   W Z ! Gal.Qab =Q/

is the class field theory isomorphism. The BC algebra is an endomotive with A D lim An , for An D QŒZ=nZ and the !n abelian semigroup action of S D N on A D QŒQ=Z. A more general class of endomotives was constructed in [11] using self-maps of algebraic varieties. One constructs a system .A; S / from a collection S of self-maps of algebraic varieties s W Y ! Y and their iterations, with s.y0 / D y0 unbranched and s of finite degree, by setting Xs D s 1 .y0 / and taking the projective limit X D lim Xs D Spec.A/ under the maps  s;s 0 W Xs 0 ! Xs ;

s;s 0 .y/ D r.y/;

s 0 D rs 2 S:

The BC endomotive is a special case in this class, with Y D Gm with self-maps u 7! uk sk W P .t; t 1 / 7! P .t k ; t k /; k;` .u.`// D u.`/k=` ;

k 2 N; P 2 QŒt; t 1 ; u.`/ D t mod t `  1:

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One then has Xk D Spec.QŒt; t 1 =.t k  1// D sk1 .1/ and X D lim Xk with k u.`/ 7! e.1=`/ 2 QŒQ=Z: x D C.Z/. y One can identify the algebras C.X.Q// 2.2 Time evolution and KMS states. An object .X; S / in the category of endomotives, constructed as above, determines the following data: • a C*-algebra A D C.X/ Ì S ; • an arithmetic subalgebra AK D A Ì S defined over K; • a state ' W A ! C from the uniform measure on the projective limit; • an action of the Galois group by automorphisms G  Aut.A/. As shown in [11], see also §4 of [15], these data suffice to apply the thermodynamic formalism of quantum statistical mechanics. In fact, Tomita–Takesaki theory shows that one obtains from the state ' a time evolution, for which ' is a KMS1 state. One starts with the GNS representation H' . The presence of a cyclic and separating vector  for this representation, so that M and M0  are both dense in H' , with M the von Neumann algebra generated by A in the representation, ensures that one has a densely defined operator S' W M ! M;

a 7! S' .a/ D a ;

S' W M 0  ! M0 ;

a0  7! S' .a0 / D a0 ;



which is closable and has a polar decomposition S' D J' 1=2 with J' a conjugate' linear involution J' D J' D J'1 and ' D S' S' a self-adjoint positive operator with J' ' J' D S' S' D 1 ' . it Tomita–Takesaki theory then shows that J' MJ' D M 0 and it ' M' D M, so that one obtains a time evolution (the modular automorphism group) it t .a/ D it ' a'

a 2 M;

for which the state ' is a KMS1 state. 2.3 The classical points of a noncommutative space. Noncommutative spaces typically do not have points in the usual sense of characters of the algebra, since noncommutative algebras tend to have very few two-sided ideals. A good way to replace characters as a notion of points on a noncommutative space is by using extremal states, which in the commutative case correspond to extremal measures supported on points. While considering all states need not lead to a good topology on this space of points, in the presence of a natural time evolution, one can look at only those states that are equilibrium states for the time evolution. The notion of KMS states provides equilibrium

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states at a fixed temperature, or inverse temperature ˇ. The extremal KMSˇ states thus give a good working notion of points on a noncommutative space, with the interesting phenomenon that the set of points becomes temperature dependent and subject to phase transitions at certain critical temperatures. In particular one can consider, depending on the inverse temperature ˇ, that subset ˇ of the extremal KMS states that are of Gibbs form, namely that are obtained from type I1 factor representations. In typical cases, these arise as low temperature KMSstates, below a certain critical temperature and are then stable when going to lower temperatures, so that one has injective maps cˇ 0 ;ˇ W ˇ ! ˇ 0 for ˇ 0 > ˇ. For a state 2 ˇ one has an irreducible representation W A ! B.H . //, where the Hilbert space of the GNS representation decomposes as H D H . / ˝ H 0 with M D fT ˝ 1 j T 2 B.H . //g. The time evolution in this representation is generated by a Hamiltonian t' . .a// D e itH .a/e itH with Tr.e ˇH / < 1, so that the state can be written in Gibbs form

.a/ D

Tr. .a/e ˇH / : Tr.e ˇH /

The Hamiltonian H is not uniquely determined, but only up to constant shifts H $ z ˇ D f."; H /g, with  ."; H / D ."; H C H C c so that one obtains a real line bundle    z log / for  2 RC . The fibration RC ! ˇ ! ˇ has a section Tr.e ˇ H / D 1, so z ˇ ' ˇ  R . it can be trivialized as  C Besides equilibrium KMS states, an algebra with a time evolution also gives rise to O  /, which is the algebra obtained by taking the crossed product with a dual system .A; the time evolution, endowed with a scaling action by the dual group. Namely, one considers the algebra ARO D AÌ R given by functions x; y 2 .R; AC / with convolution product .x ?y/.s/ D R x.t / t .y.st // dt. One equivalently writes R elements of AO formally as x.t /U t dt, where U t are the unitaries that implement the R action t . The scaling action  of  2 RC on AO is given by Z Z   x.t /U t dt D it x.t /U t dt: z ˇ determines an irreducible representation of AO by setting A point ."; H / 2  Z Z  ";H x.t /U t dt D " .x.t //e itH dt; compatibly with the scaling action: ";H B  D .";H / . When restricting to those elements x 2 AO ˇ  AO that have analytic continuation to strip of KMSˇ with rapid decay along the boundary, one obtains trace class operators [11] Z  ";H x.t / U t dt 2 L1 .H ."//:

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2.4 Restriction as a morphism of noncommutative motives. It is shown in [11] that one can define a restriction map from a noncommutative space to its classical points, where the latter are defined, as above, in terms of the low temperature extremal KMS states. This restriction map does not exist as a morphism of algebras, but it does exist as a morphism in an abelian category of noncommutative motives that contains the category of algebras, namely the category of cyclic modules described above. In fact, one can use the representations .x/."; H / and the trace class property to obtain a map  Tr z ˇ ; L1 /  z ˇ /; AO ˇ ! C. ! C.

zˇ; .x/."; H / D ";H .x/ for all ."; H / 2  under a technical hypothesis on the vanishing of obstructions; see [11] and §4 of [15]. Because this map involves taking a trace, it is not a morphism in the category of algebras. However, as we have discussed above, traces are morphisms in the category of cyclic modules, so one regards the above map as a map of the corresponding cyclic modules,  z ˇ ; L1 /\ ; AO \ˇ ! C.

\

ıD.Tr B / Q ˇ /\ : AO \ˇ ! C.

This is equivariant for the scaling action of RC . Moreover, we know by [8] that the category of cyclic modules is an abelian category. This means that the cokernel of this restriction map exists as a cyclic module, even though it does not come from an algebra. In [11] we denoted this cokernel as D.A; '/ D Coker.ı/. One can compute its cyclic homology HC0 .D.A; '//, which also has an induced scaling action of RC , as well as an induced representation of the Galois group G, coming from the Galois representation on the endomotive A. This gives a space (not a noncommutative space but a noncommutative motive) D.A; '/ whose cohomology HC0 .D.A; '// is endowed with a scaling and a Galois action. These data provide an analog in the noncommutative setting of the Frobenius action on étale cohomology in the context of motives of algebraic varieties. 2.5 The Bost–Connes endomotive and the adèles class space. In [11] and [12] the motivic setting described above was applied in particular to the case of the Bost– y Ì N with the state given by the measure '.f / D Connes endomotive A D C.Z/ R f .1;

/ d. / and the resulting time evolution recovering the original time evolution y Z of the Bost–Connes quantum statistical mechanical system, t .f /.r; / D r it f .r; /. In this case then the space of classical points is given by y   RC D CQ D A =Q Qˇ DZ  Q for small temperatures ˇ > 1.

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The dual system is the groupoid algebra of the commensurability relation on Qlattices, considered not up to scaling, AO D C  .GQ /, where one identifies Z h.r; ; / D f t .r; /it U t dt and the groupoid is parameterized by coordinates y  RC j r 2 Zg: y GQ D f.r; ; / 2 QC  Z The Bost–Connes algebra is A D C  .G / with G D GQ =RC the groupoid of the commensurability relation on 1-dimensional Q-lattices up to scaling. y   R D CQ , The combination of scaling and Galois action given an action of Z C x since in the BC case the Galois action of Gal.Q=Q/ factors through the abelianization. R y  determine projectors p D O  g.g/ dg where p is an idempoCharacters  of Z Z tent in the category of endomotives and in Endƒ D.A; '/. Thus, one can considered the cohomology HC0 .p D.A; '// of the range of this projector acting on the cokernel D.A; '/ of the restriction map. 2.6 Scaling as Frobenius in characteristic zero. The observation that a scaling action appears to provide a natural replacement for the Frobenius in characteristic zero is certainly not new to the work of [11] described above. In fact, perhaps the first very strong evidence for the parallels between scaling and Frobenius came from the comparative analysis, given in §11 of [2] of the number theoretic, characteristic p method of Harder–Narasimhan [27] and the differential geometric method of Atiyah–Bott [2]. Both methods of [27] and [2] yield a computation of Betti numbers. In the numbertheoretic setting this is achieved by counting points in the strata of a stratification, while in the Morse-theoretic approach one retracts strata onto the critical set. Both methods work because, on one side, one has a perfect Morse stratification, which essentially depends upon the fact that the strata are built out of affine spaces, and on the other hand one can effectively compute numbers of points in each stratum, for much the same reason. The explicit expressions obtained in both cases can be compared directly by a simple substitution that replaces the cardinality q by a real variable t 2 and the Frobenius eigenvalues !i by t 1 . In following this parallel between the characteristic p and the characteristic zero case, one observes then that the role played by the Frobenius in the first setting is paralleled by a scaling action in the characteristic zero world. More closely related to the specific setting of the BC endomotive, one knows from the result of [19] that there is an analog of the BC system for function fields, where one works exclusively in positive characteristic. This starts with the observation in [17] that the quantum statistical mechanical system of [14], generalizing the BC system to 2-dimensional Q-lattices, can be equivalently formulated in terms of Tate modules of elliptic curves with marked points, O T.E/ D H 1 .E; Z/

with 1 ; 2 2 T.E/;

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71

with the commensurability relation implemented by isogenies. One then has a natural analog in the function field case. In fact, for K D Fq .C /, the usual equivalence of categories between elliptic curves and 2-dimensional lattices has an analog in terms of Drinfeld modules. One can then construct a noncommutative space, which can be described in terms of Tate modules of Drinfeld modules with marked points and the isogeny relation, or in the rank one case, in terms of 1-dimensional K-lattices modulo commensurability. One constructs a convolution algebra, over a characteristic p field C1 , which is the completion of the algebraic closure of the completion K1 at a point 1 of C . One can extend to positive characteristic some of the main notions of quantum statistical mechanics, by a suitable redefinition of the notion of time evolutions and of their analytic continuations, which enter in the definition of KMS states. Over complex numbers, for  2 RC and s D x C iy one can exponentiate as s D x e iy log  . In the function field context, there is a similar exponentiation, for positive elements (with respect to a sign function) in K1 and for s D .x; y/ 2 S1 ´ C1  Zp , with s D x deg./ hiy , with deg./ D d1 v1 ./, where d1 is the degree of the point 1 2 C and v1 the corresponding valuation, and  D sign./uv11 ./ hi the decomposition analogous to the polar decomposition of complex numbers, involving a sign function and a uniformizer K1 D Fq d1 ..u1 //. The second term in the exponentiation is then given by 1   X y .hi  1/j ; hiy D j j D0

with the Zp -binomial coefficients   y y.y  1/ : : : .y  k C 1/ : D kŠ j Exponentiation is an entire function s 7! s from S1 to C1 , with sCt D s t , so one usually thinks of S1 as a function field analog of the complex line with its polar decomposition C D U.1/  RC . One can extend the above to exponentiate ideals, I s D x deg.I / hI iy . This give an associated characteristic p valued zeta function, the Goss L-function of the function field, P Z.s/ D I s ; I

which is convergent in a “half plane” of fs D .x; y/ 2 S1 j jxj1 > qg. The analog of a time evolution in this characteristic p setting is then a continuous homomorphism W Zp ! Aut.A/, where we think of Zp as the line fs D .1; y/ 2 S1 g. In the case of the convolution algebra of 1-dimensional K-lattices up to commensurability and scaling, a time evolution of this type is given using the exponentiation of ideals as hI iy f .L; L0 /; y .f /.L; L0 / D hJ iy

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for pairs L  L0 of commensurable K-lattices and the corresponding ideals. This gives a quantum statistical mechanical system in positive characteristic whose partition function is the Goss L-function. One also has a notion of KMSx functionals, which lack the positivity property of their characteristic zero version, but they have the defining property that '.ab/ D '. x .b/a/, where x is the analytic continuation of the time evolution to s D .x; 0/. Moreover, as shown in [19], one can construct a dual system in this function field setting as well, where the product on the dual algebra is constructed in terms of the momenta of the non-archimedean measure. The algebra of the dual system maps again naturally to the convolution algebra of the commensurability relation on 1-dimensional K-lattices not up to scaling, which in turn can be expressed in terms of the adèles class space AK =K of the function field. The algebra of the dual system also has a scaling action, exactly as in the characteristic zero case: Z  .X / D `.s/s Us d.s/; H

where H D G  Zp with G  C1 and

Z  jG .X / D m .X / D `.s/x d1 m Us 0; d.s/; Z  jZp .X / D hi `.s/hiy Us d.s/:

This action recovers the Frobenius action F r Z as the part  jG of the scaling action, as well as the action of the inertia group, which corresponds to the part  jZp . 2.7 The adèle class space and the Weil proof. The adèle class space is the bad quotient AK =K of the adèles of a global field by the action of K . Unlike the case of the action on the idèles AK , which gives rise to a nice classical quotient, when one takes the action on the adèles the quotient is no longer described by a nice classical space, due to the ergodic nature of the action. However, it can be treated as a noncommutative space. In fact, this is the space underlying Connes’ approach to the Riemann hypothesis via noncommutative geometry. Our purpose here is to describe the role of motivic ideas in noncommutative geometry, so we focus on the approach of [11] recalled in the previous section and we illustrate how the adèles class space relates to the algebra of the dual system of the Bost–Connes endomotive, as mentioned above for the function field analog. y Ì N D .C0 .AQ;f / Ì Q / A Morita equivalence given by compressing C.Z/ C y can be used to identify with the projection given by the characteristic function of Z the BC endomotive with the noncommutative quotient AQ;f =QC . The dual system is then identified with the noncommutative quotient AQ =Q , where AQ D AQ;f  R . The adèles class space XQ ´ AQ =Q is obtained by adding the missing point 0 2 R.

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The way in which the adèles class space entered in Connes’ work [10] on the Riemann zeta function was through a sequence of Hilbert spaces E

0 ! L2ı .AQ =Q /0 ! L2ı .CQ / ! H ! 0; P E.f /.g/ D jgj1=2 f .qg/ for all g 2 CQ ; q2Q

(2.1) (2.2)

where the space L2ı .AQ =Q /0 is defined by 0 ! L2ı .AQ =Q /0 ! L2ı .AQ =Q / ! C2 ! 0 imposing the conditions f .0/ D 0 and fO.0/ D 0. The sequence above is compatible with the CQ actions, so the operators Z h.g/Ug d  g; h 2 .CQ /; U.h/ D CQ

for compactly supported h, act on H . The Hilbert space H can be decomposed L  y H H D  according to characters  of Z , and the scaling action of RC on  H D f 2 H j Ug  D .g/g is generated by an operator D with Spec.D / D fs 2 i R j L . 12 C i s/ D 0g; where L is the L-function with Grössencharakter . In particular, the Riemann zeta function for  D 1. The approach of Connes gives a semi-local trace formula, over the adèles class space restricted to a subset of finitely many places, X Z 0 h.u1 / Tr.Rƒ U.h// D 2h.1/ log ƒ C d  u C o.1/  j1  uj Qv v2S

R0

where Rƒ is a cutoff regularization and is the principal value. The trace formula should be compared to the Weil’s explicit formula in its distributional form: X X Z 0 h.u1 / O O O h.0/ C h.1/  h. / D d  u:  j1  uj Q v  v The geometric idea behind the Connes semi-local trace formula [10] is that it comes from the contributions of the periodic orbits of the action of CQ on the complement of the classical points inside the adèles class space, XQ XCQ . These are counted according to a version of the Guillemin–Sternberg distributional trace formula, originally stated for a flow F t D exp.t v/ on manifold, implemented by transformations .U t f /.x/ D f .F t .x//;

f 2 C 1 .M /:

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Under a transversality hypothesis which gives 1  .F t / invertible, for .F t / W Tx =Rvx ! Tx =Rvx D Nx ; the distributional trace formula takes the form  Z XZ Tr distr h.t /U t dt D

I

h.u/ d  u; j1  .Fu / j

where  ranges over periodic orbits and I is the isotropy group, and d  u a meaRsure with covol.I / D 1. The distributional Rtrace for a Schwartz kernel .Tf /.x/ D k.x; y/f .y/ dy is ofR the form Tr distr .T / D k.x; x/ dx. For .Tf /.x/ D f .F .x//, this gives .Tf /.x/ D ı.y  F .x//f .y/ dy. The work of [11] and [12] presents a different but closely related approach, where one reformulates the noncommutative geometry method of [10] in a cohomological form with a motivic flavor, as we explained in the previous sections. The restriction morphism ı D .Tr B /\ from the dual system of the BC endomotive to its classical points, both seen as noncommutative motives in the category of cyclic modules, can be equivalently written as P Q P f .q. ; // D E.fQ/; f .1; n ; n/ D ı.f / D q2Q

n2N

O  RC  AQ . where fQ is an extension by zero outside of Z The Hilbert space L2ı .AQ =Q /0 is replaced here by the cyclic-module AO \ˇ;0 . This requires different analytic techniques based on nuclear spaces, as in [47]. This provides a cohomological interpretation for the map E and for the spectral realization, which is now associated to the scaling action  on the cohomology HC0 .D.A; '//, which replaces the role of the Hilbert space H of (2.1). One has an action of CQ D AQ =Q on H 1 ´ HC0 .D.A; '// by Z #.f / D f .g/#g d  g CQ

for f 2 S.CQ /, a strong Schwartz space. The Weil’s explicit formula then has a global trace formula interpretation as XZ 0 f .u1 /  Tr.#.f /jH 1 / D fO.0/ C fO.1/    f .1/  d u: j1  uj .K v ;eKv / v The term    D log jaj D  log jDj, with D the discriminant for a number field, can be thought of as a self intersection of the diagonal, with the discriminant playing a role analogous to the Euler characteristic .C / of the curve for a function field Fq .C /. Thus, summarizing briefly the main differences between the approach of [10] and that of [11], [12], we have the following situation. In the trace formula for Tr.Rƒ U.f //

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75

of [10] only the zeros on critical line are involved and the Riemann hypothesis problem is equivalent to the problem of extending the semi-local trace formula to a global trace formula. This can be thought of in physical terms as a problem of passing from finitely many degrees of freedom to infinitely many, or equivalently from a quantum mechanical system to quantum field theory. In the setting of [11] and [12], instead, one has a global trace formula for Tr.#.f /jH 1 / and all the zeros of the Riemann zeta function are involved, since one is no longer working in the Hilbert space setting that is biased in favor of the critical line. In this setting the Riemann hypothesis becomes equivalent to a positivity statement   Tr #.f ? f ] /jH 1  0 for all f 2 S.CQ /; where

Z .f1 ? f2 /.g/ D

f1 .k/f2 .k 1 g/d  g

with the multiplicative Haar measure d  g and the adjoint is given by f ] .g/ D jgj1 f .g 1 /: This second setting makes for a more direct comparison with the algebro-geometric and motivic setting of the Weil proof of the Riemann hypothesis for function fields, which is based on similar ingredients: the Weil explicit formula and a positivity statement for the trace of correspondences. In a nutshell, the structure of the Weil proof for function fields is the following. The Riemann hypothesis for function fields K D Fq .C / is the statement that the eigenvalues n of the Frobenius have jj j D q 1=2 in the zeta function K .s/ D

Y .1  q nv s /1 D †K

P .q s / ; .1   q 1s / q s /.1

Q

with P .T / D .1n T / the characteristic polynomial of the Frobenius Fr  acting on étale cohomology Het1 .Cx ; Q` /. This statement is shown to be equivalent P to a positivity statement Tr.Z ? Z 0 / > 0 for the trace of correspondences Z D n an Fr n obtained from the Frobenius. Correspondences here are divisors Z  C  C . These have a degree, codegree, and trace d.Z/ D Z  .P  C /; d 0 .Z/ D Z  .C  P /; Tr.Z/ D d.Z/ C d 0 .Z/  Z  ; with  the diagonal in C  C . One first adjusts the degree of the correspondence by trivial correspondences C  P and P  C , then one applies Riemann–Roch to the divisor on the curve P 7! Z.P / of deg D g and shows that it is linearly equivalent to an effective divisor. Then using d.Z ? Z 0 / D d.Z/d 0 .Z/ D gd 0 .Z/ D d 0 .Z ? Z 0 /,

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one gets Tr.Z ? Z 0 / D 2gd 0 .Z/ C .2g  2/d 0 .Z/  Y    .4g  2/d 0 .Z/  .4g  4/d 0 .Z/ D 2d 0 .Z/  0; where Z ? Z 0 D d 0 .Z/ C Y . In the noncommutative geometry setting of [11] and [12] the role of the Frobenius correspondences is played by the scaling action of elements g 2 CK by Zg D f.x; g 1 x/g  AK =K  AK =K (2.3) R and more generally Z.f / D CK f .g/Zg d  g with f 2 S.CK /. These correspondences also have a degree and codegree Z d.Z.f // D fO.1/ D f .u/juj d  u with d.Zg / D jgj and d .Z.f // D d.Z.fN] // D 0

Z

f .u/ d  u D fO.0/:

Adjusting degree d.Z.f // D fO.1/ is possible by adding elements h 2 V , where V is the range of the restriction map ı D Tr B , P h.u; / D .n/ n2Z

y  , where CQ D Z y   R . Indeed, one can find an element with  2 RC and u 2 Z C O h 2 V with h.1/ ¤ 0 since Fubini’s theorem fails, Z X XZ .n/ d  ¤ .n/ d  D 0: R

n

n

R

One does not have a good replacement in this setting for principal divisors and linear equivalence, although one expects that the role of Riemann–Roch should be played by an index theorem in noncommutative geometry.

3 Endomotives and F1 -geometry In trying to exploit the analogies between function fields and number fields to import some of the ideas and methods of the Weil proof to the number fields context, one of the main questions is whether one can construct a geometric object playing the role of C Fq C over which the Weil argument with correspondences Pthe product ZD an F r n is developed. We have seen in the previous section a candidate space built using noncommutative geometry, through the correspondences Zg of (2.3) on the

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adèles class space. A different approach within algebraic geometry, aims at developing a geometry “over the field with one element” that would make it possible to interpret Spec.Z/ as an analog of the curve, with a suitable space Spec.Z/ F1 Spec.Z/ playing the role of C Fq C . The whole idea about a “field with one element”, though no such thing can obviously exist in the usual sense, arises from early considerations of Tits on the behavior of the counting of points over finite fields in various examples of finite geometries. For instance, for q D p k , #.An .Fq / X f0g/ qn  1 D D Œnq ; #Gm .Fq / q1   Œnq Š n # Gr.n; j /.Fq / D #fP j .Fq /  P n .Fq /g D ; D Œj q ŠŒn  j q Š j q where one sets Œnq Š D Œnq Œn  1q : : : Œ1q ; Œ0q Š D 1: #P n1 .Fq / D

In all this cases, the expression one obtains when setting q D 1 still makes sense and it appears to suggest a geometric replacement for each object. For example one obtains P n1 .F1 / ´ finite set of cardinality n; Gr.n; j /.F1 / ´ set of subsets of cardinality j: These observations suggest the existence of something like a notion of algebraic geometry over F1 , even though one need not have a direct definition of F1 itself. Further observations along these lines by Kapranov–Smirnov enriched the picture with a notion of “field extensions” F1n of F1 , which are described in terms of actions of the monoid f0g [ n , with n the group of n-th roots of unity. In this sense, one can say that a vector space over F1n is a pointed set .V; v/ endowed with a free action of n on V X fvg and linear maps are just permutations compatible with the action. So, as observed by Soulé and Kapranov–Smirnov, although one does not define F1n and F1 directly, one can make sense of the change of coefficients from F1 to Z as F1n ˝F1 Z ´ ZŒt; t 1 =.t n  1/: Various different approaches to F1 -geometry have been developed recently by many authors: Soulé, Haran, Deitmar, Dourov, Manin, Toën–Vaquie, Connes–Consani, Borger, López-Peña and Lorscheid. Several of these viewpoints can be regarded as ways of providing descent data for rings from Z to F1 . We do not enter here into a comparative discussion of these different approaches: a good overview of the current status of the subject is given in [39]. We are interested here in some of those versions of F1 -geometry that can be directly connected with the noncommutative geometry approach described in the previous sections. We focus on the following approaches:

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• descent data determined by cyclotomic points (Soulé [49]); • descent data by ƒ-ring structures (Borger [4]); • analytic geometry over F1 (Manin [43]). Soulé introduced in [49] a notion of gadgets over F1 . These are triples of data .X; AX ; ex; /, where X W R ! Sets is a covariant functor from a category R of finitely generated flat rings, which can be taken to be the subcategory of rings generated by the group rings ZŒZ=nZ, AX is a complex algebra with evaluation maps ex; / such that, for all x 2 X.R/ and W R ! C, one has an algebra homomorphism ex; W AX ! C with ef .y/; D ey;Bf for any ring homomorphism f W R0 ! R. For example, affine varieties VZ over Z define gadgets X D G.VZ /, by setting X.R/ D Hom.O.V /; R/ and AX D O.V / ˝ C. An affine variety over F1 is then a gadget with X.R/ finite, and a variety XZ with a morphism of gadgets X ! G.XZ /, with the property that, for all morphisms X ! G.VZ /, there exists a unique algebraic morphism XZ ! VZ that functorially corresponds to the morphism of gadgets. The Soulé data can be thought of as a descent condition from Z to F1 , by regarding them as selecting among varieties defined over Z those that are determined by the data of their cyclotomic points X.R/, for R D ZŒZ=nZ. This selects varieties that are very combinatorial in nature. For example, smooth toric varieties are geometries over F1 in this and all the other currently available flavors of F1 -geometry. Borger’s approach to F1 -geometry in [4] is based on a different way of defining descent conditions from Z to F1 , using lifts of Frobenia, encoded in the algebraic structure of ƒ-rings. This was developed by Grothendieck in the context of characteristic classes and the Riemann–Roch theorem, where it relates to operations in K-theory, but it can be defined abstractly in the following way. For a ring R, whose underlying abelian group is torsion free, a ƒ-ring structure is an action of the multiplicative semigroup N of positive integers by endomorphisms lifting Frobenius, namely, such that sp .x/  x p 2 pR

for all x 2 R:

Morphisms of ƒ-rings are ring homomorphisms f W R ! R0 compatible with the actions, f B sk D sk0 B f . The Bost–Connes endomotive, which is at the basis of the noncommutative geometry approach to the Riemann hypothesis, relates directly to both of these notions of F1 geometry in a very natural way. 3.1 Endomotives and Soulé’s F1 -geometry. The relation between the BC endomotive and Soulé’s F1 geometry was investigated in [13]. One first considers a model over Z of the BC algebra. This requires eliminating denominators from the relations of the

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algebra over Q. It can be done by replacing the crossed product by ring endomorphisms by a more subtle “crossed product” by correspondences. More precisely, one considers the algebra AZ;BC generated by ZŒQ=Z and elements n , Q n with relations Q n Q m n m n Q n Q n m One has

D Q nm ; D nm ; D n; D m Q n ;

n x D n .x/n

and

.n; m/ D 1: x Q n D Q n n .x/;

where n .e.r// D e.nr/ for r 2 Q=Z. Notice that here the ring homomorphisms n .x/ D n xn are replaced by Qn .x/ D Q n xn , which are no longer ring homomorphisms, but correspondences. The resulting “crossed product” is indicated by the notation AZ;BC D ZŒQ=Z ÌQ N. One then observes that roots of unity .k/ .R/ D fx 2 R j x k D 1g D HomZ .Ak ; R/ with Ak D ZŒt; t 1 =.t k  1/ can be organized as a system of varieties over F1 in two different ways. As an inductive system they define the multiplicative group Gm as a variety over F1 by taking .n/ .R/  .m/ .R/;

njm;

Am  An ;

and by taking the complex algebra to be AX D C.S 1 /. As a projective system, which corresponds to the BC endomotive, one uses the morphisms m;n W Xn  Xm , m;n W .n/ .R/  .m/ .R/;

njm;

and obtains a pro-variety 1 .R/ D HomZ .ZŒQ=Z; R/; which arises from the projective system of affine varieties over F1 , m;n W F1n ˝F1 Z ! F1m ˝F1 Z; where the complex algebra is taken to be AX D CŒQ=Z. The affine varieties .n/ over F1 are defined by gadgets G.Spec.QŒZ=nZ//, which form a projective system of gadgets. The endomorphisms n of varieties over Z, are also endomorphisms of gadgets and of F1 -varieties.

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The extensions F1n of Kapranov–Smirnov correspond to the free actions of roots of unity y  7!  n ; n 2 N; and  7!  ˛ $ e.˛.r//; ˛ 2 Z; and these can be regarded as the Frobenius action on F11 . It should be noted that, indeed, in reductions mod p of the integral Bost–Connes endomotive these do correspond to the Frobenius, that one can consider the BC endomotive as describing the tower of extensions F1n together with the Frobenius action. As shown in [13], one can obtain characteristic p versions of the BC endomotive by separating out the parts Q=Z D Qp =Zp  .Q=Z/.p/ with denominators that are powers of p and denominators that are prime to p. One then has a crossed product algebra KŒQp =Zp  Ì p Z

C

with endomorphisms n for n D p ` and ` 2 ZC . The Frobenius 'Fp .x/ D x p of the field K in characteristic p satisfies . p` ˝ 'F`p /.f / D f p

`

for f 2 KŒQ=Z so that one has `

`

. p` ˝ 'F`p /.e.r/ ˝ x/ D e.p ` r/ ˝ x p D .e.r/ ˝ x/p : This shows that the BC endomorphisms restrict to Frobenia on the mod p reductions of the system: p` induces the Frobenius correspondence on the pro-variety 1 ˝Z K. 3.2 Endomotives and Borger’s F1 -geometry via ƒ-rings. This is also the key observation in relating the BC endomotive to Borger’s point of view [4] on F1 -geometry. One sees, in fact, that the Bost–Connes endomotive is a direct limit of ƒ-rings Rn D ZŒt; t 1 =.t n  1/;

sk .P /.t; t 1 / D P .t k ; t k /;

where the ƒ-ring structures is given by the endomorphisms action of N. The action y that combines the action of symmetries Z y  by automorphisms of the BC system of Z and the endomorphisms that give the ƒ-ring structure is given by y W . W x 7! x/ 7! . W x 7!  ˛ x/; ˛2Z y D Hom.Q=Z; Q=Z/. This agrees with the notion of Frobenius where we identify Z over F11 proposed by Haran. In fact, one can see the relation to ƒ-rings more precisely by introducing multivariable generalizations of the BC endomotive as in [45].

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One considers as varieties the algebraic tori T n D .Gm /n with endomorphisms ˛ 2 Mn .Z/C and one constructs, as in the case of the BC endomotive, the preimages X˛ D ft D .t1 ; : : : ; tn / 2 T n j s˛ .t / D t0 g organized into a projective system with maps ˛;ˇ W Xˇ ! X˛ ;

t 7! t ;

˛ D ˇ 2 Mn .Z/C ;

t 7! t D .t / D .t1 11 t2 12 : : : tn 1n ; : : : ; t1 n1 t2 n2 : : : tn nn /: The projective limit X D lim X˛ carries a semigroup action of Mn .Z/C . ˛ x One can then consider the algebra C.X.Q// Š QŒQ=Z˝n with generators e.r1 / ˝    ˝ e.rn / and the crossed product An D QŒQ=Z˝n Ì Mn .Z/C generated by e.r/ and ˛ , ˛ with

˛ .e.r// D ˛ e.r/˛ D

1 det ˛

P

e.s/;

˛.s/Dr

˛ .e.r// D ˛ e.r/˛ D e.˛.r//: This corresponds to the action of the family of endomorphisms ˛ .e.r// D ˛ e.r/˛ : These multivariable versions relate to the ƒ-rings notion of F1 -geometry through a theorem of Borger–de Smit, which shows that every torsion free finite rank ƒ-ring embeds in a finite product of copies of ZŒQ=Z, where the action of N is compatible with the diagonal action Sn;diag  Mn .Z/C in the multivariable BC endomotives. Thus, the multivariable BC endomotives are universal for ƒ-rings. 3.3 Endomotives and Manin’s analytic geometry over F1 . These multivariable generalizations of the BC endomotive introduced in [45] are also closely related to Manin’s approach to analytic geometry over F1 of [43], which is based on the Habiro ring as a ring of analytic functions of roots of unity. The Habiro ring [26] is defined as the projective limit ZŒq D lim ZŒq=..q/n /; 

b

n

where .q/n D .1  q/.1  q 2 / : : : .1  q n / and one has morphisms ZŒq=..q/n /  ZŒq=..q/k / for k  n since .q/k j.q/n . This ring has evaluation maps at roots of unity that are surjective ring homomorphisms

b

ev W ZŒq ! ZŒ;

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but which, combined, give an injective homomorphism Y ev W ZŒq ! ZŒ:

b

2Z

The elements of the Habiro ring also have Taylor series expansions at all roots of unity

b

 W ZŒq ! ZŒŒŒq  ; which also are injective ring homomorphisms. Thus, they behave like “analytic functions on roots of unity”. As argued in [45], the Habiro ring provides then another model for the noncommutative geometry of the cyclotomic tower, replacing QŒQ=Z with ZŒq. One considers endomorphisms n .f /.q/ D f .q n /, which lift P ./ 7! P . n / in ZŒ through the evaluation maps ev . This gives an action of N by endomorphisms and one can form a group crossed product

b

b

AZ;q D ZŒq1 Ì QC ;

b

where AZ;q is generated by ZŒq and by elements n and n with

b

The ring ZŒq1 D

n f D n .f /n :

n n .f / D f n ; S N

AN , with AN generated by the N f N , satisfies

b

b

b

ZŒq1 D lim. n W ZŒq ! ZŒq/: ! n

b

b

These maps are injective and determine automorphisms n W ZŒq1 ! ZŒq1 . Another way to describe this is in terms of the ring PZ of polynomials in Q-powers q r . One has POZ D lim PZ =JN ;  N

where JN is the ideal generated by .q /N D .1  q r / : : : .1  q rN /, with r 2 QC , and r

b

ZŒq1 ' POZ ;

n f n 7! f .q 1=n /;

where r .f /.q/ D f .q r /. In [43], Manin also introduced multivariable versions of the Habiro ring,

6

ZŒq1 ; : : : ; qn  D lim ZŒq1 ; : : : ; qn =In;N ;  N

where In;N is the ideal ..q1  1/.q12  1/ : : : .q1N  1/; : : : ; .qn  1/.qn2  1/ : : : .qnN  1//:

(3.1)

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These again have evaluations at roots of unity

6

ev. 1 ;:::; n / W ZŒq1 ; : : : ; qn  ! ZŒ1 ; : : : ; n  and Taylor series expansions

6

TZ W ZŒq1 ; : : : ; qn  ! ZŒ1 ; : : : ; n ŒŒq1  1 ; : : : ; qn  n  for all Z D .1 ; : : : ; n / in Zn with Z the set of all roots of unity. One can equivalently describe (3.1) as

6

ZŒq1 ; : : : ; qn  D lim ZŒq1 ; : : : ; qn ; q11 ; : : : ; qn1 =Jn;N ;  N

where Jn;N is the ideal generated by the .qi  1/ : : : .qiN  1/, for i D 1; : : : ; n and the .qi1  1/ : : : .qiN  1/. Consider then again the algebraic tori T n D .Gm /n , with algebra QŒti ; ti1 . Using the notation Q ˛ t ˛ D .ti˛ /iD1;:::;n with ti˛ D j tj ij ; we can define the semigroup action of ˛ 2 Mn .Z/C by q 7! ˛ .q/ D ˛ .q1 ; : : : ; qn / D .q1˛11 q2˛12 : : : qn˛1n ; : : : ; q1˛n1 q2˛n2 : : : qn˛nn / D q ˛ ; analogous to the case of the multivariable BC endomotives discussed above, of which these constitute an analog in the setting of analytic F1 -geometry.

4 DG-algebras and noncommutative motives The noncommutative motives we encountered so far in this overview are derived from two sources: the abelian category of cyclic modules and the category of endomotives. The latter are a very special kind of zero-dimensional noncommutative spaces combining Artin motives and endomorphism actions. More generally, one would like to incorporate higher dimensional algebraic varieties and correspondences given by algebraic cycles, together with their self-maps, and construct larger categories of noncommutative spaces that generalize whet we saw here in a zero-dimensional setting. In particular, this would be needed in order to generalize some of the results obtained so far for the Riemann zeta function using noncommutative geometry, like the trace formulae discussed above, to the more general context of L-functions of algebraic varieties and motives. When one wishes to combine higher dimensional algebraic varieties with noncommutative spaces, one needs to pay attention to the substantially different way in which one treats the rings of functions in the two settings. This is not visible in a purely zero-dimensional case where one deals only with Artin motives. When one treats noncommutative spaces as algebras, one point of view is that one essentially only needs

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to deal with the affine case. The reason behind this is the fact that the way to describe in noncommutative geometry the gluing of affine charts, or any other kind of identification, is by considering the convolution algebra of the equivalence relation that implements the identifications. So, at the expense of no longer working with commutative algebras, one gains the possibility of always working with a single algebra of functions. When one tries to combine noncommutative spaces with algebraic varieties, however, one wants to be able to deal directly with the algebro-geometric description of arbitrary quasi-projective varieties. This is where a more convenient approach is provided by switching the point of view from algebras to categories. The main result underlying the categorical approach to combining noncommutative geometry and motives is the fact that the derived category D.X / of quasicoherent sheaved on a quasiseparated quasicompact scheme X is equivalent to the derived category D.A• / of a DG-algebra A• , which is unique up to derived Morita equivalence, see [3], [35]. Thus, passing to the setting of DG-algebras and DG-category provides a good setting where algebraic varieties can be treated, up to derived Morita equivalence, as noncommutative spaces. A related question is the notion of correspondences between noncommutative spaces. We have seen in this short survey different notions of correspondences: morphisms of cyclic modules, among which one finds morphisms of algebras, bimodules, Morita equivalences, and traces. We also saw the correspondences associated to the scaling action of CK on the noncommutative adèles class space AK =K . More generally, the problem of identifying the best class of morphisms of noncommutative spaces (or better of noncommutative motives) that accounts for all the desired features remains a question that is not settled in a completely satisfactory ways. A comparison between different notions of correspondences in the analytic setting of KK-theory and in the context of derived algebraic geometry was given recently in [40], while a “motivic” category with correspondences based on noncommutative spaces defined as spectral triples and a version of smooth KK-theory was proposed in [46]. Again, a closer interplay between the analytic approach to noncommutative geometry via algebras, KK-theory, spectral triples, and such smooth differential notions, and the algebro-geometric approach via DG-categories and derived algebraic geometry is likely to play a crucial role in identifying the best notion of correspondences in noncommutative geometry.

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A short survey on pre-Lie algebras Dominique Manchon

Contents 1

Pre-Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2

Operads, pre-Lie algebras and rooted trees . . . . . . . . . . . . . . . . . .

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3

Pre-Lie algebras of vector fields . . . . . . . . . . . . . . . . . . . . . . .

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Links with other algebraic structures . . . . . . . . . . . . . . . . . . . . .

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Introduction A left pre-Lie algebra over a field k is a k-vector space A with a bilinear binary composition B that satisfies the left pre-Lie identity .a B b/ B c  a B .b B c/ D .b B a/ B c  b B .a B c/; for a; b; c 2 A. Analogously, a right pre-Lie algebra is a k-vector space A with a binary composition C that satisfies the right pre-Lie identity .a C b/ C c  a C .b C c/ D .a C c/ C b  a C .c C b/: The left pre-Lie identity rewrites as LŒa;b D ŒLa ; Lb ;

(1)

where La W A ! A is defined by La b D a B b, and where the bracket on the left-hand side is defined by Œa; b ´ a B b  b B a. As a consequence this bracket satisfies the Jacobi identity. Pre-Lie algebras are sometimes called Vinberg algebras, as they appear in the work of E. B. Vinberg [33] under the name “left-symmetric algebras” on the classification of homogeneous cones. They appear independently at the same time in the work of M. Gerstenhaber [22] on Hochschild cohomology and deformations of algebras, under the name “pre-Lie algebras” which is now the standard terminology. Note however that Gerstenhaber’s pre-Lie algebras live in the category of graded vector spaces, and then additional signs occur in the left pre-Lie identity. The term “chronological algebras” has also been sometimes used, e.g., in the fundamental work of A. Agrachev and R. Gamkrelidze [1].

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As we shall see, the notion itself can be however traced back to the work of A. Cayley [6] which, in modern language, describes the pre-Lie algebra morphism Fa from the pre-Lie algebra of rooted trees into the pre-Lie algebra of vector fields on Rn sending the one-vertex tree to a given vector field a. A very important step was indeed made at the turn of the century, with the explicit description of free pre-Lie algebras as pre-Lie algebras of decorated rooted trees endowed with grafting. F. Chapoton and M. Livernet [10] obtained this result as a byproduct of their complete description of the operad governing pre-Lie algebras in terms of labelled rooted trees. A. Dzhumadil’daev and C. Löfwall [14] give a more “elementary” proof, avoiding the language of operads, and providing in the same time the explicit description, again in terms of rooted trees, of the free NAP algebra.1 An important family of examples of pre-Lie algebras is given by any augmented operad, summing up the partial compositions and quotienting by the actions of the symmetric groups [8], [11]. Another source of examples is given by Rota–Baxter algebras, and more generally by Loday’s dendriform algebras and twisted versions of those [15], [19], [17]. The article is organized as follows: in the first part we recall the main definitions and properties, following [1]. In particular we introduce the group of formal flows of a complete filtered pre-Lie algebra and we prove a pre-Lie version of the Poincaré– Birkhoff–Witt theorem. We also recall a recent theorem by J-L. Loday and M. Ronco relating left pre-Lie algebras and right-sided commutative Hopf algebras. The second part is devoted to rooted trees and the free pre-Lie algebra. Following [10] and [8] we recall the construction of the group associated to any augmented operad as well as the associated pre-Lie algebra, and we recall the description of the free preLie algebra in terms of rooted trees. As an application we describe a second pre-Lie algebra product on rooted trees which acts on the first by derivations [5], [30]. The third part is devoted to pre-Lie algebras of vector fields on Rn , from the work of A. Cayley [6] to modern developments in numerical analysis through B-series [4], [24], [31], [12]. Finally the last part reviews the relation of pre-Lie algebras with other algebraic structures. The reader will find a complementary point of view on pre-Lie algebras in the survey article of D. Burde [3], where emphasis is made on geometric aspects through affine structures on Lie groups.

1 Pre-Lie algebras As any right pre-Lie algebra .A; C/ is also a left pre-Lie algebra with product a B b ´ b C a, one can stick to left pre-Lie algebras, which we shall do unless specifically indicated. 1 NAP stands for Non-Associative Permutative. A (left) NAP algebra is a vector space endowed with a bilinear map B such that a B .b B c/ D b B .a B c/. Right NAP algebras are defined accordingly. Left (right) NAP algebras are called “left (right) commutative” in [14].

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1.1 The group of formal flows. The following is taken from the paper of A. Agrachev and R. Gamkrelidze [1]. Suppose that A is a left pre-Lie algebra endowed with a compatible decreasing filtration, namely A D A1  A2  A3     , such that the intersection of the Aj ’s reduces to f0g, and such that Ap B Aq  ApCq . Suppose moreover that A is complete with respect to this filtration. The Baker–Campbell– Hausdorff formula 1 1 C.a; b/ D a C b C Œa; b C .Œa; Œa; b C Œb; Œb; a/ C    2 12 then endows A with a structure of pro-unipotent group. This group admits a more transparent presentation as follows: introduce a fictitious unit 1 such that 1 B a D a B 1 D a for any a 2 A, and define W W A ! A by 1 1 W .a/ ´ e La 1  1 D a C a B a C a B .a B a/ C    : 2 6 The mapping W is clearly a bijection. The inverse, denoted by , also appears under the name “pre-Lie Magnus expansion” in [18]. It satisfies the equation X L.a/ Bi Li.a/ a; .a/ D L aD e .a/  Id i0 where the Bi ’s are the Bernoulli numbers. The first few terms are 1 1 1 .a/ D a  a B a C .a B a/ B a C a B .a B a/ C    : 2 4 12 Transferring the BCH product by means of the map W , namely a # b D W .C..a/; .b///; (2)   we have W .a/ # W .b/ D W C.a; b/ D e La e Lb 1  1, hence W .a/ # W .b/ D W .a/ C e La W .b/. The product # is thus given by the simple formula a # b D a C e L.a/ b:   The inverse is given by a#1 D W  .a/ D e L.a/ 1  1. If .A; B/ and .B; B/ are two such pre-Lie algebras and W A ! B is a filtration-preserving pre-Lie algebra morphism, it is immediate to check that for any a; b 2 A we have .a # b/ D

.a/ #

.b/:

In other words, the group of formal flows is a functor from the category of complete filtered pre-Lie algebras to the category of groups. When the pre-Lie product B is associative, all this simplifies to a#b Da BbCaCb and a#1 D

X 1 .1/n an : 1D 1Ca n1

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1.2 The pre-Lie Poincaré–Birkhoff–Witt theorem Theorem 1.1. Let A be any left pre-Lie algebra, and let S.A/ be its symmetric algebra, i.e., the free commutative algebra on A. Let ALie be the underlying Lie algebra of A, i.e., the vector space A endowed with the Lie bracket given by Œa; b D a B b  b B a for any a; b 2 A, and let U.A/ be the enveloping algebra of ALie , endowed with its usual increasing filtration. Let us consider the associative algebra U.A/ as a left module over itself. There exists a left U.A/-module structure on S.A/ and a canonical left U.A/module isomorphism  W U.A/ ! .A/ such that the associated graded linear map Gr  W Gr U.A/ ! S.A/ is an isomorphism of commutative graded algebras. Proof. The Lie algebra morphism L W A ! End A;

a 7! .La W b 7! a B b/;

extends by the Leibniz rule to a unique Lie algebra morphism L W A ! Der S.A/. Now we claim that the map M W A ! End S.A/ defined by Ma u D au C La u is a Lie algebra morphism. Indeed, we have for any a; b 2 A and u 2 S.A/: Ma Mb u D Ma .bu C Lb u/ D abu C aLb u C La .bu/ C La Lb u D abu C aLb u C bLa u C .a B b/u C La Lb u: Hence ŒMa ; Mb u D .a B b  b B a/u C ŒLa ; Lb u D MŒa;b u; which proves the claim. Now M extends, by the universal property of the enveloping algebra, to a unique algebra morphism M W U.A/ ! End S.A/. The linear map  W U.A/ ! S.A/;

u 7! Mu  1;

is clearly a morphism of left U.A/-modules. It is immediately seen by induction that for any a1 ; : : : ; an 2 A we have .a1 : : : an / D a1 : : : an C v where v is a sum of terms of degree  n  1. This proves the theorem. Remark 1.2. Let us recall that the symmetrization map  W U.A/ ! S.A/, uniquely determined by  .an / D an for any a 2 A and any integer n, is an isomorphism for the two ALie -module structures given by the adjoint action. This is not the case for the map  defined above. The fact that it is possible to replace the adjoint action of U.A/ on itself by the simple left multiplication is a remarkable property of pre-Lie algebras, and makes Theorem 1.1 different from the usual Lie algebra PBW theorem.

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Let us finally notice that if p stands for the projection from S.A/ onto A, for any a1 ; : : : ; ak 2 A we easily get p B .a1 : : : ak / D La1 : : : Lak 1 D a1 B .a2 B .: : : .ak1 B ak / : : : //

(3)

by a simple induction on k. The linear isomorphism  transfers the product of the enveloping algebra U.A/ into a noncommutative product  on .A/ defined by s  t D .1 .s/1 .t //: Suppose now that A is endowed with a complete decreasing compatible filtration as in Section 1.1. This filtration induces a complete decreasing filtration S.A/ D S.A/0  S.A/1  S.A/2     , and the product  readily extends to the completion S .A/. For any a 2 A, the application of (3) gives

1

p.e a / D W .a/

1

1

as an equality in the completed symmetric algebra S .A/. According to (2) we can identify the pro-unipotent group fe a ; a 2 Ag  S .A/ and the group of formal flows of the pre-Lie algebra A by means of the projection p, namely, p.e a / # p.e b / D p.e a  e b / for any a; b 2 A. 1.3 Right-sided commutative Hopf algebras and the Loday–Ronco theorem. Let H be a commutative Hopf algebra. Following [27], we say that H is right-sided if it is free as a commutative algebra, i.e., H D S.V / for some k-vector space V , and if the coproduct satisfies .V /  H ˝ V: L Suppose moreover that V D n1 Vn is graded with finite-dimensional homogeneous components. Then the graded dual A D V 0 is a left pre-Lie algebra, and by the Milnor– Moore theorem, the graded dual H 0 is isomorphic to the enveloping algebra U.ALie / as graded Hopf algebra. Conversely, for any graded pre-Lie algebra A the graded dual U.ALie /0 is free commutative right-sided ([27] Theorem 5.3).

2 Operads, pre-Lie algebras and rooted trees We first associate a (right) pre-Lie algebra to any augmented operad, following F. Chapoton [8], and then we recall the description of the pre-Lie operad itself by F. Chapoton and M. Livernet [10], thus leading to two pre-Lie structures on the vector space generated by the rooted trees.

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2.1 Pre-Lie algebras associated to augmented operads. Recall that an augmented operad P (in the symmetric monoidal category of vector spaces over some field k) is given by a collection of vector spaces .Pn /n1 with P1 D k  e, an action of the symmetric group Sn on Pn , and a collection of partial compositions Bi W Pk ˝ Pl ! PkCl1 ;

.a; b/ 7! a Bi b;

i D 1; : : : ; k;

which, for any a 2 Pk , b 2 Pl , a 2 Pm satisfies the associativity conditions .a Bi b/ BiCj 1 c D a Bi .b Bj c/; .a Bi b/ BlCj 1 c D .a Bj c/ Bi b;

i 2 f1; : : : ; kg; j 2 f1; : : : ; lg; i; j 2 f1; : : : ; kg; i < j;

the unit axiom e B a D a;

a Bi e D a;

i D 1; : : : ; k;

as well as the equivariance condition  .a/ Bi .b/ D i .;  /.a Bi b/; where i .; / 2 SkCl1 is defined by letting  permute the set Ei D fi; i C 1; : : : ; i C l  1g of cardinality l and then by letting  permute the set f1; : : : ; i 1; Ei ; i Cl; : : : ; kCl 1g of cardinality k. The global composition is defined by W Pn ˝ Pk1 ˝    ˝ Pkn ! Pk1 CCkn ; .a; b1 ; : : : ; bn / 7! .: : : ..a Bn bn / Bn1 bn1 / : : : / B1 b1 : The free P -algebra with one generator is given by L FP W Pn =Sn : n1

C The L sum of the partial compositions yields a right pre-Lie algebra structure on FP ´ n2 Pn =Sn , namely k P a Bi b: aN C bN ´ iD1

e associated Following F. Chapoton [8] one can consider the pro-unipotent group GP C with the completion of the pre-Lie algebra FP for the filtration induced by the grading. More precisely, Chapoton’s group GP is given by the elements g 2 Fc P such that e g1 6D 0, whereas GP is the subgroup of GP formed by elements g such that g1 D e. Any element a 2 Pn gives rise through to an n-ary operation !a W FP˝n ! FP , and for any x; y1 ; : : : ; yn 2 FPC we have [30]:2

!a .y1 ; : : : ; yn / C x D 2

n P j D1

!a .y1 ; : : : ; yj C x; : : : ; yn /:

We thank Muriel Livernet for having brought this point to our attention.

(4)

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2.2 The pre-Lie operad. Recall that, for any operad O, an O-algebra is a vector space V together with a morphism of operads from O to the operad Endop.V /, where Endop.V /n ´ L.V ˝n ; V / with obvious symmetric group actions and compositions. In this sense, pre-Lie algebras are algebras over the pre-Lie operad, which has been described in detail by F. Chapoton and M. Livernet in [10] as follows: PLn is the vector space of labelled rooted trees, and partial composition s Bi t is given by summing all the possible ways of inserting the tree t inside the tree s at the vertex labelled by i . L The free right pre-Lie algebra with one generator is then given by the space T D n1 Tn of rooted trees, as quotienting with the symmetric group actions amounts to neglecting the labels. The pre-Lie operation .s; t / 7! .s t / is given by the sum of the graftings of t on sLat all vertices of s. As a consequence of (4) we have two pre-Lie operations on T 0 D n2 Tn which interact as follows [30]: .s

t / C u D .s C u/

t Cs

.t C u/:

The first pre-Lie operation C comes from the fact that PL is an augmented operad, whereas the second pre-Lie operation comes from the fact that PL is the pre-Lie operad itself! 2.3 Two Hopf algebras of rooted forests. Let us consider the two commutative Hopf algebras HCK and H associated respectively to the left pre-Lie algebras .T ; !/ and .T 0 ; B/ by the functor described in Paragraph 1.3. The first is the Connes–Kreimer Hopf algebra of rooted forests [13]. Its coproduct is described in terms of admissible cuts [20], or alternatively as follows [31], Paragraph 3.2 and Section 7: the set U of vertices of a forest u is endowed with a partial order defined by x  y if and only if there is a path from a root to y passing through x. Any subset W of the set of vertices U of u defines a subforest w of u in an obvious manner, i.e., by keeping the edges of u that link two elements of W . The coproduct is then defined by P v ˝ w: CK .u/ D V qW DU W