P. I. C. M. – 2018 Rio de Janeiro, Vol. 1 (111–118) ARITHMETIC PHYSICS – ABEL LECTURE M A 1 Introduction My Abel lecture in Rio de Janeiro was devoted to solving the riddle of the fine structure constant ˛, an outstanding problem from the world of fundamental physics. Here is what the Nobel Laureate Richard Feynman had to say about ˛ :Where does ˛ come from; is it related to ,or perhaps to e? Nobody knows, it is one of the great damn mysteries of physics: a magic number that comes to us with no understanding by man. You might say the hand of God wrote the number and we don’t know how He pushed his pencil. Within the constraints of a one-hour lecture, before an audience of a thousand math- ematicians, I sketched my explanation of where ˛ came from. The actual title of my lecture was: The Future of Mathematical Physics: new ideas in old bottles. During the ICM I listened to many brilliant lectures and talked with many clever people. As a result my ideas crystallised and, on the long flight back from Rio to Frank- furt, they took final shape. Rather than just expand on my Abel Lecture I will attempt to outline the broad programme that starts with the fine structure constant and aims to build a Theory of Arithmetic Physics. A programme with a similar title was put forward by Yuri Manin in Bonn in 1984 Atiyah [1985]. In fact Manin was not able to come and I presented his lecture for him. This gave me the opportunity to suggest that Manin’s programme was too modest, and that it should be lifted from the classical regime to the quantum regime with Hilbert spaces in the forefront. Visionary programmes take decades to mature and it was 34 years later, on the flight from the new world to the old, that the programme revealed its true self. The title of my Abel lecture suddenly acquired a new meaning, with bottles replaced by continents. In the next section I will survey the history of the “old bottles” and show how old ideas can be put to new use. MSC2010: primary 11G05; secondary 11G40, 14G25, 14H52, 14K15. Keywords: Elliptic curve. 111 112 MICHAEL ATIYAH 2 Past, Present and Future There are two stories from the past that I wish to resurrect. The first, and older of the two, is the Theory of von Neumann Algebras Murray and von Neumann [1936]. It arose from the study of operators in Hilbert space that von Neumann used to lay the foundations of quantum mechanics. It created a deep bridge between non-commutative Algebra and Analysis. In more recent years, in the hands of Alain Connes Connes [1994], it has been further refined to include Differential Geometry. The second story from the past was initiated by Fritz Hirzebruch Hirzebruch [1966], as part of the new algebraic geometry centred on Paris and Princeton. Fritz was a ma- gician, a true successor to Euler Dunham [1999], who found beautiful new applications of the Bernoulli numbers, creating a bridge between number theory and algebraic ge- ometry. My own work in collaboration with Raoul Bott Atiyah and Bott [1966] and Is Singer Atiyah and Singer [1963] extended the bridge into differential geometry and topology. The Hirzebruch era was a generation later than the von Neumann era, but by then the foundations of Arithmetic Physics were being laid, as Manin recognized. With subse- quent developments in Physics led by Edward Witten, M-theory emerged and provided strong hints that Arithmetic Physics should indeed lift to the quantum world as I had speculated. My paper on the fine structure constant, which provides the details behind the ideas sketched in my Abel Lecture, has been submitted to the Proceedings A of the Royal So- ciety Atiyah [2018a]. It was recorded in the ArXiv on 15/8/18. As advertised in Atiyah and Kouneiher [n.d.] a second paper on Newton’s constant is in preparation. In this article I will explain that (quantum) Arithmetic Physics is essentially Alge- braic Geometry over the Quaternions H. I will show that it mimics algebraic geometry over the complex field C. The only difference is that the complex field is replaced by the hyperfinite II-1 factor A(C). Moreover an algebraic number field Q0 in C is re- placed by what I will call the Hirzebruch algebra A(Q0). A careful scrutiny of Hirzebruch’s formalism shows that it essentially gives the Hilbert Nullstellensatz over the quaternions H. Since Hilbert’s theorem is the foundation stone of complex algebraic geometry, its counterpart over A will be the foundation stone of quaternionic algebraic geometry. C and its arithmetic subfields Q0 get replaced by A and its arithmetic sub-factors A(Q0). These algebraic/analytic ideas enable us to extend classical algebra/analysis from the commutative world of Q0 and C to the non-commutative world of A(Q0) and A(C). This converts large areas of classical mathematics into non-commutative counterparts. The traces of von Neumann factors, the holomorphic Euler characteristics of Hirzebruch and the index of Atiyah and Singer [1963] give stable commutative invariants from the ARITHMETIC PHYSICS – ABEL LECTURE 113 non-commutative world. This is the way gets converted to Ж. Just as is related to the circle via the equation (1) x2 + y2 = 1 where x and y are commuting variables in R C, Ж is related to the same equation with non-commuting variables in A(R) A(C). Keeping the same equation, but taking arithmetic subfields Q0 gives interesting arith- metic invariants interpolating between and Ж. As Hirzebruch showed, the Bernoulli functions and the Todd polynomials play a key role in this story, which is precisely the topic of Hirzebruch [1966]. Hirzebruch had many variants and extensions of his Todd polynomials and they are all based on this one rational case. Replacing the equation (2.1) by other polynomial equations leads to elliptic and higher genus curves whose counterparts give rise to invariants of elliptic fields or fields of higher genus. It seems likely that this will provide a novel approach to algebraic number theory, yet to be explored. In the next section I will peer into this future and describe the vista I see. 3 A new programme for the future I envisage Arithmetic Physics as a future programme ahead that will unify many fields. These include: 3.1 the Langlands programme (number theory) 3.2 Donaldson theory (differential geometry) 3.3 M–theory (theoretical physics) 3.4 Burnside problems (group theory) 3.5 Thompson theory (finite simple groups) There are already many hints that these fields are connected in some way, via mod- ular forms and L-functions. Examples of old problems that have been rejuvenated and extended by the first steps into the domain of Arithmetic Physics include: 3.6. Complex structures on the 6-sphere Atiyah [2016] Atiyah [2018b] 114 MICHAEL ATIYAH 3.7 Groups of odd order Atiyah [n.d.] 3.8 The fine structure constant Atiyah [2018a] Work in progress includes: 3.9 Newton’s constant Atiyah and Kouneiher [n.d.], joint with J.Kouneiher 3.10 New results on the restricted Burnside problem Atiyah and Zuk [n.d.], joint with A. Zuk. Going beyond the Quaternions H, to the Octonions O, deepens Arithmetic Physics in several ways: 3.11 It extends the physics to include Gravity and black holes 3.12 It extends group theory to include Exceptional groups This indicates the potential scope of the theory showing that much remains to be done. In the next section I will comment on the technical tools that will be needed. 4 New Technology The fundamental idea behind the new technology is that all the hard problems of mathe- matics or physics are non-linear, but our tools are linear. A familiar example is a CT scan which inputs linear waves and attempts to reconstruct your bones or your brain. Bom- barding a nucleus with waves and analysing the scattering is another example. These are examples from biology and physics. Representation theory is an example from group theory, while classical modular forms provide an example from the theory of elliptic curves. This was the bread and butter of Arithmetic Physics in the classical setting described by Manin in 1984. In the classical theories, as the subject developed, more elaborate processes were constructed. A simple example is the use of cohomology theory which began with H 1 but gradually moved on to H 2 and H 3. The new version of Arithmetic Physics that I envisage, replacing classical by quan- tum ideas, is infinite-dimensional and is reached by an infinite iteration of steps. Addi- tion and multiplication were handled with consummate skill, by Euler Dunham [1999] with infinite series and infinite products. But the hard operation, to iterate to infinity, is the exponential which converts addition into multiplication. In set theory the expo- nential 2S is the set of all subsets of S, and it was Emil Artin who exploited this. But it was von Neumann who went one step further and considered an infinite iteration of ex- ponentials as an analytical tool. This analysis was refined by Hirzebruch to be defined ARITHMETIC PHYSICS – ABEL LECTURE 115 over Q or over a number field Q0. This is the fundamental tool for the new theory and it encapsulates linear information from Hilbert space, such as the spectrum. Cohomology is the classical linear invariant and is the basis of much number theory, as in l-adic representations. But it was clear from the work of Grothendieck and Quillen that K-theory is a more sophisticated invariant, being filtered rather than graded.