Amidst all LHC-noise, Yuri I. Manin arXived today an interesting paper Cyclotomy and analytic geometry over .

The paper gives a nice survey of the existent literature and focusses on the crucial role of roots of unity in the algebraic geometry over the non-existent field with one element (in French called ‘F-un’). I have tried to do a couple of posts on F-un some time ago but now realize, reading Manin’s paper, I may have given up way too soon…

At several places in the paper, Manin hints at a possible noncommutative geometry over :

This is the appropriate place to stress that in a wider context of Toen-Vaqui ‘Au-dessous de Spec Z’, or eventually in noncommutative -geometry, teh spectrum of loses its privileged position as a final object of a geometric category. For example, in noncommutative geometry, or in an appropriate category of stacks, the quotient of this spectrum modulo the trivial action of a group must lie below this spectrum.
Soule’s algebras are a very important element of the structure, in particular, because they form a bridge to Arakelov geometry. Soule uses concrete choices of them in order to produce ‘just right’ supply of morphisms, without a priori constraining these choices formally. In this work, we use these algebras and their version also to pave a way to the analytic (and possibly non-commutative) geometry over .

Back when I was writing the first batch of F-un posts, I briefly contemplated the possibility of a noncommutative geometry over , but quickly forgot about it because I thought it would be forced to reduce to commutative geometry.

Here is the quick argument : noncommutative geometry is really the study of coalgebras (see for example my paper or if you prefer more trustworthy sources the Kontsevich-Soibelman paper). Now, unless I made a mistake, I think all coalgebras over must be co-commutative (even group-like), so reducing to commutative geometry.

Surely, I’m missing something…

Yesterday, Yuri Manin and Matilde Marcolli arXived their paper Modular shadows and the Levy-Mellin infinity-adic transform which is a follow-up of their previous paper Continued fractions, modular symbols, and non-commutative geometry. They motivate the title of the recent paper by :

In MaMar2, these and similar results were put in connection with the so called “holography” principle in modern theoretical physics. According to this principle, quantum field theory on a space may be faithfully reflected by an appropriate theory on the boundary of this space. When this boundary, rather than the interior, is interpreted as our observable space‚Äìtime, one can proclaim that the ancient Plato’s cave metaphor is resuscitated in this sophisticated guise. This metaphor motivated the title of the present paper.

Here’s a layout of Plato’s cave

Imagine prisoners, who have been chained since childhood deep inside an cave: not only are their limbs immobilized by the chains; their heads are chained as well, so that their gaze is fixed on a wall.
Behind the prisoners is an enormous fire, and between the fire and the prisoners is a raised walkway, along which statues of various animals, plants, and other things are carried by people. The statues cast shadows on the wall, and the prisoners watch these shadows. When one of the statue-carriers speaks, an echo against the wall causes the prisoners to believe that the words come from the shadows.
The prisoners engage in what appears to us to be a game: naming the shapes as they come by. This, however, is the only reality that they know, even though they are seeing merely shadows of images. They are thus conditioned to judge the quality of one another by their skill in quickly naming the shapes and dislike those who begin to play poorly.
Suppose a prisoner is released and compelled to stand up and turn around. At that moment his eyes will be blinded by the firelight, and the shapes passing will appear less real than their shadows.

Right, now how does the Manin-Marcolli cave look? My best guess is : like this picture, taken from Curt McMullen’s Gallery

Imagine this as the top view of a spherical cave. M&M are imprisoned in the cave, their heads chained preventing them from looking up and see the ceiling (where (or a cofinite subgroup of it) is acting on the upper-half plane via Moebius-transformations ). All they can see is the circular exit of the cave. They want to understand the complex picture going on over their heads from the only things they can observe, that is the action of (subgroups of) the modular group on the cave-exit . Now, the part of it consisting of orbits of cusps has a nice algebraic geometric description, but orbits of irrational points cannot be handled by algebraic geometry as the action of is highly non-discrete as illustrated by another picture from McMullen’s gallery

depicting the ill behaved topology of the action on the bottom real axis. Still, noncommutative differential geometry is pretty good at handling such ill behaved quotient spaces and it turns out that as a noncommutative space, this quotient is rich enough to recover many important aspects of the classical theory of modular curves. Hence, they reverse the usual NCG-picture of interpreting commutative objects as shadows of noncommutative ones. They study the _noncommutative shadow of a classical commutative object, the quotient of the action of the modular group (or a cofinite subgroup of it) on the upper half-plane.

In our noncommutative geometry course we have already seen this noncommutative shadow in action (though at a very basic level). Remember that we first described the group-structure of the modular group via the classical method of groups acting on trees. In particular, we considered the tree

and calculated the stabilizers of the end points of its fundamental domain (the thick circular edge). But later we were able to give a much shorter proof (due to Roger Alperin) by looking only at the action of on the irrational real numbers (the noncommutative shadow). Needless to say that the results obtained by Manin and Marcolli from staring at their noncommutative shadow are a lot more intriguing…

Here are my nominees for the 2006 paper of the year award in mathematics & mathematical physics : in math.RA : math.RA/0606241 : Notes on A-infinity algebras, A-infinity categories and non-commutative geometry. I by Maxim Kontsevich and Yan Soibelman. Here is the abstract :

We develop geometric approach to A-infinity algebras and A-infinity categories based on the notion of formal scheme in the category of graded vector spaces. Geometric approach clarifies several questions, e.g. the notion of homological unit or A-infinity structure on A-infinity functors. We discuss Hochschild complexes of A-infinity algebras from geometric point of view. The paper contains homological versions of the notions of properness and smoothness of projective varieties as well as the non-commutative version of Hodge-to-de Rham degeneration conjecture. We also discuss a generalization of Deligne’s conjecture which includes both Hochschild chains and cochains. We conclude the paper with the description of an action of the PROP of singular chains of the topological PROP of 2-dimensional surfaces on the Hochschild chain complex of an A-infinity algebra with the scalar product (this action is more or less equivalent to the structure of 2-dimensional Topological Field Theory associated with an “abstract” Calabi-Yau manifold).

why ? : Because this paper probably gives the correct geometric object associated to a non-commutative algebra (a huge coalgebra) and consequently the right definition of a map between noncommutative affine schemes. In a previous post (and its predecessors) I’ve tried to explain how this links up with my own interpretation and since then I’ve thought more about this, but that will have to wait for another time. in hep-th : hep-th/0611082 : Children’s Drawings From Seiberg-Witten Curves by Sujay K. Ashok, Freddy Cachazo, Eleonora Dell’Aquila. Here is the abstract :

We consider N=2 supersymmetric gauge theories perturbed by tree level superpotential terms near isolated singular points in the Coulomb moduli space. We identify the Seiberg-Witten curve at these points with polynomial equations used to construct what Grothendieck called “dessins d’enfants” or “children’s drawings” on the Riemann sphere. From a mathematical point of view, the dessins are important because the absolute Galois group Gal(\bar{Q}/Q) acts faithfully on them. We argue that the relation between the dessins and Seiberg-Witten theory is useful because gauge theory criteria used to distinguish branches of N=1 vacua can lead to mathematical invariants that help to distinguish dessins belonging to different Galois orbits. For instance, we show that the confinement index defined in hep-th/0301006 is a Galois invariant. We further make some conjectures on the relation between Grothendieck’s programme of classifying dessins into Galois orbits and the physics problem of classifying phases of N=1 gauge theories.

why ? : Because this paper gives the best introduction I’ve seen to Grothendieck’s dessins d’enfants (slightly overdoing it by giving a crash course on elementary Galois theory in appendix A) and kept me thinking about dessins and their Galois invariants ever since (again, I’ll come back to this later).

Last time we have seen that the coalgebra of distributions of an affine smooth variety is the direct sum (over all points) of the dual to the etale local algebras which are all of the form where is the dimension of the variety. Generalizing this to non-commutative manifolds, the first questions are : “What is the analogon of the power-series algebra?” and do all ‘points’ of our non-commutative manifold do have such local algebras? Surely, we no longer expect the variables to commute, so a non-commutative version of the power series algebra should be , the ring of formal power series in non-commuting variables. However, there is still another way to add non-commutativity and that is to go from an algebra to matrices over the algebra. So, in all we would expect to be our local algebras at points of our non-commutative manifold to be isomorphic to As to the second question : _qurves (that is, the coordinate rings of non-commutative manifolds) do have such algebras as local rings provided we take as the ‘points’ of the non-commutative variety the set of all simple finite dimensional representations of the qurve. This is a consequence of the tubular neighborhood theorem due to Cuntz and Quillen. In more details : If A is a qurve, then a simple -dimensional representation corresponds to an epimorphism and if we take , then is an -bimodule and the -adic completion of A is isomorphic to the completed tensor-algebra In contrast with the commutative case however where the dimension remains constant over all points, here the numbers n and d can change from simple to simple. For n this is clear as it gives the dimension of the simple representation, but also d changes (it is the local dimension of the variety classifying simple representations of the same dimension). Here an easy example : Consider the skew group algebra with the action given by sending . Then A is a qurve and its center is with . Over any point there is a unique simple 2-dimensional representation of A giving the local algebra . If the situation is more complicated as the local structure of A is given by the algebra So, over this point there are precisely 2 one-dimensional simple representations corresponding to the maximal ideals and both ideals are idempotent, that is whence the corresponding bimodule so the local algebra in either of these two points is just . Ok, so the comleted local algebra at each point is of the form , but what is the corresponding dual coalgebra. Well, is the algebra dual to the _cofree coalgebra on . As a vectorspace this is the tensor-algebra with the coalgebra structure induced by the bialgebra structure defined by taking all varaibales to be primitives, that is . That is, the coproduct on a monomial gives all different expressions such that . For example, . On the other hand, the dual coalgebra of is the matrix coalgebra which is the -dimensional vectorspace with comultiplication The coalgebra corresponding to the local algebra is then the tensor-coalgebra of the matrix coalgebra and the cofree coalgebra. Having obtained the coalgebra at each point (=simple representation) of our noncommutative manifold one might think that the _coalgebra of non-commutative distributions should be the direct sum of all this coalgebras, summed over all points, as in the commutative case. But then we would forget about a major difference between the commutative and the non-commutative world : distinct simples can have non-trivial extensions! The mental picture one might have about simples having non-trivial extensions is that these points lie ‘infinitesimally close’ together. In the example above, the two one-dimensional simples have non-trivial extensions so they should be thought of as a cluster of two infinitesimally close points corresponding to the point (that is, this commutative points splits into two non-commutative points). Btw. this is the reason why non-commutative algebras can be used to resolve commutative singularities (excessive tangents can be split over several non-commutative points). While this is still pretty harmless when the algebra is finite over its center (as in the above example where only the two one-dimensionals have extensions), the situation becomes weird over general qurves as ‘usually’ distinct simples have non-trivial extensions. For example, for the free algebra this is true for all simples… So, if we want to continue using this image of points lying closely together this immediately means that non-commutative ‘affine’ manifolds behave like compact ones (in fact, it turns out to be pretty difficult to ‘glue’ together qurves into ‘bigger’ non-commutative manifolds, apart from the quiver examples of this old paper). So, how to bring this new information into our coalgebra of distributions? Well, let’s repeat the previous argument not with just one point but with a set of finitely many points. Then we have a _semi-simple algebra quotient and taking again and , then is again an S-bimodule. Now, any S-bimodule can be encoded into a quiver Q on k points, the number of arrows from vertex i to vertex j being the number of components in M of the form . Again, it follows from the tubular neighborhood theorem that the -adic completion of A is isomorphic to the completion of an algebra Morita equivalent to the _path algebra (being the tensor algebra ). As all the local algebras of the points are quotients of this quiver-like completion, on the coalgebra level our local coalgebras will be sub coalgebras of the coalgebra which is co-Morita equivalent (and believe it or not but coalgebraists have a name for this : _Takeuchi equivalence) to the quiver coalgebra which is the vectorspace of the path algebra with multiplication induced by making all arrows from i to j skew-primitives, that is, where the are group-likes corresponding to the vertices. If all of ths is a bit too much co to take in at once, I suggest the paper by Bill Chin A brief introduction to coalgebra representation theory. The _coalgebra of noncommutative distributions we are after at is now the union of all these Takeuchi-equivalent quiver coalgebras. In easy examples such as the -example this coalgebra is still pretty small (the sum of the local coalgebras corresponding to the local algebras summed over all points summed with the quiver coalgebra of the quiver In general though this is a huge object and we would like to have a recipe to construct it from a manageable blue-print and that is what we will do next time.

Now that my non-geometry post is linked via the comments in this string-coffee-table post which in turn is available through a trackback from the Kontsevich-Soibelman paper it is perhaps useful to add a few links.

The little I’ve learned from reading about Connes-style non-commutative geometry is this : if you have a situation where a discrete group is acting with a bad orbit-space (for example, $GL2(\mathbb{Z})$ acting on the whole complex-plane, rather than just the upper half plane) you can associate to this a $C^*$-algebra and study invariants of it and interprete them as topological information about this bad orbit space. An intruiging example is the one I mentioned and where the additional noncommutative points (coming from the orbits on the real axis) seem to contain a lot of modular information as clarified by work of Manin&Marcolli and Zagier. Probably the best introduction into Connes-style non-commutative geometry from this perspective are the Lecture on Arithmetic Noncommutative Geometry by Matilde Marcolli. To algebraists : this trick is very similar to looking at the skew-group algebra $\mathbb{C}[x1,\hdots,xn] * G$ if you want to study the _orbifold for a finite group action on affine space. But as algebraist we have to stick to affine varieties and polynomials so we can only deal with the case of a finite group, analysts can be sloppier in their functions, so they can also do something when the group is infinite.

By the way, the skew-group algebra idea is also why non-commutative algebraic geometry enters string-theory via the link with orbifolds. The easiest (and best understood) example is that of Kleinian singularities. The best introduction to this idea is via the Representations of quivers, preprojective algebras and deformations of quotient singularities notes by Bill Crawley-Boevey.

Artin-style non-commutative geometry aka non-commutative projective geometry originated from the work of Artin-Tate-Van den Bergh (in the west) and Odeskii-Feigin (in the east) to understand Sklyanin algebras associated to elliptic curves and automorphisms via ‘geometric’ objects such as point- (and fat-point-) modules, line-modules and the like. An excellent survey paper on low dimensional non-commutative projective geometry is Non-commutative curves and surfaces by Toby Stafford and Michel Van den Bergh. The best introduction is the (also neverending…) book-project Non- commutative algebraic geometry by Paul Smith who maintains a noncommutative geometry and algebra resource page page (which is also available from the header).

Non-geometry started with the seminal paper ‘Algebra extensions and nonsingularity’, J. Amer. Math. Soc. 8 (1995), 251-289 by Joachim Cuntz and Daniel Quillen but which is not available online. An online introduction is Noncommutative smooth spaces by Kontsevich and Rosenberg. Surely, different people have different motivations to study non-geometry. I assume Cuntz got interested because inductive limits of separable algebras are quasi-free (aka formally smooth aka qurves). Kontsevich and Soibelman want to study morphisms and deformations of $A_{\infty}$-categories as they explain in their recent paper. My own motivation to be interested in non-geometry is the hope that in the next decades one will discover new exciting connections between finite groups, algebraic curves and arithmetic groups (monstrous moonshine being the first, and still not entirely understood, instance of this). Part of the problem is that these three topics seem to be quite different, yet by taking group-algebras of finite or arithmetic groups and coordinate rings of affine smooth curves they all turn out to be quasi-free algebras, so perhaps non-geometry is the unifying theory behind these seemingly unrelated topics.