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…

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arxiv, differential, geometry, groups, Manin, Marcolli, modular, non-commutative, noncommutative, topology

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Posted in geometry, modular
Written on Wed, 28 March 2007 at 1:29 pm
Tags: arxiv, differential, geometry, groups, Manin, Marcolli, modular, non-commutative, noncommutative, topology
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