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](http://www.arxiv.org/abs/hep-th/0201036), 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 $PSL_2(\mathbb{Z}) $ (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

$\mathbb{P}^1(\mathbb{R}) $. Now, the part of it consisting

of orbits of cusps

$\mathbb{P}^1(\mathbb{Q}) $ has a nice algebraic geometric

description, but orbits of irrational points cannot be handled by

algebraic geometry as the action of $PSL_2(\mathbb{Z}) $ 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

$\mathbb{P}^1(\mathbb{R})/PSL_2(\mathbb{Z}) $ 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_

$\mathbb{P}^1(\mathbb{R})/PSL_2(\mathbb{Z}) $ 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 $PSL_2(\mathbb{Z}) = C_2 \ast C_3 $ 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 $PSL_2(\mathbb{Z}) $ 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…