Mathematical Structures^119
4.2 Discussion
G. Horowitz At the very beginning you asked: what is quantum geometry? You
said it should involve non-commutative geometry, non-commuting coordinates.
It is not obvious to me that it has to involve that. Later on, you gave this
example of a melting crystal and it was unclear to me where non-commutative
geometry came in in that example.
R. Dijkgraaf Actually, I do not think the details have been worked out but it has
a very nice interpretation: again there is a matrix model description of this
melting crystal which is the reduction of supersymmetric Yang-Mills theory to
zero dimension. So it is a three-matrix model where the action is trX[Y,Z].
If we look at the critical point of that action, it corresponds exactly to this
kind of crystal configurations. So again in this example, there is a D-brane
interpretation. I must say that I deliberately did not make that argument
exact because usually people argue something like : “If you have to measure
space you have to concentrate energy. A little black hole will form and it will
have some uncertainty because it will evaporate”. I never felt very comfortable
with that argument.
H. Ooguri In that particular example, you are exhibiting already half of the space.
In the total space, of six dimensions, you do, in fact, see the non-commutative
structures?
R. Dijkgraaf Yes. I guess that is a good point.
H. Ooguri There are three torus directions and three directions he was exhibiting
which are non-commuting. This quantum structure of the space, the state
being represented by blocks in this three dimensions, is a reflection of this non-
commutative structure. So I think the example that Dijkgraaf was describing
exactly demonstrates the non-commutative feature of space time where the
Planck constant is replaced by e-llg, with g the coupling constant.
G. Horowitz I certainly agree, any quantum description will involve non-
commutativity, like some z and p do not commute. But were you suggesting
that it is always some sort of X with X not commuting?
R. Dijkgraaf In this case it is indeed so. To follow what Ooguri was saying: I was
talking about a six dimensional space and drawing a three dimensional picture.
In some sense I was using kind of a symplectic space and I was only showing
you the coordinates, not the momenta.
N. Seiberg You wrote that time does not commute with space? What did you
have in mind?
R. Dijkgraaf What I just said. I am just repeating folklore here: making pre-
cise space-time measurements will create small black holes that will evaporate
and give a time uncertainty. I do not have any example here. All the non-
commutativity that I was discussing here was non-commutativity in the space-
like directions. I did not have any examples.
