The Quantum Structure of Space and Time (293 pages)

(^110) The Quantum Structure of Space and Time
In general the topological string partition function (of the A model) takes the
form
ztop = exp c 9,2”-”Fg(t),
920
where the genus g contribution Fg can be expended as a sum over degree d maps
Fg (t) = c GW,,d ePdtla’.
Here GWg,d E Q denotes the Gromov-Witten invariant that “counts” the number
of holomorphic maps f : C, + X of degree d of a Riemann surface C, of genus g
into the Calabi-Yau manifold X.
To show that these invariants are very non-trivial and define some quantum
geometry structure, it suffices to look at the simplest possible CY space X = C3.
In that case only degree zero maps contribute. The corresponding Gromov-Witten
invariants have been computed and can be expressed in terms of so-called Hodge
integrals
d
B~gB2g-2
29(2g - 2)(2g - 2)!.
But in this case the full partition function Ztop simplifies considerably if it is
expressed in terms of the strong coupling variable q = e-9s instead of the weak
coupling variable gs:
9 n>O
In fact, this gives a beautiful reinterpretation in terms of a statistical mechanics
model. The partition function can be written as a weighted sum over all planar
partitions
7r
Here a planar partition 7r is a 3d version of the usual 2d partitions [25]
Clearly, this statistical model has a granular structure that is invisible in the per-
turbative limit gs + 0. In fact, these quantum crystals give a very nice model in
which the stringy and quantum geometry regimes can be distinguished. Here one
uses a toric description of C3 as a T3 bundle over the positive octant in R3. In