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A. Cole and G. Shiu
Fig. 9.7 Rigid Calabi-Yau vacua projected onto axiodilaton with L max = 150. The relative sizes
of the voids depend on number-theoretic aspects of the complex rational value of the axiodilaton at
the center
Fig. 9.8 Left: persistence diagram for rigid Calabi-Yau flux vacua projected onto axiodilaton with
L max = 150, using a lazy witness complex with 700 landmark points. Witness complexes (and
their lazy variants) are useful when one needs to subsample the data set for computational reasons,
see [33] for their definition. The orange points represent 1-cycles and the blue points represent
0-cycles. We observe many long-lived 1-cycles, corresponding to voids in the distribution, which
seem to correlate with long-lived 0-cycles. Right: we link with a red line 0-cycles and 1-cycles
whose destroying simplices overlap
the constrained minimization is achieved for a = 1, b = =
h 2
2 f 1
+
h 2
2 f 1
2 +
L max
f 1 h 2
.
Since b scales as
√
L max , we recover the previously known fact that the voids shrink
as
√
1/L max as L max is increased [34, 36].
In terms of persistent homology, we expect long-lived 1-cycles corresponding to
large voids. Additionally, we expect the death of the 1-cycle corresponding to the