9 Towards the “Shape” of Cosmological Observables and the String …
239
void will be correlated with the death of a 0-cycle corresponding to the isolated
vacuum in the center of the void. Persistence diagrams for a subregion of the rigid
Calabi-Yau axiodilaton distribution are shown in Fig. 9.8. The longer-lived 1-cycles
correspond to the larger voids in Fig. 9.7.
9.4.2.1 Dressed Persistence Diagrams
While the correlation between 1-cycle and 0-cycle deaths is suggestive, it is not
enough to reconstruct the void/isolated point structure of the distribution. To recover
more fine-grained information about the point cloud, [30] introduced a dressed persistence diagram. This representation visualizes information from the persistence
pairing that is not shown in a standard persistence diagram. Recall that for a filtration,
one generates a persistence diagram by computing a set of persistence pairs (σ, τ ).
A given homology class is created when σ is added to the filtration and destroyed
when τ is added. We might then be able to visualize correlations between different
topological features by using information contained in the persistence pairs. Note
that while elements of the homology groups are of equivalence classes of p-cycles,
the simplices in the corresponding persistence pair are unique. There is no need for
us to choose a canonical cycle. Specifically, one may draw a line between cycles
in the persistence diagram whose destroying simplices overlap, and this is a unique
procedure. In the present context, this proves useful because the death of the 0-cycle
corresponding to an isolated vacuum in the center of a void is caused by the addition
of an edge connecting that vacuum to a point on the edge of the void. This same edge
is contained in the triangles that fill in the void, causing the 1-cycle’s death. This
dressed persistence diagram is shown in Fig. 9.8, demonstrating further correlations
not evident in a standard persistence diagram.
The rigid Calabi-Yau has a moduli space of real dimension 2, and as such can
simply be examined by eye. In [30] examples of moduli spaces with 4 real dimensions were also considered, in particular the symmetric T
6
= (T
2
)
3 and a particular
hypersurface in weighted projective space.
9.5 Conclusion
Much of theoretical physics involves finding the right structures to efficiently describe
and understand a system. For example, the existence and identification of conserved
quantities can simplify one’s description of a theory greatly. Studying physical data
sets using persistent homology seems a promising avenue for discovering patterns
that allow for such a simplified description. Moreover, it has recently been shown
that persistent homology can efficiently detect and classify phases of matter, providing quantitative and interpretable order parameters for phase transitions [27]. This
suggests that persistent homology has potentially wider applications in identifying
the structures of data.
239
void will be correlated with the death of a 0-cycle corresponding to the isolated
vacuum in the center of the void. Persistence diagrams for a subregion of the rigid
Calabi-Yau axiodilaton distribution are shown in Fig. 9.8. The longer-lived 1-cycles
correspond to the larger voids in Fig. 9.7.
9.4.2.1 Dressed Persistence Diagrams
While the correlation between 1-cycle and 0-cycle deaths is suggestive, it is not
enough to reconstruct the void/isolated point structure of the distribution. To recover
more fine-grained information about the point cloud, [30] introduced a dressed persistence diagram. This representation visualizes information from the persistence
pairing that is not shown in a standard persistence diagram. Recall that for a filtration,
one generates a persistence diagram by computing a set of persistence pairs (σ, τ ).
A given homology class is created when σ is added to the filtration and destroyed
when τ is added. We might then be able to visualize correlations between different
topological features by using information contained in the persistence pairs. Note
that while elements of the homology groups are of equivalence classes of p-cycles,
the simplices in the corresponding persistence pair are unique. There is no need for
us to choose a canonical cycle. Specifically, one may draw a line between cycles
in the persistence diagram whose destroying simplices overlap, and this is a unique
procedure. In the present context, this proves useful because the death of the 0-cycle
corresponding to an isolated vacuum in the center of a void is caused by the addition
of an edge connecting that vacuum to a point on the edge of the void. This same edge
is contained in the triangles that fill in the void, causing the 1-cycle’s death. This
dressed persistence diagram is shown in Fig. 9.8, demonstrating further correlations
not evident in a standard persistence diagram.
The rigid Calabi-Yau has a moduli space of real dimension 2, and as such can
simply be examined by eye. In [30] examples of moduli spaces with 4 real dimensions were also considered, in particular the symmetric T
6
= (T
2
)
3 and a particular
hypersurface in weighted projective space.
9.5 Conclusion
Much of theoretical physics involves finding the right structures to efficiently describe
and understand a system. For example, the existence and identification of conserved
quantities can simplify one’s description of a theory greatly. Studying physical data
sets using persistent homology seems a promising avenue for discovering patterns
that allow for such a simplified description. Moreover, it has recently been shown
that persistent homology can efficiently detect and classify phases of matter, providing quantitative and interpretable order parameters for phase transitions [27]. This
suggests that persistent homology has potentially wider applications in identifying
the structures of data.