9 Towards the “Shape” of Cosmological Observables and the String …
233
the set of string vacua, called the string landscape, is vast.
5 In [14] it was noted that
the presence of multiple quantized fluxes leads to a corresponding discretuum of values for physical observables like the cosmological constant. Thus one can argue for
the existence of vacua satisfying certain criteria, but finding them remains difficult.
To understand the distribution of vacua with different properties, [40] proposed a
statistical approach. Subsequent work used statistical arguments and searched sets
of explicit constructions to make arguments about stringy naturalness, especially
regarding to the scale of supersymmetry breaking, the presence of various symmetries, and the cosmological constant [2, 8, 10, 31, 34, 35, 37–39, 41–43, 50, 60, 64,
74, 96].
More recently, advances in machine learning and data science have inspired applications of these techniques to the string landscape [1, 17, 19, 32, 59, 67, 70, 88,
100], [7, 18, 28, 54, 56, 78], see [89] for a review. In this context, we can view
persistent homology as a tool for pattern recognition and characterization. As such,
in [30] (see also [24]) persistent homology was used to study the distribution of
vacua in complex structure moduli space for Type IIB flux vacua. In this section we
review some of the results of [30]. In Sect. 9.4.1 we review the features of Type IIB
flux vacua relevant for study with persistent homology. In Sect. 9.4.2 we examine
the structure of a simple toy model, the rigid Calabi-Yau. In this context, we find
it useful to introduce a persistence diagram dressed with information regarding the
intersection of destroying simplices.
9.4.1 Flux Vacua
As previously mentioned, string theory must be compactified on a background geometry in order to make contact with 4-dimensional physics. The compactification is
well-controlled if some amount of supersymmetry is preserved. This can be achieved
by compactifying on a Calabi-Yau manifold. One then has an effective 4-dimensional
theory. In the effective theory, there are massless fields, called moduli. These arise
for example from certain deformations of the internal manifold. If the moduli remain
massless, they present problems for cosmology. Therefore, they must be stabilized,
i.e. given a potential. A particular modulus is fixed by its potential to lie at a local
minimum. More precisely, the vacuum expectation value (VEV) is fixed to lie at this
minimum.
In this section, we will consider type IIB string theory on a Calabi-Yau orientifold.
There are then several classes of moduli: the axiodilaton, a universal modulus that
includes the string coupling g s ; the complex structure moduli (which parametrize
roughly speaking the “shapes” of the internal manifold); the Kähler moduli (which
control the “sizes” of the internal (sub)manifolds); and open string moduli. The
5 For a sense of scale, it is estimated that the number of flux vacua for a typical geometry is around
10 500 [8, 34] and can be as large as 10 272,000 [98]. The number of geometries in a particular of
F-theory ensemble is bounded below by
4
3 × 2.96 × 10 755 [55].
233
the set of string vacua, called the string landscape, is vast.
5 In [14] it was noted that
the presence of multiple quantized fluxes leads to a corresponding discretuum of values for physical observables like the cosmological constant. Thus one can argue for
the existence of vacua satisfying certain criteria, but finding them remains difficult.
To understand the distribution of vacua with different properties, [40] proposed a
statistical approach. Subsequent work used statistical arguments and searched sets
of explicit constructions to make arguments about stringy naturalness, especially
regarding to the scale of supersymmetry breaking, the presence of various symmetries, and the cosmological constant [2, 8, 10, 31, 34, 35, 37–39, 41–43, 50, 60, 64,
74, 96].
More recently, advances in machine learning and data science have inspired applications of these techniques to the string landscape [1, 17, 19, 32, 59, 67, 70, 88,
100], [7, 18, 28, 54, 56, 78], see [89] for a review. In this context, we can view
persistent homology as a tool for pattern recognition and characterization. As such,
in [30] (see also [24]) persistent homology was used to study the distribution of
vacua in complex structure moduli space for Type IIB flux vacua. In this section we
review some of the results of [30]. In Sect. 9.4.1 we review the features of Type IIB
flux vacua relevant for study with persistent homology. In Sect. 9.4.2 we examine
the structure of a simple toy model, the rigid Calabi-Yau. In this context, we find
it useful to introduce a persistence diagram dressed with information regarding the
intersection of destroying simplices.
9.4.1 Flux Vacua
As previously mentioned, string theory must be compactified on a background geometry in order to make contact with 4-dimensional physics. The compactification is
well-controlled if some amount of supersymmetry is preserved. This can be achieved
by compactifying on a Calabi-Yau manifold. One then has an effective 4-dimensional
theory. In the effective theory, there are massless fields, called moduli. These arise
for example from certain deformations of the internal manifold. If the moduli remain
massless, they present problems for cosmology. Therefore, they must be stabilized,
i.e. given a potential. A particular modulus is fixed by its potential to lie at a local
minimum. More precisely, the vacuum expectation value (VEV) is fixed to lie at this
minimum.
In this section, we will consider type IIB string theory on a Calabi-Yau orientifold.
There are then several classes of moduli: the axiodilaton, a universal modulus that
includes the string coupling g s ; the complex structure moduli (which parametrize
roughly speaking the “shapes” of the internal manifold); the Kähler moduli (which
control the “sizes” of the internal (sub)manifolds); and open string moduli. The
5 For a sense of scale, it is estimated that the number of flux vacua for a typical geometry is around
10 500 [8, 34] and can be as large as 10 272,000 [98]. The number of geometries in a particular of
F-theory ensemble is bounded below by
4
3 × 2.96 × 10 755 [55].