Consideration of NISRs and FCs as Phenomenological Tools:
Recently, non-invertible selection rules (NISRs) are expected to play a quite important role in particle physics. Their characteristic behavior under quantum loop effects gives us a chance to take advantage of coupling constants to suppress the affection from some terms. As this is a great suppression, NISRs might work well on solving hierarchy problems. Going back to the origin of NISRs, it is clear that NISRs and fusion categories (FCs) are tight bound together through the correspondence between string theory (ST) and conformal field theory (CFT). With FCs, we are able to claim something advanced since NISRs only pick up algebras of FCs. After a series of categorical calculation, we can perceive that there might exist a non-trivial equivalence relation among Calabi-Yau manifolds (CY manifolds) being compact spaces of some STs which can be described by rational CFTs. This result helps us much with narrowing down the candidate CY manifolds for our spacetime.
Flux compactifications and sponteneous CP symmetry breaking:
1. CP symmetry
CP symmetry is a symmetry under which the laws of
physics remain unchanged even when particles are trans-
formed into antiparticles and space is inverted. In other
words, when the world of antiparticles is reflected in a
mirror, it obeys the same laws of physics as the original
world of particles. It is experimentally known that this CP
symmetry is violated in the SM (the Standard Model of
particle physics), and CP violation explains, among other
phenomena, the predominance of matter over antimatter
in the present Universe.
Superstring theory is a theory in which elementary par-
ticles are regarded as tiny strings, and different types
of particles, such as electrons and photons, are regarded
as the same string in different vibrational states. More-
over, because it incorporates a theory of gravity at the
quantum level, it is also regarded as a candidate for a
“theory of everything.” Superstring theory also predicts
that spacetime is ten-dimensional and that, in addition
to the four dimensions in which we live, there exists a
six-dimensional extra space. Furthermore, there are five
types of superstring theory: Type I, Type IIA, Type IIB,
heterotic E8 × E8, and heterotic SO(32) string theory.
These theories are known to be equivalent to one another.
In superstring theory, which regards spacetime as ten-
dimensional, CP symmetry, which is a symmetry of four-
dimensional spacetime, is embedded in the proper Lorentz
symmetry of ten dimensions. In other words, the La-
grangian of superstring theory must explicitly possess CP
symmetry. Therefore, investigating the origin of CP-
symmetry breaking in superstring theory is considered to
be highly significant.
Extra dimensions may be observable at high energies.
However, because we do not ordinarily see such dimen-
sions, the low-energy effective theory is used when con-
sidering the phenomenology of superstring theory. In ad-
dition to gravitational, matter, and gauge fields, the low-
energy effective Lagrangian of superstring theory contains
complex scalar fields called moduli fields, which determine
the size and shape of the extra dimensions and the string
coupling constant. These fields are generally denoted by
S, T , and U . Unless the values of these moduli are de-
termined, neither the string coupling constant nor the ge-
ometry of the extra dimensions can be determined. It
is therefore necessary to introduce fluxes into the extra
dimensions, generate a potential for the moduli, and fix
their values at a vacuum of the potential. This process
is called moduli stabilization. Furthermore, the moduli
in this vacuum state are considered to break CP symme-
try. In other words, CP is considered to be spontaneously
broken through moduli stabilization.
2. Previous studies
Previous studies have considered T 6/Z2, T 6/(Z2 × Z2),
and Calabi–Yau manifolds as structures of the extra di-
mensions and have investigated whether CP is sponta-
neously broken in Type IIB superstring theory. As a re-
sult, it was found that flat directions remain in the vac-
uum states of the moduli fields and that CP-symmetric
and CP-violating vacua are degenerate at the same en-
ergy. Consequently, these studies could not sufficiently
explain why a CP-violating vacuum is selected.
However, among the moduli fields S, T , and U , it is
known that the value of the modulus S, which determines
the string coupling constant, can also be fixed by gener-
ating a potential through methods other than the intro-
duction of fluxes, such as nonperturbative effects due to
D-brane instantons. Therefore, the fluxes are adjusted so
that no potential is generated for the modulus S, and it
is being investigated whether CP is broken as a result.
3. The present study
The fluxes are adjusted so that no potential is generated
for S, and the vacuum expectation values of the moduli
fields T and U are calculated for extra-dimensional spaces
such as T 6/Z2, T 6/(Z2 × Z2), and Calabi–Yau manifolds
in order to investigate whether CP is spontaneously bro-
ken. It is also examined whether the vacuum is an isolated
vacuum or possesses flat directions. When necessary, com-
putational tools such as Python and Julia are employed.
Algebraic methods concerning the existence of solutions
to multivariable equations, involving algebraic varieties,
ideals, and Gr¨obner bases, are also introduced in order to
advance our understanding of the structure of the solution
space of the vacuum equations. Furthermore, it is inves-
tigated whether there are modular transformations that
leave the vacuum expectation values invariant, with the
aim of clarifying the relationship between modular sym-
metry and CP violation.