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The flavor moonshine hypothesis is formulated to suppose that all particle masses (leptons, quarks, Higgs, and gauge particles—more precisely, their mass ratios) are expressed as coefficients in the Fourier expansion of some modular forms just as, in mathematics, dimensions of representations of a certain group are expressed as coefficients in the Fourier expansion of some modular forms. The mysterious hierarchical structure of the quark and lepton masses is thus attributed to that of the Fourier coefficient matrices of certain modular forms. Our intention here is not to prove this hypothesis starting from some physical assumptions but rather to demonstrate that this hypothesis is experimentally verified and, assuming that the string theory correctly describes the natural law, to calculate the geometry (Kähler potential and the metric) of the moduli space of the Calabi–Yau manifold, thus providing a way to calculate the metric of the Calabi–Yau manifold itself directly from the experimental data.

A bstract We provide evidence for partial deconfinement — the deconfinement of a SU( M ) subgroup of the SU( N ) gauge group — by using lattice Monte Carlo simulations. We take matrix models as concrete examples. By appropriately fixing the gauge, we observe that the M × M submatrices deconfine. This gives direct evidence for partial deconfinement at strong coupling. We discuss the applications to QCD and holography.

Gravity theory based on current algebra is formulated. The gauge principle rather than general covariance combined with the equivalence principle plays a pivotal role in the formalism, and the latter principles are derived as a consequence of the theory. In this approach, it turns out that gauging the Poincaré algebra is not appropriate but gauging the $SO(N,M)$ algebra gives a consistent theory. This makes it possible to have anti-de Sitter and de Sitter space-time by adopting a relation between the spin connection and the tetrad field. The Einstein equation is part of our basic equation for gravity, which is written in terms of the spin connection. When this formalism is applied to the $E(11)$ algebra in which the three-form antisymmetric tensor is part of a gravity multiplet, we have a current algebra gravity theory based on M-theory in the sense that the internal group or the connection space representations of our model are those appearing in 11D supergravity. Moreover, when our formalism in its classical limit is applied to cosmology, by introducing conformal-like modes that connect the tetrad field/current and the spin connection field/current, we can obtain an accelerating universe in the manner of the “inflating” universe at its early stage.

The flavor moonshine hypothesis is formulated to suppose that all particle masses (leptons, quarks, Higgs and gauge particles -- more precisely, their mass ratios) are expressed as coefficients in the Fourier expansion of some modular forms just as, in mathematics, dimensions of representations of a certain group are expressed as coefficients in the Fourier expansion of some modular forms. The mysterious hierarchical structure of the quark and lepton masses is thus attributed to that of the Fourier coefficient matrices of certain modular forms. Our intention here is not to prove this hypothesis starting from some physical assumptions but rather to demonstrate that this hypothesis is experimentally verified and, assuming that the string theory correctly describes the natural law, to calculate the geometry (K\"{a}hler potential and the metric) of the moduli space of the Calabi-Yau manifold, thus providing a way to calculate the metric of Calabi-Yau manifold itself directly from the experimental data.

Machine learning methods are powerful in distinguishing different phases of matter in an automated way and provide a new perspective on the study of physical phenomena. We train a Restricted Boltzmann Machine (RBM) on data constructed with spin configurations sampled from the Ising Hamiltonian at different values of temperature and external magnetic field using Monte Carlo methods. From the trained machine we obtain the flow of iterative reconstruction of spin state configurations to faithfully reproduce the observables of the physical system. We find that the flow of the trained RBM approaches the spin configurations of the maximal possible specific heat which resemble the near criticality region of the Ising model. In the special case of the vanishing magnetic field the trained RBM converges to the critical point of the Renormalization Group (RG) flow of the lattice model. Our results suggest an alternative explanation of how the machine identifies the physical phase transitions, by recognizing certain properties of the configuration like the maximization of the specific heat, instead of associating directly the recognition procedure with the RG flow and its fixed points. Then from the reconstructed data we deduce the critical exponent associated to the magnetization to find satisfactory agreement with the actual physical value. We assume no prior knowledge about the criticality of the system and its Hamiltonian.

We provide evidence for partial deconfinement -- the deconfinement of a SU(\\(M\\)) subgroup of the SU(\\(N\\)) gauge group -- by using lattice Monte Carlo simulations. We take matrix models as concrete examples. By appropriately fixing the gauge, we observe that the \\(M\\times M\\) submatrices deconfine. This gives direct evidence for partial deconfinement at strong coupling. We discuss the applications to QCD and holography.

We study the features extracted by the Restricted Boltzmann Machine (RBM) when it is trained with spin configurations of Ising model at various temperatures. Using the trained RBM, we obtain the flow of iterative reconstructions (RBM flow) of the spin configurations and find that in some cases the flow approaches the phase transition point \\(T=T_c\\) in Ising model. Since the extracted features are emphasized in the reconstructed configurations, the configurations at such a fixed point should describe nothing but the extracted features. Then we investigate the dependence of the fixed point on various parameters and conjecture the condition where the fixed point of the RBM flow is at the phase transition point. We also provide supporting evidence for the conjecture by analyzing the weight matrix of the trained RBM.

Hawking radiation of the blackhole is calculated based on the principle of local field theory. In our approach, the radiation is a unitary process, therefore no information loss will be recorded. In fact, observers in different regions of the space communicate using the Hawking radiation, when the systems in the different regions are entangled with each other. The entanglement entropy of the blackhole is also calculated in the local field theory. We found that the entanglement entropy of the systems separated by the blackhole horizon is closely connected to the Hawking radiation in our approach. Our calculation shows that the entanglement entropy of the systems separated by the horizon of a blackhole is just a pure number \\(\\frac{\\pi^3 + 270 \\zeta(3)}{360 \\pi^2}\\), independent of any parameter of the blackhole, and its relation to the Hawking radiation is given by \\(S_{EE} = \\frac{8 \\pi}{3} \\frac{\\pi^3 + 270 \\zeta(3)}{\\pi^3 + 240 \\zeta(3)} {\\cal A} R_H\\), where \\(S_{EE}\\) is the entanglement entropy, \\(\\cal A\\) is the area of the horizon, and \\(R_H\\) is the Hawking radiation.