International Journal of Modern Physics A

In this paper, we briefly overview how, historically, string theory led theoretical physics first to precise problems in algebraic and differential geometry, and thence to computational geometry in the last decade or so, and now, in the last few years, to data science. Using the Calabi–Yau landscape — accumulated by the collaboration of physicists, mathematicians and computer scientists over the last four decades — as a starting-point and concrete playground, we review some recent progress in machine-learning applied to the sifting through of possible universes from compactification, as well as wider problems in geometrical engineering of quantum field theories. In parallel, we discuss the program in machine-learning mathematical structures and address the tantalizing question of how it helps doing mathematics, ranging from mathematical physics, to geometry, to representation theory, to combinatorics and to number theory.

In this study, the bound state solutions of the Dirac equation (DE) have been determined with the generalized Cornell potential model (GCPM) under the condition of spin symmetry. The GCPM includes the Cornell potential plus a combination of the harmonic and inversely quadratic potentials. In the framework of the Nikiforov–Uvarov (NU) method, the relativistic and nonrelativistic energy eigenvalues for the GCPM have been obtained. The energies spectra of the Kratzer potential (KP) and the modified Kratzer potential (MKP) have been derived as particular cases of the GCPM. The present results have been applied to some diatomic molecules (DMs) as well as heavy and heavy-light mesons. The energy eigenvalues of the KP and MKP have been computed for several DMs, and they are fully consistent with the results found in the literature. In addition, the energy eigenvalues of the GCPM have been employed for predicting the spin-averaged mass spectra of heavy and heavy-light mesons. One can note that our predictions are in close agreement with the experimental data as well as enhanced compared to the recent studies.

With appropriate modifications, the multi-spin Klein–Gordon (KG) equation of quantum field theory can be adapted to curved space–time for spins 0, 1, 1/2. The associated particles in the microworld then move as a wave at all space–time coordinates. From the existence in a Lorentzian space–time of a line element field $(X^{\beta },-X^{\beta })$, the spin-1 KG equation ${\nabla }_{\mu }{\nabla }^{\mu }{X}^{\beta }=k^2{X}^{\beta }$ is derived from an action functional involving $X^{\beta }$ and its covariant derivative. The spin-0 KG equation and the KG equation of the outer product of a spin-1/2 Dirac spinor and its Hermitian conjugate are then constructed. Thus, $X^{\beta }$ acts as a fundamental quantum vector field. The symmetric part of the spin-1 KG equation, ${\tilde{\mathrm{\Psi }}}_{\alpha \beta }$, is the Lie derivative of the metric. That links the multi-spin KG equation to Modified General Relativity (MGR) through its energy–momentum tensor of the gravitational field. From the invariance of the action functionals under the diffeomorphism group Diff(M), which is not restricted to the Lorentz group, ${\tilde{\mathrm{\Psi }}}_{\alpha \beta }$ can instantaneously transmit information along $X^{\beta }$. That establishes the concept of entanglement within a Lorentzian formalism. The respective local/nonlocal characteristics of MGR and quantum theory no longer present an insurmountable problem to unify the theories.

Modified gravity models are popular among cosmologists, as they can describe the cosmological evolution quite efficiently. Reconstruction of newly introduced $f(P)$ gravity, with the help of ordinary, power-law entropy corrected and logarithmic entropy-corrected versions of Holographic dark energy (HDE) and Pilgrim dark energy (PDE) models have been studied in this work. For such reconstruction, we have considered the power-law scale factor $a(t)=a_0{t}^k,k>1$. Further, the classical stabilities (the squared speed of sound method) of such reconstructions and their implications on the nature of the equation of state (EoS) parameters and deceleration parameter with respect to red-shift have also been examined. Finally, we have computed the age of the universe for reconstructed models.

In the background of homogeneous and isotropic flat FLRW model, a complete cosmic scenario from nonsingular emergent scenario to the present late time acceleration through inflationary era and matter-dominated epoch has been presented in this work with cosmic matter in the form of diffusive barotropic fluid. By proper choices of the diffusion parameter and using Friedmann equations, it is possible to show the transitions: Emergent scenario$\to$Inflationary era$\to$matter-dominated phase$\to$Late time acceleration epoch. In analogy with analytic continuation, it is found that the above evolution will be continuous for suitable values of the parameters involved. Finally, possible first-order thermodynamic phase transition has been analyzed for such cosmic evolution.

We propose a new class of vector fields to construct a conserved charge in a general field theory whose energy–momentum tensor is covariantly conserved. We show that there always exists such a vector field in a given field theory even without global symmetry. We also argue that the conserved current constructed from the (asymptotically) timelike vector field can be identified with the entropy current of the system. As a piece of evidence we show that the conserved charge defined therefrom satisfies the first law of thermodynamics for an isotropic system with a suitable definition of temperature. We apply our formulation to several gravitational systems such as the expanding universe, Schwarzschild and Banãdos, Teitelboim and Zanelli (BTZ) black holes, and gravitational plane waves. We confirm the conservation of the proposed entropy density under any homogeneous and isotropic expansion of the universe, the precise reproduction of the Bekenstein–Hawking entropy incorporating the first law of thermodynamics, and the existence of gravitational plane wave carrying no charge, respectively. We also comment on the energy conservation during gravitational collapse in simple models.

The effects of uniform rotation on a relativistic scalar particle that interacts with a Cornell-type potential in background space–time described by the Kaluza–Klein theory are analyzed and the gravitational analogue of the Aharonov–Bohm effect is observed. Furthermore, linear confinement of a relativistic scalar particle was also discussed. We see a coupling between the angular velocity of the rotating frame $\omega$ and the angular momentum eigenvalue $l$ which shows the Sagnac-type effect.

In the context of precise measurements in some of the CKM elements and angles of the unitarity triangle coupled with latest theoretical improvements of lattice QCD calculations of “hadronic factors,” we have explored the implication of these and unitarity on some of the CKM parameters which are ridden with ambiguities. In particular, we have undertaken a thorough scrutiny of the implications of the precise measurements of the angles of the unitarity triangle along with the element $\left|V_{us}\right|$, on the $\left|V_{ub}\right|$, $\left|V_{td}\right|$, $\left|V_{ts}\right|$, as well as the CP violating parameter ${𝜖}_K$. Interestingly, we find a unitarity-based analysis while unambiguously favoring $\left|V_{ub}\right|$ exclusive-over-inclusive value which is also in agreement with the present data pertaining to ${𝜖}_K$, $\left|V_{td}\right|$, $\left|V_{ts}\right|$, $\mathrm{\Delta }{M}_d/\mathrm{\Delta }{M}_s$, etc. The present analysis also shows that the presence of NP in $K-\bar{K}$ and $B_d-\overline{B_d}$ systems, if any, would be less than a few percent only.

In this paper, we construct the foliation of a space associated to correlation functions of vertex operator algebras, considered on Riemann surfaces. We prove that the computation of general genus $g$ correlation functions determines a foliation on the space associated to these correlation functions a sewn Riemann surface. Certain further applications of the definition are proposed.

We discuss the nilpotent Becchi–Rouet–Stora–Tyutin (BRST), anti-BRST and (anti-) co-BRST symmetry transformations and derive their corresponding conserved charges in the case of a two $(1+1)$-dimensional (2D) self-interacting non-Abelian gauge theory (without any interaction with matter fields). We point out a set of novel features that emerge out in the BRST and co-BRST analysis of the above 2D gauge theory. The algebraic structures of the symmetry operators (and corresponding conserved charges) and their relationship with the cohomological operators of differential geometry are established too. To be more precise, we demonstrate the existence of a single Lagrangian density that respects the continuous symmetries which obey proper algebraic structure of the cohomological operators of differential geometry. In the literature, such observations have been made for the coupled (but equivalent) Lagrangian densities of the 4D non-Abelian gauge theory. We lay emphasis on the existence and properties of the Curci–Ferrari (CF)-type restrictions in the context of (anti-)BRST and (anti-)co-BRST symmetry transformations and pinpoint their key differences and similarities. All the observations, connected with the (anti-)co-BRST symmetries, are completely novel.

Witten–Sakai–Sugimoto model is used to study Yang–Mills theory with flavors and large number of colors at finite temperature and in the presence of chemical potential for baryon number and isospin. Sources for $U{(1)}_B$ and $U{(1)}_3$ gauge fields on the flavor 8-branes are D4-branes wrapped on $S^4$ part of the background. Here, gauge symmetry on the flavor branes has been decomposed as $U(2)\equiv U{(1)}_B\times SU(2)$ and $U{(1)}_3$ is within $SU(2)$ and generated by the diagonal generator. We show various brane configurations, along with the phases in the boundary theory they correspond to, and explore the possibility of phase transition between various pairs of phases.

In this work, the Dirac–Kratzer problem with spin and pseudo-spin symmetries in a deformed nucleus is analyzed. Thus, the Dirac equation in curved space–time was considered, with a line element given by $d{s}^2={(1+{\alpha }^{2}U(r))}^2(d{t}^2-d{r}^2)-r^{2}d{𝜃}^2-r^2{sin}^2𝜃d{\varphi }^2$, where $U(r)$ is a scalar potential, coupled to vector $V(r)$ and tensor $A(r)$ potentials. Defining the vector and scalar potentials of the Kratzer type and the tensor potential given by a term centrifugal-type term plus a term cubic singular at the origin, we obtain the Dirac spinor in a quasi-exact way and the eigenenergies numerically for the spin and pseudo-spin symmetries, so that these symmetries are removed due to the coupling of an Coulomb-type effective tensor potential coming from the curvature of space, however, when such potential is null the symmetries return. The probability densities were analyzed using graphs to compare the behavior of the system with and without spin and pseudo-spin symmetries.

Using a lattice equation of state combined with the D-dimensional Tolman–Oppenheimer–Volkoff equation and the Friedmann equations, we investigate the possibility of the formation of compact objects as well as the time evolution of the scale factor and the density profile of a self-gravitating material cluster. The numerical results show that in a ($2+1$)-dimensional space–time, the mass is independent of the central pressure. Hence, the formation of only compact objects with a finite constant mass similar to the white dwarf is possible. However, in a ($3+1$)-dimensional space–time, self-gravity leads to the formation of compact objects with a large gap of mass and the corresponding phase diagram has the same structure as the one for Neutron Star. The results also show that beyond certain critical central pressure, the star is unstable against gravitational collapse, and it may end in a black hole. Analysis of space–times of higher dimensions shows that gravity has the stronger effect in $3+1$ dimensions. Numerical solutions of the Friedmann equations show that the effect of the curvature of space–time increases with the increasing temperature, but decreases with the increasing dimensionality beyond $D=3$.

In this paper, we study real-time classical matrix mechanics of a simplified $2\times 2$ matrix model inspired by the black hole evaporation problem. This is a step towards making a quantitative model of real-time evaporation of a black hole, which is realized as a bound state of D0-branes in string theory. The model we study is the reduction of Yang–Mills in $2+1$ dimension to $0+1$ dimensions, which has been corrected with an additional potential that can be interpreted as a zero-point energy for fermions. Our goal is to understand the lifetime of such a classical bound state object in the classical regime. To do so, we pay particular attention to when $D$-particles separate to check that the “off-diagonal modes” of the matrices become adiabatic and use that information to improve on existing models of evaporation. It turns out that the naive expectation value of the lifetime with the fermionic correction is infinite. This is a logarithmic divergence that arises from very large excursions in the separation between the branes near the threshold for classical evaporation. The adiabatic behavior lets us get some analytic control of the dynamics in this regime to get this estimate. This divergence is cutoff in the quantum theory due to quantization of the adiabatic parameter, resulting in a long lifetime of the bound state, with a parametric dependence of order $log(1/\hslash )$.

In this paper, we apply decomposition to orbifolds with quantum symmetries to resolve anomalies. Briefly, it has been argued by, e.g. Wang–Wen–Witten, Tachikawa that an anomalous orbifold can sometimes be resolved by enlarging the orbifold group so that the pullback of the anomaly to the larger orbifold group is trivial. For this procedure to resolve the anomaly, one must specify a set of phases in the larger orbifold, whose form is implicit in the extension construction. There are multiple choices of consistent phases, which give rise to physically distinct resolutions. We apply decomposition, and find that theories with enlarged orbifold groups are equivalent to (disjoint unions of copies of) orbifolds by nonanomalous subgroups of the original orbifold group. In effect, decomposition implies that enlarging the orbifold group is equivalent to making it smaller. We provide a general conjecture for such descriptions, which we check in a number of examples.

For one-dimensional power-like potentials $|x{|}^m$, $m>0$, the Bohr–Sommerfeld energies (BSE) extracted explicitly from the Bohr–Sommerfeld quantization condition are compared with the exact energies. It is shown that for the ground state as well as for all positive parity states the BSE are always above the exact ones as opposed to the negative parity states where the BSE remain above the exact ones for $m>2$ but below them for $m<2$. The ground state BSE as function of $m$ are of the same order of magnitude as the exact energies for linear $(m=1)$, quartic $(m=4)$ and sextic $(m=6)$ oscillators but their relative deviation grows with $m$, reaching the value 4 at $m=\infty$. For physically important cases $m=1,4,6$, for the 100th excited state BSE coincide with exact ones in 5–6 figures.

It is demonstrated that by modifying the right-hand side of the Bohr–Sommerfeld quantization condition by introducing the so-called WKB correction $\gamma$ (coming from the sum of higher-order WKB terms taken at the exact energies or from the accurate boundary condition at turning points) to the so-called exact WKB condition one can reproduce the exact energies. It is shown that the WKB correction is a small, bounded function $\left|\gamma \right|<1/2$ for all $m\ge 1$. It grows slowly with increasing $m$ for fixed quantum number $N$, while it decays with quantum number growth at fixed $m$. It is the first time when for quartic and sextic oscillators the WKB correction and energy spectra (and eigenfunctions) are found in explicit analytic form with a relative accuracy of $10^{-9}\text{–}10^{-11}$ (and $10^{-6}$).

In this paper, to understand space–time dynamics in the canonical tensor model of quantum gravity for the positive cosmological constant case, we analytically and numerically study the phase profile of its exact wave function in a coordinate representation, instead of the momentum representation analyzed so far. A saddle point analysis shows that Lie group symmetric space–times are strongly favored due to abundance of continuously existing saddle points, giving an emergent fluid picture. The phase profile suggests that spatial sizes grow in “time,” where sizes are measured by the tensor-geometry correspondence previously introduced using tensor rank decomposition. Monte Carlo simulations are also performed for a few small $N$ cases by applying a re-weighting procedure to an oscillatory integral which expresses the wave function. The results agree well with the saddle point analysis, but the phase profile is subject to disturbances in a large space–time region, suggesting existence of light modes there and motivating future computations of primordial fluctuations from the perspective of canonical tensor model.

Using the approach based on Bogoliubov compensation principle is applied to the calculation of a contribution to the muon $g-2$. Using the previous results on spontaneous generation of the effective anomalous three-boson interaction we calculate the contribution, which proves to agree with the well-known discrepancy. The calculated quantity contains no adjusting parameters but the experimental values for the muon and the $W$-boson masses. The result can be considered as a confirmation of the approach.