International Journal of Modern Physics A

This paper summarizes recent developments in the theory of Bogomol'nyi–Prasad–Sommerfield (BPS) state counting and the wall crossing phenomena, emphasizing in particular the role of the statistical mechanical model of crystal melting.

This paper is divided into two parts, which are closely related to each other. In the first part, we discuss the statistical mechanical model of crystal melting counting BPS states. Each of the BPS states contributing to the BPS index is in one-to-one correspondence with a configuration of a molten crystal, and the statistical partition function of the melting crystal gives the BPS partition function. We also show that smooth geometry of the Calabi–Yau manifold emerges in the thermodynamic limit of the crystal. This suggests a remarkable interpretation that an atom in the crystal is a discretization of the classical geometry, giving an important clue as such to the geometry at the Planck scale.

In the second part, we discuss the wall crossing phenomena. Wall crossing phenomena states that the BPS index depends on the value of the moduli of the Calabi–Yau manifold, and jumps along real codimension one subspaces in the moduli space. We show that by using type IIA/M-theory duality, we can provide a simple and an intuitive derivation of the wall crossing phenomena, furthermore clarifying the connection with the topological string theory. This derivation is consistent with another derivation from the wall crossing formula, motivated by multicentered BPS extremal black holes. We also explain the representation of the wall crossing phenomena in terms of crystal melting, and the generalization of the counting problem and the wall crossing to the open BPS invariants.

Recently, Kazakov, Gromov and Vieira applied the discrete Hirota dynamics to study the finite size spectra of integrable two dimensional quantum field theories. The method has been tested from large values of the size L down to moderate values using the SU(2) × SU(2) principal chiral model as a theoretical laboratory. We continue the numerical analysis of the proposed nonlinear integral equations showing that the deep ultraviolet region L → 0 is numerically accessible. To this aim, we introduce a relaxed iterative algorithm for the numerical computation of the low-lying part of the spectrum in the U(1) sector. We discuss in detail the systematic errors involved in the computation. When a comparison is possible, full agreement is found with previous thermodynamical Bethe ansatz computations.

According to the helicity conservation (HCns) theorem, the sum of the helicities should be conserved, in any 2-to-2 processes in MSSM with R-parity conservation, at high energies, i.e. all amplitudes violating this rule must vanish asymptotically. The realization of HCns in , is studied, at the one-loop electroweak order, and simple high energy expressions are derived for the nonvanishing helicity conserving (HC) amplitudes. These are very similar to the corresponding expressions for gg → W⁺W^-, ZZ, γZ, γγ derived before. Asymptotic relations among observable unpolarized cross-sections for many such processes are then obtained, some of which may hold at LHC-type energies.

Motivated by the measured large branching ratio of (the so-called ππ puzzle), we investigate the effects of a family nonuniversal Z′ model on the tree-dominated B → ππ decays. We find that the Z′ coupling parameter with a nontrivial new weak phase , which is relevant to the Z′ contributions to the QCD penguin sector △C₅, is needed to reconcile the observed discrepancy. Combined with the recent fitting results from B → πK, πK* and ρK decays, the Z′ parameter spaces are severely reduced but still not excluded entirely, implying that both the "ππ" and "πK" puzzles could be accommodated simultaneously within such a family nonuniversal Z′ model.

Quark interactions inside baryons depend on the spin states of the quarks. We investigated the contribution of the spin–spin interactions of quarks in comparison with other factors necessary for diquark formation in baryons, concluding that the effects of spin–spin interactions are correlated with the state of angular momentum excitation.

We study various possible Supersymmetric Left–Right (SUSYLR) models with Higgs doublets carrying B-L charge ±1: with single step symmetry breaking down to the Minimal Supersymmetric Standard Model (MSSM) as well as multistep symmetry breaking. Single step symmetry breaking can be achieved with the minimal field content of just Higgs doublet and bidoublets whereas multistep symmetry breaking can be realized only at the cost of including additional Higgs superfields. However, going beyond the minimal field content comes up with the exciting possibility of TeV scale intermediate symmetry which can have important implications in the ongoing collider experiments. We show that spontaneous parity violation can be achieved naturally in all these models and R-parity is spontaneously broken by the vacuum expectation value of B-L odd Higgs doublets. The tiny neutrino mass can arise from a double seesaw mechanism in the presence of additional singlet or triplet fermions. We show that gauge coupling unification can be achieved in these models with the possibility of TeV scale intermediate symmetry in some specific nonminimal versions.

A numerical algorithm is used to solve the bare and the effective potential for the scalar ϕ⁴ model in the local potential approximation. An approximate dynamical Maxwell-cut is found which reveals itself in the degeneracy of the action for modes at some scale. This result indicates that the potential develop singular field dependence as far as one can see it by an algorithm of limited numerical accuracy.

Any local field theory can be equivalently reformulated in the so-called unfolded form. General unfolded equations are non-Lagrangian even though the original theory is Lagrangian. Making use of the unfolded massless scalar field equations as a basic example, the concept of Lagrange anchor is applied to perform a consistent path-integral quantization of unfolded dynamics. It is shown that the unfolded representation for the canonical Lagrange anchor of the d'Alembert equation inevitably involves an infinite number of space–time derivatives.

In this paper, we present exact solutions of the Dirac equation with the pseudoharmonic potential including linear as well as Coulomb-like tensor potential with arbitrary spin–orbit coupling number κ under spin and pseudospin symmetry limits. The Nikiforov–Uvarov method is used to obtain energy eigenvalues and corresponding eigenfunctions in closed forms. We show that tensor interaction removes degeneracies between spin and pseudospin doublets. Some numerical results are also given.

In this paper, we show that in some important cases, four-dimensional data can be extracted from superstring theory such that (i) the data are Euclidean four-geometries embedded in standard ℝ⁴, (ii) these data depend on NS and D brane charges of some string backgrounds, (iii) it is of potential relevance to four-dimensional physics, (iv) the compactification and stabilization techniques are not in use, but are rather replaced. We analyze certain configurations of NS and D-branes in the context of SU(2) WZW model and find the correlations with different exotic smoothings of ℝ⁴. First, the dynamics of D-branes in SU(2) WZW model at finite k, i.e. the charges of the branes, refers to the exoticness of ambient ℝ⁴. Next, the correspondence between exotic smoothness on four-space, transversal to the world volume of NS5 branes in type IIA, and the number of these NS5 branes follows. Finally, the translation of ten-dimensional string backgrounds to Euclidean four-spaces embedded as open subsets in the standard ℝ⁴ is achieved.

In the present paper, we show that for a low frequency limit the wave equation of massless scalar field in the background of rotating NS5-branes can be written as the Casimir of SL(2, R) symmetry. Our result shows that the entropy of the rotating NS5-brane is reproduced by the Cardy formula, also the absorption cross-section is in consistence with the finite temperature absorption cross-section for a 2D CFT.

It is shown that the E₈ trigonometric Olshanetsky–Perelomov Hamiltonian, when written in terms of the fundamental trigonometric invariants, is in algebraic form, i.e. it has polynomial coefficients, and preserves two infinite flags of polynomial spaces marked by the Weyl (co)-vector and E₈ highest root (both in the basis of simple roots) as characteristic vectors. The explicit form of the Hamiltonian in new variables has been obtained both by direct calculation and by means of the orbit function technique. It is shown the triangularity of the Hamiltonian in the bases of orbit functions and of algebraic monomials ordered through Weyl heights. Examples of first eigenfunctions are presented.

We investigate the properties of κ-Minkowski space–time by using representations of the corresponding deformed algebra in terms of undeformed Heisenberg–Weyl algebra. The deformed algebra consists of κ-Poincaré algebra extended with the generators of the deformed Weyl algebra. The part of deformed algebra, generated by rotation, boost and momentum generators, is described by the Hopf algebra structure. The approach used in our considerations is completely Lorentz covariant. We further use an advantage of this approach to consistently construct a star product, which has a property that under integration sign, it can be replaced by a standard pointwise multiplication, a property that was since known to hold for Moyal but not for κ-Minkowski space–time. This star product also has generalized trace and cyclic properties, and the construction alone is accomplished by considering a classical Dirac operator representation of deformed algebra and requiring it to be Hermitian. We find that the obtained star product is not translationally invariant, leading to a conclusion that the classical Dirac operator representation is the one where translation invariance cannot simultaneously be implemented along with hermiticity. However, due to the integral property satisfied by the star product, noncommutative free scalar field theory does not have a problem with translation symmetry breaking and can be shown to reduce to an ordinary free scalar field theory without nonlocal features and tachyonic modes and basically of the very same form. The issue of Lorentz invariance of the theory is also discussed.