A new large N phase transition in YM2

Inspired by the interpretation of two-dimensional Yang–Mills theory on a cylinder as a random walk on the gauge group, we point out the existence of a large N transition which is the gauge theory analogue of the cutoff transition in random walks. The transition occurs in the strong coupling region, with the 't Hooft coupling scaling as $\alpha \log N$, at a critical value of α ($\alpha =4$ on the sphere). The two phases below and above the transition are studied in detail. The effective number of degrees of freedom and the free energy are found to be proportional to $N^{2-\frac{\alpha }{2}}$ below the transition and to vanish altogether above it. The expectation value of a Wilson loop is calculated to the leading order and found to coincide in both phases with the strong coupling value.     

Klein–Gordon equation for a charged particle in space-varying electromagnetic fields–A systematic study via the Laplace transform

Exact solutions of the Klein–Gordon equation for a charged particle in the presence of three spatially varying electromagnetic fields, namely, (i) $\overrightarrow{E}=\alpha {\beta }_0{e}^{?\alpha x_2}{\widehat{x}}_2,$$\overrightarrow{B}=\alpha {\beta }_1{e}^{?\alpha x_2}{\widehat{x}}_3$ (ii) $\overrightarrow{E}=\frac{{\beta }_0^{{}^{\prime }}}{x_2^2}{\widehat{x}}_2,$$\overrightarrow{B}=\frac{{\beta }_1^{{}^{\prime }}}{x_2^2}{\widehat{x}}_3,$ and (iii) $\overrightarrow{E}=\frac{2{\beta }_0^{{}^{\prime }}}{x_2^3}{\widehat{x}}_2,$$\overrightarrow{B}=\frac{2{\beta }_1^{{}^{\prime }}}{x_2^3}{\widehat{x}}_3,$ are studied. All these fields are generated from a systematic study of a particular type of differential equation whose coefficients are linear in the independent variable. The Laplace transform approach is used to find the solutions, and the corresponding eigenfunctions are expressed in terms of the hypergeometric functions ?₁F₁(a′, b′; x) for the first two cases of the above configurations, while the same are expressed in terms of the Bessel functions of first kind, J_n(x), for the last case.     

The theory of C-axis elastic moduli of graphite and their strain dependence using simple models of the interplanar interaction

Theoretical values for the c-axis elastic moduli C₃₃, C₄₄ and their strain derivatives $\frac{1}{C_{33}}\frac{?C_{33}}{?{\in }_3}$, $\frac{1}{C_{44}}\frac{?C_{44}}{?{\in }_3}$ in hexagonal graphite have been computed as a function of c-axis strain using the Lennard-Jones and the exponential core pairwise interplanar potentials. Fair agreement with experiment was found in the compressive behavior for the Lennard-Jones potential, good aggrement for the exponential core potential. The largest measured shear moduli in graphite, probably representative of the ideal hexagonal material, are larger than the theoretical shear moduli by a factor of at least ten. The theoretical shear c-axis strain derivatives are in somewhat better agreement. The notable discrepancy between elementary interplanar potential theory and experiment for the c-axis shear modulus in graphite appears to be an intrinsic flaw of such theories.     

Supersymmetric extensions of Schrödinger-invariance

The set of dynamic symmetries of the scalar free Schrödinger equation in d space dimensions gives a realization of the Schrödinger algebra that may be extended into a representation of the conformal algebra in $d+2$ dimensions, which yields the set of dynamic symmetries of the same equation where the mass is not viewed as a constant, but as an additional coordinate. An analogous construction also holds for the spin-$\frac{1}{2}$ Lévy-Leblond equation. An $N=2$ supersymmetric extension of these equations leads, respectively, to a ‘super-Schrödinger’ model and to the $(3|2)$-supersymmetric model. Their dynamic supersymmetries form the Lie superalgebras $\mathfrak{osp}(2|2)⋉\mathfrak{sh}(2|2)$ and $\mathfrak{osp}(2|4)$, respectively. The Schrödinger algebra and its supersymmetric counterparts are found to be the largest finite-dimensional Lie subalgebras of a family of infinite-dimensional Lie superalgebras that are systematically constructed in a Poisson algebra setting, including the Schrödinger–Neveu–Schwarz algebra ${\mathfrak{sns}}^{(N)}$ with N supercharges. Covariant two-point functions of quasiprimary superfields are calculated for several subalgebras of $\mathfrak{osp}(2|4)$. If one includes both $N=2$ supercharges and time-inversions, then the sum of the scaling dimensions is restricted to a finite set of possible values.