Flow Equations for Uplifting Half-Flat to Spin(7) Manifolds

In this short supplement to [1], we discuss the uplift of half-flat six-folds to Spin(7) eight-folds by fibration of the former over a product of two intervals. We show that the same can be done in two ways - one, such that the required Spin(7) eight-fold is a double G_2 seven-fold fibration over an interval, the G_2 seven-fold itself being the half-flat six-fold fibered over the other interval, and second, by simply considering the fibration of the half-flat six-fold over a product of two intervals. The flow equations one gets are an obvious generalization of the Hitchin's flow equations (to obtain seven-folds of G_2 holonomy from half-flat six-folds [2]). We explicitly show the uplift of the Iwasawa using both methods, thereby proposing the form of the new Spin(7) metrics. We give a plausibility argument ruling out the uplift of the Iwasawa manifold to a Spin(7) eight fold at the "edge", using the second method. For $Spin(7)$ eight-folds of the type $X_7\times S^1$, $X_7$ being a seven-fold of SU(3) structure, we motivate the possibility of including elliptic functions into the "shape deformation" functions of seven-folds of SU(3) structure of [1] via some connections between elliptic functions, the Heisenberg group, theta functions, the already known $D7$-brane metric [3] and hyper-K\"{a}hler metrics obtained in twistor spaces by deformations of Atiyah-Hitchin manifolds by a Legendre transform in [4].Comment: 12 pages, LaTeX; v3: (JMP) journal version which includes clarifying remarks related to connection between Spin(7)-folds and SU(3)structur

Static, massive fields and vacuum polarization potential in Rindler space

In Rindler space, we determine in terms of special functions the expression of the static, massive scalar or vector field generated by a point source. We find also an explicit integral expression of the induced electrostatic potential resulting from the vacuum polarization due to an electric charge at rest in the Rindler coordinates. For a weak acceleration, we give then an approximate expression in the Fermi coordinates associated with the uniformly accelerated observer.Comment: 11 pages, latex, no figure

A precise description of the p-adic valuation of the number of alternating sign matrices

Following Sun and Moll, we study v_p(T(N)), the p-adic valuation of the counting function of the alternating sign matrices. We find an exact analytic expression for it that exhibits the fluctuating behaviour, by means of Fourier coefficients. The method is the Mellin-Perron technique, which is familiar in the analysis of the sum-of-digits function and related quantities

Nonexistence of an integral of the 6th degree in momenta for the Zipoy-Voorhees metric

We prove nonexistence of a nontrivial integral that is polynomial in momenta of degree less than 7 for the Zipoy-Voorhees spacetime with the parameter $\delta=2$Comment: 7 pages, no figure

PT-Symmetric Sinusoidal Optical Lattices at the Symmetry-Breaking Threshold

The $PT$ symmetric potential $V_0[\cos(2\pi x/a)+i\lambda\sin(2\pi x/a)]$ has a completely real spectrum for $\lambda\le 1$, and begins to develop complex eigenvalues for $\lambda>1$. At the symmetry-breaking threshold $\lambda=1$ some of the eigenvectors become degenerate, giving rise to a Jordan-block structure for each degenerate eigenvector. In general this is expected to result in a secular growth in the amplitude of the wave. However, it has been shown in a recent paper by Longhi, by numerical simulation and by the use of perturbation theory, that for a broad initial wave packet this growth is suppressed, and instead a saturation leading to a constant maximum amplitude is observed. We revisit this problem by explicitly constructing the Bloch wave-functions and the associated Jordan functions and using the method of stationary states to find the dependence on the longitudinal distance $z$ for a variety of different initial wave packets. This allows us to show in detail how the saturation of the linear growth arises from the close connection between the contributions of the Jordan functions and those of the neighbouring Bloch waves.Comment: 15 pages, 7 figures Minor corrections, additional reference

Adiabatic-Impulse approximation for avoided level crossings: from phase transition dynamics to Landau-Zener evolutions and back again

We show that a simple approximation based on concepts underlying the Kibble-Zurek theory of second order phase transition dynamics can be used to treat avoided level crossing problems. The approach discussed in this paper provides an intuitive insight into quantum dynamics of two level systems, and may serve as a link between the theory of dynamics of classical and quantum phase transitions. To illustrate these ideas we analyze dynamics of a paramagnet-ferromagnet quantum phase transition in the Ising model. We also present exact unpublished solutions of the Landau-Zener like problems.Comment: 12 pages & 6 figures, minor corrections, version accepted in Phys. Rev.

On a generalization of Jacobi's elliptic functions and the Double Sine-Gordon kink chain

A generalization of Jacobi's elliptic functions is introduced as inversions of hyperelliptic integrals. We discuss the special properties of these functions, present addition theorems and give a list of indefinite integrals. As a physical application we show that periodic kink solutions (kink chains) of the double sine-Gordon model can be described in a canonical form in terms of generalized Jacobi functions.Comment: 18 pages, 9 figures, 3 table

Exponential beams of electromagnetic radiation

We show that in addition to well known Bessel, Hermite-Gauss, and Laguerre-Gauss beams of electromagnetic radiation, one may also construct exponential beams. These beams are characterized by a fall-off in the transverse direction described by an exponential function of rho. Exponential beams, like Bessel beams, carry definite angular momentum and are periodic along the direction of propagation, but unlike Bessel beams they have a finite energy per unit beam length. The analysis of these beams is greatly simplified by an extensive use of the Riemann-Silberstein vector and the Whittaker representation of the solutions of the Maxwell equations in terms of just one complex function. The connection between the Bessel beams and the exponential beams is made explicit by constructing the exponential beams as wave packets of Bessel beams.Comment: Dedicated to the memory of Edwin Powe

Harmonic Maa{\ss}-Jacobi forms of degree 1 with higher rank indices

We define and investigate real analytic weak Jacobi forms of degree 1 and arbitrary rank. En route we calculate the Casimir operator associated to the maximal central extension of the real Jacobi group, which for rank exceeding 1 is of order 4. In ranks exceeding 1, the notions of H-harmonicity and semi-holomorphicity are the same.Comment: 28 page