On the complete integrability of the discrete Nahm equations

The discrete Nahm equations, a system of matrix valued difference equations, arose in the work of Braam and Austin on half-integral mass hyperbolic monopoles. We show that the discrete Nahm equations are completely integrable in a natural sense: to any solution we can associate a spectral curve and a holomorphic line-bundle over the spectral curve, such that the discrete-time DN evolution corresponds to walking in the Jacobian of the spectral curve in a straight line through the line-bundle with steps of a fixed size. Some of the implications for hyperbolic monopoles are also discussed

The performance of Seventh District food processing

Federal Reserve District, 7th ; Food industry and trade

An L^2-Index Theorem for Dirac Operators on S^1 * R^3

An expression is found for the $L^2$-index of a Dirac operator coupled to a connection on a $U_n$ vector bundle over $S^1\times{\mathbb R}^3$. Boundary conditions for the connection are given which ensure the coupled Dirac operator is Fredholm. Callias' index theorem is used to calculate the index when the connection is independent of the coordinate on $S^1$. An excision theorem due to Gromov, Lawson, and Anghel reduces the index theorem to this special case. The index formula can be expressed using the adiabatic limit of the $\eta$-invariant of a Dirac operator canonically associated to the boundary. An application of the theorem is to count the zero modes of the Dirac operator in the background of a caloron (periodic instanton).Comment: 14 pages, Latex, to appear in the Journal of Functional Analysi

Positive Einstein metrics with small Ln/2-norm of the Weyl tensor

AbstractA gravitational analogue is given of Min-Oo's gap theorem for Yang-Mills fields

Analytic approximations to the phase diagram of the Jaynes-Cummings-Hubbard model with application to ion chains

We discuss analytic approximations to the ground state phase diagram of the homogeneous Jaynes-Cummings-Hubbard (JCH) Hamiltonian with general short-range hopping. The JCH model describes e.g. radial phonon excitations of a linear chain of ions coupled to an external laser field tuned to the red motional sideband with Coulomb mediated hopping or an array of high-$Q$ coupled cavities containing a two-level atom and photons. Specifically we consider the cases of a linear array of coupled cavities and a linear ion chain. We derive approximate analytic expressions for the boundaries between Mott-insulating and superfluid phases and give explicit expressions for the critical value of the hopping amplitude within the different approximation schemes. In the case of an array of cavities, which is represented by the standard JCH model we compare both approximations to numerical data from density-matrix renormalization group (DMRG) calculations.Comment: 9 pages, 5 figures, extended and corrected second versio