Zero curvature conditions and conformal covariance

Two-dimensional zero curvature conditions with special emphasis on conformal properties are investigated in detail and the appearance of covariant higher order differential operators constructed in terms of a projective connection is elucidated. The analysis is based on the Kostant decomposition of simple Lie algebras in terms of representations with respect to their ``principal'' SL(2) subalgebra. Journal of Mathematical Physics is copyrighted by The American Institute of Physics

The GENGA Code: Gravitational Encounters in N-body simulations with GPU Acceleration

We describe an open source GPU implementation of a hybrid symplectic N-body integrator, GENGA (Gravitational ENcounters with Gpu Acceleration), designed to integrate planet and planetesimal dynamics in the late stage of planet formation and stability analyses of planetary systems. GENGA uses a hybrid symplectic integrator to handle close encounters with very good energy conservation, which is essential in long-term planetary system integration. We extended the second order hybrid integration scheme to higher orders. The GENGA code supports three simulation modes: Integration of up to 2048 massive bodies, integration with up to a million test particles, or parallel integration of a large number of individual planetary systems. We compare the results of GENGA to Mercury and pkdgrav2 in respect of energy conservation and performance, and find that the energy conservation of GENGA is comparable to Mercury and around two orders of magnitude better than pkdgrav2. GENGA runs up to 30 times faster than Mercury and up to eight times faster than pkdgrav2. GENGA is written in CUDA C and runs on all NVIDIA GPUs with compute capability of at least 2.0.Comment: Accepted by ApJ. 18 pages, 17 figures, 4 table

Spectrum of a duality-twisted Ising quantum chain

The Ising quantum chain with a peculiar twisted boundary condition is considered. This boundary condition, first introduced in the framework of the spin-1/2 XXZ Heisenberg quantum chain, is related to the duality transformation, which becomes a symmetry of the model at the critical point. Thus, at the critical point, the Ising quantum chain with the duality-twisted boundary is translationally invariant, similar as in the case of the usual periodic or antiperiodic boundary conditions. The complete energy spectrum of the Ising quantum chain is calculated analytically for finite systems, and the conformal properties of the scaling limit are investigated. This provides an explicit example of a conformal twisted boundary condition and a corresponding generalised twisted partition function.Comment: LaTeX, 7 pages, using IOP style

The effective action of warped M-theory reductions with higher-derivative terms - Part II

We study the three-dimensional effective action obtained by reducing eleven-dimensional supergravity with higher-derivative terms on a background solution including a warp-factor, an eight-dimensional compact manifold, and fluxes. The dynamical fields are K\"ahler deformations and vectors from the M-theory three-form. We show that the potential is only induced by fluxes and the naive contributions obtained from higher-curvature terms on a Calabi-Yau background vanish once the back-reaction to the full solution is taken into account. For the resulting three-dimensional action we analyse the K\"ahler potential and complex coordinates and show compatibility with N=2 supersymmetry. We argue that the higher-order result is also compatible with a no-scale condition. We find that the complex coordinates should be formulated as divisor integrals for which a non-trivial interplay between the warp-factor terms and the higher-curvature terms allow a derivation of the moduli space metric. This leads us to discuss higher-derivative corrections to the M5-brane action.Comment: 26 page

On M-theory fourfold vacua with higher curvature terms

We study solutions to the eleven-dimensional supergravity action, including terms quartic and cubic in the Riemann curvature, that admit an eight-dimensional compact space. The internal background is found to be a conformally Kahler manifold with vanishing first Chern class. The metric solution, however, is non-Ricci-flat even when allowing for a conformal rescaling including the warp factor. This deviation is due to the possible non-harmonicity of the third Chern-form in the leading order Ricci-flat metric. We present a systematic derivation of the background solution by solving the Killing spinor conditions including higher curvature terms. These are translated into first-order differential equations for a globally defined real two-form and complex four-form on the fourfold. We comment on the supersymmetry properties of the described solutions.Comment: 14 page

Non-Supersymmetric F-Theory Compactifications on Spin(7) Manifolds

We propose a novel approach to obtain non-supersymmetric four-dimensional effective actions by considering F-theory on manifolds with special holonomy Spin(7). To perform such studies we suggest that a duality relating M-theory on a certain class of Spin(7) manifolds with F-theory on the same manifolds times an interval exists. The Spin(7) geometries under consideration are constructed as quotients of elliptically fibered Calabi-Yau fourfolds by an anti-holomorphic and isometric involution. The three-dimensional minimally supersymmetric effective action of M-theory on a general Spin(7) manifold with fluxes is determined and specialized to the aforementioned geometries. This effective theory is compared with an interval Kaluza-Klein reduction of a non-supersymmetric four-dimensional theory with definite boundary conditions for all fields. Using this strategy a minimal set of couplings of the four-dimensional low-energy effective actions is obtained in terms of the Spin(7) geometric data. We also discuss briefly the string interpretation in the Type IIB weak coupling limit.Comment: 39 pages, 4 figures, v2: improvements and clarifications on the 4d interpretation and weak coupling limit; typos correcte

Low Price Equilibrium in Multi-Unit Auctions: The GSM Spectrum Auction in Germany

The second-generation GSM spectrum auction in Germany is probably the most clear cut example of a low price outcome in a simultaneous ascending-bid auction.The present paper gives an account of the events, describes the auction rules and market conditions, and provides a theoretical explanation of low price equilibria in simultaneous, ascending-bid auctions.In particular it is shown that the low price equilibrium that implements the efficient allocation is the unique perfect equilibrium of that game.Multi-unit auctions, spectrum auctions, telecomm-unications, industrial organization, game theory

Implementing Efficient Market Structure

This article studies the design of optimal mechanisms to regulate entry in natural oligopoly markets, assuming the regulator is unable to control the behavior of firms once they are in the market. We adapt the Clark-Groves mechanism, characterize the optimal mechanism that maximizes the weighted sum of expected social surplus and expected tax revenue, and show that these mechanisms generally avoid budget deficits and prevent excessive entry.Mechanism design, natural oligopoly, auctions, entry