Non-commutative tori and Fourier-Mukai duality

The classical Fourier-Mukai duality establishes an equivalence of categories between the derived categories of sheaves on dual complex tori. In this article we show that this equivalence extends to an equivalence between two dual objects. Both of these are generalized deformations of the complex tori. In one case, a complex torus is deformed formally in a non-commutative direction specified by a holomorphic Poisson structure. In the other, the dual complex torus is deformed in a B-field direction to a formal gerbe. We show these two deformations are Fourier-Mukai equivalent.Comment: 80 pages, LaTeX2

R\'enyi entanglement entropy of critical SU(NN) spin chains

We present a study of the scaling behavior of the R\'{e}nyi entanglement entropy (REE) in SU($N$) spin chain Hamiltonians, in which all the spins transform under the fundamental representation. These SU($N$) spin chains are known to be quantum critical and described by a well known Wess-Zumino-Witten (WZW) non-linear sigma model in the continuum limit. Numerical results from our lattice Hamiltonian are obtained using stochastic series expansion (SSE) quantum Monte Carlo for both closed and open boundary conditions. As expected for this 1D critical system, the REE shows a logarithmic dependence on the subsystem size with a prefector given by the central charge of the SU($N$) WZW model. We study in detail the sub-leading oscillatory terms in the REE under both periodic and open boundaries. Each oscillatory term is associated with a WZW field and decays as a power law with an exponent proportional to the scaling dimension of the corresponding field. We find that the use of periodic boundaries (where oscillations are less prominent) allows for a better estimate of the central charge, while using open boundaries allows for a better estimate of the scaling dimensions. For completeness we also present numerical data on the thermal R\'{e}nyi entropy which equally allows for extraction of the central charge.Comment: 8 pages, 13 figure

Harmonization of space-borne infra-red sensors measuring sea surface temperature

Sea surface temperature (SST) is observed by a constellation of sensors, and SST retrievals are commonly combined into gridded SST analyses and climate data records (CDRs). Differential biases between SSTs from different sensors cause errors in such products, including feature artefacts. We introduce a new method for reducing differential biases across the SST constellation, by reconciling the brightness temperature (BT) calibration and SST retrieval parameters between sensors. We use the Advanced Along-Track Scanning Radiometer (AATSR) and the Sea and Land Surface Temperature Radiometer (SLSTR) as reference sensors, and the Advanced Very High Resolution Radiometer (AVHRR) of the MetOp-A mission to bridge the gap between these references. Observations across a range of AVHRR zenith angles are matched with dual-view three-channel skin SST retrievals from the AATSR and SLSTR. These skin SSTs act as the harmonization reference for AVHRR retrievals by optimal estimation (OE). Parameters for the harmonized AVHRR OE are iteratively determined, including BT bias corrections and observation error covariance matrices as functions of water-vapor path. The OE SSTs obtained from AVHRR are shown to be closely consistent with the reference sensor SSTs. Independent validation against drifting buoy SSTs shows that the AVHRR OE retrieval is stable across the reference-sensor gap. We discuss that this method is suitable to improve consistency across the whole constellation of SST sensors. The approach will help stabilize and reduce errors in future SST CDRs, as well as having application to other domains of remote sensing