Intersecting Solitons, Amoeba and Tropical Geometry

We study generic intersection (or web) of vortices with instantons inside, which is a 1/4 BPS state in the Higgs phase of five-dimensional N=1 supersymmetric U(Nc) gauge theory on R_t \times (C^\ast)^2 \simeq R^{2,1} \times T^2 with Nf=Nc Higgs scalars in the fundamental representation. In the case of the Abelian-Higgs model (Nf=Nc=1), the intersecting vortex sheets can be beautifully understood in a mathematical framework of amoeba and tropical geometry, and we propose a dictionary relating solitons and gauge theory to amoeba and tropical geometry. A projective shape of vortex sheets is described by the amoeba. Vortex charge density is uniformly distributed among vortex sheets, and negative contribution to instanton charge density is understood as the complex Monge-Ampere measure with respect to a plurisubharmonic function on (C^\ast)^2. The Wilson loops in T^2 are related with derivatives of the Ronkin function. The general form of the Kahler potential and the asymptotic metric of the moduli space of a vortex loop are obtained as a by-product. Our discussion works generally in non-Abelian gauge theories, which suggests a non-Abelian generalization of the amoeba and tropical geometry.Comment: 39 pages, 11 figure

Calibration of ATLAS Tile Calorimeter at Electromagnetic Scale

In this paper we summarize the measurement of the electromagnetic (EM) scale calibration constant for 11% of the Tile Calorimeter modules exposed to electron and muon test beams at CERN SPS accelerator. The Tile Calorimeter modules are currently installed in the ATLAS detector. The analysis presented in this paper takes into account the recent improvements in the Tile Calorimeter cesium calibration, charge injection system calibration and Fit Method energy reconstruction. The overall conversion factor between the measured charge and the energy deposited by measured particles for Tile Calorimeter cells is $1.050pm0.003$~pC/GeV, with spread of $2.4pm0.1$%. We discuss in detail the sources of uncertainties of EM scale calibration constant. We also show, that after inter-calibrating all the Tile Calorimeter cells with a~radioactive cesium source and setting the EM scale in the first calorimeter sampling with electron beams, the values of signals measured in the second and third calorimeter sampling need to be increased by 1--9% to keep the EM scale uniform in the whole calorimeter

Image formation modeling in cryo-electron microscopy

Microscopic imaging and technolog