Monte Carlo Simulation of Comptonization in Inhomogeneous Media

Comptonization is the process in which photon spectrum changes due to multiple Compton scatterings in the electronic plasma. It plays an important role in the spectral formation of astrophysical X-ray and gamma-ray sources. There are several intrinsic limitations for the analytical method in dealing with the Comptonization problem and Monte Carlo simulation is one of the few alternatives. We describe an efficient Monte Carlo method that can solve the Comptonization problem in a fully relativistic way. We expanded the method so that it is capable of simulating Comptonization in the media where electron density and temperature varies discontinuously from one region to the other and in the isothermal media where density varies continuously along photon paths. The algorithms are presented in detail to facilitate computer code implementation. We also present a few examples of its application to the astrophysical research.Comment: 12 pages, 4 figures, Postscript file, in press ("Computers in Physics", Vol. 11, No. 6

Blowup Equations for Refined Topological Strings

G\"{o}ttsche-Nakajima-Yoshioka K-theoretic blowup equations characterize the Nekrasov partition function of five dimensional $\mathcal{N}=1$ supersymmetric gauge theories compactified on a circle, which via geometric engineering correspond to the refined topological string theory on $SU(N)$ geometries. In this paper, we study the K-theoretic blowup equations for general local Calabi-Yau threefolds. We find that both vanishing and unity blowup equations exist for the partition function of refined topological string, and the crucial ingredients are the $\bf r$ fields introduced in our previous paper. These blowup equations are in fact the functional equations for the partition function and each of them results in infinite identities among the refined free energies. Evidences show that they can be used to determine the full refined BPS invariants of local Calabi-Yau threefolds. This serves an independent and sometimes more powerful way to compute the partition function other than the refined topological vertex in the A-model and the refined holomorphic anomaly equations in the B-model. We study the modular properties of the blowup equations and provide a procedure to determine all the vanishing and unity $\bf r$ fields from the polynomial part of refined topological string at large radius point. We also find that certain form of blowup equations exist at generic loci of the moduli space.Comment: 85 pages. v2: Journal versio

Topological Strings and Quantum Spectral Problems

We consider certain quantum spectral problems appearing in the study of local Calabi-Yau geometries. The quantum spectrum can be computed by the Bohr-Sommerfeld quantization condition for a period integral. For the case of small Planck constant, the periods are computed perturbatively by deformation of the Omega background parameters in the Nekrasov-Shatashvili limit. We compare the calculations with the results from the standard perturbation theory for the quantum Hamiltonian. There have been proposals in the literature for the non-perturbative contributions based on singularity cancellation with the perturbative contributions. We compute the quantum spectrum numerically with some high precisions for many cases of Planck constant. We find that there are also some higher order non-singular non-perturbative contributions, which are not captured by the singularity cancellation mechanism. We fix the first few orders formulas of such corrections for some well known local Calabi-Yau models.Comment: 47 pages, 3 figures. v2: journal version, typos correcte