Black hole formation at the correspondence point

We study the process of bound state formation in a D-brane collision. We consider two mechanisms for bound state formation. The first, operative at weak coupling in the worldvolume gauge theory, is pair creation of W-bosons. The second, operative at strong coupling, corresponds to formation of a large black hole in the dual supergravity. These two processes agree qualitatively at intermediate coupling, in accord with the correspondence principle of Horowitz and Polchinski. We show that the size of the bound state and time scale for formation of a bound state agree at the correspondence point. The time scale involves matching a parametric resonance in the gauge theory to a quasinormal mode in supergravity.

Large scale features and assessment of spatial scale correspondence between TMPA and IMD rainfall datasets over Indian landmass

Daily rainfall datasets of 10 years (1998-2007) of Tropical Rainfall Measuring Mission (TRMM) Multi-satellite Precipitation Analysis (TMPA) version 6 and India Meteorological Department (IMD) gridded rain gauge have been compared over the Indian landmass, both in large and small spatial scales. On the larger spatial scale, the pattern correlation between the two datasets on daily scales during individual years of the study period is ranging from 0.4 to 0.7. The correlation improved significantly (similar to 0.9) when the study was confined to specific wet and dry spells each of about 5-8 days. Wavelet analysis of intraseasonal oscillations (ISO) of the southwest monsoon rainfall show the percentage contribution of the major two modes (30-50 days and 10-20 days), to be ranging respectively between similar to 30-40% and 5-10% for the various years. Analysis of inter-annual variability shows the satellite data to be underestimating seasonal rainfall by similar to 110 mm during southwest monsoon and overestimating by similar to 150 mm during northeast monsoon season. At high spatio-temporal scales, viz., 1 degrees x1 degrees grid, TMPA data do not correspond to ground truth. We have proposed here a new analysis procedure to assess the minimum spatial scale at which the two datasets are compatible with each other. This has been done by studying the contribution to total seasonal rainfall from different rainfall rate windows (at 1 mm intervals) on different spatial scales (at daily time scale). The compatibility spatial scale is seen to be beyond 5 degrees x5 degrees average spatial scale over the Indian landmass. This will help to decide the usability of TMPA products, if averaged at appropriate spatial scales, for specific process studies, e.g., cloud scale, meso scale or synoptic scale.

Correspondence between neutron depolarization and higher order magnetic susceptibility to investigate ferromagnetic clusters in phase separated systems

It is a tough task to distinguish a short-range ferromagnetically correlated cluster-glass phase from a canonical spin-glass-like phase in many magnetic oxide systems using conventional magnetometry measurements. As a case study, we investigate the magnetic ground state of La0.85Sr0.15CoO3, which is often debated based on phase separation issues. We report the results of two samples of La0.85Sr0.15CoO3 (S-1 and S-2) prepared under different conditions. Neutron depolarization, higher harmonic ac susceptibility and magnetic relaxation studies were carried out along with conventional magnetometry measurements to differentiate subtle changes at the microscopic level. There is no evidence of ferromagnetic correlation in the sample S-2 attributed to a spin-glass phase, and this is compounded by the lack of existence of a second order component of higher harmonic ac susceptibility and neutron depolarization. A magnetic relaxation experiment at different temperatures complements the spin glass characteristic in S-2. All these signal a sharp variance when we consider the cluster-glass-like phase (phase separated) in S-1, especially when prepared from an improper chemical synthesis process. This shows that the nonlinear ac susceptibility is a viable tool to detect ferromagnetic clusters such as those the neutron depolarization study can reveal.

Lipschitz Correspondence between Metric Measure Spaces and Random Distance Matrices

Given a metric space with a Borel probability measure, for each integer N, we obtain a probability distribution on N x N distance matrices by considering the distances between pairs of points in a sample consisting of N points chosen independently from the metric space with respect to the given measure. We show that this gives an asymptotically bi-Lipschitz relation between metric measure spaces and the corresponding distance matrices. This is an effective version of a result of Vershik that metric measure spaces are determined by associated distributions on infinite random matrices.

Moduli of equivariant sheaves and Kronecker-McKay modules

We extend Alvarez-Consul and King description of moduli of sheaves over projective schemes to moduli of equivariant sheaves over projective Gamma-schemes, for a finite group Gamma. We introduce the notion of Kronecker-McKay modules and construct the moduli of equivariant sheaves using a natural functor from the category of equivariant sheaves to the category of Kronecker-McKay modules. Following Alvarez-Consul and King, we also study theta functions and homogeneous co-ordinates of moduli of equivariant sheaves.

Three-dimensional superconformal field theory, Chern-Simons theory, and their correspondence

In this thesis, we discuss 3d-3d correspondence between Chern-Simons theory and three-dimensional N = 2 superconformal field theory. In the 3d-3d correspondence proposed by Dimofte-Gaiotto-Gukov information of abelian flat connection in Chern-Simons theory was not captured. However, considering M-theory configuration giving the 3d-3d correspondence and also other several developments, the abelian flat connection should be taken into account in 3d-3d correspondence. With help of the homological knot invariants, we construct 3d N = 2 theories on knot complement in 3-sphere for several simple knots. Previous theories obtained by Dimofte-Gaiotto-Gukov can be obtained by Higgsing of the full theories. We also discuss the importance of all flat connections in the 3d-3d correspondence by considering boundary conditions in 3d N = 2 theories and 3-manifold.