Aspects of supersymmetric mechanics

Abstract

Supersymmetry is a space time symmetry which relates bosons to fermions or vice verca. It requires that for each particle there has to be an anti- particle with the same mass. Along this thesis some supersymmetric Lagrangian models and their properties are discussed. One of them is a Quiver Lagrangian model of the classical system of D-particles connected to each other by light strings. Quiver mechanics is used in a quantum description of black holes. The Quiver quantum mechanical model is one of the key ways to understand the thermodynamic properties of large N=2 black holes in 4 dimensions from the point of view of string theory. In the thesis some quantum approximations are given to have an e ective Quiver Lagrangian similar to the Lagrangian in a background magnetic eld with a Dirac monopole term. Following it, we discuss the classical properties of the Higgs and Coulomb branches of N=4 Quiver mechanics and derive Coulomb and Higgs minima of vacua in this thesis. However, we are addressing the question whether a stable Coulomb Branch can also be obtained, classically. We use separation of scales and quasi-classical expansion methods to derive the same e ective Lagrangian in a classical way. Separation of scales is the method of determining the fast and slow elds in the Lagrangian and eliminating the e ect of fastly oscillating elds by averaging them in a long time interval. We aim to get some time independent e ective potentials for slowly changing elds by using the adiabatic invariant theorem. Quasi-classical expansion describes the bosonic classical dynamics along with fermionic degrees of freedom. It tells us that the classical solution always involves Grassmann terms with a quasi-classical solution when the coupling between bosons and fermions appears in the equations of motion. Therefore, it is a Grassmann valued function.