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Title:  Aspects of Supersymmetric Black Holes and Galilean Conformal Algebras
Author:  Ipsita Mandal
As Contributor:   Ipsita Mandal
Type: Ph.D.
Field: Physics
  • High-Energy Physics - Theory
Approach: Theoretical
Degree granting institution:  Homi Bhabha National Institute
Supervisor(s): Prof. Ashoke Sen
Defense date:  2011-08-23


The first part of my thesis is concerned with black holes in string theory. Black holes are classical solutions of the equations of motion of general theory of relativity. Each black hole is surrounded by an event horizon that acts as a one way membrane. Nothing, including light, can escape a black hole horizon. Thus classically the horizon of a black hole behaves as a perfect black body at zero temperature. This picture undergoes a dramatic modification in quantum theory. There a black hole behaves as a thermodynamic system with definite temperature, entropy etc. In particular, the temperature and the Bekenstein-Hawking entropy of a black hole is given by the simple formul\ae: \[ T = {\kappa \over 2\pi }\,, \qquad S_{BH} = {A \over 4 \, G_N }\,, \] where $\kappa$ is the surface gravity -- acceleration due to gravity at the horizon of the black hole (measured by an observer at infinity), $ A $ is the area of the event horizon and $ G_N $ is the Newton's gravitational constant. We have set $ \hbar = c =k_B = 1$. Now, for ordinary objects, the entropy of a system has a microscopic interpretation. If we fix the macroscopic parameters and count the number of quantum states (dubbed microstates), each of which has the same charge, energy etc., then we can define the microscopic (statistical) entropy as: \[ S_{micro} = \ln \, d_{micro} \,, \] where $d_{micro}$ is the number of such microstates. This naturally leads to the question whether the entropy of a black hole has a similar statistical interpretation. In order to investigate the statistical origin of black hole entropy, we need a quantum theory of gravity. Since string theory gives a framework for studying classical and quantum properties of black holes, we shall carry out our investigation in string theory. Now, even though there is a unique string (M)-theory, it can exist in many different stable and metastable phases. However, there are some issues like those involving black hole thermodynamics, which are universal, and hence can be addressed in any phase of string theory. We shall make use of this freedom to study these issues in a special class of phases of string theory with a large amount of unbroken supersymmetry. One of my research projects focusses on the identification of the hair degrees of freedom for an extremal black hole. Macroscopic entropy of an extremal black hole is expected to be determined completely by its near horizon geometry. Thus two black holes with identical near horizon geometries should have identical macroscopic entropy, and the expected equality between macroscopic and microscopic entropies will then imply that they have identical degeneracies of microstates. An apparent counterexample is provided by the 4D-5D lift relating BMPV black hole to a four dimensional black hole. The two black holes have identical near horizon geometries but different microscopic spectrum. We suggest that this discrepancy can be accounted for by black hole hair, -- degrees of freedom living outside the horizon and contributing to the degeneracies. We identify these degrees of freedom for both the four and the five dimensional black holes and show that after their contributions are removed from the microscopic degeneracies of the respective systems, the result for the four and five dimensional black holes match exactly. The second part of my thesis deals with the Galilean Conformal Algebras (GCA), which correspond to the generators of a non-relativistic conformal symmetry obtained by a parametric contraction of the relativistic conformal group. In the paper ``Supersymmetric Extension of Galilean Conformal Algebras", we extend the analysis to include supersymmetry in four spacetime dimensions. We work at the level of the co-ordinates in superspace to construct the ${\mathcal{N}} = 1$ Super Galilean conformal algebra. One of the interesting outcomes of the analysis is that one is able to naturally extend the finite algebra to an infinite one. We also comment on the extension of our construction to cases of higher ${\mathcal{N}}$. In a subsequent work, ``Supersymmetric Extension of GCA in 2d", we derive the infinite dimensional Supersymmetric Galilean Conformal Algebra (SGCA) in the case of two spacetime dimensions by performing group contraction on 2d superconformal algebra. We also obtain the representations of the generators in terms of superspace coordinates. Here we find realisations of the SGCA by considering scaling limits of certain 2d SCFTs which are non-unitary and have their left and right central charges become large in magnitude and opposite in sign. We focus on the Neveu-Schwarz sector of the parent SCFTs and develop the representation theory based on SGCA primaries, Ward identities for their correlation functions and their descendants which are null states.

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