Split attractor flow trees and black hole entropy in type II string theory
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Abstract:
This thesis is about black holes in string theory. More specifically examine mathematical techniques herein and devel kkeld which lead to a better understanding of the entropy of black holes in four-dimensional compactifications of type II string theory. Three of the five chapters the reader from the basics of e snaartheori escorted to a point where the examinations of the author terminal. In two chapters, the results of the research and studies by the author are presented. The general context of type II string theory and the classical low-energy supergravity theories are briefly explained put. Com pactificatie is presented from ten to four dimensions, and there is e special attention to the situation where compact dimensions are a Calabi-Ya you variety. There also is a brief introduction of strings and branes, as well as an outline of the correspondence between D-branes in string theory and p-brane n supergravity. Black holes are introduced in terms of string theory, starting with the well-known Schwarzschild black hole in the G eneral Relativity, then load and optionally torque evoegd leis, and supersymmetric, extremal black holes are explained complying with a mass-charge-parity : the BPS-states of the theory. For BPS black holes in type II string theory and supergravity introduced the attrac tor mechanism, and the extension to multi-BP S bound states of black holes: Split attractor flow, waarv an a suspicion that they have an existence criterion for BPD bound states. Other key ingredients, of which it was made using research conducted by the author, are then explained: a brief introduction of topological string theory, oriented towards Donaldson T homas invariants, a BPS wall-crossing index, black hole partition functions and a special kind of states called polar states (these determine the full partition functions), and aspects of mirror symmetry, e and equality postulates of type IIA string theory on a Calabi-Yau variety and type IIB string theory to another, so-called mirror-Calabi-Ya you variety. The bulk of the research presented in this thesis consists of classifying and counting BPS bound send anden. This is accomplished by the etabliëren of states food flows through attractor Diff, and counting the corresponding states with BPD wall-crossing index to factoring and components to count separately, Donaldson-Thomas invariants degenerations. The counting of all polar conditions in an ensemble of D-branes is sufficient to determine the entire partition function of a D-particle (a low loaded nephew va n is a black hole), an elliptical genus. This is more directly with the M-lift theory of this D-brane systems associated. For several D-brane syste one on algebraic Calabi-Yau manifolds, embedded as hyperplanes in projective spaces ?? ?? ighted, elliptic genera are derived exactly. Because po polar states always BPS bound states prove their existences can ence of split flow trees are deduced, and counting painted edt by the BPS wall-crossing model. Counting bound states really need to be refined. A sophisticated BPS wall-crossing index is calculated for the examined samples, and various algebraic technique and developed it. As a byproduct invariants are partitions Donaldson-Thomas discovered. These make possible a more refined tell ing from (D6-D2-D0) the constituents of a bound state. In a slightly different context, a relationship demonstrated between the four-dimensional formalism of multi-black holes, po polar states and split flow trees, and the five-dimensional form alisme the fuzzball program Mathur. Finally there is a brief overview of the fuzzball research, which a prese orientation follows an explicit mapping between a class of 4D multi-center black hole solutions in N = 8 supergravity, and 5d fuzzball geometries, is given to the interpretation of micro-states a small black ring.- black hole: entropy
- flow: attractor
- string model
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