Perz, Jan (2007): Black hole attractors and the entropy function in four and fivedimensional N=2 supergravity. Dissertation, LMU München: Faculty of Physics 

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Abstract
Extremal black holes in theories of gravity coupled to abelian gauge fields and neutral scalars, such as those arising in the lowenergy description of compactifications of string theory on CalabiYau manifolds, exhibit the attractor phenomenon: on the event horizon the scalars settle to values determined by the charges carried by the black hole and independent of the values at infinity. It is so, because on the horizon the energy contained in vector fields acts as an effective potential (the black hole potential), driving the scalars towards its minima. For spherically symmetric black holes in theories where gauge potentials appear in the Lagrangian solely through field strengths, the attractor phenomenon can be alternatively described by a variational principle based on the socalled entropy function, defined as the Legendre transform with respect to electric fields of the Lagrangian density integrated over the horizon. Stationarity conditions for the entropy function then take the form of attractor equations relating the horizon values of the scalars to the black hole charges, while the stationary value itself yields the entropy of the black hole. In this study we examine the relationship between the entropy function and the black hole potential in fourdimensional N=2 supergravity and demonstrate that in the absence of higherorder corrections to the Lagrangian these two notions are equivalent. We also exemplify their practical application by finding a supersymmetric and a nonsupersymmetric solution to the attractor equations for a conifold prepotential. Exploiting a connection between four and fivedimensional black holes we then extend the definition of the entropy function to a class of rotating black holes in fivedimensional N=2 supergravity with cubic prepotentials, to which the original formulation did not apply because of broken spherical symmetry and explicit dependence of the Lagrangian on the gauge potentials in the ChernSimons term. We also display two types of solutions to the respective attractor equations. The link between four and fivedimensional black holes allows us further to derive fivedimensional firstorder differential flow equations governing the profile of the fields from infinity to the horizon and construct nonsupersymmetric solutions in four dimensions by dimensional reduction. Finally, fourdimensional extremal black holes in N=2 supergravity can be also viewed as certain twodimensional string compactifications with fluxes. Motivated by this fact the recently proposed entropic principle postulates as a probability measure on the space of these string compactifications the exponentiated entropy of the corresponding black holes. Invoking the conifold example we find that the entropic principle would favor compactifications that result in infraredfree gauge theories.
Item Type:  Thesis (Dissertation, LMU Munich) 

Keywords:  black hole attractor mechanism, entropy function, N=2 supergravity 
Subjects:  600 Natural sciences and mathematics 600 Natural sciences and mathematics > 530 Physics 
Faculties:  Faculty of Physics 
Language:  English 
Date Accepted:  20. September 2007 
1. Referee:  Lüst, Dieter 
Persistent Identifier (URN):  urn:nbn:de:bvb:1974466 
MD5 Checksum of the PDFfile:  f38bb83351fd2fbd003c80db386bacbe 
Signature of the printed copy:  0001/UMC 16500 
ID Code:  7446 
Deposited On:  10. Oct 2007 
Last Modified:  16. Oct 2012 08:09 