Resumen de La función de partición de la cuerda topológica como función de onda agujeros negros y dualidades de "Gran N"
Topological strings, introduced by Witten more than 15 years ago, have led not only to very interesting mathematical results, but also to physical applications beyond those that originally motivated their construction. In addition they can also be considered as ``toy models'' in order to trying to understand some properties of physical string theory. In fact, it was precisely Witten \cite{Witten:1993ed} who, trying to study the problem of background-dependence in this simplified setting, found the following interesting result: {\bf closed topological string partition function is not really a function, but a wavefuntion}. This wavefunction corresponds to a (background-dependent) coherent state representation of a (background-independent) state $|\psi_{\rm closed}\ra$ of the quantization of the third cohomology $H^3$ of the Calabi-Yau target space. This fact gives us strong evidence that there is a underlying (unknown, background independent) integrable structure behind the (known, background dependent) worldsheet formulation of topological string theory. In this PhD thesis I address this question by \begin{itemize} \item studying in detail the quantization of $H^3$, the different polarizations (K\""ahler, real, holomorphic, non-linear) at which it can be done, the quantum mechanical meaning of the changes of polarization and the properties of the different wavefunctions associated with $|\psi_{\rm closed}\ra$.
\item mapping the 4d attractor black hole entropy to quantum distribution functions on $H^3$. This has led us to study the quantum mechanical meaning of the OSV corrections to the entropy \cite{Ooguri:2004zv} as the quantum information lost through ``coarse graining'' effects \cite{Gomez:2006gq}.
\item studying what happens with this wavefunction property in large $N$ dualities \cite{Gopakumar:1998ki,Ooguri:2002gx}, in particular, Dijkgraaf-Vafa large $N$ duality \cite{Dijkgraaf:2002fc}. The conclusions of this part of the thesis \cite{Montanez:2007pj} are
