Partition Functions for Supersymmetric Black Holes

Abstract

This thesis presents a number of results on partition functions for four-dimensional supersymmetric black holes. These partition functions are important tools to explain the entropy of black holes from a microscopic point of view. Such a microscopic explanation was desired after the association of a macroscopic entropy to black holes in the 70's, based on the analogies between black hole physics and thermodynamics. The correct microscopic account of black hole entropy was achieved in string theory and M-theory during the 90's, and a crucial role is played by D-branes and M-branes. The black holes, which are studied in this thesis, are supersymmetric solutions of four-dimensional $\mathcal{N}=2$ supergravity, which carry both electric and magnetic charges. An important feature of the global geometry is the the near-horizon geometry, which is AdS$_2\times S^2$ and where the K\ahler moduli are fixed at their attractor values. The horizon area of this class of black holes is given by $S_\mathrm{BH}=\pi|Z|^2=\pi \sqrt{\frac{2}{3}p^3 (q_{\bar 0}+\frac{1}{2}q^2)}$, where $p^a$ and $(q_{\bar 0},q_a)$ are respectively the electric and magnetic charges, and $Z$ is the central charge. The combination $\hat q_{\bar 0}=q_{\bar 0}+\frac{1}{2}q^2$ is required to be positive for black holes. The first motivation to introduce a black hole partition function $\mathcal{Z}_\mathrm{BH}$, is to explain $S_\mathrm{BH}$ microscopically. An analysis of the attractor equations suggest that $\mathcal{Z}_\mathrm{BH}$ is naturally expanded in $q_{\bar 0}$ and $q_a$, while the $p^a$ are kept fixed. In other more physical words, the electric charges are in a macrocanonical ensemble and the magnetic charges in a microcanonical ensemble. If higher derivative contributions are included in the supergravity action, the entropy receives corrections. The form of these corrections suggests that $\mathcal{Z}_\mathrm{BH}$ is well approximated by the square of the topological string partition function $|\mathcal{Z}_\mathrm{top}|^2$. Topological string theory is a simplified version of string theory, which allows the computation of many quantities using elaborate mathematical techniques, for example $\mathcal{Z}_\mathrm{top}$ basically enumerates the holomorphic maps of a Riemann surface into a Calabi-Yau threefold. The conjecture that $\mathcal{Z}_\mathrm{BH}=|\mathcal{Z}_\mathrm{top}|^2$, is the second motivation for this thesis. The third motivation is the correspondence between a theory including gravity in Anti-de Sitter (AdS) space and a conformal field theory (CFT) on the boundary of the AdS-space. Part of the near-horizon geometry of the black holes in the eleven dimensions is AdS$_3$, whose boundary is a two-dimensional torus. The correspondence suggests that the CFT$_2$ partition function equals the one of the theory in the bulk of AdS$_3$. Therefore, the CFT$_2$ partition function should admit an expansion which is natural for an AdS$_3$-(super)gravity partition function. Dijkgraaf {\it et al.} proposed in 2000 that an SCFT partition function can be rewritten as a Poincar\'e series, which is a sum over the coset $\Gamma_\infty\backslash\Gamma$. Every element in the coset corresponds to a semi-classical saddle point geometry, providing therefore evidence for the AdS/CFT correspondence. Chapter \ref{chap:bheinstein} explains these notions rather heuristically in a bosonic setting, subsequent chapters are more precise. Chapter \ref{chap:bhmtheory} explains how the black holes arise as a solution of 11-dimensional M-theory, and how this can account for the entropy microscopically. Four-dimensional supergravity appears in this context as the reduction of M-theory on a six-dimensional Calabi-Yau $X$ times a circle $S^1_\mathrm{M}$. The heavy objects, which source the black holes, are M5-branes. These correspond to the magnetic charges whereas the electric charges are generated by fluxes on the worldvolume of the M5-brane and momentum around $S^1_\mathrm{M}$. The six-dimensional M5-branes wrap a four-dimensional divisor $P$ of $X$ together with $S^1_\mathrm{M}$ and the Euclidean time circle $S^1_\mathrm{t}$. The parameters of the theory can be chosen such that the low energy approximation to M-theory is valid. The M5-brane low energy degrees of freedom can be reduced to the $T^2$, which is formed by the two circles. There, the degrees of freedom combine to an $\CN=(4,0)$ superconformal field theory. The central charges of the holomorphic and anti-holomorphic sector can be determined using index formulas. The relevant partition function for this SCFT is a (modified) elliptic genus. The symmetries of the theory determine that the elliptic genus transforms covariantly under modular transformations, which makes it possible to derive the Cardy formula for the entropy. This gives the correct leading behavior of the entropy. For a specific identification of the parameters, the CFT partition function is equal to the (divergent) black hole partition function. An important property of the elliptic genera is the decomposition into theta functions and a vector-valued modular form. The principal part of the Laurent expansion of the vector-valued modular form gives rise to the definition of the ``polar spectrum'' of the SCFT. Chapter \ref{chap:vectforms} is devoted to an analysis of meromorphic vector-valued modular forms. It is shown how the Fourier coefficients can be expressed as an infinite sum over the coset $\Gamma_\infty\backslash \Gamma$. If the polar degeneracies are known, the non-polar degeneracies can be determined with an arbitrary accuracy, improving on the leading order estimate by the Cardy formula. In addition is shown how the vector-valued modular form can be written as a sum over $\Gamma_\infty\backslash \Gamma$. The sum is a regularized Poincar\'e series, and an improvement of the proposal by Dijkgraaf {\it et al.}. The regularization leads in general to an anomaly, which is canceled if the polar degeneracies satisfy a number of constraints. This number can be determined using the Selberg trace formula. The dimension of the space of vector-valued modular forms is simply given by the number of polar terms minus the number of constraints. With the results of Chapter \ref{chap:vectforms}, Chapter \ref{chap:interpretation} revisits the motivations of Chapter \ref{chap:bheinstein}. The regularized Poincar\'e series confirms the AdS$_3$/CFT$_2$ correspondence, since the sum over $\Gamma_\infty\backslash \Gamma$ is suggestive of a semi-classical sum over AdS$_3$-geometries. If the complex structure $\tau$ is varied, the most contributing geometry to $\mathcal{Z}_{\mathrm{BH}}$ might jump, which is a nice manifestation of Hawking-Page phase transitions. The Poincar\'e series are essentially a sum over $\Gamma_\infty\backslash \Gamma$ of the polar spectrum, which lies classically below the cosmic censorship bound. Therefore, one can view the series heuristically as a sum over all geometries (including black holes) of the states which do not collapse into a black hole. This is also how the connection between black holes and topological strings can be understood. The degeneracies of charged BPS-particles (M2-branes) in the near-horizon geometry are enumerated by $|\mathcal{Z}_\mathrm{top}|^2$. Therefore, $|\mathcal{Z}_\mathrm{top}|^2$ appears for every saddle point geometry in $\mathcal{Z}_\mathrm{BH}$. The conjecture is now elucidated for strong topological string coupling constant ($\hat q_{\bar 0}\gg p^3$), since then a single AdS$_3$-geometry dominates the partition function. Also the case of weak topological string couple is discussed. It is shown that for this part of the spectrum, the elliptic genus confirms the leading behavior of two-centered solutions. The approximation $\mathcal{Z}_\mathrm{BH}\sim |\mathcal{Z}_\mathrm{top}|^2$ is however no longer valid. The constraints on the polar degeneracies by modularity are strongest if the number of polar terms is small. This is generically not the case for the black hole and AdS$_3$ applications, which are discussed before. The vector-valued modular forms appear however at many places in theoretical physics, for example rational conformal field theory and $\CN=4$ supersymmetric gauge theory on a four-manifolds $M$. The partition functions of such gauge theories with gauge group $U(N)$ are closely related to M5-brane elliptic genera. If certain conditions are satisfied, the partition function of this theory is the generating function of the Euler characteristic of instanton moduli spaces. Section \ref{sec:n4ym} performs an analysis of the partition functions of the $U(N)$ theories on $\mathbb{CP}^2$. This confirms the older results in the literature for $N=1$ and $2$. A new generating function is proposed for the Euler numbers of $SU(3)$ moduli spaces.