Arithmetic, Aesthetics, and the Taming of Infinities

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.

Due to the failure of thermodynamics for low temperature near-extremal black holes, it has long been conjectured that a ‘thermodynamic mass gap’ exists between an extremal black hole and the lightest near-extremal state. For non-supersymmetric near-extremal black holes in Einstein gravity with an AdS 2 throat, no such gap was found. Rather, at that energy scale, the spectrum exhibits a continuum of states, up to non-perturbative corrections. In this paper, we compute the partition function of near-BPS black holes in supergravity where the emergent, broken, symmetry is PSU(1, 1|2). To reliably compute this partition function, we show that the gravitational path integral can be reduced to that of a N=4 supersymmetric extension of the Schwarzian theory, which we define and exactly quantize. In contrast to the non-supersymmetric case, we find that black holes in supergravity have a mass gap and a large extremal black hole degeneracy consistent with the Bekenstein–Hawking area. Our results verify a plethora of string theory conjectures, concerning the scale of the mass gap and the counting of extremal micro-states.

We compute the instanton partition function for ${\cal N}=4$ U(N) gauge theories living on toric varieties, mainly of type $\R^4/\Gamma_{p,q}$ including $A_{p-1}$ or $O_{\PP_1}(-p)$ surfaces. The results provide microscopic formulas for the partition functions of black holes made out of D4-D2-D0 bound states wrapping four-dimensional toric varieties inside a Calabi-Yau. The partition function gets contributions from regular and fractional instantons. Regular instantons are described in terms of symmetric products of the four-dimensional variety. Fractional instantons are built out of elementary self-dual connections with no moduli carrying non-trivial fluxes along the exceptional cycles of the variety. The fractional instanton contribution agrees with recent results based on 2d SYM analysis. The partition function, in the large charge limit, reproduces the supergravity macroscopic formulae for the D4-D2-D0 black hole entropy.

It is well known that the thermodynamics of certain near-extremal black holes in asymptotically flat space can be lifted to an effective string description created from the intersection of D-branes. In this paper we present evidence that the semiclassical thermodynamics of near-extremal R-charged black holes in AdS5 × S5 is described in a similar manner by effective strings created from the intersection of giant gravitons on the S5. We also present a free fermion description of the supersymmetric limit of the one-charge black hole, and we give a crude catalog of the microstates of the two and three-charge black holes in terms of operators in the dual conformal field theory.

In this paper, we investigate the minimal length effects on the thermodynamics and weak cosmic censorship conjecture in a RN-AdS black hole via charged particle absorption. We first use the generalized uncertainty principle (GUP) to investigate the minimal length effect on the Hamilton-Jacobi equation. After the deformed Hamilton-Jacobi equation is derived, we use it to study the variations of the thermodynamic quantities of a RN-Ads black hole via absorbing a charged particle. Furthermore, we check the second law of thermodynamics and the weak cosmic censorship conjecture in two phase spaces. In the normal phase space, the second law of thermodynamics and the weak cosmic censorship conjecture are satisfied in the usual and GUP-deformed cases, and the minimal length effect makes the increase of entropy faster than the usual case. After the charge particle absorption, the extremal RN-AdS black hole becomes nonextremal. In the extended phase space, the black hole entropy can either increase or decrease. When T>2Pr+, the second law is satisfied. When T<2Pr+, the second law of thermodynamics is violated for the extremal or near-extremal black hole. Finally, we find that the weak cosmic censorship conjecture is legal for extremal and near-extremal RN-Ads black holes in the GUP-deformed case.

One interesting property of black holes is that they obey thermodynamic laws. The entropy of black holes is given by the area of the horizon and it is naturally expected to be understood from the statistical mechanical viewpoint. However there has not been many things known about black hole microstates in the gravity, even though their holographic understanding has been well- known. In microstate geometry program which is a conjecture about black hole microstates, typical microstates are described as smooth and entropyless (i.e., horizonless) supergravity so- lutions which have the same mass, angular momentum, and charges as that of black holes. In this thesis, a new class of black hole microstates are suggested and studied in addition to the known microstate solutions and discussed in the context of microstate geometry program.

Freudenthal duality (F-duality), an anti-involution of charge vectors, keeps the entropy and attractor solutions invariant for an extremal supersymmetric black hole. We analyze the effect of F-duality on the entropy of a near-extremal STU black hole in $\mathcal{N}=2$ ungauged, four-dimensional supergravity. We consider double-extremal black holes, whose attractor solutions are fixed in terms of the black hole charges throughout the moduli space. It is well known that Jackiw-Teitelboim (JT) gravity governs the dynamics of the near-horizon regions of higher-dimensional, near-extremal black holes. Owing to this fact, we reduce the four-dimensional supergravity theory to two dimensions to construct a JT gravity--like model and compute the near-extremal entropy. We then analyze the effect of F-duality on this entropy. We show that the F-duality breaks down for the case of near-extremal solutions if one considers the duality operation generated through near-extremal entropy rather than the extremal one.

In a recent publication we studied the decay rate of primordial black holes perceiving the dark dimension, an innovative five-dimensional (5D) scenario that has a compact space with characteristic length scale in the micron range. We demonstrated that the rate of Hawking radiation of 5D black holes slows down compared to 4D black holes of the same mass. Armed with our findings we showed that for a species scale of O(1010 GeV), an all-dark-matter interpretation in terms of primordial black holes should be feasible for black hole masses in the range 1014≲M/g≲1021. As a natural outgrowth of our recent study, herein we calculate the Hawking evaporation of near-extremal 5D black holes. Using generic entropy arguments we demonstrate that Hawking evaporation of higher-dimensional near-extremal black holes proceeds at a slower rate than the corresponding Schwarzschild black holes of the same mass. Assisted by this result we show that if there were 5D primordial near-extremal black holes in nature, then a primordial black hole all-dark-matter interpretation would be possible in the mass range 105β≲M/g≲1021, where β is a parameter that controls the difference between mass and charge of the associated near-extremal black hole. Published by the American Physical Society 2024

The thermodynamics and weak cosmic censorship conjecture in Reissner-Nordström anti-de Sitter black holes are investigated by the scattering of the scalar field. The first law of thermodynamics in the non-extremal Reissner-Nordström anti-de Sitter black hole is recovered by the scattering. The increase of the horizon radius indicates that the singularity is not naked in this black hole. For the near-extremal and extremal black holes, the validity is tested by the minimum values of the function f at their final states. It is found that both of the near-extremal and extremal black holes can not be overcharged. When ω=qϕ, the final state of the extremal black hole is still an extremal black hole. When ω≠qϕ, it becomes a near-extremal black hole with new mass and charge.

We formulate a refined version of the Ooguri-Strominger-Vafa (OSV) conjecture. The OSV conjecture that Z_{BH} = |Z_{top}|^2 relates the BPS black hole partition function to the topological string partition function Z_{top}. In the refined conjecture, Z_{BH} is the partition function of BPS black holes counted with spin, or more precisely the protected spin character. Z_{top} becomes the partition function of the refined topological string, which is itself an index. Both the original and the refined conjecture are examples of large N duality in the 't Hooft sense. The refined conjecture applies to non-compact Calabi-Yau manifolds only, so the black holes are really BPS particles with large entropy, of order N^2. The refined OSV conjecture states that the refined BPS partition function has a large N dual which is captured by the refined topological string. We provide evidence that the conjecture holds by studying local Calabi-Yau threefolds consisting of line bundles over a genus g Riemann surface. We show that the refined topological string partition function on these geometries is computed by a two-dimensional TQFT. We also study the refined black hole partition function arising from N D4 branes on the Calabi-Yau, and argue that it reduces to a (q,t)-deformed version of two-dimensional SU(N) Yang-Mills. Finally, we show that in the large N limit this theory factorizes to the square of the refined topological string in accordance with the refined OSV conjecture.

Canonical quantum statistics of Schwarzschild black holes and Ising droplet nucleation

Recent work on the quantum mechanics of near-extremal non-supersymmetric black holes has identified a characteristic T3/2 scaling of the low temperature black hole partition function. This result has only been derived using the path integral in the near-horizon region and relies on many assumptions. We discuss how to derive the T3/2 scaling for the near-extremal rotating BTZ black hole from a calculation in the full black hole background using the Denef-Hartnoll-Sachdev (DHS) formula, which expresses the 1-loop determinant of a thermal geometry in terms of a product over the quasinormal mode spectrum. We also derive the spectral measure for fields of any spin in Euclidean BTZ and use it to provide a new proof of the DHS formula and a new, direct derivation of the BTZ heat kernel. The computations suggest a path to proving the T3/2 scaling for the asymptotically flat 4d Kerr black hole.

Supersymmetric black holes provide an excellent theoretical laboratory to test ideas about quantum gravity in general and black hole entropy in particular. When four-dimensional supergravity is interpreted as the low-energy approximation of ten-dimensional string theory or eleven-dimensional M-theory, one has a microscopic description of the black hole which allows one to count microstates and to compare the result to the macroscopic (geometrical) entropy. Recently it has been conjectured that there is a very direct connection between the partition function of the topological string and a partition function for supersymmetric black holes. We review this idea and propose a modification which makes it compatible with electric-magnetic duality.

The weak cosmic censorship conjecture in the near-extremal BTZ black hole has been tested using test particles and fields. It has been claimed that such a black hole can be overspun. In this paper, we review the thermodynamics and weak cosmic censorship conjecture in BTZ black holes using the scattering of a scalar field. The first law of thermodynamics in the non-extremal BTZ black hole is recovered. For the extremal and near-extremal black holes, due to the divergence of the variation of entropy, we test the weak cosmic censorship conjecture by evaluating the minimum of the function f, and find that both the extremal and near-extremal black holes cannot be overspun.

The macroscopic entropy and the attractor equations for BPS black holes in four-dimensional N=2 supergravity theories follow from a variational principle for a certain `entropy function'. We present this function in the presence of R^2-interactions and non-holomorphic corrections. The variational principle identifies the entropy as a Legendre transform and this motivates the definition of various partition functions corresponding to different ensembles and a hierarchy of corresponding duality invariant inverse Laplace integral representations for the microscopic degeneracies. Whenever the microscopic degeneracies are known the partition functions can be evaluated directly. This is the case for N=4 heterotic CHL black holes, where we demonstrate that the partition functions are consistent with the results obtained on the macroscopic side for black holes that have a non-vanishing classical area. In this way we confirm the presence of a measure in the duality invariant inverse Laplace integrals. Most, but not all, of these results are obtained in the context of semiclassical approximations. For black holes whose area vanishes classically, there remain discrepancies at the semiclassical level and beyond, the nature of which is not fully understood at present.