Spin-Statistics for Black Hole Microstates

We can extend the previous calculation to black holes in the presence of a Maxwell field, in a background of fixed electric charge $Q$, which we take to be positive. The classical solution of Einstein-Maxwell theory we would need in this case is the Kerr-Newman black hole. We normalize the charge $Q$ such that the thermodynamic potentials derived from the Kerr-Newman solution are

while the mass is given in terms of $r_{+}$ as $E=\frac{r_{+}^{2}+a^{2}+Q^{2}}{2r_{+}}$ and $J=aE$. The saddle contributing to the grandcanonical partition function, with $\Omega=0$ and therefore no $(-1)^{\sf F}$ insertion, has $a=0$. This leads to the Euclidean section of the Reissner-Nordstrom black hole. At fixed temperature, the on-shell action has an explicit but complicated expression at fixed temperature. The result simplifies when written in the fixed energy ensemble and results in an entropy

the standard Bekenstein-Hawking area law for the Reissner-Nordstrom black hole. We only consider the black hole solutions with $E\geq Q$.

We can use the gravitational path integral to compute the difference between the number of bosonic and fermionic black hole microstates. To do this, for the same reasons as outlined before, we need to set $\beta\Omega=2\pi{\mathrm{i}}$. This implies, using equation (19) and writing $a={\mathrm{i}}{\sf a}$, that $2{\sf a}r_{+}=r_{+}^{2}+{\sf a}^{2}-Q^{2}$ and therefore ${\sf a}=r_{+}-Q$. The horizon radius is fixed in terms of the inverse temperature as $r_{+}=\frac{Q(\beta-2\pi Q)}{\beta-4\pi Q}.$⁴⁴4The radius of the horizon is negative for $2\pi Q<\beta<4\pi Q$. One option to deal with temperatures in this range is to define the geometry along a complex contour in the $r$-plane that approaches $r_{+}$ from infinity without going close to the singularity at $r=0$. This type of complex geometry would violate the criterion put forth in Witten:2021nzp . For $\beta<2\pi Q$, $r_{+}$ is positive again but a new horizon appears with a larger value of $r$, which again has to be avoided along the complex plane. If we consider the classical limit of $\mathcal{N}=2$ supergravity in 4d, these complex geometries seem necessary to ensure that the index is temperature independent. In this case the on-shell action appropriate to the fixed temperature ensemble has a very simple form $I=-\pi Q^{2}+\beta Q$, implying that

which is the result found in Iliesiu:2021are .

This solution has two issues. The first is that it does not have a smooth $Q\to 0$ limit since $r_{+}\to 0$. The second is that for this solution the energy is $E=Q$ independently of $\beta$. Instead we should be able to compute the difference between bosonic and fermionic microstates in a fixed energy ensemble with any $E\neq Q$. This requires finding a different saddle in the fixed temperature ensemble. To do this set $\beta\Omega=2\pi{\mathrm{i}}(1-\varepsilon)$ and take the limit $\varepsilon\to 0$ more carefully. This way we discover a new solution such that the horizon radius is

This solution is similar to the one we found in the uncharged case. Now the average ADM energy is given by $E=\beta/4\pi$ and becomes a free parameter set by the temperature (and happens to be independent of the charge). The action in the fixed temperature ensemble is now $I=\pi Q^{2}+\beta^{2}/8\pi$, or equivalently

Notice the sign change of the temperature independent term between $F_{\mathrm{spin}}$ and $F_{\mathrm{spin}}^{\prime}$. The true free energy should be the minimum between $F_{\mathrm{spin}}$ and $F^{\prime}_{\mathrm{spin}}$, which happens to be $F_{\mathrm{spin}}$ for any temperature.

The goal of Iliesiu:2021are was to use this type of solution to compute the index of the black hole in $\mathcal{N}=2$ supergravity. In this case we expect the solution with $E=Q$ to be the only contribution. The boundary conditions in this context are supersymmetric and any solution that breaks supersymmetry (if $E\neq Q$ there is no Killing spinor) will have a fermionic zero-mode that makes this contribution vanish.⁵⁵5Even if $E=Q$ some supersymmetries are broken since the Reissner-Nordstrom preserves only four supercharges, and the contribution to the index also vanishes. One can fix this considering the helicity supertrace instead of the index, which can be also computed with the gravitational path integral. The second solution above with $r_{+}\to\infty$ therefore makes a vanishing contribution to the index in supergravity due to this fermion zero-modes. If we are working with a non-supersymmetric theory of gravity, then both solutions should be in principle included. For any temperature, the black hole with $r_{+}$ finite has always a lower free energy and is dominant. Nevertheless, when working at fixed energy $E>Q$ the solution with finite $r_{+}$ disappears and only the solution with $r_{+}\to\infty$ remains.

Let us then consider entropy in the fixed energy ensemble with the insertion of $(-1)^{\sf F}$, using the relation between energy and temperature derived above, $\beta=4\pi E$. The action, including the boundary terms necessary to change ensemble, is given by $\pi(2E^{2}-Q^{2})$ and therefore the entropy counting the difference between bosonic and fermionic microstates is

We do not expect any black hole microstate with $E<Q$ regardless of its statistics, so this result should only be considered for $E>Q$. The ultimate reason why a complex saddle with $E<Q$ is not allowed has to be tracked back to the integration contour of the gravitational path integral.

In the uncharged case we found $S_{\mathrm{spin}}(E,Q=0)=\frac{1}{2}\cdot S(E,Q=0)$. We can now evaluate the ratio between the entropy with and without a $(-1)^{\sf F}$ insertion and we find the following charge dependent result

where $0\leq q\leq 1$. We plot this function in figure 2. When $q=0$ we recover the $1/2$ we found for the uncharged black hole. When $q=1$ we reproduce the expectation that number of states counted with a $(-1)^{\sf F}$ is the same as the total number of states, and therefore to leading order all states have the same statistics. (This interpretation is not correct after including quantum effects which imply the zero-temperature entropy at $E=Q$ does not correspond to a real degeneracy Ghosh:2019rcj ; Iliesiu:2020qvm ; Heydeman:2020hhw ; Turiaci:2023wrh .) As indicated above, this ratio is always smaller than one. This is consistent with unitarity since the total number of states cannot be smaller than the number of bosonic minus fermionic states.

In this section we analyze the quantum statistics of black hole microstates in asymptotically AdS₄ spaces. Through AdS/CFT this puts a constraint on the quantum statistics of high energy states of holographic CFT₃’s.

As explained for example in Chamblin:1999tk it is possible to construct backgrounds in string theory where the low energy theory is described by Einstein-Maxwell in asymptotically AdS₄ coupled to matter

We work in units with $G_{N}=1$. We parametrize the cosmological constant by $\Lambda=-3/\ell^{2}$ and $I_{\rm bdy}$ are the boundary terms necessary depending on the ensemble. The black hole solutions are given by the following generalization of the Kerr-Newman black hole

where

We take the charge to be electric and the gauge field is $A=-\frac{qr}{\rho^{2}}({\mathrm{d}}t-\frac{a\sin^{2}\theta}{\Xi}{\mathrm{d}}\phi)+\alpha{\mathrm{d}}t$, where $\alpha$ is a constant chosen to make the gauge connection smooth at the horizon. The solution is parametrized by $m$, $a$, and $q$, which roughly correspond to the mass/energy (or temperature), the angular momentum (or angular velocity), and the charge (or chemical potential).

In the Euclidean section, one can find the inverse temperature and angular velocity by demanding smoothness at the horizon, located at $\Delta_{r}(r_{+})=0$, which gives

We can trade the parameter $m$ by $r_{+}$ using the relation $m=\frac{(r_{+}^{2}+a^{2})(1+\frac{r_{+}^{2}}{\ell^{2}})+q^{2}}{2r_{+}}$. The ADM charges, the energy $E$, angular momentum $J$, and $\text{U}(1)$ charge $Q$ are given by

Finally, the area of the even horizon is given by $A=4\pi(r_{+}^{2}+a^{2})/\Xi$.

Below we consider in detail the uncharged case $q=0$ (the generalization to $q\neq 0$ is striaghtfoward). The analysis is very similar to the one presented in the previous section so we will be brief, emphasizing mainly the differences.

As before, we begin by reminding the reader the results for the free energy and entropy without an insertion of the fermion parity operator. In the fixed temperature ensemble we can sum over all states by taking $\Omega=0$ and therefore the black hole solution has $a=0$. The size of the horizon is determined through the (inverse) temperature by

This solution only exists for $\beta\leq 2\pi\ell/\sqrt{3}$. The free energy, computed from the on-shell action $I=\beta E-A/4-\beta\Omega J$ is given by

From the first line we can derive that the black hole dominates over thermal AdS for temperatures higher than $\beta<\pi\ell$. The second line shows the result in the high temperature limit. There is a second solution for $r_{+}$ which has higher free energy and therefore never dominates the ensemble.

In the fixed energy ensemble, the black hole radius is constrained by $E=\frac{r_{+}(r_{+}^{2}+\ell^{2})}{2\ell^{2}}$. There are three solutions of this equation, one real and two complex conjugate ones. The two complex solutions have negative real part of the entropy. The real solution has an entropy given by

where we also indicated the behavior at large energies. In this limit the radius of the black hole is $r_{+}\sim 2^{1/3}\ell^{2/3}E^{1/3}$.

We are now ready to consider the distribution between fermionic and bosonic black hole microstates. We begin by computing the free energy $F_{\mathrm{spin}}$ in the presence of a fermion parity operator insertion $(-1)^{\sf F}$. Again, we implement this by imposing $\beta\Omega=2\pi{\mathrm{i}}$. The first observation we can make is that nothing interesting happens that requires regulating the limit $\beta\Omega\to 2\pi{\mathrm{i}}$, as opposed to the flat space analysis. We find two solutions for $a={\mathrm{i}}{\sf a}$ given by

The first solution has $\beta=0$ and $E=0$ so it is the same as ${\sf a}=r_{+}$ in flat space. The second solution has temperature and energy

To compute $F_{\mathrm{spin}}$ we focus on the first relation and solve for $r_{+}(\beta)$. For general $\beta$ and $\ell$, we find four solutions for $r_{+}$. Two are real and two are complex conjugated, and they are all finite. In the flat space limit, one of the solution for the black hole radius becomes the one found in the previous section with $r_{+}\to\infty$. We see therefore that the background cosmological constant regulates the large $r_{+}$ limit of the flat space black hole, without the need to go away from $\beta\Omega=2\pi{\mathrm{i}}$. We find that all these four solutions satisfy, for any $\beta$ and $\ell$,

and therefore thermal AdS always dominates the ensemble computing $F_{\mathrm{spin}}$. The possible presence of other saddles which are neither thermal AdS nor black holes and support periodic fermion boundary conditions is not ruled out. Assuming this is not the case and the free energy is indeed of order one, controlled by thermal AdS, it suggests a large cancellation in $F_{\mathrm{spin}}$ compared with $F$ since there are actually a large number of black hole states in a given energy window.

Next we consider the entropy $S_{\mathrm{spin}}$, counting black hole states in the presence of a $(-1)^{\sf F}$ insertion. Using equation (37) we solve for $r_{+}(E)$. It is clear by looking at the equation for $E$ that it can be rewritten as the roots of an eight order polynomial. There are therefore eight solutions (that cannot be written analytically). We find for positive values of the energy that the solutions always come in four pairs of complex conjugated values of $r_{+}$. Out of these four pairs, two pairs have negative real part of the action while two pairs have positive real part of the action. We denote the action of these solutions by $I_{1}$ and $I_{2}$, such that ${\rm Re}(I_{1,2})<0$ and ${\rm Im}(I_{1,2})>0$, and their complex conjugate. Out of the two pairs with negative real part of the action, one pair has the smallest real part which we choose to be $I_{2}$.

In general, in the presence of two competing saddles with the same real part we should add their contributions in the total partition function. The one-loop determinants arising from each saddle are important in writing the total contribution. In our case the two saddles are related by ${\mathrm{i}}\to-{\mathrm{i}}$ and therefore the one-loop determinants are related by complex conjugation. The total answer is

This applies whether $Z$ computes the partition function in the fixed energy or temperature ensemble. As a function of either, the real part of the action and the absolute value of the one-loop determinant determine the envelope of the partition function while the imaginary part of the action leads to rapid oscillations around the envelope. The overall phase of these oscillations requires knowledge of the one-loop determinants which we do not evaluate here, but the frequency to leading order is determined by ${\rm Im}(I)$. A similar feature takes place in the context of the WKB approximation in quantum mechanics. This conclusion applies to both $I_{1}$ and $I_{2}$.

The pair of solutions with minimal ${\rm Re}(I)$, with an action $I_{2}$, would naively be the one that dominates the ensemble. Nevertheless, we find it to have strange properties. For example, it satisfies ${\rm Re}(r_{+})<0$ ⁶⁶6One can check that they do not satisfy the Kontsevich-Segal-Witten criterion Kontsevich:2021dmb ; Witten:2021nzp , see Appendix A for more discussion. and moreover if we continue this solution smoothly from $\beta\Omega=2\pi{\mathrm{i}}$ down to $\beta\Omega=0$ it does not become the familiar black hole that computes the partition function and instead remains complex. On the other hand the other solution with action $I_{1}$ is less dominant but has a more reasonable behavior. For example, ${\rm Re}(r_{+})>0$ and becomes the black hole computing the partition function when continued to $\beta\Omega=0$. Moreover, only this solution becomes the one studied in Section 2.1 in the flat space limit. For these reasons we conjecture that the integration contour in the gravitational path integral is such that the solutions $I_{2}$ do not contribute and instead the pair of complex conjugate solutions with action $I_{1}$ and ${\rm Re}(r_{+})>0$ dominate the ensemble.

To get some insight on these results we can find the black hole solution analytically in the large energy limit. Looking at equation (37) the solution for large $E$ is such that the denominator vanishes $9r_{\star}^{4}-2\ell^{2}r_{\star}^{2}+\ell^{4}=0$. (As opposed to the partition function without a $(-1)^{\sf F}$ insertion, large $r_{+}$ leads to small energies.) This equation has four solutions $r_{\star}=\pm\frac{\ell}{3}\sqrt{1\pm{\mathrm{i}}2^{3/2}}$ for each choice of signs. Then we expand $r_{+}=r_{\star}+\delta r$ and solve for $\delta r$ as a function of $E$. For large $E$, $\delta r$ will be small, so we can solve for $\delta r$ perturbatively. The solution we propose to keep is the one that arises from a small fluctuation around $r_{\star}=\frac{\ell}{3}\sqrt{1\pm{\mathrm{i}}2^{3/2}}\approx 0.47\ell\pm{\mathrm{i}}0.33\ell$. The other solutions have negative real part for $r_{\star}$. In this limit the action is given by

where the dots denote terms subleading in the large energy limit. The other saddle in the pair has action $\overline{I}_{1}=c\sqrt{\ell^{3}E}-2\pi{\mathrm{i}}\ell E$. Since these two solutions have the same real part of the action they both contribute and should be added up in the path integral. The result for the entropy in the presence of $(-1)^{\sf F}$ is therefore

in the large energy limit. As explained above, knowing the actual phase of the oscillatory term requires including quantum effects, a problem we leave for future work. This result explains the reason why the black hole states do not contribute to the canonical ensemble with the insertion of $(-1)^{\sf F}$. We find within each energy window the difference between bosonic and fermionic states is large in magnitude, of order $e^{c\sqrt{\ell^{3}E}}$.⁷⁷7This expression looks like the Cardy formula in 2d and would be interesting if there was a two dimensional interpretation of this result from the CFT side. It might very well be a coincidence. Nevertheless whether most states are bosonic or fermionic depends also on the energy due to the oscillatory factor coming from the imaginary part of the action, making the sum over energies (involved in the fixed temperature ensemble) smaller than each individual term.

The contribution to the difference between fermionic and bosonic states, coming from the black hole studied here, is qualitatively consistent with Cherman:2018mya . The authors of Cherman:2018mya propose that without supersymmetry, large $N$ adjoint QCD in four dimensions has a large cancellation between bosonic and fermionic states. The partition function with an insertion of $(-1)^{\sf F}$ is found to grow with energy, with a rate slower than the partition function. This is similar to the prediction we find here for three dimensional conformal field theories.

In this section we make some comments regarding the generalization to five dimensional AdS, relevant to four dimensional conformal field theories. We consider a low energy theory with an action given by Einstein-Hilbert with cosmological constant $\Lambda=-6/\ell^{2}$, with $\ell$ being the five-dimensional AdS radius. For simplicity we consider the case without charge. The black hole metric with mass and angular momentum is given by

where we define $\rho^{2}=r^{2}+a^{2}\cos^{2}\theta+b^{2}\sin^{2}\theta$ and the functions

The boundary has the topology of $S^{1}\times S^{3}$ with $S^{1}$ being thermal time. The spatial three-sphere is parametrized by $\phi$ and $\psi$ which are $2\pi$ periodic and $\theta\in[0,\pi/2]$. When considering a solution with a Lorenzian interpretation one has to restrict $a^{2}<\ell^{2}$ and $b^{2}<\ell^{2}$. In the Euclidean path integral involved in computing the free energy or entropy, we allow these parameters to be complex. The two angular momenta $J_{1}$ and $J_{2}$ generate rotations around $\phi$ and $\psi$ respectively. The energy and angular momenta for this black hole are given by

The thermodynamic potentials, the temperature and the two angular velocities conjugated to the two angular momenta $J_{1}$ and $J_{2}$, are derived from imposing smoothness at the Euclidean horizon and are given by

The area of the event horizon that appears in the Bekenstein-Hawking entropy is given by $A=\frac{\pi^{2}(r_{+}^{2}+a^{2})(r_{+}^{2}+b^{2})}{2r_{+}\Xi_{a}\Xi_{b}}$. The on-shell action in the fixed temperature and angular velocity ensemble is $I=\beta E-A/4-\beta\Omega_{1}J_{1}-\beta\Omega_{2}J_{2}$. In the fixed energy and angular velocity ensemble, the on-shell action is instead $I=-A/4-\beta\Omega_{1}J_{1}-\beta\Omega_{2}J_{2}$.

In the fixed temperature ensemble we can compute the free energy with no insertion of $(-1)^{\sf F}$. The saddle point in this case contributing to the partition function is a black hole with $a=b=0$ and $r_{+}$ is computed as a function of inverse temperature $\beta$. The solution is only real for $\beta<\pi\ell/2$. Moreover the black hole solution dominates the ensemble only for even higher temperatures $\beta<2\pi\ell/3$. In the range $2\pi\ell/3<\beta$ thermal AdS dominate the ensemble and therefore $\beta_{\rm HP}=2\pi\ell/3$ is the Hawking-Page transition corresponding to the confinement-deconfinement transition of the dual gauge theory Witten:1998zw . At very high temperatures, the size of the black hole is $r_{+}\approx\ell^{2}\pi/\beta$ and the free energy is large and negative

and in a fixed energy ensemble the entropy grows as $S(E)\sim E^{3/4}$.

We now move on to the computation of $F_{\mathrm{spin}}$ and $S_{\mathrm{spin}}$. In this case (and really for black holes in any number of dimensions bigger than four) we have more freedom to decide how to implement the $(-1)^{\sf F}$ since there are more than one angular momenta. In our definition $J_{1}$ and $J_{2}$ are two generators that each rotate along an $\mathbb{R}^{2}\subset\mathbb{R}^{4}$. From the point of view of the field theory living in the four dimensional boundary of AdS₅, Lorentz invariance on $\mathbb{R}^{4}$ implies that $J_{1}$ and $J_{2}$ are either both half-integer (corresponding to fermionic states) or both integer (corresponding to bosonic states). In other words $2J_{1}=2J_{2}~{}{\rm mod}~{}2\mathbb{Z}$. Therefore we could implement the fermion parity operator either by $(-1)^{\sf F}=e^{2\pi{\mathrm{i}}J_{1}}$ or $(-1)^{\sf F}=e^{2\pi{\mathrm{i}}J_{2}}$. For concreteness, we choose to use $J_{1}$ and therefore we can set $\Omega_{2}=J_{2}=0$ on our black hole.

There are other notions of fermion parity operator in five dimensions. The group of rotations of $S^{3}$ can be written as ${\rm SO}(4)\approx{\rm SU}(2)_{L}\times{\rm SU}(2)_{R}$ and the two ${\rm SU}(2)$ factors are generated by $J_{L}=\frac{J_{1}+J_{2}}{2}$ and $J_{R}=\frac{J_{1}-J_{2}}{2}$. Therefore one could also define other parity operators such as $(-1)^{{\sf F}_{\sf L}}=e^{2\pi{\mathrm{i}}J_{L}}$ and $(-1)^{{\sf F}_{\sf R}}=e^{2\pi{\mathrm{i}}J_{R}}$. The free energy and entropy in the presence of such insertions can be computed in a similar way, but they would not be probing the quantum statistics of the black hole microstate which requires the $(-1)^{\sf F}$ insertion of the previous paragraph.

After these clarifications we implement the insertion of $(-1)^{\sf F}$ by setting $\beta\Omega_{1}=2\pi{\mathrm{i}}$ and $\beta\Omega_{2}=0$ for concreteness. The second condition implies $b=0$ and the solution for $a={\mathrm{i}}{\sf a}$ is

(There is also another solution with ${\sf a}=r_{+}$ and with $\beta=E=0$ which we ignore.) With (47), the temperature and energy are given by