From multi-gravitons to Black holes: The role of complex saddles

The Hamiltonian definition of the $\mathcal{N}=1$ superconformal index on $S_{3}$ Romelsberger:2005eg ; Kinney:2005ej is

The $\mathcal{Q}$ and $\overline{\mathcal{Q}}$ are complex conjugated supercharges. $F$ is the Fermionic number. $J_{1}$ and $J_{2}$ are the left and right angular momenta on $S_{3}$. $Q$ is the $U(1)$ R-charge. The $J_{1,2}\,+\,\frac{Q}{2}$ are the two combinations out of $J_{1,2}$ and $Q$ that commute with $\mathcal{Q}$ and $\overline{\mathcal{Q}}$. The $Q_{g}$ denotes a generic global charge. The $p\,=\,{\bf e}(\sigma)$, $q\,=\,{\bf e}(\tau)$ and $t\,=\,{\bf e}(\Delta)$ are the rapidities associated to the charges $J_{1,2}\,+\,\frac{Q}{2}$ and $Q_{g}$, respectively.

The series coefficients of (1) in an expansion around $p=q=0$ (and for $t=1$) count the difference between bosonic and fermionic gauge invariant states in the cohomology of $\mathcal{Q}$, with fixed values of energy $E$. In an abuse of terminology, we will call the energy $E$ charge, as that is justified by the BPS relation that all relevant BPS states obey

For a theory carrying a representation $R$ of a gauge group $G$, which we assume to be a product of unitary groups

The $\mathcal{I}_{sl}$ denotes the same index (1) but computed after restricting the trace tr to run over gauge invariant operators that include only single fundamental fields ¹¹¹¹11For example, supposing the theory is Lagrangian these are the fundamental fields in the free limit of such Lagrangian. , and any number of derivative insertions, instead of over the entire Hilbert space of the theory. $[dM]$ is the invariant measure of integration over $G$ which is the product of Haar measures of each of the simple factors in $M$. $\chi_{R(M)}$ is the character of the representation $R$. If $\{{\bf e}{(u)}\}$ denotes the set of eigenvalues of $M$ then $\chi_{R}(U)\,=\,\sum_{\rho}{\bf e}(\rho(u))$ where $\{\rho\}$ is the set of weights of the representation $R$. From (3) one can recover the index of a generic theory from the single-letter indices of chiral and vector multiplets

For example, consider a $U(N)$ $\mathcal{N}=1$ SCFT composed by one vector multiplet and three adjoint chiral multiplets. The vector and chiral multiplets are assumed to be charged under the $U(1)_{R}$ superconformal $R$-charge. The chiral multiplets are assumed to carry the fundamental representation of the $SU(3)$ flavour symmetry group. In this case the superconformal index is a unitary matrix integral

where the variables $t$, $p$ and $q$ are constrained by $\frac{t_{1}t_{2}t_{3}}{pq}=1$ and initially one must assume $0<|t_{1,2,3}|,\,|p|,|q|\,<1$ in order to guaranty this representation of $\mathcal{I}$ to be well defined. The $\mathcal{I}$ can be analytically continued in $t$’s, $p$ and $q$ to larger regions. Interestingly, it turns out that $\mathcal{I}$ has singularities when $p=q$ becomes a root of unit. These singularities are important in the large-$N$ counting of BPS operators. Their neighborhood encodes the essential information needed to extract the asymptotic form of such numbers.

After integrating over angular variables, the integral (5) can be recast in terms of integrals of products of elliptic functions Dolan:2008qi Spiridonov:2010em

where for $G=U(N)$, $\text{rk}(G)\,=\,N$, and for $G=SU(N)$, $\text{rk}(G)=N-1$. Moreover, for $G=~{}SU(N)$ the $\xi^{-1}_{N}=\prod_{i=1}^{N-1}\xi_{i}$.  ¹³¹³13For generic theories similar expressions exist. From this expression and after algebraic manipulations it follows that $$\mathcal{I}_{SU(N)}\,=\,(p;p)^{-1}(q;q)^{-1}\,\mathcal{I}_{U(N)}\,.$$ (7) In particular, this implies that the answer at leading order in large $N$ expansion, is the same for both gauge groups. It is useful to note that the vector multiplet contribution can also be expressed in terms of elliptic Gamma functions by using the following identities

Sometimes it is convenient to express (6) in terms of the potentials $\underline{u}$, $\sigma$, $\tau$ and $\Delta$’s

where $\underline{u}=(u^{1},u^{2},\ldots)$ is an array that collects the $N$ Cartan angles. For the moment, we can consider the vector $\rho$ to be a weight of the adjoint of either $U(N)$ or $SU(N)$. For the case of $SU(N)$ we can consider $\rho$ to be a weight of $U(N)$ upon the imposition of the constraint $u^{N}\,=\,-\sum^{N}_{i=1}u^{i}$.

The $\Delta$’s encode the dependence on $R$-charges, and chemical potentials dual to global symmetries. They can be used to recover refined versions of the superconformal index. ¹⁴¹⁴14The point at which the flavour fugacities vanish $\Delta_{g}=0$ corresponds to $\Delta_{1}=\Delta_{2}=\Delta_{3}=-\frac{\tau\,+\,\sigma}{6}+\mathbb{Z}$. As pointed out in Cabo-Bizet:2019eaf for this value of the parameters there is a pole in $A\times B$ and thus in principle one would need to use the ABBV formula plus contributions coming from the non-vanishing residues. This correction seems to be essential at finite $N$. However, at large-$N$ we argue that in principle one can compute the integral (6) before imposing the constraint (10) and then recover the constrained result after using meromorphic flow in the $\Delta$’s. This is the approach we will follow in this paper. In forthcoming work, we will comeback to analyze the details of the contributions that come from residues. In these variables the constraint below (5) takes the form

As mentioned before, for a generic theory with gauge group $G$ of rank $\text{rk}(G)$, the index can be written as elliptic hypergeometric integrals as well Dolan:2008qi Spiridonov:2010em . The generic answer can be recast in the form

where the integration measure is defined as $d{\underline{u}}\equiv\prod_{i=1}^{\text{rk}(G)}du_{i}$. The $|\mathcal{W}|$ is the dimension of the corresponding Weyl group. The $S(u)$ is a zero-dimensional analog of QFT quantum effective action, and thus we will call it effective action from now on. The label $\alpha$ runs over all the multiplets in the theory. We define $z_{\alpha}\,\equiv\,\rho_{\alpha}(u)\,+\,\Delta_{\alpha}\,$ where $\rho_{\alpha}$ is the vector of charges, i.e a weight, that the multiplet $\alpha$ carries (under the Cartan elements of $G$). The complex number $\Delta_{\alpha}$ encodes both the dependence on the R-charges of $\alpha$ and the dependence on the set of potentials associated to global charges $Q_{g}$, as defined in (1). This parameter can be used to recover more refined versions of the superconformal index that are obtained by turning on global symmetry holonomies.

We find convenient to analyze the generic integrals like (11), without imposing constraints like (10). Namely for each multiplet $\alpha$ we associate a generic $\Delta_{\alpha}$ (even for the vector multiplets) and only after the evaluation of the observable one wishes to study, the $\Delta_{\alpha}$ can be fixed to specific values. For example, in the case of (6) we can work by default with $\Delta_{\alpha}\,=\,\{\Delta_{(1)},\Delta_{(2)},\Delta_{(3)},\Delta_{(4)}\}$ and in the very end, once a convergent result is obtained one can fix $\Delta_{(3)}$ from condition (10).

Let $A\times B$ be a real torus with $A$ and $B$ cycles. Let $e^{-S_{\lambda}}$ be a family of smooth complex functions in $A\times B$ labelled by a real parameter $\lambda\,$. Assume without loss of generality that $\lambda$ flows from zero to one.

Suppose that as $\lambda\,\to\,1^{-}$ the $e^{-S_{\lambda}}$ develops a discontinuity at an $A$-cycle of the torus. We can rephrase the previous statement by saying that in the limit $\lambda\,\to\,1^{-}$ the domain on which $S_{\lambda}$ was smooth, which used to be a torus $A\times B$ before the limit, becomes a cylinder as sketched in Figure 1. If the limit of $e^{-S_{\lambda}}$ is holomorphic, its integral over the cylinder collapses to a middle-dimensional integral over any path homologous to an $A$-period. Under such conditions, as it will be clear from the explanation we will give below, the limit of the integral over the original torus equals the integral of the limit function over any of the A-cycles. ¹⁶¹⁶16This last statement follows from the Lebesgue’s dominated convergence theorem (DCT) Baxter1996 . We assume integration to be defined in the Lebesgue’s sense. In summary, this theorem states that the limit of a Lebesgue integral coincides with the Lebesgue integral of the limit if the limit of the integrand is pointwise convergent, and the integral over the same domain of the absolute value of the limit is convergent. In particular, this implies that if the limit of the integrand is bound and the domain of integration is compact, the limit can always be commuted with the Lebesgue integral.

Now, think of the $A$-cycle in the previous paragraph as the Cartan torus of a gauge group. Let the torus $A\times B$ be a double copy of $A$. Suppose there exists a family of $e^{-S_{\lambda}}$ on $A\times B$ that flows to a holomorphic function living in the cylinder obtained by slicing $A\times B$ across an $A$-cycle. If the limit of $e^{-S_{\lambda}}$ matches the integrand of the superconformal index (or the integrand of any other integrals that one wishes to compute) at one of these $A$-cycles, any of them, we conclude that in such limit the integral over $A\times B$ converges to the superconformal index.

A trivial example of how an integral over a slicing of $A\times B$ collapses to a middle dimensional one over $A$ (assuming $\text{Im}(\Delta)>0$) is

From inspection of the position of poles of $\Gamma_{\text{e}}$ that are given in appendix A, one reaches the conclusion that there are no poles in the domain of integration of the left-hand side of (12). That implies that the integral over $u_{2}$ is spurious. The next obvious example is the $n=2$ one in the following family

where $u_{1,2ij}=u_{1,2}^{i}\,-\,u_{1,2}^{j}\,$. This can be generalized further to integrals of the following kind (assuming $\text{Im}(\Delta_{\alpha})>0$)

with $\rho_{\alpha}$ being weights of a quiver gauge theory with matter in the fundamental, adjoint and/or bi-fundamental of say $U(N)$ gauge nodes. The analysis for $SU(N)$ nodes is more subtle, but one can always obtain it from $U(N)$ by using the identity (7).

What is the limitation of the previous approach?

As we will explain in due time, integrals over $A\times B$ of the smooth function $e^{-S_{\lambda}}$ can be computed by using the Atiyah-Bott-Berline-Vergne formula. The expressions take the following form

Here $n=\text{rk}(G)$ and the volume form $\int\frac{1}{n!}\omega^{n}\,=\,\int^{1}_{0}d\underline{u}_{1}\int^{1}_{0}d\underline{u}_{2}\,$, is defined in terms of the two-form $\omega\,=\,\sum_{i=1}^{n}du^{i}_{1}\wedge du^{i}_{2}$. The remaining objects will be properly defined in due time. Here we just want to draw attention to a specific point.

The main claim of the previous subsection is summarized by the following equality

However, as stated by the inequality in parenthesis, the limit $\lambda\,\to\,1^{-}$ of the previous subsection does not commute with the sum over $P$’s. The reason being that

This can be understood before doing computations. The quantity $E(\nu_{P})$, which will be properly defined below, is proportional to the square root of the determinant of the Hessian of $S_{\lambda}$, and inversely proportional to the determinant of the matrix components of the two-form $\omega$. As in that limit $S_{\lambda}$ becomes holomorphic, the Hessian develops anti-holomorphic zero modes. Thus, as $\det\omega$ is constant, equality (17) follows through.

Let us wrap up and move forward. The relation (33) can be used to compute unitary matrix integrals like (11) or (3) with the help of the Atiyah-Bott-Berline-Vergne integration formula. The first step is to define the following sequences of integrals

As said earlier, the action $S_{\lambda}$ will be defined in equation (82). The deformation parameter $\lambda$ ranges from $0$ to $1$, including $0$ but not $1$! As we will explain in details in section 2 the sequence of integrals (34) is computed exactly by the ABBV equivariant integration formula i.e.

where the $P$’s are configurations of complex eigenvalues that are fixed points of the moment maps defined by the real and imaginary parts of the action $S_{\lambda}$. For $0\,\leq\,\lambda\,<\,1$ they are also saddle points (or stationary phase points) of $S_{\lambda}$.

The factor in the denominator is an equivariant Euler class that equates to

For the examples we will analyze, half of the fixed-point (saddle-point) conditions can be written in terms of the objects $B^{i}(u)$ (See definition (84)). They take the form of Bethe Ansatz equations $B^{i}(P)=1$. This object does not seem to be the same as the Bethe operator of Benini:2018mlo , but it is closely related. ¹⁹¹⁹19A more detailed comparison will be left as future work.

We will assume $G=\otimes SU(N)$ and choose $S_{\lambda}$ and $\omega_{\lambda}$ in such a way that

Here $\partial/\partial\lambda$ means derivative with respect to $\lambda$ at fixed values of $P$.

For fixed points $P$ that are not continuously connected to other, the proof will go in two steps. First, we will show that for any such $P$

and thus $\frac{\partial\mathcal{I}_{\lambda}(P)}{\partial\lambda}\,=\,0$. The first equation follows from the fact that $\partial_{\lambda}S_{\lambda}(u)$ is proportional to a linear combination of fixed point conditions and thus it vanishes at any $P$ (not just for those with non-vanishing Hessian). The second condition follows trivially from the fact $\partial_{\lambda}B^{i}(u)\,=\,0$. Second, we will show that the position of that type of fixed points $P$ can not depend on $\lambda$ in virtue of ABBV theorem. That will complete the proof of (37) for such cases.

For a fixed point whose Hessian of the action vanishes, and thus it is continuously connected to others, the variation in $\lambda$ generates motion in the corresponding connected manifold of fixed points. As the ABBV formula includes integration over such manifolds, it follows that even in that case there is no dependence on $\lambda$.

The final conclusion is that in appropriate chambers of chemical potentials  $\dfrac{d\mathcal{I}_{\lambda}}{d\lambda}\,=\,0$ and thus in virtue of (33)

where the superconformal index $\mathcal{I}$, was defined in (11). At last, plugging (35) into (39) we obtain

As just mentioned, the $P$’s can be part of connected manifolds and in that case one needs to integrate over such components with the measure induced by the inclusion map of the corresponding submanifold in $A\times B$ upon the Liouville measure of $A\times B$. We will elaborate on this later on.

In the second part of the paper, the ABBV formula will be combined with Picard-Lefschetz method to study how complex eigenvalue configurations compete at leading order in the large-$N$ expansion of the microcanonical index. So far, that is the only case where numerical results have been reported in the literature Murthy:2020rbd ; Agarwal:2020zwm . The analysis will focus on the toy example of the case of $\mathcal{N}=4$ SYM with $\sigma=\tau$ and no flavour potentials. More general cases $\sigma\neq\tau$ can be analyzed by using the results in appendix B.

Before entering in details let us explain in simple terms the conclusions of our study. As we will explain below, the perturbative parameter controlling the competition among saddles will be the exponential of minus the Bekenstein-Hawking entropy of the Gutowski-Reall black holes. To ease presentation let us denote this parameter as $g\,\equiv\,e^{-\frac{A_{H}}{4}}$, where $A_{H}$ is a function of the single independent charge carried by this BPS black hole, and which from now on we identify with the very same charge of the dual operators in the field theory side. For charges of order $N^{2}$, it turns out that $g<<1$ for large enough values of $N$, and thus only the two saddle contributions dominate the complete sum over $P$. The contributions from other complex saddles to the microcanonical index are exponentially suppressed and then can be safely truncated when counting large operators. For small enough charges, the $g$ starts to approach one and indeed for charges of order $N^{\frac{2}{3}}$, $\log g$ becomes of order one. When the charges become of order one in the large-$N$ expansion ((.e for multi-gravitons) the $g\sim 1$ and then every known complex saddle contributes.

Next, we introduce some notation and explain our conclusions in more detail. Further details are postponed until section 5.3.

We will find that at large $N$ the microcanonical index

defined as the trace over $\frac{1}{16}$-th BPS operators with charge $E=N^{2}\,\mathfrak{q}$ for $\mathfrak{q}$ finite, is defined by the competition of only two fixed points $(1,0)$ and $(1,1)$ out of the many $(m,n)$’s. We should mention that the associated grand canonical index is a function of a single chemical potential $\tau$, namely

This quantity can be obtained from the more general ensemble $\mathcal{I}\,=\,\mathcal{I}(\Delta_{1},\Delta_{2},\Delta_{3},\sigma,\tau)$ (See (6)) after imposing the following linear constraint among chemical potentials $\Delta_{1}+\Delta_{2}+\Delta_{3}+\frac{\sigma+\tau}{2}\,=\,p$, $p\,\in\,\mathbb{Z}$. For instance $p$ can be fixed to an integer that we denote as $n_{0}=\pm 1$ e.g. $p=n_{0}=\pm 1$. The ensemble to be compared versus the numerical results obtained in references Murthy:2020rbd ; Agarwal:2020zwm corresponds to fixing $\Delta_{1}\,=\,\Delta_{2}\,=\,\Delta_{3}$, $\sigma\,=\,\tau$ and finally $p$ to a specific integer value. We will fix $p\,=\,n_{0}=-1$ and proceed.

The formula (35) predicts that the microcanonical index is a linear combination of contributions from each fixed point $P$ i.e.

The single contributions $d_{P}$ are extracted out of the grand canonical index by a contour integral in $\tau$-plane. We denote such integral as $C_{\eta}$. For each $P$, this integral can be decomposed in an integer combination of $6$ Lefschetz-thimbles in $\tau$-plane. To understand whether a fixed point $P$ contributes or not to $d(\mathfrak{q})$, we need to compute the intersection numbers of $C_{\eta}$ with its $6$ thimbels.  In that way one reaches a large-$N$ expansion of competing exponential terms the form

where the magnitude of the $(m,n)$-th exponential is controlled by the Morse function

and $I_{(m,n)}$ are two real functions. $H_{(m,n)}$ is the would-be entropy of the supposed-to-exist gravitational configuration dual to $(m,n)$ fixed points. ²²²²22As computed in section 4.3 of Cabo-Bizet:2019eaf . The $\ldots$ denote sub-leading contributions in the large-$N$ expansion at finite $\mathfrak{q}$. It denotes also potential contributions from other large-$N$ fixed points which are unknown to us at the moment.

The integer intersection numbers $\eta_{(m,n)}$ take three possible values

For instance for $(m,n)=(1,0)$ or $(1,1),$ $\eta_{(m,n)}\,=\,1$. ²³²³23The $(0,1)$ does not contribute at leading order in large $N$, so effectively we can take $\eta_{(0,1)}=0$. The sufficient condition for $(m,n)\neq(0,1)$ and $\chi_{1}(m+n)\neq 0$ to contribute, i.e for $\eta_{m,n}\neq 0$, is that $C_{\eta}$ must enclose its corresponding Cady-like point $\tau=-\frac{n}{m}$ Cabo-Bizet:2019eaf . For the chosen $C_{\eta}$ that condition is equivalent to

In that case the intersection numbers will be non vanishing i.e. $|\eta_{(m,n)}|=1$. The $(m,n)$ fixed points for which (48) does not hold, have vanishing intersection number $\eta_{(m,n)}=0$, and thus they do not contribute to the counting. That implies that replica ²⁴²⁴24The grand canonical index we study has periodicity $\tau\to\tau+3$. This implies that the effective action of saddle $(m,n)$ is mapped to the effective action of $(m,n+3m)$ by such transformation. We say then that they are replica of each other. fixed points of $(1,0)$ e.g. like $(1,3)$, do not contribute to the microcanonical index expansion defined by the contour $C_{\eta}$, as expected.

Then, at large enough values of $N$ with fixed and finite $\mathfrak{q}\,=\,\frac{E}{N^{2}}\,=\,\frac{\ell}{3N^{2}}$, and up to exponentially suppressed contributions, the prediction coming from the ABBV formula to the microcanonical index (45) is

Here the $\ldots$ denote sub-leading corrections in large-$N$ expansion. Note that in virtue of (46), the $(m,n)$ with $m>1$ are exponentially suppressed. We will call (49) the two-saddles approximation to the microcanonical index.

The Morse function $H_{(1,0)}$ coincides with the Bekenstein-Hawking entropy of the dual AdS₅ black hole Gutowski:2004ez . The cosine modulation

explains the oscillations of $d(\frac{\ell}{3N^{2}})$ as a function of $\ell$ at relatively large but still finite values of $N$ e.g. $N\sim 10$. These oscillations have been recently observed in the $U(N)$ index ²⁵²⁵25At leading order in large $N$ approximation the $U(N)$ and $SU(N)$ indices are equivalent. Differences come in at finite values of $N$. The two saddle approximation can be used to approximate the microcanonical index of both, $U(N)$ and $SU(N)$, finite values of $N$ e.g. $N\sim 10$. The approximation should be even better for larger values of the charge $\ell\,$, or equivalently for finite values of $\mathfrak{q}=\frac{\ell}{3N^{2}}$. This is consistent with the outcome of the fit made in Agarwal:2020zwm to their numerical results Agarwal:2020zwm ; Murthy:2020rbd . The reason is that the small parameter that controls the order of the corrections coming from other fixed points is $e^{-H_{(1,0)}(\mathfrak{q})}$; and this parameter increases when $\mathfrak{q}$ decreases, at fixed $N$. In particular it approaches $e^{-\pi N^{2}\mathfrak{q}^{3/2}}\,\sim 1$ when $\mathfrak{q}\to 0$. Roughly speaking, for values of charges $\ell$ (resp. $\mathfrak{q}$) smaller enough than $N^{2/3}$ (resp. $\frac{1}{N^{\frac{4}{3}}}$) we need to start considering contributions from other fixed points in order to increase precision. Implementing this procedure to refine the counting lies beyond the scope of this paper. Here we always assume that charges are large enough in such a way that the two-fixed point approximation to the oscillations is a good approximation (e.g. $\mathfrak{q}$ finite at large $N$). via explicit evaluation of the Fourier coefficients up to rank $N=10$ Murthy:2020rbd ; Agarwal:2020zwm .