Following Black Hole States

TL;DR

This work develops a continuous-$N$ framework for tracking the spectrum of the one-loop dilatation operator in $\mathcal{N}=4$ SYM from the integrable large-$N$ limit to finite $N$, enabling a unified view of 1/16-BPS multigraviton states and non-MG quantum black hole states. By constructing the lightest non-MG BH at $N=2$ and following its evolution in $N$ at weak coupling—and conjecturally at strong coupling—the paper reveals a dichotomy: MG cohomology yields protected, $N$-independent dimensions, while BHs acquire nonzero energy at large $N$ that typically vanishes at specific integer $N$ due to trace relations. The analysis combines the continuous-$N$ dilatation operator $\mathbb{H}$, the Wick contraction matrix $\mathbb{W}$, and explicit constructions such as an entanglement operator $\overleftrightarrow{\mathbf{E}}$ to express BH states, notably a lightest BH that decomposes into Konishi dressed by gravitons at large $N$. The results illuminate how heavy operators interpolate between integrable string-like regimes and gravity-dominated regimes, offering insights into AdS/CFT black-hole physics, non-planar effects, and the possible strong-coupling fate of BH-like states, while suggesting several avenues for further exploration, including higher-$N$ BHs, index analysis, and localization techniques for strong coupling.

Abstract

We study $\mathcal{N}=4$ SYM at non-integer number of colours. By varying $N$ we can continuously follow states all the way from $N=\infty$ where integrability reigns to finite $N$ where quantum gravity effects dominate. As an application we consider classically $1/16$ BPS states. Quantum mechanically, these states are generically non-supersymmetric but some special states - at special values of $N$ - become super-symmetric at the quantum level as well. They are the so-called quantum black hole states studied recently using cohomology. We write down the form of the lightest BH state at $N=2$ - and follow it in $N$, both at weak coupling and - more speculatively - at strong coupling as well. At weak coupling this state has protected dimension $Δ=19/2$ at $N=2$ and becomes a triple trace made out of Konishi and two light BPS operators at infinite $N$ with $Δ=19/2+12λ+\dots$. At strong coupling we suspect it becomes a quadruple trace with dimension $Δ\simeq 19/2+\text{integer}$.

Figures (10)

• Figure 1: At large $N$ there are 3 MGs and 3 non-protected states. This counting persists for all $N\ge 6$. As we decrease $N$ some of these states start decoupling (drawn as dashed lines in the figure) due to trace relations. At $N=2,3,4$ we have $1,3,5$ physical states respectively -- with a positive norm and non-negative energy.
• Figure 2: Energies of the 69 states in the BH sector described in the main text. There are several level repulsions (and some accidental crossings which will likely get resolved at two-loops, see Appendix \ref{['crossingSU2']}). Some things are similar to the two scalars toy example -- for instance, states annihilating extremely close to integers where the trace relations kick in -- but there is an obvious novelty here: There is a state whose energy goes to zero at $N=2$. This is the so-called $1/16$-th $N=2$ black hole state identified by the blue circle in the figure. This state flows to an entangled triple trace of a Konishi multiplet operator and two gravitons at large $N$ (\ref{['konishiBH']}). The inset shows the same plot in a bigger range and interestingly the black hole annihilates with the second excited state -- which is a single trace at large $N$ -- a bit to the left of $N=1$.
• Figure 3: At $N=2$ the BH state is a $1/16$-BPS state whose energy is zero at any coupling (blue line). The green line is a sketch of the weak coupling interpolation in section \ref{['sec:BH']}. At infinite $N$ the BH state is a triple trace state made of Konishi and two gravitons. As we crank up the coupling at infinite $N$, the anomalous dimension follows Konishi Gromov:2009zb (red solid line). If we stick to large but finite$N$ as we go to strong coupling, the state will probably follow a stair-case pattern like figure \ref{['stairs']} and eventually plateau at an approximate MG state (dashed red line). The energy at strong coupling will be a finite integer away from the classical dimension $19/2$ of the BH state. Then, if we follow the state to small $N$ it should again dive to zero at $N=2$ as depicted by the dotted pink line; it would be important to find a new idea which would allow us to say something quantitative about this strong coupling interpolation.
• Figure 4: On the left cartoon, the blue solid lines represent the planar dimensions of non-protected states -- whose energy blows up at strong coupling -- and the red dashed lines correspond to products of protected traces (gravitons). At large but finite $N$ mixing will resolve the level crossing and we end up with something as sketched on the right. (Not all horizontal lines get repelled - some stay protected by SUSY and those are precisely the MGs; we are not drawing those.)
• Figure 5: The gray lines are the one-loop anomalous dimensions and the orange lines are the eigenvalues of $H+\epsilon H_2$ for $\epsilon=0.01$. There is an accidental one-loop level crossing, highlighted by the dashed red circle. At two-loops, the levels are instead repelled as seen in the inset.