Instanton condensation and a new phase of BPS black holes

Abstract

We analyse the 1/16-BPS superconformal index for BPS black holes at equal charge in $AdS_5 \times S_5$, uncovering evidence for a new instability in the microcanonical ensemble along the small black hole saddle. This is indicated by instanton condensation in the matrix model description of the index. This instability occurs for black holes of radius close to, but below, the scale at which black holes become `small', and implies a new dominant phase in this region. We propose a connection to the partially deconfined phase in the field theory dual description. This would resolve recent confusion about the location of the partially deconfined phase in the BPS phase diagram and promises new avenues for understanding confinement, partial deconfinement, and the encoding of colour degrees of freedom under the holographic map. We also motivate the importance of instantons in partial deconfinement from a matrix model perspective.

Figures (19)

• Figure 1: Cartoon depiction of the partially deconfined phase at large $N$. The colour degrees of freedom are split between $M$ confined and $N-M$ deconfined colours. There is effectively an $SU(N-M)$ confined sector and an $SU(M)$ deconfined sector.
• Figure 2: The three standard phases of large-$N$ gauge theories shown by Polyakov loop eigenvalue distributions, $\rho(\theta)$, at real coupling. At zero chemical potential, the uniform distribution (\ref{['fig:gww-uniform']}) corresponds to the confined phase. The distribution (\ref{['fig:gww-gapped']}) corresponds to the deconfined phase. The distribution (\ref{['fig:gww-ungapped']}) is described as non-uniform ungapped and corresponds to the partially deconfined phase. At nonzero chemical potential, the interpretation of these distributions can change Hanada:2025rca.
• Figure 3: The critical distribution marking the Gross-Witten-Wadia transition between gapped and ungapped phases. At zero chemical potential, this marks the boundary between partial and complete deconfinement. At finite chemical potential, this critical point might instead correspond to the condensation of particles with non-trivial gauge orbits, such as baryons.
• Figure 4: The confined (blue), deconfined (red), and partially deconfined (orange) saddles in the situation where the partially deconfined phase has negative heat capacity. This shows (\ref{['fig:pdec-energy-temperature']}) energy and (\ref{['fig:pdec-freeEnergy-temperature']}) free energy, respectively, against temperature. This depicts a first order phase transition in the canonical ensemble, from the confined to deconfined phases. In the microcanonical saddle, where we fix energy, the partially deconfined phase becomes stable. The partially deconfined saddle necessarily connects the confined and deconfined saddles.
• Figure 5: Phase diagram of the BPS black hole free energy against supersymmetric temperature $\tau$, made using the results of Ezroura_2022. The cusp around $\tau \approx 2.4$ marks the point at which the large and small black hole branches meet.