Abstract
We study the Cardy-like asymptotics of the 4d mathcal{N} = 4 index and demonstrate the existence of partially deconfined phases where the asymptotic growth of the index is not as rapid as in the fully deconfined case. We then take the large-N limit after the Cardy-like limit and make a conjecture for the leading asymptotics of the index. While the Cardy-like behavior is derived using the integral representation of the index, we demonstrate how the same results can be obtained using the Bethe ansatz type approach as well. In doing so, we discover new non-standard solutions to the elliptic Bethe ansatz equations including continuous families of solutions for SU(N ) theory with N ≥ 3. We argue that the existence of both standard and continuous non-standard solutions has a natural interpretation in terms of vacua of mathcal{N} = 1∗ theory on ℝ3× S1.
Highlights
For the first time it has become possible to study a black hole-counting index [1, 2] both in a Cardy-like limit [3,4,5] and in a large-N limit [6]
We study the Cardy-like asymptotics of the 4d N = 4 index and demonstrate the existence of partially deconfined phases where the asymptotic growth of the index is not as rapid as in the fully deconfined case
We argue that by varying the chemical potentials in the index, its Cardy-like asymptotics displays “infinite-temperature” Roberge-Weiss-type first-order phase transitions [16] between the fully-deconfined phase associated to black holes [10,11,12,13,14,15], and confined or partially-deconfined phases, with the latter possibly associated to new multi-center black objects
Summary
For the first time it has become possible to study a black hole-counting index [1, 2] both in a Cardy-like limit [3,4,5] and in a large-N limit [6]. We consider the SU(N > 4) cases, and in particular argue that taking the large-N limit after the Cardy-like limit should yield various partially-deconfined infinite-temperature phases Our investigation of this double-scaling limit leads up to a conjecture for the leading asymptotics of the index as displayed in (3.19). Despite the said shortcoming of the Bethe Ansatz approach for N > 2, we will still utilize it in section 4 by temporarily neglecting the continua of eBAE solutions This way we study the CKKN limit (with b = 1) of the index, and will compare the result with that obtained in section 3 from the integral expression.
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