Solving the Hagedorn temperature of AdS5/CFT4 via the Quantum Spectral Curve: chemical potentials and deformations

Abstract

We describe how to calculate the Hagedorn temperature of $$ \mathcal{N} $$ N = 4 SYM theory and type IIB superstring theory on AdS5 × S5 via the Quantum Spectral Curve (QSC) — providing further details on our previous letters [1] and [2]. We solve the QSC equations perturbatively at weak ‘t Hooft coupling λ up to seven-loop order and numerically at finite coupling, finding that the perturbative results can be expressed in terms of single-valued harmonic polylogarithms. Moreover, we generalize the QSC to describe the Hagedorn temperature in the presence of chemical potentials. Finally, we show that the Hagedorn temperature in certain deformations of $$ \mathcal{N} $$ N = 4 SYM theory (real-β and γi deformation) agrees with the one in $$ \mathcal{N} $$ N = 4 SYM theory at any value of λ.

The asymptotic form of the Hagedorn temperature in planar super Yang-Mills

“…Here we will review this analysis, mainly following the discussion in [24]. We then show that in the plane-wave limit, one finds the same first order correction for the Hagedorn temperature found numerically in [8] and using the supergravity dual in [14,15]. We further show that the zero-point energy leads to a correction to the second-order term in the expansion, but not the third-order term.…”

“…From the Y-system they computed T H up to the two-loop level at weak coupling [6], which previously was only known to one loop [9]. Switching over to the quantum spectral curve formalism [10][11][12] they extended their results to seven loops at weak coupling and found the asymptotic behavior numerically at strong coupling [7,8].…”

“…Recently, Harmark and Wilhelm have exploited the integrability of planar N = 4 SYM to make substantial progress in finding the coupling dependence of T H [6][7][8]. From the Y-system they computed T H up to the two-loop level at weak coupling [6], which previously was only known to one loop [9].…”

“…The first coefficient matches within the error bars the flat-space prediction c 0 = 1 √ 2π ≈ 0.39894, which follows from the original computation of the superstring Hagedorn temperature [13] and the AdS/CFT dictionary (cf [1] and references therein). After (1.1) in [8] was announced, the second term was derived analytically by Maldacena and Urbach [14,15], where they found that c 1 = 1 2π ≈ 0.159 155. We will rederive this last result and also conjecture the analytic result for the next two terms in the expansion.…”

“…To test the conjecture we take the QSC prescription described in [8] and go to higher orders in the series expansions for the Q-functions, as described in section 4 and explicitly given in (4.10), to improve the numerical estimates. We find…”