On branches of the KS black hole

The Klebanov-Strassler black hole is a holographic dual to ${\cal N}=1$ supersymmetric $SU(N)\times SU(N+M)$ cascading gauge theory plasma with spontaneously broken chiral symmetry. The chiral symmetry breaking sector of the cascading gauge theory contains two dimension-3 operators and a single dimension-7 operator. The black hole solution constructed in arXiv:1809.08484 represents the end point of the instability triggered by the condensation of one of the dimension-3 operators. We study here all three branches of the quasinormal modes of the chiral symmetry breaking sector -- there are no additional instabilities beyond the one identified in arXiv:1012.2404. Thus, the Klebanov-Strassler black hole solution of arXiv:1809.08484 is the only one with homogeneous and isotropic horizon, perturbatively connected to the chirally symmetric Klebanov-Tseytlin black hole arXiv:0706.1768.


Introduction and summary
The Klebanov-Strassler (KS) black hole [1] is a holographic dual to a cascading gauge theory 1 [5] plasma in the deconfined homogeneous and isotropic state with spontaneously broken chiral symmetry. It is a "branch" of the Klebanov-Tseytlin (KT) black hole [3,[6][7][8][9]] -a holographic dual to a deconfined thermal equilibrium state of the cascading gauge theory with unbroken chiral symmetry -in that it is associated with the perturbative spontaneous chiral symmetry breaking (χSB). In other word, there is a critical temperature [2] T χSB = 0.54195 Λ , (1.1) with Λ being the strong coupling scale of the cascading gauge theory [3,10], below which the χSB fluctuations become perturbatively unstable. The endpoint of condensation of these fluctuations is the KS black hole [1]. The constructed KS black hole has some unusual features, when interpreted as a holographic dual of a thermal equilibrium state of the renormalizable 2 four-dimensional gauge theory: • (i) To begin, the KS black hole is actually neither thermodynamically nor dynamically stable -it has a negative specific heat, and correspondingly [11] an imaginary speed of the sound waves. The latter implies that the small inhomogeneities in the energy density and the pressure in the chiral symmetry broken phase of the cascading plasma amplify, destroying the homogeneity and isotropy of the corresponding thermal state.
• (ii) The phase transitions in the cascading gauge theory plasma look very different in canonical and microcanonical ensembles. From the canonical ensemble perspective, the KS black hole is an exotic object 3 -it exists only for T ≥ T χSB and has a higher free energy density as compared to that of the KT black at the corresponding temperature. It is thus irrelevant in the canonical ensemble. On the other hand, in the microcanonical ensemble, it is a dominant (more entropic) configuration below some critical energy density E χSB (associated with T χSB )in a constraint (to spatial homogeneity and isotropy) dynamical evolution it is the end-point of the relaxation associated with the χSB.
• (iii) The χSB instability occurs at a lower temperatures/lower energy densities than the confinement/deconfinement phase transition in the cascading gauge theory [3], Here the (canonical ensemble) transition proceeds from the deconfined chirally symmetric phase to a confined phase with spontaneous χSB. Still, T χSB is above the temperature of the thermodynamic/dynamic instability in the KT black hole plasma [13], While unusual, (i)-(iii) can be incorporated into a consistent physics story.
(i) and (ii) suggests that the end point of the χSB in the cascading plasma can simply be an inhomogeneous state; it would proceed in two steps: spontaneous symmetry breaking as in [2], followed by the relaxation to a spatially inhomogeneous equilibrium state (so far unknown). There are plenty examples of such phenomena (albeit in the presence of the chemical potential/charge density), see [14] for examples in the Nambu-Jona-Lasino-type models, and [15] for a holographic model.
(iii) suggests that the confinement and the chiral symmetry breaking can be two separate transitions, see [16] for an example in a phenomenological QCD model and [17] for a holographic example. Despite (1.2), the cascading gauge theory χSB phase transition can still be important as it is the second-order (thus being a perturbative one) phase transition, while the confinement transition of [3] is of the first-order, proceeding through the bubble nucleation, which is strongly suppressed for a large number of colors.
Alternatively, it is possible that there are other KS black holes, beyond the construction of [1], with the more standard thermodynamics and the pattern of the phase transitions. This is the question we would like to address in this paper. was identified in [2].
In what follows, we identify all the three branch of the χSB QNMs: B 3u , B 3s and B 7 . We find that the former two branches in the limit T ≫ Λ combine into a single branch representing ∆ = 3 QNMs of the AdS 5 -Schwarzschild black hole [18]. At high temperatures, the degeneracy between the two branches is broken by O(1/ ln T Λ ) effects. Following these branches to low temperatures, we recover the instability on the B 3u branch at T χSB (1.1), originally found in [2]. The other branch, B 3s , corresponds to the QNMs that remain stable for T > T χSB . The B 7 branch at high temperatures represents ∆ = 7 QNMs of the AdS 5 -Schwarzschild black hole [18]. The dispersion relation of its QNMs differ from that of the conformal modes by O(1/ ln T Λ ) effects. Similar to the B 3s branch, the B 7 branch corresponds to the QNMs that remain stable for T > T χSB . Thus, we conclude that the χSB instability of [2] is the dominant perturbative instability of the Klebanov-Tseytlin black hole for T ≥ T χSB . Of course, this is the instability of the lowest QNM on the B 3u branch -as one further reduces the temperature below T χSB , one expects developing instabilities of the excited QNMs on B 3u , as well as on the other two branches of the χSB sector.
In the rest of this section we present the numerical results for the spectrum of the QNMs on the three branches of the χSB sector. Specifically, we analyze q 2 ≡ k 2 /(2πT ) 2 of the QNMs at zero frequency, ω = 0, as one varies the Hawking temperature of the (green curves) at zero frequency ω = 0 and high temperature, see section 2.3. In the The fact that q 2 < 0 for all the QNMs implies that the chiral symmetry breaking In fig. 3 we follow the QNM branch B 3u to low temperatures. The results reported reproduce the analysis presented in [2]. We perform analysis in two computation schemes (see section 2. 2 Effective action, equations of motion and the boundary asymptotes for χSB QNMs of the KT black hole The five-dimensional effective action (KS) describing SU(2) × SU(2) × Z 2 states of the cascading gauge theory has been derived in [2]. This effective action contains as an on-shell solution the Klebanov-Strassler black hole constructed in [1]. The KS effective action allows for a consistent truncation to a U(1) chirally symmetric sector, reproducing the Klebanov-Tseytlin (KT) effective action [9]. The KT effective action contains as an on-shell solution the KT black hole, constructed 4 in [3]. The effective action for the chiral symmetry breaking sector, i.e., for the U(1) chiral symmetry breaking linearized fluctuations on top of the KT states of the cascading gauge theory, was derived in [2]. Only one branch (out of the total three branches) of the QNMs was identified and analyzed in [2]. The QNM instability found in [2] identified a bifurcation point for a branch of the Klebanov-Strassler black holes, perturbatively connected to the KT black hole. In this section we review the relevant effective actions, the equations of motion, and the boundary asymptotes for the χSB QNMs of the KT black hole. Next, we proceed to a careful analysis of the QNMs in the conformal (or high-temperature) limit, P 2 → 0. Although it is straightforward to identify the three branches of the QNMs (two for the operators of ∆ = 3 and a single one for the operator of ∆ = 7) in the strict P 2 = 0 (conformal) limit, constructing perturbative in P 2 expansions for ∆ = 3 branches is rather subtle: while the KT BH is a series expansion 5 in P 2 of the AdS 5 Schwarzschild black hole, we find here that the pair of ∆ = 3 QNM branches realize a series expansion 6 in √ P 2 -the appearance of the square root branch point, i.e., ± √ P 2 , naturally leads to breaking of the spectral degeneracy in the two ∆ = 3 QNM branches at finite P 2 . Once the QNM branches are constructed in the high-temperature (conformal) limit, it is straightforward to follow them to low-temperatures. These results are reported in section 1: see figs. 1-4.

The effective actions
The effective gravitational action describing SU(2)×SU(2)×Z 2 states of the cascading gauge theory is given by [2] S 5 = 108 is the Ricci scalar of the ten-dimensional background geometry with the one-forms g i are the usual forms of the T 1,1 (see [2] for explicit expressions), and R 5 is the five-dimensional Ricci scalar of See [13] for the KT black hole construction up to O(P 8 ) inclusive. 6 The same phenomenon was observed also for the normal modes of the χSB sector of the cascading gauge theory vacuum on S 3 in [19].

KT black hole and χSB quasinormal modes
The Klebanov-Tseytlin black hole [3] is a chirally symmetric (see (2.8)) solution of the effective action (2.1). The five-dimensional metric is where (2.18) The gravitational bulk scalars 9 {K, h, f 2 , f 3 , g} are functions of the radial coordinate x ∈ (0, 1). Asymptotically near the boundary (x → 0 + ), characterized by {f 3,2,0 , f 3,3,0 , f 3,4,0 , g 2,0 } (in addition to P, g 0 , a 0 ); while near the horizon The thermodynamics of the KT black hole was discussed in [3]; in what follows we will need only the expression for the Hawking temperature and the strong coupling scale of the theory, see [3,10], Using the KT background (2.17), and the plane-wave ansatz for the chiral symmetry breaking fluctuations, To determine the spectrum of the QNMs, one needs to solve (2.27)-(2.29) with "normalizable only" conditions at the bulk gravitational boundary, and the incoming-wave boundary conditions are the horizon [20]. Fixing (without the loss of generality) the normalizable coefficient of F to one, see (2.36) below, the above boundary conditions determine the spectrum of the QNMs: As in [2], we are not interested in the numerical data for the spectrum per se, rather, we would like to identify QNMs at the threshold of instability. To this end, we solve (2.33) (2.34) characterized (in addition toq) by Given a KT black hole solution at a certain temperature, we expect three distinct branches of the quasinormal modes -the χSB sector mixes in three gravitational modes {δf, δk 1 , δk 2 }, dual to a pair of dimension ∆ = 3 operators (the normalizable coefficients f 3,0 and k 2,3,0 ), and a single dimension ∆ = 7 operator (the normalizable coefficient k 2,7,0 ). Furthermore, each QNM branch has its own tower of excitations, with increasingly higherq, characterized by the number of nodes in the radial wavefunction.
In section 2.3 we identify each QNM branch at high temperatures, where particular linear combinations of {δf, δk 1 , δk 2 } decouple. We then follow the lowest QNM of each branch numerically to low temperatures. We employ two different computational schemes: SchemeA and SchemeB.
• SchemeA: as in [2], we set P = a 0 = g 0 = 1 as vary k s -this is a convenient regime to reach low temperatures.
• SchemeB: alternatively, we set a 0 = g 0 = 1 and k s = 1 b , P 2 = b -this is a convenient regime to reach high temperatures.
Results of these computations are collected in Figs. 1-4. We plot dimensionless quantities: as a function of , (2.40) in the computational SchemeA, or as a function of b in the computational SchemeB.

High-temperature (near-conformal) limit
There are three branches of the quasinormal modes associated with the χSB sector of the cascading gauge theory. In this section we explain how these branches are identified Exactly at b = 0 the KT BH is just the AdS 5 -Schwarzschild black hole: where the AdS 5 radius is The leading order O(b) corrections were discussed in [8]; here we follow the state-of-the-art construction of [13], done to O(b 4 ) inclusive: (2.43) The equations for {κ 2n , ξ 2n , η 2n , λ 2n , ζ 2n } decouple at each order n, see eqs. (2.16)-(2.20) of [13]. At order n = 1, κ 2 and ξ 2 can be determined analytically 10 : while the remaining functions η 2 , λ 2 , ζ 2 has to be determined numerically. We will need the asymptotes of these functions: in the UV, i.e., as x → 0 + , (2.45) in the IR, i.e., as y = 1 − x → 0 + , additionally, from (2.40), Assuming, at fixed ω, we find from (2.27)-(2.29) that at each order n ≥ 1 the equations of motion for {F n , K 3,n−1 , K 7,n−1 } decouple: Thus, we identify F 1 and K 3,0 with the gravitational duals to a pair of the dimension ∆ = 3 operators, and K 7,0 with the gravitational dual to the dimension ∆ = 7 operator.
We can now identify distinct branches of the quasinormal modes of the KT black hole: The B 7 branch is defined by the boundary conditions where we fixed to unity the normalizable coefficient of K 7,0 . This branch is analytic in the conformal deformation parameter b: for all k ≥ 0, F 2k+1 , K 3,2k+1 and K 7,2k+1 vanish identically. We construct the B 7 branch to order O(b) in appendix A.
A pair of ∆ = 3 QNM branches is defined by the boundary conditions where α 0 = 0 is a constant, which is fixed at the subleading order. As in (2.54), we fixed to unity the normalizable coefficient of K 3,0 . As we explain in appendix B, there are two possible choices of α 0 , differing by the overall sign -this leads to two possible values of q 1 ,

KS branches from explicit χSB fluctuations of the KT black hole
There is an alternative approach to identify spontaneous symmetry broken phases in holographic duals advocated in [21]. Unlike the analysis of the QNMs in the symmetry breaking sector, it does no identify exactly what mode becomes unstable, but it does determine the onset of the instability. The idea is simple. The homogeneous and isotropic "branches" of the symmetry-broken KS black hole connect to the KT black hole "trunk" wherever the parameters of the latter allow for a linearized normalizable fluctuations in the symmetry breaking sector. If, exactly at the onset of the instability, one turns on a non-normalizable coefficient for the fluctuations, their normalizable coefficients will necessarily diverge. Thus, a way to identify onset of the χSB instabilities of the KT black hole is to monitor for the divergence of the expectation values of the condensates (such as parameters {f 3,0 , k 2,3,0 , k 2,7,0 } in (2.33)-(2.35) ) as the KT hole temperature varies, provided we turn on the non-normalizable coefficient -here, a gravitational dual to one of the gaugino mass parameters [2].
The relevant set of equations is (2.27)-(2.29) with ω = k = 0. We use the computational SchemeA (see section 2.3), i.e., we set P = a 0 = g 0 = 1 and vary k s parameter of the KT black hole. It is possible to turn on two independent non-normalizable coefficients µ 1 and µ 2 in the chiral symmetry breaking sector: (3.1) where the normalizable near the boundary, i.e., as x → 0 + , coefficients are In the IR we have the asymptotic expansion as in (2.38).
Precisely how we turn on the non-normalizable parameters is irrelevant -what matters is that µ i 's are not simultaneously zero. We set µ 1 = 0 and µ 2 = 1. The numerical results for the normalizable coefficients dashed blue line (see the right panel) -this is the χSB instability identified in [2].
There are no other divergences for T > T χSB . Once again, we conclude that there are no additional KS black hole branches beyond the one identified in [2] and constructed in [1].
A B 7 branch of the QNMs of the KT black hole in the b → 0
The asymptotic expansions near the boundary, i.e., as x → 0 + , Note that where we used (2.47). The high-temperature approximation (A.13) to the QNM branch B 7 is shown as a dashed green line on the right panel of fig. 2. B B 3u and B 3s branches of the KT black hole QNMs in the √ b → 0 conformal limit We construct here B 3u and B 3s branches to order O( √ b). The relevant equations of motion at ω = 0 are Note that The high-temperature approximations (B.16) to the QNM branches B 3u /B 3s are shown as dashed black/red lines on the right panel of fig. 1.