Effective Action of the Baryonic Branch in String Theory Flux Throats

We discuss consistent truncations of type IIB supergravity on resolved warped deformed conifolds with fluxes. These actions represent the gravitational duals to the baryonic branch deformation of the Klebanov-Strassler cascading gauge theory. As an application, we demonstrate that the baryonic branch is lifted in cascading gauge theory plasma.


Introduction and Summary
A conifold Y 6 is a simplest non-compact Calabi-Yau three-fold [1]. It is a cone over a homogeneous five dimensional Einstein manifold T 1,1 = (SU(2) × SU(2))/U(1), with the U(1) being a diagonal subgroup of the maximal torus of SU(2) × SU (2). When a large number N ≫ 1 of D3-branes are placed at its tip, for large 't Hooft coupling g s N ≫ 1 their backreaction warps the conifold: Along with N-units of 5-form flux through T 1,1 , the resulting geometry is a consistent background of type IIB string theory, holographically dual to N = 1 four-dimensional superconformal SU(N) × SU(N) gauge theory [2]. The warped conifold can be deformed (without further breaking the supersymmetry) by wrapping M ≫ 1 D5-branes over the two-cycle of T 1,1 . In this case the supergravity background realizes the holographic dual to non-conformal N = 1 supersymmetric SU(N + M) × SU(N) cascading gauge theory [3] (KS). One the geometry side, the SU(2) × SU(2) × U(1) global symmetry of T 1,1 is broken to SU(2) × SU(2) × Z 2 . The conifold deformation parameter breaking U(1) → Z 2 represents the spontaneous chiral symmetry breaking in the confining vacuum of cascading gauge theory. The vacuum structure of N = 1 cascading gauge theories was studied in [4]. Precisely when N is an integer multiple of M, there is a baryonic branch of confining vacua. In fact, the KS vacuum (without mobile D3branes) corresponds to a special Z 2 symmetric point on this branch. A generic point on the baryonic branch breaks Z 2 . The supergravity dual to the baryonic branch of cascading gauge theory was constructed in [5] (BGMPZ): moving away from the Z 2 symmetric solution corresponds to a resolution of the KS warped deformed conifold.
The type IIB supergravity backgrounds constructed in [3] and [5] are supersymmetric, and thus are not suitable to address nonsupersymmetric questions in cascading gauge theory. Likewise, given the prominent role the KS warped throat geometries play in constructing de-Sitter vacua in string theory [6], one needs to understand generic nonsupersymmetric deformations of BGMPZ resolved warped deformed conifolds. The first step in this direction was taken in [7], where a five dimensional effective action describing the SU(2) × SU(2) × U(1) invariant sector of the warped conifold was constructed. This action includes five dimensional metric coupled to four bulk scalar fields. It was used to prove the renomalizability of cascading gauge theory [7], and detailed studies of thermodynamics and hydrodynamics of chirally symmetric phase of cascading gauge theory plasma [8][9][10]. In [9] it was shown that cascading gauge theory undergoes the first order confinement-deconfinement phase transition at a certain critical temperature T c . Furthermore, there is a critical point at T u = 0.8749(0)T c where the chirally symmetric phase becomes perturbatively unstable towards condensation of hydrodynamic (sound) modes [10]. To understand chiral symmetry breaking in cascading gauge theory plasma, in [11] we derived effective action corresponding to SU(2) × SU(2) × Z 2 invariant sector of the warped deformed conifold -here, three additional scalar fields are included compare to [7]. This effective action 1 was used to establish that chiral symmetry breaking fluctuations in cascading gauge theory plasma become tachyonic at T χSB = 0.882503(0)T c ; as a result, both confinement and the chiral symmetry breaking in cascading plasma occur simultaneously via the first-order phase transition at T c .
Comparing to the warped deformed conifold consistent truncation [11], the BGMPZ supersymmetric holographic renormalization group (RG) flow contains two additional scalar fields (a mode dual to a dimension two operator and a mode mode dual to a dimension four operator of the boundary cascading gauge theory). It is straightforward to perform Kaluza-Klein reduction of this enlarged gravity-scalar sector and produce a five-dimensional truncation of the resolved warped deformed conifold [15] 2 . Unfortunately, this action is not a consistent truncation away from the origin of the baryonic branch [15] 3 ; at the origin of the baryonic branch the truncation is consistent and is identical to [11].
The fully consistent SU(2) × SU(2) truncation of type IIB supergravity on resolved warped deformed conifold was constructed in [17] 4 (CF). In this paper we reproduce the derivation of the effective action [17], and point further consistent truncation to effective action [11]. We further discuss linearized fluctuations of CF effective action about SU(2) × SU(2) × U(1) symmetric warped conifold with fluxes consistent truncations of [7]. We recover consistent truncation of chiral symmetry breaking sector in cascading gauge theory plasma [11]. Lastly, we present linearized effective action describing baryonic branch deformation about SU(2)×SU (2)

Effective action
In this section, following [17] and [19] 5 , we reproduce the derivation of CF effective action of the resolved warped deformed conifold with fluxes. The offshoot is that the effective action derived in [17] is correct; moreover, we did not find any typos in the presentation.
We will work in the gravitational approximation to type IIB string theory, using 2 See also [16]. 3 I would like to thank Davide Cassani and Anton Faedo for bringing reference [15] to my attention, and pointing out the inconsistency of the truncation [16]. 4 Related discussion appeared in [18]. We will not attempt to verify [18] and relate it to earlier work, partly because the authors did not present the Chern-Simons part of the action in full generality. 5 Related work appeared also in [20].
the type IIB supergravity action. This action takes the form (in the Einstein frame) where M 10 is the ten dimensional bulk space-time, κ 10 is the ten dimensional gravitational constant. The form-field strengths, determined from the potentials {C 0 , B 2 , C 2 , C 4 }, satisfy the Bianchi identities: The equations of motion following from the action (2.1) have to be supplemented by the self-duality condition It is important to remember that the self-duality condition (2.3) can not be imposed at the level of the action, as this would lead to wrong equations of motion.
Appendix A contains our conventions regarding differential forms.

Metric ansatz and its dimensional reduction
We take the ten-dimensional spacetime M 10 to be a direct warped product M 5 × T 1,1 .
SU(2) × SU(2) symmetry requires that both the dilaton φ and the axion C 0 are 0-forms on M 5 . Their reduction on T 1,1 is trivial:

3-forms ansatz and their dimensional reduction
Most general SU(2) × SU(2) symmetric ansatz of NSNS 3-form field strength H 3 (solving the Bianchi identity (2.2)) is parameterized by a 2-form b 2 , a one form b 1 , two real 0-forms b J and b Φ , a complex 0-form b Ω on M 5 and a constant p, [17]: The field strength H 3 can be decomposed in a basis of left-invariant forms on T 1,1 (2.5): where we defined (2.14) Similarly, most general SU(2)×SU(2) symmetric ansatz of RR 3-form field strength F 3 (solving the Bianchi identity (2.2)) is parameterized by a 2-form c 2 , a one form c 1 , two real 0-forms c J and c Φ , a complex 0-form c Ω on M 5 and a constant q, [17]: The field strength F 3 can be decomposed in a basis of left-invariant forms on T 1,1 (2.5): where we defined Reducing RR 3-form contribution in (2.1) on T 1,1 results in expression equivalent to the RHS of (2.14) with the obvious substitutions:

5-form ansatz and its reduction reduction
Because of the self-duality condition (2. 3), special care should be taken in dealing with the reduction of the 5-form; furthermore, to reproduce correct type IIB supergravity equations of motion the 5-form topological term (the second line in (2.1)) must be replaced with [17] S IIB,top = − 1 8κ 2 where the third line is used to define L 5 and L 10 , and for a constant k. Note that neither L 5 nor L 10 contain 5-form degrees of freedom. The proper strategy in dealing with the 5-form self-duality condition was developed in [19], which we apply here.
Let's focus first on 5-from degrees of freedom. 5-form Bianchi identity (2.2) is solved with 21) and the 5-form part of the action (2.1) can be written as As with 3-forms, we can decompose 5-from into the basis of left invariant forms on T 1,1 : where we defined (2.34) The last identities in (2.24)-(2.33) are used to define (2.40) We can not substitute (2.36)-(2.40) directly into (2.22); rather, we supplement it with the following term 6 : (2.41) In the modified action S F 5 + S ′ F 5 , the self-duality constraints (2.36)-(2.40) arise as equations of motion: where (up to total derivatives) S kinetic

44)
6 This term is a total derivative on-shell.
where we defined Additional contribution to five-dimensional topological couplings comes from L 10 term in (2.19), which, up to total derivatives, takes form: (2.47)

CF effective action
Collecting ( (2.53) The equations of motion obtained from (2.48) are equivalent to type IIB supergravity equations of motion [21]. Thus, SU(2) × SU(2) symmetric effective action (2.48) provides consistent truncation of type IIB supergravity on resolved warped deformed conifolds with fluxes.

Consistent truncations to KS/KT effective actions
There is a consistent truncation of the SU(2)×SU(2) symmetric CF action to SU(2)× SU(2) × Z 2 sector describing warped deformed conifold with fluxes obtained in [11,12,15] with the non-vanishing CF fields identified as where the superscript KS corresponds to the parametrization of fields in [11].

Decoupling of linearized fluctuations of CF action around KT action
Here we characterize decoupled linearized fluctuation sectors about SU(2) × SU(2) × U(1) truncation of CF effective action: In what follows we focus on the last two fluctuation sets: the chiral symmetry breaking sector, (2.57) and the baryonic branch deformation sector, (2.58) We explicitly verified that with the identifications the effective action S χcb is equivalent to the effective action obtained in [11].
Effective action S baryonic is a new result. Remarkably, consistent truncation of the baryonic branch deformations around generic SU(2)×SU(2)×U(1) states of cascading gauge theory requires inclusion of a vector field δb 1 , in addition to the supersymmetric scalar modes δw and δb J identified in [5]. We also verified that effective action (2.58), reduced 7 with δb 1 = 0, is equivalent to the one discussed in [16]. Notice that S baryonic is invariant under the λ-gauge symmetry: for an arbitrary 0-form λ on M 5 . This gauge symmetry is simply a restriction of general λ-gauge transformations discussed in [17] to linearized (decoupled) fluctuations {δω, δb J , δb 1 } about SU(2) × SU(2) × U(1) states of cascading gauge theory. Gauge symmetry (2.60) can be used to completely eliminate δb J fluctuations. 7 As we emphasized earlier, such a reduction is not a consistent truncation.

Baryonic branch in cascading gauge theory plasma
As an application of the effective action (2.58), we study stability of the baryonic branch fluctuations in cascading gauge theory plasma [9]. We focus on geometries dual to thermal states of cascading plasma, and study the spectrum of the baryonic branch quasinormal modes of Klebanov-Tseytlin black hole [9,10]. We show that these modes remain massive for all accessible temperatures, i.e., for T ≥ T u .
First, we rewrite effective action (2.58) using the KS background metric (see (2.54)): As a result of a Weyl rescaling (3.1), for any p-forms A (p) and B (p) on M 5 . Thus, (2.58) is modified tô The background geometry dual to the deconfined homogeneous and isotropic phase of the cascading plasma is given by with {f 1 ,f 2 , K, h, f 2 , f 3 , g s ≡ e φ } being functions of r only. We focus on modes at the threshold of instability, thus, without loss of generality we assume 8 asymptotic solution where we presented the expansions only to leading order in the normalizable UV coefficients z 1 , b 2,0 .
where we presented the expansions only to leading order in the normalizable IR coef- In what follows we consider only the lowest quasinormal mode, which has monotonic radial profiles. We find that over all range of temperatures, the fluctuations (solid blue line) have q 2 < 0 -as a result, they are massive. The red dashed line represents the best fit to (the high-temperature tail of) the data. Notice that in the limit T ≫ Λ the cascading theory approaches a conformal theory with temperature being the only relevant scale, thus, in agreement with (3.16), q 2 must approach a constant in this limit.

A Conventions
A differential p-form A (p) in ten dimensions is defined as where A (p) I 1 ···Ip are form components in orthonormal ten-dimensional vielbein {E I } basis. A Hodge dual is defined according to ⋆ 10 E I 1 ∧ · · · ∧ E Ip = 1 (10 − p)! ǫ I 1 ···Ip I p+1 ···I 10 E I p+1 ∧ · · · ∧ E I 10 , Similarly, a differential p-form A (p) in five dimensions is defined as where A (p) i 1 ···ip are form components in orthonormal five-dimensional vielbein {E i } basis. A Hodge dual is defined according to in ten and five dimensions correspondingly.