Abstract

We extend and complete our analysis arXiv:1608.02380 and study the induced world volume metrics and Hawking temperatures of all type IIB rotating probe Dp-branes, dual to the temperature of different flavors at finite R-charge, in the Ben Ami–Kuperstein–Sonnenschein holographic models including the effects of spontaneous conformal and chiral flavor symmetry breakdown. The model embeds type IIB probe flavor Dp-branes into the Klebanov-Witten gravity dual of conformal gauge theory, with the embedding parameter, given by the minimal radial extension of the probes, dual to the IR scale of conformal and chiral flavor symmetry breakdown. We show that when the minimal extension is positive definite, the induced world volume metrics of type IIB rotating probe branes admit thermal horizons and Hawking temperatures despite the absence of black holes in the bulk subject to the world volume and topology of the nontrivial internal cycle wrapped by the probe. We also derive the energy–stress tensor of the thermal probes and study their backreaction and energy dissipation. We show that at the IR scale the backreaction is nonnegligible and find the energy can flow from the probes to the bulk, dual to the energy dissipation from the flavor sectors into the gauge theory.

1. Introduction

Nonequilibrium steady states are of great interest in many branches of physics, such as heavy-ion physics (thermalization of the quark-gluon plasma), condensed matter physics (quenches of cold atom systems), and cosmology (nonequilibrium phase transitions and the Kibble-Zurek mechanism). Studying such states by standard field theory techniques is prohibitively difficult (except for special cases involving conformal symmetry or integrability); however, they can be easily constructed in string theory by using gauge/gravity duality with D-brane dynamics. Nonetheless, nonequilibrium states have been mainly constructed in gravity duals of conformal gauge theories and more general and realistic examples of such states including the effects of spontaneous conformal and chiral symmetry breakdown remain scarce.

In gauge/gravity duality, ref. [1–3] (see also ref. [4, 5]), the pure gravity dual of conformal gauge theory is produced by taking the near horizon limit of backreacting D3-branes at a smooth point of the transverse space described by the background metric. The extension of this solution to holographic QCD-like solutions includes the addition of flavored quarks to the pure gauge theory, by embedding additional flavor branes into the pure gravity dual, taking the probe limit, , ref. [6] (see also ref. [4, 5]). Such holographic setup including probes, ref. [7, 8], has been used to model flavor physics, ref. [6, 9–16], and quantum critical phenomena, ref. [17–24], complementary to other works on charged adS black holes, ref. [25–30]. Such phenomena have also been studied at zero temperature in refs. [31–33]. In order to construct nonequilibrium systems in such flavored holographic models, one then notes that when the flavored quark sector (dual to the probe flavor branes) admits finite temperature, while the pure gauge theory itself (dual to the pure gravity solution) is at zero temperature, the system describes nonequilibrium steady states. The prime examples of such nonequilibrium steady states have been constructed in ref. [34] (see also refs. [35–53]), by embedding type IIB probe flavor Dp-branes into the gravity dual of conformal gauge theory. (For , the probe fills all the dual spacetime directions and the gauge theory is given by conformal field theory (CFT). For , the probes are localized in some dual spacetime directions and the gauge theory is given by defect conformal field theory (dCFT)). In ref. [34] (see also refs. [35–53]) rotating probe -brane solutions have been constructed and shown that when spin is turned on, the induced world volume metrics on all type IIB rotating probe Dp-branes in admit thermal horizons and Hawking temperatures despite the absence of black holes in the bulk. It was then noted in ref. [34] that by taking into account the backreaction of the rotating Dp-branes solutions to the gravity dual of (d)CFT, one naturally expects all such type IIB branes to form mini black holes in the bulk, in the probe limit. By gauge/gravity duality, this means that when the flavor sector of the gauge theory gets R-charged, it becomes thermal and thereby the gauge theory couples to thermal objects and describes locally thermal vacua. Since the gauge theory itself is at zero temperature, while its flavor sector, given, for instance, by a magnetic monopole or a quark , is at finite temperature, it was concluded in ref. [34] such systems describe nonequilibrium steady states. However, by computing the energy-stress tensor, it has been shown in ref. [34] that the energy of the probe (defect) flavor brane will eventually dissipate into the bulk and form, with the large backreaction in the IR, a (localized) black hole in the bulk. By gauge/gravity duality, this means that the energy from the (defect) flavor sector will eventually dissipate into the conformal gauge theory ( SYM).

This analysis has been extended in refs. [54, 55], by studying the induced world volume metrics on rotating probe D-branes embedded in more general and realistic gauge/gravity dualities, including warped Calabi-Yau throats, refs. [56–58] (see also refs. [59–63]). The warped throat geometry is produced by placing regular D3-branes and D5-branes (fractional D3-branes) at the generic Calabi-Yau singularity—the conifold tip point, [64], and describes the gravity dual of gauge theory which breaks some supersymmetry and conformal invariance. The duality includes confinement and chiral symmetry breakdown in the regular IR, where it describes the Klebanov-Strassler (KS) solution, [56], and admits RG cascade in the singular UV, where it describes the Klebanov-Tseytlin (KT) solution, [57]. The duality is conformal in the singular UV without fractional D3-branes, where it describes the Klebanov-Witten (KW) solution, [58]. In refs. [54, 55] it has been shown that when spin is turned on, the induced worldvolume metrics on rotating probe Dp-branes () in gravity duals of gauge theories with conformal and chiral symmetry breakdown, refs. [56–58], also admit thermal horizons and Hawking temperatures despite the absence of black holes in the throat. However, it has been shown there that the scale and behavior of the horizon and temperature change dramatically with the scale of conformal and chiral symmetry breakdown, implying that horizons and temperatures of expected features form in the singular UV and not in the regular confining IR. It has been then noted in refs. [54, 55] that by taking into account the backreaction of the rotating brane solutions to the throat background, one expects such solutions to form mini black holes in the bulk, in the probe limit. By gauge/gravity duality, this means that when the flavor sector of the gauge theory gets R-charged, it becomes thermal and thereby the gauge theory couples to thermal objects. Since the gauge theory itself is at zero temperature, while its flavor sector is at finite temperature, it was concluded in refs. [54, 55] that such systems describe nonequilibrium steady states in the gauge theory conformal and chiral symmetry breakdown. However, by computing the energy-stress tensor of the rotating branes, it has been found there that the energy of the branes will eventually dissipate into the bulk and form, with the large backreaction at small radii, a black hole in the throat. By gauge/gravity duality, it was then concluded in refs. [54, 55] that the energy from the flavor sector will eventually dissipate into the gauge theory conformal and chiral symmetry breakdown ( SYM).

The aim of this work is to generalize and complete such previous analysis by studying thermalization in new embeddings dual to new gauge field theories. Our particular aim is to complete the analysis of ref. [55] by studying in its configuration thermalization for new embeddings involving the rest of probe branes in the theory. Thus by duality, our aim is to find novel examples and features of thermalization in new field theories with spontaneous conformal and chiral flavor symmetry breakdown.

The holographic model we consider is the Ben-Ami–Kuperstein–Sonnenschein (BKS) embedding model of D3- and D5-branes, [65] (see also ref. [66, 67]), dual to dCFTs. The model follows the earlier Kuperstein–Sonnenschein holographic embedding model of D7-branes, ref. [68] (see also ref. [69, 70] and ref. [71, 72]), dual to CFT, which was motivated by the Sakai–Sugimoto model of D8-branes, ref. [73, 74]. These models, ref. [65, 68], embed the probe Dp-branes and probe anti-Dp-branes into the KW gravity dual, ref. [58], of gauge theory (dCFTS and CFT) including fundamental quarks and antiquarks (see also refs. [75–77]—considering holomorphic embeddings of D7-branes into the KS solution dual to supersymmetric gauge theory without flavor chiral symmetry breaking, and refs. [78–83]—studying the backreaction of D7-branes in the KS & KW solutions). The quarks and antiquarks transform, respectively, in the fundamental representation of and , resulting in a non-Abelian gauge symmetry on the gravity side, corresponding to a global chiral symmetry on the field theory side. This chiral symmetry, together with the conformal flavor symmetry, get spontaneously broken, when on the gravity side the Dp-branes and anti-Dp-branes extend from the UV and merge smoothly into a single curved -brane in the IR where only one factor survives. The embedding then describes a probe -brane of U-shape where the probe extends from the infinite far UV boundary to the minimal extension in the IR from where bends back to the boundary. The minimal extension measures the minimal separation between the probe and the background branes in the near-horizon limit of at the conifold point and is dual to the IR scale conformal and chiral flavor symmetry breakdown which sets the VEV deformations in the dual gauge theory. In the U-shape configuration, the induced metric on the probe has no factor, and the embedding is dual to gauge theory conformal and chiral flavor symmetry breakdown. When the minimal extension shrinks to zero, the embedding forms a V-shape as it appears as a disconnected -brane/anti--brane pair, the induced metric on the probe is -like and the embedding is dual to gauge theory conformal and chiral symmetry restoration. The probe also describes a SUSY breaking -brane/anti--brane pair, merged in the bulk at minimal extension, and guarantees tadpole cancellation on the transverse space because the -brane and anti--brane differ only in orientation and the total charge annihilates on the transverse sphere. As discussed in §2, the BKS setup cannot be realized in the well-used background, but instead it can be accomplished in the KW background, . Furthermore, we modify the BKS models by turning on additional spin on the probes dual to exciting R-charge in the flavor sectors of the gauge theory.

The motivation of this work is the fact that in such new embeddings the induced worldvolume metrics on rotating probe D3- and D5-branes in KW, if describing black hole geometries, are then expected to give the Hawking temperatures dual to the temperatures of flavors now in dCFTs (unlike D7 in KW which is a CFT) with spontaneous conformal and chiral flavor symmetry breakdown. Since the gauge field theory itself is at zero temperature while its defect flavor sector is at finite temperature, such systems exemplify novel nonequilibrium steady states in dCFTs with conformal and chiral flavor symmetry breakdown. The motivation is thus to construct and classify all such novel examples of nonequilibrium systems in KW.

The main results we find are as follows. We find that when the embedding is U-like and additional spin is turned on, the induced world volume metrics on rotating probes describe the black hole geometry subject to the worldvolume and topology of the nontrivial internal cycle wrapped by the probe. By gauge/gravity duality, we then find that when the conformal and chiral flavor symmetry gets broken and the flavor sector gets in addition R-charged, it becomes thermal subject to the type of flavor and codimension of dual spacetime where the flavors fields reside. We thus find that such nonequilibrium states are classified according to the kind of field theory (CFT or dCFT) and type of flavor inserted into the theory. Nonetheless, by computing the energy-stress tensor of the U-like embedded thermal probes, we find there that the energy of the brane will eventually dissipate into the bulk and form, with the large backreaction in the IR, a black hole in the KW throat. By gauge/gravity duality, we thus find that the energy from the flavor sector will eventually dissipate into the gauge theory conformal and chiral flavor symmetry breakdown.

The paper is organized as follows. In Sec. 2, we review the type IIB BKS models. In Sec. 3, we modify the BKS models and compute the induced world volume metrics and Hawking temperatures of rotating probes. We also compute the energy–stress tensors of the thermal probes and analyse their backreaction and energy dissipation. In Sec. 4, we summarize and discuss our results with future outlook.

2. Review of the BKS Models

In this section, we first review the relevant aspects of the KW gravity dual of conformal gauge theory, ref. [58]. We then review the Kuperstein–Sonnenschein and BKS embeddings of U-like probe flavor branes in the KW gravity dual of CFT and dCFTs with spontaneous conformal and chiral flavor symmetry breakdown ref. [65, 68].

2.1. The KW Solution

The particular ten-dimensional holographic string solution we are interested in is the KW solution, [58], resulting from taking the near horizon limit of a stack of background D3-branes on the conifold point. The conifold is a singular and noncompact Calabi-Yau three-fold described, in terms of complex coordinates , by the constraint and three-form as:

The constraint equation in (1) describes a singular noncompact cone with symmetry , and the three-form in (1) is charged under the R-symmetry. The base of the cone is a five-dimensional Einstein manifold, , of topology , which can be identified by intersecting the constraint in (1) with the unit three-sphere .

The constraint equation in (1) can recast its form as:

Here, denotes the Pauli matrices with and . To solve Eq. (2), one may use unconstrained variables , where denotes complex scalars with . These scalars poses an additional global symmetry, which is quotiented by the symmetry via and , so that the global symmetry is .

To construct the gauge field theory of the solution, one may use and note the following points. First, since the conifold is a Calabi-Yau manifold, it preserves by the virtue of Killing spinor equations, one-quarter of the original supersymmetry. Thus, the solution has supersymmetry and one may associate chiral superfields with to the complex scalars . Second, further to the global symmetry discussed above, the solution has a symmetry. Third, the field theory exemplifies a quiver gauge theory. Namely, the field theory of D3-branes placed on the conifold tip point has a gauge symmetry where and transform in the () and () representation, respectively. Fourth, anomaly cancellation in requires and have R-charge , respectively. Thus, the gauge theory also includes a marginal superpotential which is uniquely fixed by the symmetries modulo an overall factor.

To construct the gravity dual of this gauge theory, one notes that Eq. (2) describes a cone over the base of coset space . One also notes that the can be explicitly parameterized by five Euler angels (, ) with (not written out here) whereby the metric on the conifold then takes the form [64]:

Here, , and . It is clear from (3) that the base is a bundle over . It is also clear from (3) that the base has the topology where the three-cycle can be parameterized by and the two-cycle by , , , with both and shrinking to zero size at the tip of the cone.

The alternative way of writing the metric on the conifold is [84]:

Here, is the radial coordinate of the conifold (1), and parameterize the , and are one-forms parameterizing the . The s can be represented by Maurer-Cartan one-forms via two matrices (not written out here) parameterized by and , respectively, which show that the is fibered trivially over the .

Taking the near-horizon limit of D3-branes at the tip of the cone over , gives the gravity dual, , of the superconformal gauge field theory described by the superfields and the superpotential , as above. The full ten-dimensional warped background metric takes the form [58]:

Here, the first term in (5) is the usual four-dimensional Minkowski spacetime metric, the second term is the metric on the conifold, given either by (3) or by (4), and . In order to have a valid supergravity solution, (5), the number of D3-branes, , has to be large and the string coupling, , has to be small, so that ; denotes the string scale. Here, the dilaton is constant, and the other nontrivial background field is a self-dual R–R five-form flux (not written out here).

To add flavored quarks to the pure conformal gauge field theory, one embeds probe flavor -brane(s) () into the gravity dual above. The action of a -brane is the sum of the DBI and CS actions and takes the general form:

Here, are the IIB R–R background fields coupling to the -brane world volume; is the gauge-invariant field strength with the world volume gauge field and the pullback of the NS–NS two-form background field, , onto the world volume, , the pullback of the ten-dimensional metric in string frame, and . Lastly, denotes the world volume coordinates, and is the -brane tension.

2.2. U-like Embedded Probe D7-branes Wrapping

As the first example, we consider the U-like brane embedding configuration of ref. [68], embedding U-like probe D7-branes into the KW gravity dual, corresponding to adding flavored quarks to its dual gauge theory with spontaneous breakdown of the conformal and chiral flavor symmetry. In the KW gravity dual, the probe D7-brane wraps the entire dual spacetime coordinates of in the 01234-directions, and the three-sphere parameterized by the forms in the 567-directions. Thus, the probe D7-brane wraps the spacetime, and the transverse space is given by the two-sphere parameterized by the coordinates and in the 89-directions. This all can be summarized and represented by the array:

Here, we note that, as are left-invariant forms, the ansatzes preserves one of the factors of the global symmetry group of the conifold, . Thus, one may assume that the coordinates and are independent of the coordinates. The embedding breaks one of , but by expanding the action around the solution, it can be shown that contribution from the nontrivial shows up only at the second-order fluctuations. Thus, one can assume, in classical sense, that and depend only on the radial coordinate, , of the conifold geometry.

The Kuperstein–Sonnenschein embedding of the D7-branes includes two choices. There is one choice where the D7-branes are placed on two separate points of the , with both of them stretched down to the tip of the conifold at where both the and shrink to zero size. The configuration produced, in this case, is V-like. There is another choice where the D7-branes are placed on the with both of them smoothly merged into a single stack somewhere at above the conifold tip point. The configuration produced, in this case, is U-like. In both of the U-like and V-like configurations, the D7-brane(s) wrap the spacetime, as in the array above. However, on the transversal , here are two different pictures. In the V-like configuration, the D7-branes look like two separate fixed points, while the U-like configuration results an arc along the equator. The position of the two points, setting the position of the D7-branes, depends on and with them smoothly connected in the midpoint arc at . In the gravity dual, this configuration produces a one-parameter family of D-brane profiles with the parameter setting the minimal radial extension of the probe—dual to the IR scale of symmetry breakdown.

By the choice of the world volume fields and , it is easy to derive the induced world volume metric from (5) in terms of (4) and obtain from (6) the action of the form: where with . The Lagrangian in (8) is invariant, and therefore one can restrict motion to the equator of the parameterized by and . In writing down the action (8) from (6), one has set and noted that since the KW gravity dual contains only R–R four-form fluxes, so there is no contribution from the CS part and therefore the D7-brane action is just the DBI action.

The solution of the equation of motion from the action (8) yields a one-parameter family of D7-brane profiles of the form:

The solution (9) describes two separate branches, a disconnected D7 and an anti-D7-brane pair, including the V-like configuration, when . On contrary, the solution (9) describes a single branch, including the U-like configuration, when the two branches merge at . Furthermore, when the configuration is U-like, taking the limit implies .

The solution (9) contains several important features. First, when the configuration is V-like, it is clear from that the induced world volume metric is that of , and the configuration describes the conformal and chiral symmetric phase. On contrary, when the configuration is U-like, the induced world volume metric has no factor and the conformal and chiral flavor symmetry of the dual gauge theory must be broken spontaneously. Second, by taking the asymptotic UV limit, , (9) describes two constant solutions . This gives an asymptotic UV separation between the branes as , and an asymptotic expansion as . The asymptotic UV separation between the branes is independent. The dual gauge field theory is a CFT, in which corresponds to a normalizable mode, a vacuum expectation value (VEV). The fact that is a modulus, or a flat direction, implies the spontaneous breaking of the conformal symmetry. The expansion reveals that a marginal operator has a VEV fixed by as , the fluctuations of which gives the Goldstone boson associated with the conformal symmetry breakdown. Third, the solutions , giving an -independent UV separation, make the brane antibrane interpretation natural. This is because the brane world volume admits two opposite orientations once the asymptotic points are approached. Fourth, the presence of both the D7 and anti-D7-brane guarantees tadpole cancellation and annihilation of total charge on the transverse , and it breaks supersymmetry explicitly with the embedding being nonholomorphic.

To this end, one also notes that the above setup cannot be embedded in the solution. This is because the contains no nontrivial cycle, and therefore the D7-brane will shrink to a point on the . This problem may be fixed by a specific choice of boundary conditions at infinity, but this turns out to be incompatible with the U-shape configuration of interest. In addition, tadpole cancellation by an anti-D7-brane is not required, since one has no 2-cycle as in the conifold framework.

2.3. U-like Embedded Probe D5-branes Wrapping

As the second example, we consider the U-like brane embedding configuration of ref. [65], embedding U-like probe D5-branes into the KW gravity dual, corresponding to adding defect flavored quarks to its dual gauge theory with spontaneous breakdown of the conformal and chiral flavor symmetry. In the KW gravity dual, the probe D5-brane wraps some of the dual spacetime coordinates, including , given by in the 014-directions, and the three-sphere parameterized by the forms in the 567-directions. Thus, in this example, the probe D5-brane wraps the spacetime, and the transverse space is given by the two-sphere parameterized by the coordinates and in the 89-directions. This all can be summarized and represented by the array:

By the choice of the world volume fields similar to the D7-brane above, and , it is easy to derive the induced world volume metric from (5) in terms of (4) and obtain from (6) the action of the form: where . Here, we note that the probe D5-brane wraps the same as the probe D7-brane above, and the Lagrangian density in (11) is the same as the Lagrangian density in (8) modulo .

The solution of the equation of motion from the action (11) yields a one-parameter family of D5-brane profiles of the form:

The solution (12) has the same general features and satisfies the same boundary conditions, as before. However, in this example, the UV separation is , instead of , in the previous example. Thus, (12) represents another U-shape solution, though this time, dual to dCFT, with the additional quark fields residing in just two out of the four spacetime dimensions, producing a codimension two defects. The expected near boundary behavior of (12) is that of a VEV deformation of a marginal () operator in one dimension.

2.4. U-like Embedded Probe D5-branes Wrapping

As the third example, we consider the U-like brane embedding configuration of ref. [65], embedding U-like probe D5-branes alternatively, refs. [66, 67, 85], into the KW gravity dual, corresponding to adding defect flavored quarks to its dual gauge theory with spontaneous breakdown of the conformal and chiral flavor symmetry. In the KW gravity dual, the probe D5-brane wraps some of the dual spacetime coordinates, including , given by () in the 0124-directions, and the two-sphere parametrized by , in the 56-directions. Thus, in this example, the probe D5-brane wraps the spacetime, and the transverse space is given by the three-sphere parametrized by the coordinates , , and in the 789-directions. This all can be summarized and represented by the array:

By the choice of the world volume fields , , and , it is easy to derive the induced world volume metric from the metric (5) in terms of (3) and obtain from (6) the action of the form: where . Here, we note that the probe D5-brane wraps this time the , unlike in the previous example, and the Lagrangian density in (14) is not the same as the Lagrangian density in (11).

The solution of the equation of motion from the action (14) yields a one-parameter family of D5-brane profiles of the form:

The solution (15) has the same of the general features and meets the same boundary conditions, as before. However, in this example, the UV separation is given by , with both corresponding to supersymmetric D5-brane embedding and preserving different supersymmetries. Thus, (15) represents another U-shape solution, though this time, supersymmetric and dual to dCFT, with the additional quark fields residing in three out of the four spacetime dimensions, producing a codimension one defect. The expected near boundary behavior of (15) is that of a VEV deformation of a marginal operator.

2.5. U-like Embedded Probe D3-branes Wrapping

As the final example, we consider the U-like brane embedding configuration of ref. [65], embedding U-like probe D3-branes into the KW gravity dual, corresponding to adding defect flavored quarks to its dual gauge theory with spontaneous breakdown of the conformal and chiral flavor symmetry. In the KW gravity dual, the probe D3-brane wraps some of the dual spacetime coordinates, including , given by in the 04-directions, and the two-sphere parameterized by , in the 56-directions. Thus, the probe D3-brane wraps the spacetime, and the transverse space is given by the three-sphere parameterized by the coordinates , , and in the 789-directions. This all can be summarized and represented by the array:

By the choice of the world volume fields similar to the probe D5-brane above, and , it is easy to derive the induced world volume metric from (5) in terms of (3) and obtain from (6) the action of the form: where . We also note that the Lagrangian density in (17) is, up to the radial factor, , the same as the density of the probe D5-brane in (14).

The solution of the equation of motion from the action (17) yields a one-parameter family of D3-brane profiles of the form:

The solution (18) has the same general features and satisfies the same boundary conditions, as before. However, in this example, the UV separation is , whereby the brane never wraps the internal cycle more than once, as . Thus, (18) represents another U-shape solution dual to dCFT, with the additional quark fields being “quantum mechanical,” depending only on time and not on space. The expected near boundary behavior of (18) is that of a VEV deformation of a marginal () operator in one dimension.

3. Induced Metric and Temperature on U-like Embedded Probes

In this section, we first modify the Kuperstein–Sonnenschein and BKS models reviewed above, by turning on additional spin on the U-like embedded probes dual to exciting R-charge in the flavor sector of the CFT and dCFTs with spontaneous conformal and chiral flavor symmetry breakdown. We do this by noting, in general, that global rotational symmetry implies conserved angular momentum. This means, in all our examples below, that the RHS of our field equations vanishes because the time derivative of our ansatzes solutions is constant, making our solutions stationary. We also note that on the conifold background (see Sec. 2) with global symmetry parametrized by (, ), such ansäze for the transverse world-volume scalars constitute exclusively the directions and consistent with rotations; with the other “oscillatory” directions for the transverse scalars fixed. We then consider the simplest case of single field ansäze for the world-volume scalars, with world-volume gauge fields turned off, , and compute the related induced world volume metrics and temperatures on the U-like embedded rotating probes.

3.1. Induced Metric and Temperature on U-like Embedded Probe D7-brane Wrapping and Spinning about

As the first and warm-up example in our study, we consider the U-like probe D7-brane model wrapping in reviewed in Sec. 2, and turn on additional spin for the probe, following closely [55]. We include additional spin degrees of freedom, dual to finite R–charge chemical potential, by allowing in our system the D7-brane rotate in the direction of the transverse with conserved angular momentum. Thus, in our setup, we allow to have time-dependence, such that , with denoting the angular velocity of the probe. This way, we construct rotating solutions. Hence, the world-volume field is given by , with other directions fixed.

In this section, therefore, we will study U-like embedded probe D7-branes spinning with an angular frequency . Thus, we will consider the ansatzes for the D7-brane world-volume field of the form , and . Using this ansatzes and the metric (5) in terms of (4), it is easy to derive the components of the induced world-volume metric on the D7-brane, , and compute the determinant, , resulting the DBI action (6), as:

Here, we note that by setting , our action (19) reduces to that of the probe D7-brane action in the Kuperstein–Sonnenschein model, (8). As in the Kuperstein–Sonnenschein model reviewed in Sec. 2, we restrict brane motion to the -direction of the transverse sphere and fix other directions constant. Thus, in our setup we let, in addition, the probe rotate about the .

From (19), the D7-brane equation of motion and solution read:

Here, we set and note that when , our solution (21) integrates to that of probe D7-brane in the Kuperstein–Sonnenschein model, [68], reviewed in Sec. 2, with the probe wrapping in . The solution (21) is parameterized by and describes probe D7-brane motion with angular velocity about the transverse , starting and ending up at the boundary. The probe descends from the UV boundary at infinity to the minimal extension dual to the IR scale where it bends back up the boundary. We also note that inspection of (21) shows that in the limit , the -dependent term is subleading and therefore the behavior of does not depend on . This means that in the large radii limit the solution in (21) gives the world-volume field of the Kuperstein–Sonnenschein model (see Sec. 2) with the boundary values in the asymptotic UV limit, , independent from . Nonetheless, we note that in the IR limit, , the -dependent term becomes large, and thus the behavior of may depend on . However, inspection of (21) shows that one can always choose so that in the IR the behavior of with is comparable to that of with . This means that for certain in the small radii limit in the IR the behavior of does compare to that of in the Kuperstein–Sonnenschein model (see Sec. 2), with in the IR limit , consistent with U-like embedding (see Figure 1).

Figure 1 

The behavior of the derivative of the D7-brane world volume field with respect to with , (black-solid), (blue-dashed), (gray-dashed), and (black-dashed) (up-left). The behavior of the component of the D7-brane induced world volume metric with , , (black-solid), , (gray-dashed), and , (blue-dashed) (up-right). The behavior of the world volume Hawking temperature of the probe D7-brane with and (black-solid), (black-dashed), (gray-dashed), (blue-dashed) (down-left). The behavior of the energy density of the probe D7-brane with and (black-solid), (blue-dashed), and (gray-dashed) (down-right).

To derive the induced metric on the D7-brane, we put the rotating solution (21) into the background metric (5) in terms of (4) and obtain:

Here, we note that by setting , our induced world-volume metric (22) reduces to that of the Kuperstein–Sonnenschein model, [68], reviewed in Sec. 2. In this case, for , the induced world-volume metric is that of , and the dual gauge theory describes the conformal and chiral symmetric phase. On the contrary, for , the induced world volume metric has no factor, and the conformal invariance of the dual gauge theory must be broken in such case. In order to find the world-volume horizon and Hawking temperature, we first eliminate the relevant cross term. To eliminate the relevant cross-term in (22), we consider a coordinate transformation:

The induced metric on the rotating D7-brane(s) then takes the form: