Holographic QCD₃ and Chern-Simons theory from anisotropic supergravity

In this subsection, we review and collect the relevant content of the anisotropic solution in ten-dimensional type IIB supergravity in [12]. The remarkable anisotropic solution describes the bulk dynamics of $N_{c}$ D3-branes with $N_{\mathrm{D7}}$ D7-branes dissolved in the spacetime in the large $N_{c}$ limit and the D-brane configuration is given in Table 1.

As our concern would be the holographic duality, let us start with the type IIB supergravity action in string frame,

where the index $M$ runs over 0 to 9, $\kappa_{10}$ is the ten-dimensional gravitational coupling constant $2\kappa_{10}^{2}=\left(2\pi\right)^{7}l_{s}^{8}$. To obtain an anisotropic solution, the associated equation of motion to (2.1) can be solved by the following anisotropic ansatz in string frame,

where $\chi,\phi,\Omega_{S^{5}}$ refers to the axion, dilaton and the unit volume form of a five-sphere $S^{5}$. The parameters in the solution are given as follows,

where $L,g_{s},\lambda$ represents the radius of the bulk, the string coupling and the ’t Hooft coupling constant respectively. The solution (2.2) describes the black branes with a horizon at $u=u_{H}$ and the anisotropy in $z$ direction. There would not be new field in the boundary which is located at $u=0$ because the $N_{\mathrm{D7}}$ D7-branes do not extend along the holographic direction $u$. Since the dynamic of the axion $\chi$, which magnetically couples to $N_{\mathrm{D7}}$ D7-branes, is taken into account, it is clear that the supergravity solution (2.2) includes the backreaction of $N_{\mathrm{D7}}$ D7-branes to the D3-brane bulk geometry. We note that the $N_{\mathrm{D7}}$ D7-branes are distributed along $z$ direction with the constant distribution density $n_{\mathrm{D7}}=dN_{\mathrm{D7}}/dz$ according to the solution for $\chi$. So once the backreaction of $N_{\mathrm{D7}}$ D7-branes to the background geometry is included, it implies that $N_{\mathrm{D7}}/N_{c}$ is fixed in the large $N_{c}$ limit.

The regular functions $\mathcal{F},\mathcal{B},\phi$ in (2.2) depend on the holographic coordinate $u$ which must be determined by their equations of motion. However they are non-analytical in general. In order to avoid the conical singularities in the bulk, the Euclidean version of the bulk metric near the horizon,

must impose the period $\delta t_{E}$ to be $2\pi$. Hence it reduces to the formula of the Hawking temperature $T$ as,

Suppose the temperature is sufficiently large $T\rightarrow\infty$ (or equivalently $\delta t_{E}\rightarrow 0$), the functions $\mathcal{F},\mathcal{B},\phi$ can be analytically written as the series of $a$ which are given in the Appendix. We note this high-temperature analysis would be remarkably useful in the following sections of this work.

As it is known that the type IIB supergravity theory holographically corresponds to the $\mathcal{N}=4$ super Yang-Mills theory on D3-brane, it would be very straightforward to construct the D3-brane configuration or the $\mathcal{N}=4$ super Yang-Mills theory in order to obtain a non-supersymmetric and non-conformal dual theory by following the steps in the well-known [35, 36]. Specifically the first step is to take the spatial dimensions $y$ of the D3-brane to be compactified on a circle $S^{1}$ with a period $\delta y$. Therefore the dual theory is effectively three-dimensional below the Kaluza-Klein energy scale defined as $M_{KK}=2\pi/\delta y$. The second step is going to get rid of all massless fields other than the gauge fields, which is to impose respectively the periodic and anti-periodic boundary condition on bosonic and fermionic fields along $S^{1}$. Afterwards the supersymmetric fermions and scalars acquire mass of order $M_{KK}$ thus they are decoupled in the low-energy dynamics. So the dual theory below $M_{KK}$ becomes three-dimensional pure gauge theory. By keeping these in mind, the next step is to identify the bulk geometry that corresponds to this gauge theory. The answer can be found by interchanging the roles of $t$ and $y$ i.e. performing a double Wick rotation $t\rightarrow-iy,y\rightarrow-it$ to the metric presented in (2.2), which is

where we have renamed $y,z$ after the double Wick rotation⁴⁴4Performing $t\rightarrow-iy,y\rightarrow-it$ to the metric presented in (2.2) reduces to $$ds^{2}=\frac{L^{2}}{u^{2}}\left(-dt^{2}+dx^{2}+\mathcal{F}\mathcal{B}dy^{2}+\mathcal{H}dz^{2}+\frac{du^{2}}{\mathcal{F}}\right)+L^{2}\mathcal{Z}d\Omega_{5}^{2}.$$ Then we rename $y,z$ by $y,z\rightarrow z,y$ in order to obtain (2.6). . We note that in this notation the axion field becomes,

while $\phi,F_{5}$ remains and now $z$ is periodic as,

Using the formulas given in the appendix, we can obtain

The solution (2.6) represents a bubble geometry of the bulk which is anisotropic on $\left\{x,y\right\}$ plane and defined only for $0\leq u\leq u_{KK}$. The D-brane configuration for the bubble solution (2.6) is given in Table 2.

Here we have renamed $u_{H}$ as $u_{KK}$ since there is not a horizon in the bulk as it is illustrated in Figure 1.

Since the wrap factor $L^{2}/u^{2}$ never goes to zero, the dual theory would exhibit confinement according to the behavior of the Wilson loop in this bulk geometry. Besides, we can notice that the gauge theory on D3-brane becomes purely three-dimensional theory if the compactified direction $z$ shrinks to zero i.e. $\delta z\rightarrow 0$ or $M_{KK}\rightarrow\infty$. And this limit exactly corresponds to the high temperature limit in the black brane solution (2.2) so that in the limit of $\delta z\rightarrow 0$, the functions $\mathcal{F},\mathcal{B},\phi$ in (2.6) are also analytical as they are given in the appendix. Accordingly we are going to consider the case in the limit $\delta z\rightarrow 0$ throughout this work since our concern is the three-dimensional dual theory exactly⁵⁵5A safe statement is to further require $N_{\mathrm{D7}}/N_{c}\ll 1$ in the construction of the bubble geometry since the limit of $N_{\mathrm{D7}}/N_{c}\gg 1$would lead to pure D7-brane background which may have many issues in a holographic approach. To avoid those issues, we may consider that $N_{\mathrm{D7}}/N_{c}\ll 1$ with fixed $N_{\mathrm{D7}}/N_{c}\sim a$ in the large $N_{c}$ for the gravity background produced by multiple D-branes as in [59, 60] which means the anisotropy may not be very large in this setup..

The dual theory with respect to the bulk solution (2.6) is defined at the zero temperature limit $T\rightarrow 0$ since there is not a horizon i.e. $\delta t\rightarrow\infty$. To examine the dual theory, let us introduce a single probe D3-brane located at the boundary $u\rightarrow 0$. As we have discussed, in the bubble geometry (2.6), the low-energy modes in the dual field theory contain the gauge field only, so the effective action for such a probe D3-brane is given as,

where the tension of the D3-brane is given as $T_{3}=\left(2\pi\right)^{-3}l_{s}^{-4}g_{s}^{-1}$. And $g_{ab},\mathcal{F}_{ab}=2\pi\alpha^{\prime}F_{ab}$ is the induced metric and the gauge field strength on the worldvolume of the D3-brane. Assuming $\mathcal{F}_{ab}$ does not have components along $z$ and does not depend on $z$, then the quadratic expansion of the action (2.10) is

where $\lambda_{3}=\lambda M_{KK}/\left(2\pi\right)$ is the three-dimensional ’t Hooft coupling constant and

is the Chern-Simons three-form. We use $A$ to denote the gauge potential and have imposed the boundary value $\phi_{\mathrm{bdry}}=0,\mathcal{F}_{\mathrm{bdry}}=\mathcal{B}_{\mathrm{bdry}}=1$ to (2.11). Clearly the dual theory on the D3-brane is effectively three-dimensional Yang-Mills plus Chern-Simons theory below the energy scale $M_{KK}$. Notice that the dual theory is expected to be purely three-dimensional theory if we take the limit $\delta z\rightarrow 0$ or equivalently $M_{KK}\rightarrow\infty$.

It is remarkable to notice that in this holographic setup, the level number of the Chern-Simons term $N_{\mathrm{D7}}$ is integer automatically since the level number is exactly the number of D7-branes in the gravity side. This leads to a proof of the quantization of the Chern-Simons level via holography. On the other hand, when the backreaction of the $N_{\mathrm{D7}}$ D7-branes is included in the bulk, the dual theory is equivalently a topological massive theory. This can be confirmed once we derive the formula of the propagator with respect to action (2.11) which is [38],

where $\kappa=N_{\mathrm{D7}}/\left(4\pi\right)$ and we use $p$ to denote the momentum in 2+1 dimensional spacetime. Therefore, the propagator defines the topological mass of the gauge field via the pole $p^{2}=\kappa^{2}g_{YM}^{2}$ which is determined by the numbers of D7-branes. Thus the presented D7-branes involve the topological properties of the dual theory and, we will see, they contribute to the various vacuum configurations in the dual theory.

In the holographic dictionary, one of the basic entries is the relation between the renormalized on-shell supergravity action and partition function of the dual field theory [1, 2, 35, 36]. Therefore the ground-state energy density $f$ of the dual field theory can be obtained through the relation

where $V_{3}$ refers to the infinite three-dimensional Euclidean spacetime volume. $S_{E,\mathrm{on-shell}}^{ren}$ is the renormalized on-shell action for type IIB supergravity, which is given by

where $S_{\mathrm{GH}},S_{\mathrm{CT}}$ refers to the Gibbons-Hawking term and holographic counterterm for the type IIB supergravity. $S_{\mathrm{IIB}}^{E}$ refers to the Euclidean version of (2.1) which in Einstein frame is given as,

Since only $F_{5}$ has components on $\Omega_{5}$, the onshell action $S_{\mathrm{IIB}}^{E}$ can be integrated out over $\Omega_{5}$ to become an effective five-dimensional action as,

where $M$ runs over 0 to 4. The cosmological constant is $\Lambda=-6/L^{2}$, $\mathcal{R}^{\left(5\right)}$ is the five-dimensional scalar curvature and $\kappa_{5}$ is the five-dimensional gravitational coupling constant. The action (3.4) is nothing but the five-dimensional axion-dilaton-gravity action. Hence the holographic counterterm can be chosen as [12],

where $h_{\mu\nu}$ refers to the boundary metric and $\mathcal{A}\left(h_{\mu\nu},\phi,\chi\right)$ refers to the conformal anomaly in the axion-dilaton-gravity system ⁶⁶6The counterterm in axion-dilaton-gravity system can also be found in [49, 50] . In this sense, the metric near the boundary is required to take the form as,

in which the coordinate $v$ is the standard Fefferman-Graham (FG) coordinate. The relation between the coordinate $u$ presented in (2.6) and the FG coordinate is collected as,

We also note that the standard Gibbons-Hawking term is given by the trace of the extrinsic curvature $K$ of the boundary as

Plugging (3.4) (3.5) (3.8) into (3.2) and using the relation (3.1), the resultant free energy density of the dual theory is computed up to $\mathcal{O}\left(a^{2}\right)$ as,

The free energy density depending on $a$ implies the ground-state is degenerate to the Chern-Simons coupling coefficient $\kappa=N_{\mathrm{D7}}/\left(4\pi\right)$. Since $a\sim dN_{\mathrm{D7}}/dy$ is the distribution density of the $N_{\mathrm{D7}}$ D7-branes, the value of $a$ could be same for different $\kappa$. Therefore the (3.9) describes the vacuum energy in various branches characterized by its level number $4\pi\kappa$. To any given interval, we fix the length $1/\left(4\pi\right)$ of $a$ for possible values of $\kappa$, then the ground-state free energy (3.9), which implies the vacuum with distinct $\kappa$, is shown in Figure 2.

And the true vacuum should minimize the free energy (3.9).

Figure 2 also illustrates that the behavior of ground-state energy is similar to the same observable in QCD₄ with finite theta angle, especially in some holographic approaches [24, 25, 26]. The reason is as follows. First, the black brane solution (2.2) and bubble solution (2.6) share same value of the onshell action in gravity side since the difference here between them is just the double Wick rotation. Then, on the other hand, the dual theory in the the black brane and bubble background is respectively four-dimensional theta-depended gauge theory and three-dimensional Yang-Mills-Chern-Simons given in (2.11). Therefore it is not surprised that their dual theories also share similar behavior of the ground-state energy according to the AdS/CFT dictionary (3.1) ⁷⁷7One may consider the dualities in string theory to study this similarity in another way, since the holographic investigation in [24, 25, 26] is based on the D4-brane approach in IIA string theory which is a T-duality version of IIB string theory.. In addition, we may find in (2.10) (2.11), the Chern-Simons three-form comes from the integral by part to the four-dimensional Wess-Zumino term in which the axion field plays exactly the role of the theta term as in QCD₄. Thus the similarity in the behaviors of the ground-state energy with respect to QCD₃ and QCD₄ may also be understood by this integral relation.

Besides, the degeneracy to the Chern-Simons level number in the ground-state free energy via holography may also have an interpretation in terms of quantum field theory (QFT). As we have analyzed, the dual theory on D3-brane is the three-dimensional Yang-Mills-Chern-Simons whose action is given by (2.11). So under the local $SU\left(N_{c}\right)$ gauge transformation

the Yang-Mills-Chern-Simons action $S_{\mathrm{YMCS}}$ presented in (2.11) transforms as,

where $W$ is the winding number given by

Therefore, under the gauge transformation, the partition function given by the Euclidean path integral is

up to a phase $e^{-8i\pi^{2}\kappa W}$. Comparing (3.13) with the AdS/CFT dictionary (3.1), it means the free energy must be degenerate to $\kappa$. This property of Chern-Simons theory also leads to an additive renormalization condition to the renormalized and bare Chern-Simons coupling coefficient

up to the one-loop order calculation at least, according to [38]. It also implies the renormalized parameter (denoted by $a$ in our system) is degenerate to the bare parameter (denoted by $\kappa$) up to a finite integer-valued shift.

In holography, the vacuum expectation value (VEV) of Wilson loop on a contour $\mathcal{C}$ corresponds to the renormalized Nambu-Goto on-shell action of a fundamental open string whose endpoints span the contour $\mathcal{C}$ [51] which is,

The static quark-antiquark potential $V$ can be obtained by evaluating the Nambu-Goto action as,

where $m$ refers to the bare mass of quark which is also the counterterm in the Nambu-Goto action. In the background (2.2), the bare mass can be evaluated by putting the fundamental open string extending along $u$ direction. Choosing $\tau=t,\sigma=u$, the induced metric on the worldsheet parametrized by $\left\{\tau,\sigma\right\}$ is written as,

Hence the bare mass is obtained by the Nambu-Goto action with respected to (3.17) as,

Next in order to compute the quark-antiquark potential, we consider a rectangular contour with sides of length $\mathcal{T},L$ along $t$ and one spatial direction $x$ or $y$. Notice the background metric is anisotropic in $\left\{x,y\right\}$ plane, the calculation of Wilson loop would be a little different when the string extends along $x$ and $y$ direction.

Since the entanglement entropy is expected to be a probe to characterize the phase transition in the field theory, in this subsection let us compute the entanglement entropy in the anisotropic background (2.6).

We will take into account the region A and its complement, the region B, as two physically disjoint spatial regions in the dual theory. Based on the AdS/CFT correspondence [39, 52], in the dual theory the quantum entanglement entropy between region A and B is identified to be the surface $\gamma$ stretched in the bulk whose boundary coincides with the boundary of A. In general the classical area of surface $\gamma$ in the correspondence of AdS_d+2/CFT_d+1 is given as,

where $G_{N}^{\left(d+2\right)}$ is the Newton constant in $d+2$ dimensional spacetime and $g_{\mathrm{ind}}$ is the induced metric on $\gamma$. In order to represent the entanglement entropy at a fixed time, surface $\gamma$ must be space-like. A natural generalization of (3.31) to the ten-dimensional geometry in string theory is

where the induced metric $g_{\mathrm{ind}}$ should be given in the string frame and the entanglement entropy can be obtained by minimizing the action (3.32). As the most simple case, we consider the “slab” geometry of A as $\mathbb{R}\times l$, however it would be straightforward to realize that the resultant entanglement entropy given by (3.32) depends on the configuration of the slab as we have seen in the cases of studying the Wilson loop, since the background metric (2.6) is anisotropic in the $\left\{x,y\right\}$ plane. Therefore let us proceed the calculations to obtain the entanglement entropy in two cases: parallel and perpendicular case, which is similar as what we have analyzed with the setup of Wilson loop in the previous section⁹⁹9It also provides a parallel setup to compute the holographic entanglement entropy in anisotropic supergravity background in [53, 54]. .

In the gauge-gravity duality, the baryon vertex is identified as a probe D-brane wrapped on the additional dimensions denoted by the spherical coordinates with $N_{c}$ open strings [43], and it can be treated as operator to create the baryon state¹⁰¹⁰10In the gauge-gravity duality, the baryon state is created by quantizing the baryonic brane somehow. While it may be a little tricky in IIB string theory, one could review the quantization of the baryonic brane in IIA string theory by the approach of instanton [61] and matrix model [62].. Accordingly, in the type IIB supergravity on $\mathrm{AdS}_{5}\times S^{5}$, the baryon vertex is a D5-brane wrapped on $S^{5}$.

To search for the stable wrapped configuration of D5-brane in the background (2.6), it would be straightforward to investigate the condition of the force balance for the baryon vertex. To begin with, let us decompose the metric on $\Omega_{5}$ by the coordinates of polar angle $\eta$ and $\Omega_{4}$, then define the radius coordinate $\xi$ as,

Therefore the metric (2.6) becomes,

where

which implies $\xi$ and $\eta$ refers respectively to the radius and polar angle in $\left\{\xi,\eta\right\}$ plane. Since the baryon vertex D5-brane extends along the directions of $\left\{t,\eta,\Omega_{4}\right\}$, the induced metric on a probe D5-brane is,

where the derivatives “ ^′ ” are with respect to $\eta$. To include the baryon potential, we turn on a single component of the gauge field on the D5-brane as $A=A_{t}\left(\eta\right)dt$, then the effective action for such a D5-brane is,