Sakai–Sugimoto Model in Holographic QCD

• The Sakai–Sugimoto model is a holographic dual framework that geometrically realizes chiral symmetry and its spontaneous breaking in low-energy QCD.
• It employs a D4-brane background with probe D8–D8-branes to capture phase transitions, magnetic catalysis, and effective quark mass generation.
• The model leverages the DBI action to analyze electromagnetic responses, including nonlinear conductivity and Schwinger pair production in distinct phases.

The Sakai–Sugimoto model is a top-down holographic dual designed to capture essential features of large-$N_c$ $SU(N_c)$ gauge theories with chiral symmetry and its spontaneous breaking, reproducing many qualitative and quantitative aspects of low-energy QCD. In this construction, color degrees of freedom are realized via a stack of $N_c$ D4-branes wrapped on a spatial circle with antiperiodic boundary conditions for fermions, while $N_f$ probe D8- and $\overline{\text{D8}}$-branes are introduced to carry fundamental flavor. The model is notable for its geometrical realization of chiral symmetry breaking, its ability to incorporate external fields, and its adaptability to paper phase transitions, baryonic and mesonic structures, transport, and non-equilibrium phenomena.

1. Holographic Construction and Model Architecture

The Sakai–Sugimoto model is formulated in Type IIA string theory with $N_c$ D4-branes wrapping a thermal circle (size $M_{\mathrm{KK}}^{-1}$) and $N_f \ll N_c$ pairs of D8- and $\overline{\text{D8}}$-branes transverse to this circle. The low-energy dynamics of the color sector is governed by an effective five-dimensional gauge theory, and the introduction of the D8-branes defines the flavor sector in the quenched (probe) limit, allowing treatment without backreaction.

The background geometry is characterized by the (string-frame) metric: $$ds^2 = \left( \frac{u}{R_{D4}} \right)^{3/2} \left[ -dt^2 + dx_i dx^i + f(u)(dx^4)^2 \right] + \left( \frac{u}{R_{D4}} \right)^{-3/2} \left[ \frac{du^2}{f(u)} + u^2 d\Omega_4^2 \right],$$ with

$f(u) = 1 - \left( \frac{U_{KK}}{u} \right)^3,$

where $x^4$ is the compact direction (size $2\pi/M_{\mathrm{KK}}$). Chiral symmetry is encoded in the relative positioning and embedding profile of the D8–$\overline{\text{D8}}$-branes. Their smooth joining in the bulk indicates spontaneous breaking of $U(N_f)_L \times U(N_f)_R$ to the diagonal subgroup. The embedding dynamics is governed by the Dirac–Born–Infeld (DBI) action, which, in a generic background with external $B$-field, reads

$$S_{D8} = \mu_8 \int d^9\xi \, e^{-\phi} \sqrt{ \det(P[G_{\mu\nu} + B_{\mu\nu}]) }.$$

2. Flavor Branes, Chiral Symmetry, and the DBI Action

Chiral symmetry, both its spontaneous breaking and restoration, arises from the geometry of the D8-brane embeddings. In the low-temperature (confined) phase, only connected (U-shaped) embeddings are allowed by topology, enforcing broken chiral symmetry. At high temperature (deconfined phase), both connected (bent) and disconnected (straight) embeddings exist, corresponding to chirally broken and symmetric phases respectively.

Flavored dynamics is captured by the DBI action, where the probe brane embedding profile $u(\tau)$ (or, equivalently, $\tau(u)$, with $\tau$ parameterizing the $x^4$ direction) is determined by solving the classical extremality equations. The presence of background fields or worldvolume gauge fields modifies this embedding via their appearance inside the DBI square root. The energy of a string stretching from the tip of the brane ($U_0$) to $U_{KK}$ provides an effective constituent quark mass, which becomes a dynamic, field-dependent quantity,

$$M_q = \frac{1}{2\pi\alpha'} \int_{U_{KK}}^{U_0} du \sqrt{g_{tt}\, g_{uu}}.$$

3. External Magnetic and Electric Fields: Catalysis and Transport

Introducing a constant external magnetic field $H$ in the $x^2$–$x^3$ plane is implemented by turning on the worldvolume gauge field $A_3 = H x^2$ or, equivalently, the NS–NS two-form $B_2 = H dx^2 \wedge dx^3$. The resulting DBI action is modified with a magnetic field dependence inside the square root: $\sqrt{1 + H^2 \left( \frac{R_{D4}}{u} \right)^3}.$ The embedding profile is now determined by a first integral (constant of motion), for example as

$$u^4 \sqrt{1 + H^2 \left( \frac{R_{D4}}{u} \right)^3 } \frac{ \sqrt{f(u)} }{ \sqrt{ f(u) + (R_{D4}/u)^3 u'^2 / f(u) } } = U_0^4 \sqrt{1 + H^2 \left( \frac{R_{D4}}{U_0} \right)^3 } \sqrt{ f(U_0) }.$$

Applying an external magnetic field "bends" the branes further, increasing $U_0$ at fixed separation, thereby enhancing the chiral condensate and constituent quark mass. This effect, called magnetic catalysis, persists universally for all such Sakai–Sugimoto-type brane constructions.

External electric fields can be analyzed similarly, with gauge fields $A_1(t,u) = -Et + h(u)$ producing nonlinear pair-production and associated current in the disconnected phase. The field-dependent conductivity, extracted from the DBI action, is nonlinear in $E$: $$\sigma = \frac{4}{27} \lambda N_f N_c T^2 \left(1 + \frac{27}{32} \frac{E}{ \lambda \pi^3 T^3 } \right)^{1/3}.$$ In the chirally broken (connected) phase, the current vanishes unless additional charged sources are present.

4. Phase Diagram, Magnetic Catalysis, and Critical Temperatures

The interplay between temperature and magnetic field yields a rich phase diagram:

• At low $T$, the confined geometry enforces connected embeddings and broken chiral symmetry; the value of $U_0$ increases with $H$, continuously enhancing chiral symmetry breaking.
• At high $T$, both connected and disconnected solutions are available. The transition between these (chiral restoration) is first order and controlled by the difference in their regularized DBI actions,

$$\Delta S = S_{\mathrm{curved}} - S_{\mathrm{straight}}.$$

The line $\Delta S = 0$ defines the critical temperature $T_{\chi SB}(H)$. Numerically, $T_{\chi SB}$ increases monotonically with $H$ and saturates at large $H$, identifying a maximal critical temperature (for chiral restoration) determined by the field. The system thus exhibits a temperature–magnetic field ($T$–$H$) phase diagram with chiral restoration occurring at higher temperatures as $H$ increases, a clear signature of magnetic catalysis.

For other brane configurations (e.g., with different probe brane dimensions $p$ and quark worldvolume dimensions $n$), this behavior is universal, though the scaling of $T_{\chi SB}$ with $H$ depends on $p$ and $n$ via DBI scaling.

5. Generalizations: Electric Field Dynamics and Higher-Dimensional Probes

Inclusion of external electric fields requires worldvolume gauge potentials $A_1(t,u) = -E t + h(u)$. The DBI action encompasses both the profile and electromagnetic response: $$S_{D8} = C \int du\, dt\, e^{-\phi}\, G_{xx} \sqrt{ G_{tt}G_{xx}g_{uu} + G_{tt} h'(u)^2 - E^2 g_{uu} }.$$ In the deconfined phase, disconnected branes lead to nonzero currents via Schwinger pair production, and the resulting conductivity is highly non-linear. For connected (chirally broken) embeddings, current is suppressed except possibly by baryonic sources.

General Sakai–Sugimoto-type models, with $Dp$-brane probes, yield similar results: magnetic fields universally increase the brane "bending" and augment chiral symmetry breaking, and phase diagrams exhibit analogous qualitative structure albeit with parameter-dependent scaling in the critical temperatures.

6. Summary and Physical Implications

• The Sakai–Sugimoto model, via probe branes and DBI dynamics in a D4-background, provides a geometric and calculable realization of chiral symmetry breaking/restoration and its response to external electromagnetic fields.
• Magnetic fields catalyze increased chiral condensate and constituent quark mass, and raise the chiral symmetry restoration temperature; chiral symmetry is never restored in the confined phase, but a clear transition line exists in the deconfined phase, saturating at high $H$.
• Electric fields induce non-linear conductivity in the chirally symmetric phase and are suppressed in the broken-symmetry phase without baryonic charge.
• All these features extend to wider classes of brane probe models, confirming the universality of magnetic catalysis and the field-affected phase structure dictated by DBI dynamics.
• This holographic approach links geometrical brane embeddings with non-perturbative QCD phenomena, enabling computation of field-dependent phase transitions, transport, and mass generation effects in the strong coupling, large-$N_c$ regime.