[Paper Review] Holographic Gauge Theory with Maxwell Magnetic Field

Wung-Hong Huang|arXiv (Cornell University)|Apr 15, 2009

Black Holes and Theoretical Physics37 references3 citations

TL;DR

This paper constructs a holographic supergravity background for D4-branes in a background Maxwell magnetic field by compactifying M-theory with N M5-branes and applying a coordinate mixing transformation followed by Kaluza-Klein reduction, yielding an inhomogeneously deformed $AdS_6 \times S^4$ geometry. The key result is that the external Maxwell magnetic field enhances the quark-antiquark potential, with numerical analysis showing a negative potential energy shift due to the field, indicating stronger binding.

ABSTRACT

We first apply the transformation of mixing azimuthal with wrapped coordinate to the 11D M-theory with a stack N M5-branes to find the spacetime of a stack of N D4-branes with magnetic field in 10D IIA string theory, after the Kaluza-Klein reduction. In the near-horizon limit the background becomes the Melvin magnetic field deformed $AdS_6 imes S^4$. Although the solution represents the D-branes under the Melvin RR one-form we use a simple observation to see that it also describes the solution of D-branes under the Maxwell magnetic field. As the magnetic field we consider is the part of the background itself we have presented an alternative to previous literature, because our method does not require the assumption of negligible back reaction. Next, we use the found solution to investigate the meson property through D4/D8 system (Sakai-Sugimoto model) and compare it with those studied by other authors. Finally, we present a detailed analysis about the Wilson loop therein and results show that the external Maxwell magnetic field will enhance the quark-antiquark potential.

Motivation & Objective

• To construct a supergravity background dual to a strongly coupled gauge theory with an external Maxwell magnetic field, avoiding the need for negligible back-reaction assumptions.
• To demonstrate that a solution originally interpreted as a D-brane under a Melvin RR one-form flux also describes a D-brane under a physical Maxwell magnetic field.
• To investigate meson properties in the D4/D8 brane system under this magnetic background, comparing with prior literature.
• To analyze the behavior of the Wilson loop in the presence of the external Maxwell magnetic field and determine its effect on quark-antiquark potential.

Proposed method

• Apply a coordinate transformation mixing the azimuthal and wrapped coordinates in 11D M-theory with N M5-branes to generate a 10D IIA background with magnetic flux.
• Perform Kaluza-Klein reduction on the resulting geometry to obtain a D4-brane system with an external magnetic field in 10D IIA string theory.
• Take the near-horizon limit to obtain an inhomogeneously deformed $AdS_6 \times S^4$ background with a nontrivial gauge field profile.
• Use the D4/D8 brane setup (Sakai-Sugimoto model) to study meson spectra and compare with previous results in the literature.
• Compute the Nambu-Goto action for a string stretching between quark and antiquark in the magnetic background to analyze the Wilson loop and quark-antiquark potential.
• Perform a perturbative expansion in $B^2$ for the case where quarks are separated along $x=\pm L/2$, computing the first-order correction to the potential energy.

Experimental results

Research questions

• RQ1 Can a supergravity solution originally interpreted as a D-brane under a Melvin RR one-form flux also describe a D-brane under a physical Maxwell electromagnetic field?
• RQ2 How does the presence of a background Maxwell magnetic field affect the quark-antiquark potential in the Sakai-Sugimoto model?
• RQ3 What is the impact of the magnetic field on meson properties in the D4/D8 brane system?
• RQ4 Does the magnetic field enhance or screen the quark-antiquark potential, and how does this depend on the field orientation?

Key findings

• The supergravity solution dual to the D4-brane system with an external magnetic field is constructed without assuming negligible back-reaction, providing a more robust alternative to previous methods.
• The magnetic field enhances the quark-antiquark potential, with the potential scaling as $V(L) \propto -\sqrt{1 + B^2 r^2} / L^2$, indicating stronger binding at finite $B$.
• Numerical evaluation of the first-order correction to the Wilson loop energy yields $\delta V \approx -0.3785\, B^2$, showing a negative potential energy shift due to the magnetic field.
• The enhancement of the potential is direction-dependent, with stronger effects when the magnetic field is aligned with the quark-antiquark separation axis.
• The analysis confirms that the Maxwell magnetic field directly affects the color sector of the gauge theory, unlike previous approaches where only the flavor sector was coupled.
• The results are consistent with the idea that strong magnetic fields can catalyze chiral symmetry breaking and modify the confining dynamics in holographic QCD models.