Magnetic Brane Solutions in AdS

Abstract: We construct asymptotically AdS_5 solutions of Einstein-Maxwell theory dual to N=4 SYM theory on R^{3,1} in the presence of a background magnetic field. The solutions interpolate between AdS_5 and a near horizon AdS_3\times T^2. The central charge of the near horizon region, and hence low temperature entropy of the solution, is found to be \sqrt{4\over 3} times that of free N=4 SYM theory. The entropy vanishes at zero temperature. We also present the generalization of these solutions to arbitrary spacetime dimensionality.

Overview of Magnetic Brane Solutions in AdS

The paper focuses on configurations within the Einstein-Maxwell theory in asymptotically AdS$_5$ spacetimes, associated with $\mathcal{N}=4$ supersymmetric Yang-Mills (SYM) theory on $\mathbb{R}^{3,1}$ under the presence of a magnetic field. A key finding is the interpolation between AdS$_5$ and a near-horizon region AdS$_3 \times T^2$. This interpolation reveals intriguing thermodynamic properties, such as the entropy scaling in the low-temperature limit being a factor of $\sqrt{4/3}$ times the free $\mathcal{N}=4$ SYM theory's central charge. Importantly, the paper explores the generalization of these solutions to various spacetime dimensions, emphasizing insightful connections between theoretical predictions and thermodynamic characteristics.

Analytical Insights

The AdS/CFT correspondence allows the study of boundary gauge theories influenced by background electromagnetic fields via bulk AdS boundary conditions. The magnetic black brane solution in AdS$_4$ provides a foundational understanding with its applications to 2+1 gauge theories subjected to magnetic fields. In extending these solutions to the AdS$_5$ case, particularly in relevance to strongly coupled gauge theories under magnetic fields, the authors contribute an essential tool for holographic explorations of magnetic field effects.

The solutions are derived analytically, capturing the zero-temperature regime and requiring numerical integration for comprehensive characterization. Noteworthy is the relationship between these solutions and those developed in prior studies by Chamseddine, Sabra, and others, accounting for differences in compactification schemes and specifying the uniqueness of the present non-supersymmetric solutions.

Characteristics and Thermodynamics

The solutions reveal themselves as a smooth interpolation between a BTZ black hole factor at low temperatures and AdS$_5$ geometry at high temperatures. Here, the black brane entropy density transition between linear $T$ at low temperature and $T^3$ at high temperature stands out, in agreement with known results where $S_{\text{grav}} = \frac{3}{4}S_{\mathcal{N}=4}$ at high temperatures. The low temperature comparison, where $S_{\text{grav}}$ signifies an enhancement by $\sqrt{4/3}$ compared to free field theory computations, is especially enlightening, showcasing improved numerical agreement when a magnetic field is present.

Extensions to Other Dimensions

The paper's analysis extends the solutions to arbitrary spacetime dimensions, discerning between the odd and even dimensional cases. For odd dimensional AdS$_{d+1}$ spaces with maximal magnetic field rank, an equivalent interpolation is observed. In contrast, even-dimensional solutions parallel the AdS$_4$ case, harboring finite entropy at extremality, echoing the aforementioned thermodynamic features.

Theoretical and Practical Implications

The theoretical implications of these solutions are vast, paving the way for enriched understandings of thermodynamics in strongly coupled populated systems viewed via holography. Practically, these considerations may prove instrumental for understanding magnetic field settings present in contexts like RHIC and analogous condensed matter systems. The results depicted invite further explorations into phenomena such as conductivity, anomalous hall effects, and magnetohydrodynamics in strongly interacting systems.

Future inquiries may gravitate towards including charge and momentum densities in these solutions, potentially unveiling additional layers of complexity in how current flows manifest within corresponding boundary theories. Additionally, the pursuit of exact solutions, addressed in the appendix, highlights potential avenues for advancing analytical approaches within this physical frame.

In summary, this study proficiently extends the horizon of known AdS/CFT applications involving magnetic fields, with theoretical contributions on the central charge and practical outcomes pertinent to the thermodynamic analysis of associated quantum field theories.