Exact half-BPS Type IIB interface solutions II: Flux solutions and multi-Janus (0705.0024v1)

Abstract: Regularity and topology conditions are imposed on the exact Type IIB solutions on $AdS_4 \times S^2 \times S^2 \times \Sigma $ with 16 supersymmetries, which were derived in a companion paper. We construct an infinite class of regular solutions with varying dilaton, and non-zero 3-form fluxes. Our solutions may be viewed as the fully back-reacted geometries of $AdS_5 \times S^5$ (or more generally, Janus) doped with D5 and/or NS5 branes. The solutions are parametrized by the choice of an arbitrary genus $g$ hyper-elliptic Riemann surface $\Sigma $ with boundary, all of whose branch points are restricted to lie on a line. For genus 0, the Janus solution with 16 supersymmetries and 6 real parameters is recovered; its topology coincides with that of $AdS_5 \times S^5$. The genus $g\geq 1$ solutions are parametrized by a total of $4g+6$ real numbers, $2g-1$ of which are the real moduli of $\Sigma$. The solutions have $2g+2$ asymptotic $AdS_5 \times S^5$ regions, $g$ three-spheres with RR 3-form charge, and another $g$ with NSNS 3-form charge. Collapse of consecutive branch points of $\Sigma $ yields singularities which correspond to D5 and NS5 branes in the probe limit. It is argued that the AdS/CFT dual gauge theory to each of our solutions consists of a 2+1-dimensional planar interface on which terminate $2g+2$ half-Minkowski 3+1-dimensional space-time $\mathcal{N}=4$ super-Yang-Mills theories. Generally, the $\mathcal{N}=4$ theory in each Minkowski half-space-time may have an independent value of the gauge coupling, and the interface may support various operators, whose interface couplings are further free parameters of the dual gauge theory.

• The paper establishes a new family of regular multi-Janus half-BPS solutions by imposing rigorous boundary conditions on harmonic functions on a Riemann surface.
• It demonstrates an infinite class of solutions parameterized by the genus of hyperelliptic surfaces, featuring multiple asymptotic AdS₅×S⁵ regions.
• The study reveals novel dual CFT interfaces linking 3+1-dimensional gauge theories, enriching the AdS/CFT correspondence through detailed supergravity embeddings.

Overview of the Research on Exact Half-BPS Type IIB Interface Solutions

The paper under consideration presents a detailed study of solutions to the Type IIB supergravity field equations with exact half-BPS symmetry, specifically those on the $AdS_4 \times S^2 \times S^2 \times \Sigma$ (where $\Sigma$ is a Riemann surface with boundary) manifold. These solutions exhibit an $SO(2,3) \times SO(3) \times SO(3)$ symmetry and are characterized by regularity and topology conditions imposed on the exact solutions. The primary focus is the construction of a family of regular solutions using a hyperelliptic Riemann surface, leading to what the authors term "multi-Janus" solutions. These solutions generalize the well-known Janus and $AdS_5 \times S^5$ geometries.

Key Contributions

1. Regularity and Topology Conditions: The study rigorously outlines conditions ensuring that the solutions are non-singular. The form of these solutions is derived from two locally harmonic functions on the Riemann surface $\Sigma$. The authors impose Dirichlet and Neumann boundary conditions on $h_1$ and $h_2$, the harmonic functions defining the solution, to ensure physical acceptability and regularity on the boundary $\partial \Sigma$.
2. Infinite Class of Solutions: The authors construct an infinite class of solutions parameterized by the genus $g$ of the hyperelliptic surface. These solutions are the fully back-reacted geometries that generalize known solutions by integrating new topological structures, such as D5 and NS5 branes, into the framework.
3. Parametrization and Period Relations: For genus $g \geq 1$, the solutions are parameterized by branch points on the real axis and specific ordering conditions between these points and the real zeros of the Abelian differentials. A set of period relations further constrains the parameters of the complex zeros ensuring clean extensions to higher genus while adhering to the regularity criteria set forth by the authors.
4. Multifarious Asymptotic Regions: Each solution has $2g+2$ asymptotic $AdS_5 \times S^5$ regions, representing half-Minkowski spacetime embeddings. These regions are separated by a planar interface that serves as a boundary to each of these regions.
5. AdS/CFT Dual Gauge Theory: The researchers propose a dual conformal field theory (CFT) for each solution that comprises $2+1$-dimensional interfaces with $3+1$-dimensional $\mathcal{N}=4$ Super-Yang-Mills theories. These interfaces serve as termination planes of the half-$\mathbb{R}^{1,3}$, thereby enabling independent gauge couplings.

Implications and Future Directions

The solutions illuminated in this study have several significant implications for string theory and the AdS/CFT correspondence. The presence of multiple asymptotic $AdS_5 \times S^5$ regions in a single solution brings a novel perspective to understanding how different conformal field theories can interact through interfaces defined by these geometries. These solutions point towards rich structures in multi-boundary holography where several field theories may be connected through intricate supergravity backgrounds.

Moreover, since the solutions have homology spheres equipped with non-trivial RR and NSNS 3-form charges, this opens pathways to new interpretations of topologically and geometrically enriched dual CFTs. The connection with probe D5 and NS5 branes provides intriguing avenues for examining how these solutions can be seen as fully back-reacted versions of known supergravity embeddings.

Future work may explore the mathematical analysis of the moduli space of these hyperelliptic solutions, akin to the studies on instantons or magnetic monopoles. Furthermore, investigating the dual gauge theories could deepen the understanding of interface CFTs and their role in holography. The methodology and findings of this paper could stimulate further explorations into related supergravity configurations and the broadening of the AdS/CFT dictionary.