Supergravity and D-branes Wrapping Supersymmetric 3-Cycles

TL;DR

This work builds holographic descriptions for D3-branes wrapping supersymmetric 3-cycles, focusing on associative cycles in $G_2$ manifolds and their twisted worldvolume theories. Using a five-dimensional ${\cal N}=4$ gauged supergravity truncation with gauge group $SU(2)\times U(1)$ to realize the twist, it derives BPS equations and shows supersymmetric $AdS_2\times L$ backgrounds require $L$ to be a constant negative curvature space, notably $H^3$ and its quotients. It constructs a regular domain-wall solution interpolating between $AdS_5$ in the UV and $AdS_2\times H^3$ in the IR, and analyzes the IR fixed point at the intersection of the Higgs and Coulomb branches, with singularities classified as 'good' or 'bad' via the Gubser criterion. For negative curvature $L$ the Higgs and Coulomb sectors decouple appropriately, while for positive curvature ($S^3$) all singularities are 'bad' and a Higgs branch is absent, revealing geometry-driven differences in IR dynamics of twisted D3-branes. Overall, the paper elucidates how the ambient cycle geometry governs IR fixed points and moduli-space structure in twisted D3-brane theories, linking $G_2$ and Thurston geometries to holographic domain-wall solutions.

Abstract

We construct dual supergravity descriptions of D3-branes wrapping associative 3-cycles $L$. We analyse the conditions for having five-dimensional background solutions of the form $AdS_2 \times L$ and show that they require $L$ to be of constant negative curvature type. This provides $AdS_2$ background solutions when $L$ is the hyperbolic space $H^3$ or its quotients by subgroups of its isometry group. We construct a regular numerical solution interpolating between $AdS_5$ in the UV and $AdS_2 \times H^3$ in the IR. The IR fixed point exists at the ``intersection'' of the Coulomb and Higgs branches. We analyse the singular supergravity solutions which correspond to moving into the Higgs and the Coulomb branches. For negative constant curvature spaces the singularity is of a ``good'' type in the Higgs branch and of a ``bad'' type in the Coulomb branch. For positive constant curvature spaces such as $S^3$ the singularity is of a ``bad'' type in both the Higgs and the Coulomb branches. We discuss the meaning of these results.

Figures (9)

• Figure 1: In this figure we plot $e^{2 f(r)}$ as obtained from a numerical solution of equations (\ref{['eq:sol3']}) when $C=\frac{1}{4}\left(3-\log{4}\right)$ in comparison to the expected behaviour in the IR, $e^{2f}=\frac{1}{4^{\frac{4}{3}}r^2}$ in equation (\ref{['eq:csol1']}). We see that the two coincide in the IR.
• Figure 2: In this figure we plot $e^{2 g(r)}$ as obtained from a numerical solution in comparison to the expected behaviour in the IR, $e^{2g}=4^{-\frac{1}{3}}$ in equation (\ref{['eq:csol1']}). We see that the two coincide in the IR.
• Figure 3: The numerical behaviour of $g(r)$ approaching a "bad" singularity in the Coulomb branch, $C_{\varphi}>C_{\varphi}^{crit.}$
• Figure 4: $\varphi(r)$ blows up near a "bad" singularity in the Coulomb branch, $C_{\varphi}>C_{\varphi}^{crit.}$
• Figure 5: The radius $e^{2f(r)}$ is shrinking in the IR, in the Higgs branch $C_{\varphi}<C_{\varphi}^{crit.}$. However the effective potential is bounded from above and the singularity is of a "good" type.