Supersymmetric quantum mechanics from wrapped D4-branes

TL;DR

The paper develops gravity duals of supersymmetric quantum mechanics from D4-branes wrapped on four-manifolds by employing six-dimensional maximal gauged supergravity with gaugings $\mathrm{CSO}(p,q,5-p-q)$ and $\mathrm{CSO}(p,q,4-p-q)\ltimes\mathbb{R}^4$. It constructs $t\times\mathcal{M}_4$-sliced domain-wall solutions, derives their BPS equations under a variety of twists (including $SO(4)$, $SO(2)\times SO(2)$, $SO(3)$, $SU(2)$, and $SO(2)$) for different $\mathcal{M}_4$ (Riemannian, products of Riemann surfaces, and Kahler four-cycles), and analyzes IR singularities via uplift to type IIA using Maldacena’s nogo criterion. The work provides analytic solutions in several subtruncations and a comprehensive numerical survey across multiple gauge groups, identifying physically acceptable IR singularities for many cases and mapping out when the dual quantum mechanics arises from twisted five-dimensional SYM. It discusses prospects for holographic correlators, the interpretation of the dual QMs, and the construction of complete higher-dimensional truncations to uplift all solutions to IIA string theory. Overall, the results offer a broad, structured set of gravity duals for twisted D4-brane compactifications leading to supersymmetric quantum mechanics with varying amounts of preserved supersymmetry.

Abstract

We find a large class of holographic solutions describing D4-branes wrapped on 4-manifolds $\mathcal{M}_4$ with constant curvature leading to gravity duals of supersymmetric quantum mechanics in the IR via twisted compactifications. The manifolds $\mathcal{M}_4$ considered here are four-dimensional spheres and hyperbolic spaces, products of two Riemann surfaces, and Kahler four-cycles. The solutions are obtained from the maximal gauged supergravity in six dimensions with $CSO(p,q,5-p-q)$ and $CSO(p,q,4-p-q)\ltimes \mathbb{R}^4$ gauge groups. These gauged supergravities can be embedded in type IIA theory via consistent truncations on $H^{p,q}\times \mathbb{R}^{5-p-q}$ and $H^{p,q}\times\mathbb{R}^{4-p-q}\times S^1$, respectively. The solutions take the form of $t\times \mathcal{M}_4$-sliced domain walls interpolating between locally flat domain walls and singular geometries in the IR. Upon uplifted to type IIA theory, many solutions admit physical IR singularities and could holographically describe supersymmetric quantum mechanics arising from twisted compactifications of D4-branes on $\mathcal{M}_4$.

Figures (65)

• Figure 1: Interpolating solutions between the locally $SO(5)$ flat domain wall as $r\rightarrow+\infty$ and $t\times S^4$-sliced curved domain walls for $SO(4)$ twist in $SO(5)$ gauge group. The blue, orange, green, red, and purple curves refer to $g=0.15, 0.50, 1, 2$, and $4.04$, respectively.
• Figure 2: Interpolating solutions between the locally $SO(5)$ flat domain wall as $r\rightarrow+\infty$ and $t\times H^4$-sliced curved domain walls for $SO(4)$ twist in $SO(5)$ gauge group. The blue, orange, green, red, and purple curves refer to $g=0.20, 0.45, 1, 2$, and $4$, respectively.
• Figure 3: The behavior of $\hat{g}_{00}$ along the RG flows given in figures \ref{['15_S4_SO(4)_special_SO(5)gg_flows']} and \ref{['15_H4_SO(4)_special_SO(5)gg_flows']} respectively for $\Sigma_1^4=S^4$ and $\Sigma_{-1}^4=H^4$ in $SO(5)$ gauge group.
• Figure 4: Interpolating solutions between the locally $SO(5)$ flat domain wall as $\rho\rightarrow+\infty$ and $t\times S^2\times S^2$-sliced curved domain walls for $SO(2)\times SO(2)$ twist in $SO(5)$ gauge group. The blue, orange, green, red, purple, brown, and black curves refer to $(z_1,z_2)=(0,0), (1.7,0), (0,-1.9), (-2,0), (3,-1), (4,3), (0,-16)$.
• Figure 5: Interpolating solutions between the locally $SO(5)$ flat domain wall as $\rho\rightarrow+\infty$ and $t\times S^2\times H^2$-sliced curved domain walls for $SO(2)\times SO(2)$ twist in $SO(5)$ gauge group. The blue, orange, green, red, purple, brown, and black curves refer to $(z_1,z_2)=(0,0), (0.01,0), (0.50,-0.50), (-2,2), (3,-1), (8,-8), (0,-16)$.