Notes on SU(3) Structures in Type IIB Supergravity

TL;DR

This work analyzes type IIB supergravity solutions with an $SU(3)$ structure and four-dimensional Poincaré invariance by translating fermionic supersymmetry variations into concrete relations among bosonic fields, warp factors, and fluxes. It proves that the complex structure is integrable under an $SU(3)$ structure and derives four independent scalar relations linking $A$, $\phi$, $C$, $h$, and the angles $\alpha$, $\beta$, as well as differential constraints encoded in $dJ$ and $d\Omega$. It then solves the radial-flow case, obtaining explicit expressions such as $\cos\alpha = \sin\theta'\cot\theta\tan\beta$, $e^\phi = g_s \frac{\sin(2\theta)}{\sin(2\beta)}$, $C = \frac{\tan\theta}{g_s}\left(1-\frac{\sin^2\beta}{\sin^2\theta}\right)$, $e^{2A} = \frac{\sin\theta\cot\theta'}{\sin\beta\tan\alpha}$, and $h = \frac{1}{g_s}\cot^2\theta'\sin\theta'\cot^2\alpha + h_0$. Using these, the paper constructs the explicit supergravity solution for a bound state of $(p,q)$5-branes and D3-branes, deriving fluxes, warp factors, and angular relations that fix the asymptotic D3 charge to $n$ via $\tan\theta' = \frac{n g_s}{\sqrt{p^2+q^2 g_s^2}}$ and showing agreement with duality-based solutions. Overall, the results provide a direct, structured route to generating supersymmetric $SU(3)$-structure solutions and their holographic RG-flow interpretations, with a concrete bound-state example illustrating the method’s power.

Abstract

We study solutions of type IIB supergravity with an SU(3) structure group and four dimensional Poincare invariance and present relations among the bosonic fields which follow from the supersymmetry variations. We make explicit some results which also follow from the more general case of an SU(2) structure and give some short comments applicable to general supersymmetric solutions. We also provide simplified relations appropriate for duals of gauge theory renormalization group flows, and use these to derive the supergravity solution for a bound state of (p,q)5-branes and D3-branes.