Domain walls from ten dimensions

TL;DR

The paper derives ten-dimensional conditions for ${\cal N}=1$ domain walls in type II string theory using generalized complex geometry, revealing flow equations that couple radial evolution to internal geometry. By a controlled truncation to a finite set of four-dimensional fields, these ten-dimensional equations map to a four-dimensional gradient flow with a generating function ${\cal C}$, reproducing familiar ${\cal N}=1$ supergravity structures and connecting to AdS$_4$/CFT$_3$ holography. It provides explicit realizations on SU(3)-structure coset and nearly Kähler manifolds, demonstrating how to compute the four-dimensional superpotential $W$ and Kähler potential ${\cal K}$ from ten-dimensional data and how branes can be consistently included. A key result is the D-brane modified c-theorem, showing how D-brane junctions contribute positively to the monotonic flow of ${\cal C}$ and how the total domain-wall tension remains positive, with brane-induced jumps in flux driving transitions between AdS$_4$ vacua. Overall, the work offers a robust framework to study holographic RG flows and brane dynamics directly from ten dimensions, with concrete truncations enabling explicit, testable effective theories.

Abstract

We write down the general conditions for N=1 supersymmetric four-dimensional domain walls, deriving them from a ten-dimensional point of view using generalized complex geometry. In cases where the compactification allows for a truncation to a finite number of fields, we make contact with a four-dimensional effective description. In the context of the AdS/CFT correspondence, the equations can be applied to renormalization-group flows of three-dimensional field theories. We allow for the presence of explicit brane sources and show how supersymmetry restricts their location in a natural way.