D-brane networks in flux vacua, generalized cycles and calibrations

TL;DR

This work develops a generalized homology for D-brane networks in flux vacua by introducing generalized chains and a boundary operator $\hat{\partial}$ that pairs with the twisted differential $d_H$. Generalized currents dual to these chains define a consistent framework in which D-brane networks (including branes dissolving in higher-dimensional branes and MMS-like transitions) wrap generalized cycles, ensuring RR-gauge invariance. The authors then extend this topological structure with generalized calibrations, identifying energy-minimizing (supersymmetric) networks inside a given generalized-homology class, and apply the formalism to ${\cal N}=1$ flux compactifications, including warped Calabi–Yau geometries and explicit examples on $T^6/\mathbb{Z}_2$ and Klebanov–Strassler backgrounds. The approach provides a natural intermediate between ordinary homology and twisted K-theory, enabling concrete analysis of brane networks, their stability, and RR-flux quantization in flux vacua. It also yields explicit BPS equations and geometry for composite domain walls, clarifying how brane junctions and D3-brane tadpoles are organized within generalized cycles.

Abstract

We consider chains of generalized submanifolds, as defined by Gualtieri in the context of generalized complex geometry, and define a boundary operator that acts on them. This allows us to define generalized cycles and the corresponding homology theory. Gauge invariance demands that D-brane networks on flux vacua must wrap these generalized cycles, while deformations of generalized cycles inside of a certain homology class describe physical processes such as the dissolution of D-branes in higher-dimensional D-branes and MMS-like instantonic transitions. We introduce calibrations that identify the supersymmetric D-brane networks, which minimize their energy inside of the corresponding homology class of generalized cycles. Such a calibration is explicitly presented for type II N=1 flux compactifications to four dimensions. In particular networks of walls and strings in compactifications on warped Calabi-Yau's are treated, with explicit examples on a toroidal orientifold vacuum and on the Klebanov-Strassler geometry.

Figures (4)

• Figure 1: Here we see two dynamical processes in which a network transforms into a different network in the same $\hat{\partial}$-homology class. In the southern hemisphere a brane wrapping the cycle $C$ with gauge bundle ${\cal F}$ moves, sweeping out the chain $B$. As it moves the gauge bundle $\hat{{\cal F}}$ also changes, although it always satisfies $\text{d}\hat{{\cal F}}=H|_B$. Finally it arrives at the cycle $C{}^{\prime}$ where the gauge bundle is $\hat{{\cal F}}|_{C{}^{\prime}}={\cal F}{}^{\prime}$. In the northern hemisphere 3 D0-branes swell to form a D2-brane wrapping a trivial cycle but with 3 worldvolume magnetic vortices. They sweep out the chain $(\tilde{B},\tilde{{\cal F}}_{\{ p_i\}})$, whose worldvolume flux $\tilde{{\cal F}}_{\{ p_i\}}$ is sourced at the positions $p_i$ of the original D0-branes.
• Figure 2: 3 D0-branes represent a trivial homology class. They therefore can decay via an MMS instanton, which is a D2-brane that sweeps out a 3-cycle $\Sigma$ such that the endpoints of the D0-branes are Poincaré dual in $\Sigma$ to the pullback of the $H$ flux. The D0's inflate into a spherical D2-brane via the Myers effect, sweep out the 3-cycle and then disappear as the sphere shrinks to nothing at the north pole. This process creates a residual RR 6-form field strength.
• Figure 3: In general the energy of a network is infinite if the lower-dimensional D-brane tendrils are semi-infinite. This infinity must be regularized, for example imposing an IR cutoff in the background, which may be automatic if the tendrils end on a horizon or the end of the world. On the left we see a network which ends on a background brane but has infinite energy because the D$p$-brane's Dirac monopole is localized on a codimension 3 surface. On the right a finite-energy BIon solution is drawn, which is in the same $\hat{\partial}$-homology class as that on the left, but the D$p$-D$(p-2)$ system has been replaced by a single continuous D$p$ that ends on the same background brane.
• Figure 4: The $(x^2,x^3)$ cross-section of two examples of networks of domain walls are drawn, indicating explicitly the contribution to the total wall tensions coming from each D5-brane in units where $\text{Vol}(T^6)/g_s^2=1$. On the left there are three walls of tension equal to one in the four-dimensional theory, which in the full theory are D5-branes that wrap the cycles $\Gamma_1=A^0$, $\Gamma_2=A^1$ and a third cycle $\Gamma_3$ which is homologous to $-A^0-A^1$. These extend at relative angles of 120 degrees in the $(x^2,x^3)$-plane, and so their tensions cancel in the four-dimensional sense leaving a stable configuration, in agreement with the general formula (\ref{['anglecond']}). On the right 7 intersecting domain walls are drawn, which extend in four distinct directions in this 2-dimensional cross-section, although they all wrap distinct cycles on the internal toroidal orientifold. These walls all end on a string which lifts to a D7-brane. The fact that $\hat{\Omega}\wedge H$ integrates to zero, combined with the fact that the total cycle wrapped by the walls is equal to $-H$ according to the Freed-Witten anomaly, guarantees that the tensions of these walls also cancel (see Eq. (\ref{['Hangle']}) and related discussion), as can be verified explicitly in the figure.