It is well known that the worldvolume theory of $N$ coincident D$p$-branes is given by the $U(N)$ Yang-Mills theory in $(p+1)$-dimensions. One important feature of this setup is the possibility of producing a $U(N) \rightarrow U(1)^{N}$ symmetry breaking pattern by separating the branes along a common transverse direction.

There is, however a case of a slight generalization of this setup that puzzles me; concretely, how to localize to the Higgs branch of a system $N$ coincident D4-branes wrapping a 4-cycle inside a toric Calabi-Yau manifold threefold. I have read several times about the possibility to produce a BRST-exact mass deformation of the D4 worldvolume theory to deform the 4-cycle $\mathcal{C}$ to a torus-equivariant configuration (see for example section 2.1 in Crystals and Intersecting Branes or Two Dimensional Yang-Mills, Black Holes and Topological Strings).

Problem: I have not a rigorous understanding about how this mechanism actually works beyond heuristics, nor a precise list of the conditions under this procedure is valid. The papers I posted discuss interesting examples, but I'm interested in the discussion in more generality, if possible.

The following cite Crystals and Intersecting Branes (third paragraph in page 8, section 3) suggests that the requirement of being very ample (talking about a representative of the homology class of $\mathcal{C}$) is the only one needed.

Any collection of intersecting D4 branes whose total homology class is very ample can be deformed in the normal directions, generically producing a single D4 brane wrapping a complicated surface.

A very remarkable statement!

Question: How can I check that explicitly?


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