M-theory geometries from five-brane webs, seven-branes, and T-branes

We track the chain of dualities relating five-brane webs in Type IIB to M-theory on noncompact Calabi-Yau threefolds, and follow the effect of adding $(p,q)$ seven-branes. T-dualizing along the seven-brane, a junction of D5-branes ending on a D7-brane becomes a single smooth D6-brane wrapping a smooth holomorphic curve, which uplifts directly to an M-theory geometry. Treating the branes as coherent sheaves (tachyon condensation), we obtain a spectral-curve dictionary that maps non-Abelian (T-brane) partition data to explicit complex-structure deformations of the threefold. As applications we (i) give a physical derivation of these deformations, (ii) exhibit a simple geometric prototype of an s-rule violation, and (iii) link T-brane data to geometry.