Can componentwise D7-tadpole cancellation yield a chiral Standard Model? A no-go for smooth O7$^{-}$ orientifolds

Abstract

We study a componentwise prescription for D7-tadpole cancellation that includes, as a special case, the local cancellation used in several moduli-stabilisation constructions. We work in smooth Type IIB Calabi--Yau orientifolds with O7$^{-}$ components of pairwise distinct divisor classes. For each component class $[R_i]$, eight covering-space D7-branes are distributed among smooth irreducible representatives of that class in an orientifold-compatible way. Each component sector may therefore contain several orthogonal, unitary, or symplectic parent factors. We show that these configurations cannot supply the chiral Standard Model through their perturbative D7 spectrum. Without symplectic factors, the entire parent algebra of each component sector embeds into $\mathfrak{so}(8)$ and cannot contain commuting color and weak-isospin subalgebras. Disjointness excludes massless fermions connecting different component sectors, so quark-doublet fermions are absent. Configurations containing symplectic factors require a separate argument: massless hypercharge, together with the same disjointness, forces the net quark-doublet chirality to vanish. Both arguments allow admissible D7 gauge backgrounds and diagonal gauge embeddings, with no restriction on $h^{1,1}_{-}(X)$. We also give a geometric condition under which the tadpole forces purely local cancellation. For a specified finite set of allowed D7 support classes, a separate test identifies component sectors in which the tadpole forces componentwise cancellation. We discuss applications to toric constructions and axion potentials.