Codimension-two matter and codimension-three Yukawa dictionary

Adopt the following conditional dictionary for a specified resolved F-theory model over a smooth complex threefold $B_3$. Fix one resolution chamber $(\rho ,\mathcal{C})$ as in Crepant resolution and matter surface, where $\rho$ is a specified smooth projective crepant resolution and $\mathcal{C}$ is a specified collection of irreducible fiber-curve components. Let $D\subset B_3$ be the model's specified codimension-one gauge divisor. A matter curve ${\mathrm{\Sigma }}_R\subset D$ is the exact fiber-change datum of the same supplier: a specified reduced irreducible closed curve of complex dimension one for which the geometric generic resolved fibers over the generic points of ${\mathrm{\Sigma }}_R$ and $D$, after base change along any chosen pair of $\mathbb{C}$-embeddings of their residue-field algebraic closures into a common algebraically closed field, are nonisomorphic. A selected curve family supplies a matter surface $S_R$ only as the supplier's specified reduced irreducible surface with a proper flat map $q_R:S_R\to {\mathrm{\Sigma }}_R$, nonempty pure one-dimensional geometric fibers, and geometric generic fiber isomorphic to ${\mathbb{P}}^1$; these are hypotheses in an application, not consequences of this postulate. Codimension here is complex codimension, so ${\mathrm{\Sigma }}_R$ has codimension two in $B_3$.

The codimension-one gauge divisor and complex Lie algebra must be fixed in the same model under the restricted transverse-slice and local-minimal-Weierstrass conditions of 7-brane monodromy and codimension-one gauge-algebra dictionary. A selected fibral curve is a specified irreducible rational component in the resolved fiber over the generic point of a matter curve. For a chosen orientation of its wrapped-M2 state, write $[C_{\mathrm{o}\mathrm{r}}]$ for the resulting oriented curve class. With the source convention of Weigand, §5.2, equation (5.8), its candidate Cartan weight vector is

where $E_i$ are the specified resolution divisors associated with the simple Cartan generators. Reversing the M2 orientation replaces $[C_{\mathrm{o}\mathrm{r}}]$ and $\beta$ by their negatives. These are candidate weights for the selected modes; the geometry and the action interface do not derive which oriented states occur in the effective theory. Distinct resolution chambers or selected curve components are distinct data.

A codimension-three enhancement point is an isolated point of $B_3$ where specified enhancement loci meet and the resolved fiber further splits or degenerates. A finite-dimensional complex representation of the supplied complex gauge Lie algebra ${\mathfrak{g}}_{\mathbb{C}}$ is a pair $(V,\rho )$, where $V$ is a finite-dimensional complex vector space and $\rho :{\mathfrak{g}}_{\mathbb{C}}\to {\mathrm{End}}_{\mathbb{C}}(V)$ is a Lie-algebra homomorphism. A gauge-invariant trilinear in this convention is a nonzero ${\mathfrak{g}}_{\mathbb{C}}$-equivariant map $V_1\otimes V_2\otimes V_3\to \mathbb{C}$, with the trivial action on $\mathbb{C}$; the actual global gauge group and the descent of representations to it remain separate model data. Where the chosen four-dimensional theory is $\mathcal{N}=1$, let $\mathcal{U}$ be a complex coordinate patch of chiral field and moduli values and let $W:\mathcal{U}\to \mathbb{C}$ be its holomorphic superpotential. A holomorphic Yukawa coupling is the coefficient $y(m)$ of a cubic Taylor term $y(m){\mathrm{\Phi }}_1{\mathrm{\Phi }}_2{\mathrm{\Phi }}_3$ in $W$, where ${\mathrm{\Phi }}_j$ are chiral fields and $m$ are chiral moduli. A codimension-three point with an allowed trilinear is only a candidate locus: it does not establish localized modes, a nonzero coefficient, a numerical Yukawa value, or a complete zero-mode spectrum.

The physical wrapped-M2 interpretation is conditional on the classical action interface in [L1]. That current carrier is present as a draft with a local format precheck pass, which is not independent acceptance, and its two direct CJS postulate prerequisites are also drafts; the component-action precheck is pending and the superspace-constraints precheck passes. The action definition supplies no derivation of oriented membrane states, a complete spectrum, or nonzero couplings.

Crepant resolution and matter surface

7-brane monodromy and codimension-one gauge-algebra dictionary

M2 Brane Worldvolume Action

Katz–Vafa, §§1–2 and §5

Krause–Mayrhofer–Weigand, §§3.1–3.3 and Appendix A.2–A.3

Weigand, §§5.1–6.1

This is an adopted physical postulate, not a theorem that experiments or the cited papers prove for every compactification. The local source treatments support the conditional enhancement dictionary and the chamber-specific example on the companion page; source statements do not establish resolution-independent matter, total vectorlike multiplicities, or nonzero Yukawa coefficients. The M2 action item is present as a draft with a local format precheck pass, which is not independent acceptance, and its two direct CJS postulate prerequisites are drafts; presence does not establish acceptance. The wrapped-M2 interpretation remains conditional and does not derive oriented states from the action alone.