SU(5) enhancement curves and Yukawa points

Given: The analytification of the affine algebraic chart $B_{\mathrm{l}\mathrm{o}\mathrm{c}}=\mathrm{Spec}\mathbb{C}[w,u,v]$, the SU(5) Tate vanishing orders on $W=\{w=0\}$, the polynomial coefficient functions below, and a specified smooth proper crepant T11 resolution chamber with the source-reported fibral curves and their Cartan intersections, together with the local transverse models in [A5].

Take the chart to be $B_{\mathrm{l}\mathrm{o}\mathrm{c}}=\mathrm{Spec}\mathbb{C}[w,u,v]$ and trivialize the Tate coefficient line bundles there. Let $a_1=u$, $a_{2,1}=v$, $a_{3,2}=1+u+v$, $a_{4,3}=1+2u+3v$, and $a_6=0$, with $a_2=a_{2,1}w$, $a_3=a_{3,2}{w}^2$, and $a_4=a_{4,3}{w}^3$. On the affine fiber patch the Tate hypersurface is $y^2+a_{1}xy+a_{3}y=x^3+a_2{x}^2+a_{4}x+a_6$. These regular local-model coefficients are not asserted to extend to a compact base.

Assume the chosen T11 chamber is a smooth proper algebraic crepant resolution as in Crepant resolution and matter surface and supplies the listed rational fiber curves and the fiber splittings at the three displayed points. Here T11 means the proper transform (3.4), scaling data of Table 1, and Stanley–Reisner exclusions (3.6) of Krause–Mayrhofer–Weigand, with the additional exclusions $z{e}_4,z{e}_1,z{e}_2,s{e}_2,e_0{e}_3,e_1{e}_3$ chosen from (3.7). An excluded monomial means its coordinate factors cannot all vanish simultaneously. The generic fiber over $W$ is the source-reported split five-cycle of rational curves, and the geometric generic fibers over the three selected matter curves have the distinct component configurations of Appendix A.2. For each of the three matter curves, assume the selected closed reduced irreducible rational fiber curve extends to a proper flat family with nonempty pure one-dimensional geometric fibers and geometric generic fiber ${\mathbb{P}}^1$; its reduced irreducible surface closure is the matter surface in this chamber. The example does not prove resolution existence, smoothness, crepancy, or flatness from the coefficient functions alone; it uses the source's explicit blow-up/chamber data conditionally.

The physical interpretation uses only the local, geometric, large-radius, low-energy F-theory dictionary of Local geometric engineering of gauge and matter sectors, with its local transverse-disc input supplied in [A5]. No numerical control of string-loop or ${\alpha }^{\mathrm{\prime }}$ corrections, flux, backreaction, or compact completion is supplied; only candidate loci and invariant tensors are asserted. Extra gauge factors, flux-dependent mode counts and additional selection rules must be supplied before asserting any actual interaction.

Use the mathematical definitions in Crepant resolution and matter surface: a matter curve is a reduced irreducible curve in the gauge divisor whose geometric generic resolved fiber differs from that over the divisor, after comparison over a common algebraically closed extension field. A matter surface is the selected reduced irreducible surface with proper flat map to that curve, nonempty pure one-dimensional geometric fibers and geometric generic fiber ${\mathbb{P}}^1$. These are the inputs in [A2]. For a chosen orientation, let $[C_{\mathrm{o}\mathrm{r}}]$ be the corresponding signed class of a proper rational fibral curve. Adopt the source convention of Weigand §5.2, (5.8)–(5.13): its candidate Dynkin weight has components ${\beta }_i=E_i\cdot [C_{\mathrm{o}\mathrm{r}}]$, where the $E_i$ are the four Cartan resolution divisors in the source order. The intersection is the degree of $\mathcal{O}(E_i)$ on the curve, extended by sign to the reversed class. Reversing orientation negates the weight; this convention alone counts no modes and proves no nonzero coupling.

Local geometric engineering of gauge and matter sectors: “A codimension-two enhancement curve is a complex curve in $B_3$ where the singular-fiber data over $D$ enhance relative to their generic values.” Its mode and interaction interpretation is conditional on the actual local geometry, background, resolution, and mode content.

The current item M2 Brane Worldvolume Action is placed on the st-m-theory-foundations page and has status draft with a local format precheck pass, which is not independent acceptance. For a smooth embedding $X:{\mathrm{\Sigma }}_3\to Y^{11}$ of a closed oriented three-manifold with nondegenerate Lorentzian induced metric $g_{ij}={\mathrm{\partial }}_i{X}^m{\mathrm{\partial }}_j{X}^n{G}_{mn}(X)$, it defines the classical bosonic action

with independent positive tension $T_{\mathrm{M}2}$ and source convention $C_3=-6B_3$. In $\mathrm{\hslash }=c=1$ units, ${\xi }^i$ are dimensionless, target coordinates are length-valued with dimensionless metric components, $T_{\mathrm{M}2}$ has units of ${\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}}^{-3}$, and the normalization gives $\int C_3$ units of ${\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}}^3$. Its Green–Schwarz completion uses a smooth superembedding into an on-shell $11\mathrm{\mid }32$ CJS superspace and the pulled-back supervielbein. Its direct prerequisites, post-eleven-dimensional-cjs-supergravity-action and post-m2-cjs-superspace-constraints, are draft postulates; the component-action precheck is pending and the superspace-constraints precheck passes. The current definition supplies an action interface, not a derivation of oriented membrane states, a complete spectrum, or nonzero Yukawa couplings.

Krause–Mayrhofer–Weigand, §§3.1–3.3 and Appendix A.2–A.3, supply the following model-specific data: the leading discriminant factor on $w=0$ is proportional to $a_1^4{a}_{3,2}(-a_1{a}_{4,3}+a_{2,1}{a}_{3,2})$, with selected point strata $a_1=a_{2,1}=0$, $a_1=a_{3,2}=0$, and $a_{3,2}=a_{4,3}=0$. In the T11 chamber, tables (A.18), (A.26), (A.32) and (3.21) give the respective selected weights ${\mu }_{10}-{\alpha }_2-{\alpha }_3-{\alpha }_4$ for $P_{4D}^1$, ${\mu }_5-{\alpha }_1-{\alpha }_2-{\alpha }_3$ for $P_{3F}^1$, and $-{\mu }_5+{\alpha }_1+{\alpha }_2$ for $P_{3G}^1$. Appendix A.3 supplies their further splittings at the point strata. These data are imposed in [A2]; no general resolution-existence theorem or spectrum is inferred from them.

Set $V={\mathbb{C}}^5$ with its defining action of $\mathrm{S}\mathrm{U}(5)=\{g\in M_5(\mathbb{C}):g^{\ast }g=I,\det g=1\}$. Write $\mathbf{5}=V$, $\mathbf{10}={\wedge }^{2}V$, $\bar{\mathbf{5}}=V^{\ast }$ with contragredient action, and $\mathbf{1}=\mathbb{C}$ with trivial SU(5) action; the singlet can still carry additional abelian charge. On the traceless diagonal Cartan subalgebra of ${\mathfrak{s}\mathfrak{l}}_5(\mathbb{C})$, put ${ϵ}_i(\mathrm{diag}(h_1,\dots ,h_5))=h_i$, ${\alpha }_i={ϵ}_i-{ϵ}_{i+1}$, and ${\alpha }_i^{\vee }=\mathrm{diag}(0,\dots ,1,-1,\dots ,0)$, $1\le i\le 4$. The Cartan matrix $A_{ij}=⟨{\alpha }_i,{\alpha }_j^{\vee }⟩$ has diagonal $2$, adjacent entries $-1$, and other entries $0$. The fundamental weights ${\omega }_i={ϵ}_1+\cdots +{ϵ}_i$ obey $⟨{\omega }_i,{\alpha }_j^{\vee }⟩={\delta }_{ij}$; ${\mu }_{10}={\omega }_2$ and ${\mu }_5={\omega }_1$. Dynkin coordinates are these four ordered coroot pairings. An allowed SU(5) trilinear means a nonzero complex-linear invariant map from the tensor product of the three displayed representations to $\mathbb{C}$.

Supply the local slice data required by Local geometric engineering of gauge and matter sectors. Let $W^{\circ }\subset W$ be the dense open where $u(1+u+v)(-u(1+2u+3v)+v(1+u+v))\mathrm{\ne }0$. For each general $p=(0,u_0,v_0)\in W^{\circ }$, choose $\epsilon \mathrm{\ne }0$ small enough that ${\mathrm{ȷ}}_p(t)=(\epsilon t,u_0,v_0)$, $\mathrm{\mid }t\mathrm{\mid }<1$, meets no residual discriminant component. Assume the restriction of the chosen resolved family to this disc is a smooth complex surface $S_p$ with a proper flat relatively minimal elliptic fibration ${\pi }_p:S_p\to \mathbb{D}$ and a regular section, birational over $\mathbb{D}$ to the Tate cubic and smooth away from $t=0$, whose central fiber is the split $I_5$ configuration: five multiplicity-one ${\mathbb{P}}^1$ components in a cycle, with adjacent transverse intersections and self-intersection $-2$ in this surface. These slice properties are supplied local resolution data, not a claim that every resolution has them. For the corresponding short Weierstrass coefficients put $b_2=a_1^2+4a_2$, $b_4=a_1{a}_3+2a_4$, $b_6=a_3^2$, $f=-(b_2^2-24b_4)\mathrm{/}48$, and $g=(b_2^3-36b_2{b}_4+216b_6)\mathrm{/}864$. Completing the square and translating $x$ converts the Tate cubic to $y^{2}z=x^3+fx{z}^2+g{z}^3$ with section $[0:1:0]$. Its coefficients are algebraic on the affine base, so the splitness test is over $K=\mathbb{C}(W)=\mathbb{C}(u,v)$, not the residue field at $p$.