New Aspects of Heterotic--F Theory Duality

TL;DR

The paper develops a detailed framework for describing chiral matter and Yukawa couplings in F-theory via its heterotic dual, introducing covering matter curves to handle singularities that arise in wedge representations. It provides explicit constructions of the relevant sheaves from spectral data, derives chirality formulas in terms of four-form fluxes, and refines the Het–F duality dictionary to map spectral moduli to F-theory complex-structure data. By translating heterotic results into F-theory language, the authors illuminate how codimension-3 singularities on discriminant loci correspond to line bundles on matter curves and govern Yukawa couplings, enabling local model-building prospects in F-theory. The work offers a coherent, flux-driven account of chirality across representations and ranks, and emphasizes how Yukawa interactions arise at specific singular loci, with potential implications for texture and doublet–triplet splitting in SU(5) GUTs.

Abstract

In order to understand both up-type and down-type Yukawa couplings, F-theory is a better framework than the perturbative Type IIB string theory. The duality between the Heterotic and F-theory is a powerful tool in gaining more insights into F-theory description of low-energy chiral multiplets. Because chiral multiplets from bundles /\^2 V and /\^2 V^x as well as those from a bundle V are all involved in Yukawa couplings in Heterotic compactification, we need to translate descriptions of all those kinds of matter multiplets into F-theory language through the duality. We find that chiral matter multiplets in F-theory are global holomorphic sections of line bundles on what we call covering matter curves. The covering matter curves are formulated in Heterotic theory in association with normalization of spectral surface, while they are where M2-branes wrapped on a vanishing two-cycle propagate in F-theory. Chirality formulae are given purely in terms of (possibly primitive) four-form flux. In order to complete the translation, the dictionary of the Heterotic--F theory duality has to be refined in some aspects. A precise map of spectral surface and complex structure moduli is obtained, and with the map, we find that divisors specifying the line bundles correspond precisely to codimension-3 singularities in F-theory.

Figures (8)

• Figure 1: (i) is a local picture of spectral surface $C_V$ around a point where $D$ and $r$ intersect as (a) in the classification in the text. Among the three local coordinates $(\xi, u, v)$ ($x$ is used as axes label in the figures instead of $\xi$), $(u,v)$ are for the base 2-fold $B_2$, and $\xi$ is for the elliptic fiber. $(u,v)=(0,0)$ corresponds to a type (a) point in $\bar{c}_{\wedge^2 V} = \bar{c}_V$, and $\xi=0$ to the zero section $\sigma$. (ii) and (iii) cut out $u \geq 0$ and $u \leq 0$ parts of (i), so that curves $\bar{c}_V = \bar{c}_{\wedge^2 V}$ (thick green), $D$ (thin blue) and $r$ (thin red) are clearly visible. See the appendix \ref{['ssec:r-D']} for more details.
• Figure 2: Coordinates $(u,v)$ describe base manifold $B_2$, and $\xi$ the fiber direction, with a zero section corresponding to $\xi=0$. The left panel shows a local picture of $C_V$ for a rank-4 vector bundle around type (a) points on $\bar{c}_{\wedge^2 V}$. Local coordinates $(u,v)$ on the base is chosen so that $a_3 = u$ and $a_4 = v$. $\bar{c}_V$ (thick yellow) and $\bar{c}_{\wedge^2 V}$ (thick blue) in the zero section are given by $a_4 = 0$ and $a_3 = 0$, respectively. Curves $D$ and $r$ on $C_V$ are also shown by thin blue and thin red curves, respectively. Two right panels show local pictures of $C_{\wedge^2 V}$ around type (c) points viewed from opposite directions. Local coordinates are chosen so that $a_3 = u$ (a direction transverse to $\bar{c}_{\wedge^2 V}$), and $a_2^2 - 4a_0 a_4 = v$ on the base manifold. The matter curve $\bar{c}_{\wedge^2 V}$ (thick blue) is a double curve in $C_{\wedge^2 V}$, and a more complicated singularity---called a pinch point---is developed at a type (c) point.
• Figure 3: A schematic picture showing relations between various curves and points that are used in the text.
• Figure 4: The left panel (i) shows how the two matter curves $\bar{c}_V$ (thick yellow) and $\bar{c}_{\wedge^2 V}$ (thick blue) intersect in the zero section for cases with rank-5 bundles $V$. The right panel (ii) shows geometry of a curve $D$ (thin blue) and $\bar{c}_{\wedge^2 V}$ (thick blue) associated with a type (c1) point. Only a degree-2 part of degree-5 spectral cover surface $C_V$ is shown here. $C_V$ is a ramified cover over $B_2$ along a ramification divisor $r$ (thin red), but $D$ is not a ramified cover over $\bar{c}_{\wedge^2 V}$.
• Figure 5: Extended Dynkin diagrams of (i) $E_7$ and (ii) $E_6$. By removing one node from the diagrams, one can see that these groups have subgroups $\mathop{\rm SU}(4) \times \mathop{\rm SU}(2) \times \mathop{\rm SU}(4) \subset E_7$, and $\mathop{\rm SU}(3) \times\mathop{\rm SU}(3) \times \mathop{\rm SU}(3) \subset E_6$.