F-theory and 2d (0,2) Theories

TL;DR

This work inaugurates the study of F-theory compactifications on elliptically fibered Calabi–Yau five-folds $Y_5$ to two dimensions, yielding $N=(0,2)$ gauge theories. By combining a partially twisted 8d SYM analysis on the 7-brane worldvolume with a global M/F-theory perspective, the authors construct a precise dictionary between geometric data (fiber types, singularities, fluxes) and 2d field-theoretic data (vector, chiral, and Fermi multiplets, plus $E$ and $J$ couplings). They systematically derive global consistency conditions, including gauge and abelian anomalies, tadpole cancellations, and Chern–Simons couplings, and validate these against explicit global fibrations for ADE-type gauge groups and abelian factors. The paper also establishes a heterotic/GLSM interpretation, relating GLSM phases to F-theory Higgs-bundle configurations (T-branes) and outlining potential SCFT points in the IR. Overall, the results provide a robust framework for constructing and analyzing rich 2d $(0,2)$ vacua from F-theory, with deep connections to M-theory, heterotic theories, and GLSMs, and with clear avenues for further exploration of IR fixed points and GLSM phase structure.

Abstract

F-theory compactified on singular, elliptically fibered Calabi-Yau five-folds gives rise to two-dimensional gauge theories preserving N=(0,2) supersymmetry. In this paper we initiate the study of such compactifications and determine the dictionary between the geometric data of the elliptic fibration and the 2d gauge theory such as the matter content in terms of (0,2) superfields and their supersymmetric couplings. We study this setup both from a gauge-theoretic point of view, in terms of the partially twisted 7-brane theory, and provide a global geometric description based on the structure of the elliptic fibration and its singularities. Global consistency conditions are determined and checked against the dual M-theory compactification to one dimension. This includes a discussion of gauge anomalies, the structure of the Green-Schwarz terms and the Chern-Simons couplings in the dual M-theory supersymmetric quantum mechanics. Furthermore, by interpreting the resulting 2d (0,2) theories as heterotic worldsheet theories, we propose a correspondence between the geometric data of elliptically fibered Calabi-Yau five-folds and the target space of a heterotic gauged linear sigma-model (GLSM). In particular the correspondence between the Landau-Ginsburg and sigma-model phase of a 2d (0,2) GLSM is realized via different T-branes or gluing data in F-theory.

Figures (5)

• Figure 1: Codimension one to four fibers of an F-theory model with gauge group $SU(5)$ realized by an $I_5$ fiber in codimension one. Lines correspond to rational curves, and multiple lines indicate the multiplicities of the fiber components. In codimension two, the fibers correspond to local enhancements to $SU(6)$ and $SO(10)$, respectively, and are given in terms of Kodaira fibers. All higher codimension fibers have monodromy reduction: compared to the standard Kodaira fiber, components are absent due to monodromies. The resolution shown here is encoded in the box graph in figure \ref{['fig:BoxGraphSU5']}, and is realized in terms of the blowup sequence (\ref{['ResSeq']}).
• Figure 2: Box Graph for the ${\bf 5}$ (on the left) and ${\bf 10}$ (right) representation of $SU(5)$ corresponding to the fiber in codimension two of the resolution discussed in section \ref{['sec:SU5Ex']}. Each box corresponds to a weight of the representations. The action of the roots $\alpha_i$ connects the weights into this representation graph. The coloring corresponds to the signs blue $\varepsilon=+$ and yellow $\varepsilon=-$, indicating that $C^{\varepsilon(\lambda)}_{\lambda}$ is an effective curve.
• Figure 3: Box Graph for the ${\bf 10}$ representation of $SO(10)$ corresponding to the fiber in codimension two of the $SO(10)$ model. Each box corresponds to a weight of the ${\bf 10}$. The action of the roots $\alpha_i$ connects the weights into this representation graph. The coloring corresponds to the signs blue $\varepsilon=+$ and yellow $\varepsilon=-$, indicating that $C^{\varepsilon(\lambda)}_{\lambda}$ is an effective curve.
• Figure 4: Box Graph for the ${\bf 27}$ representation of $E_6$ corresponding to the fiber in codimension two of the $E_6$ model. Each box corresponds to a weight of the ${\bf 27}$, as listed in (\ref{['E627Weights']}). The action of the roots $\alpha_i$ connects the weights into this representation graph. The coloring corresponds to the signs blue $\varepsilon=+$ and yellow $\varepsilon=-$, indicating that $C^{\varepsilon(\lambda)}_{\lambda}$ is an effective curve.
• Figure 5: The codimension two $I_2$ fibers, realizing matter with charges $q= \mp 5, 0, \pm 1$. The left-most picture shows the $I_2$ fiber, with the two rational curves $C^\pm$ intersecting in two points. The remaining fiber diagrams show how the charges are realized in terms of sections intersecting or containing the curves $C^\pm$. Blue/red corresponds to the sections $\sigma_0$ and $\sigma_1$, respectively. The numbers next to fiber components contained (colored) in sections are the degrees of the normal bundle of the curve in the section.