Matter From Geometry Without Resolution

TL;DR

The paper develops a deformation-theoretic framework to realize charged matter in elliptically fibered Calabi–Yau compactifications, arguing that complex-structure deformations of ADE singularities yield the physical two-cycles associated with matter in F-theory. It builds a precise dictionary from string junctions to Lie-algebra weights using a topological intersection form $I$ and Picard–Lefschetz monodromy, enabling the application of Freudenthal’s recursion to obtain weight multiplicities. Explicit one-parameter deformations for $A_r$, $D_r$, $E_6$, $E_7$, and $E_8$ reproduce root and higher representations (e.g., the 126 of $SO(10)$ and 43,758 of $E_6$) and show how non-simply-laced algebras arise via O-monodromy. The work clarifies codimension-two localization of massless matter through branching and monodromy in the deformation picture and provides computational tools, with a companion mathematical paper to formalize underpinnings.”

Abstract

We utilize the deformation theory of algebraic singularities to study charged matter in compactifications of M-theory, F-theory, and type IIa string theory on elliptically fibered Calabi-Yau manifolds. In F-theory, this description is more physical than that of resolution. We describe how two-cycles can be identified and systematically studied after deformation. For ADE singularities, we realize non-trivial ADE representations as sublattices of Z^N, where N is the multiplicity of the codimension one singularity before deformation. We give a method for the determination of Picard-Lefschetz vanishing cycles in this context and utilize this method for one-parameter smooth deformations of ADE singularities. We give a general map from junctions to weights and demonstrate that Freudenthal's recursion formula applied to junctions correctly reproduces the structure of high-dimensional ADE representations, including the 126 of SO(10) and the 43,758 of E_6. We identify the Weyl group action in some examples, and verify its order in others. We describe the codimension two localization of matter in F-theory in the case of heterotic duality or simple normal crossing and demonstrate the branching of adjoint representations. Finally, we demonstrate geometrically that deformations correctly reproduce the appearance of non-simply-laced algebras induced by monodromy around codimension two singularities, showing the reduction of D_4 to G_2 in an example. A companion mathematical paper will follow.

Figures (2)

• Figure 1: On the left is a four-pronged junction in $B$. The picture on the right is of the corresponding real two-manifold in the total space $X$. The loose end wraps the one-cycle $a(J)$ in the elliptic fiber; i.e. the asymptotic charge.
• Figure 2: An explicit depiction of the topology of a three-pronged junction. See appendix \ref{['app:junction']} for an explicit description of this two-sphere.