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String Junctions and Non-Simply Connected Gauge Groups

Zachary Guralnik

TL;DR

This work addresses how to determine the global gauge-group structure in elliptic F-theory by using string-junctions and the Mordell-Weil lattice. It develops a framework where fractional null junctions encode the center structure and yield $\pi^1(\tilde{G})$ for elliptic surfaces and K3s, and extends to Calabi-Yau threefolds. It provides explicit examples for rational elliptic surfaces and elliptic K3s, including U(1) factors, and outlines a procedure to generalize to CY3-folds despite discriminant-curve subtleties. The results offer a practical method to compute the full gauge group $G$ in these geometric backgrounds and connect to heterotic/Mordell-Weil data.

Abstract

Relations between the global structure of the gauge group in elliptic F-theory compactifications, fractional null string junctions, and the Mordell-Weil lattice of rational sections are discussed. We extend results in the literature, which pertain primarily to rational elliptic surfaces and obtain pi^1(G) where G is the semi-simple part of the gauge group. We show how to obtain the full global structure of the gauge group, including all U(1) factors. Our methods are not restricted to rational elliptic surfaces. We also consider elliptic K3's and K3-fibered Calabi-Yau three-folds.

String Junctions and Non-Simply Connected Gauge Groups

TL;DR

This work addresses how to determine the global gauge-group structure in elliptic F-theory by using string-junctions and the Mordell-Weil lattice. It develops a framework where fractional null junctions encode the center structure and yield for elliptic surfaces and K3s, and extends to Calabi-Yau threefolds. It provides explicit examples for rational elliptic surfaces and elliptic K3s, including U(1) factors, and outlines a procedure to generalize to CY3-folds despite discriminant-curve subtleties. The results offer a practical method to compute the full gauge group in these geometric backgrounds and connect to heterotic/Mordell-Weil data.

Abstract

Relations between the global structure of the gauge group in elliptic F-theory compactifications, fractional null string junctions, and the Mordell-Weil lattice of rational sections are discussed. We extend results in the literature, which pertain primarily to rational elliptic surfaces and obtain pi^1(G) where G is the semi-simple part of the gauge group. We show how to obtain the full global structure of the gauge group, including all U(1) factors. Our methods are not restricted to rational elliptic surfaces. We also consider elliptic K3's and K3-fibered Calabi-Yau three-folds.

Paper Structure

This paper contains 11 sections, 89 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Review of string junctions
  3. Non-simply connected gauge groups for elliptic surfaces
  4. Junctions and representations of the center; an $I_3$ example.
  5. Junctions and representations of the center for a general A-D-E singularity
  6. $\pi^1(G)$ for elliptic surfaces.
  7. Examples for rational elliptic surfaces
  8. Gauge group for an elliptic K3.
  9. The global structure including $U(1)$ factors.
  10. Elliptically fibered Calabi-Yau three-folds and $\pi^1(G)$.
  11. Appendix.

Figures (8)

  • Figure 1: An example of a string junctions in $P^1$. The crosses indicate 7-branes.
  • Figure 2: Two equivalent string junctions related by a Hanany-Witten transition. Figure 2(b) is a canonical representation of the equivalence class, in which the junction lies entirely above the real axis. The dashed lines are branch cuts going down from each 7-brane.
  • Figure 3: The simple root junction of an $I_2$ Kodaira fiber, which has been split into two $I_1$ fibers, corresponding to 7-branes with the same charge. The associated algebra is $SU(2)$.
  • Figure 4: A proper null junction, which is related by Hanany Witten transitions to a contractable closed string loop. The loop surrounds all the 7-branes, around which the total $SL(2,Z)$ monodromy is $1$.
  • Figure 5: proper null junctions of a rational elliptic surface.
  • ...and 3 more figures