Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

M-Theory Versus F-Theory Pictures of the Heterotic String

Paul S. Aspinwall

TL;DR

The paper investigates how M-theory and F-theory portrayals of the heterotic string on T^4 relate through two distinct degenerations of a K3 surface. It establishes a differential-geometric path (a squashed K3) yielding M-theory on a line segment with end-point E8 degrees of freedom, and an algebraic-geometric path (stable degeneration) yielding F-theory on two rational elliptic surfaces glued along an elliptic curve. By linking these degenerations, the authors derive the M-theory description of point-like E8-instantons directly from the F-theory picture. The work thus provides a concrete geometric bridge between M-theory and F-theory descriptions of heterotic compactifications, clarifying how 5-brane dynamics on the M-theory side emerge from F-theory stable degenerations.

Abstract

If one begins with the assertion that the type IIA string compactified on a K3 surface is equivalent to the heterotic string on a four-torus one may try to find a statement about duality in ten dimensions by decompactifying the four-torus. Such a decompactification renders the K3 surface highly singular. The resultant K3 surface may be analyzed in two quite different ways - one of which is natural from the point of view of differential geometry and the other from the point of view of algebraic geometry. We see how the former leads to a "squashed K3 surface" and reproduces the Horava-Witten picture of the heterotic string in M-theory. The latter produces a "stable degeneration" and is tied more closely to F-theory. We use the relationship between these degenerations to obtain the M-theory picture of a point-like E8-instanton directly from the F-theory picture of the same object.

M-Theory Versus F-Theory Pictures of the Heterotic String

TL;DR

The paper investigates how M-theory and F-theory portrayals of the heterotic string on T^4 relate through two distinct degenerations of a K3 surface. It establishes a differential-geometric path (a squashed K3) yielding M-theory on a line segment with end-point E8 degrees of freedom, and an algebraic-geometric path (stable degeneration) yielding F-theory on two rational elliptic surfaces glued along an elliptic curve. By linking these degenerations, the authors derive the M-theory description of point-like E8-instantons directly from the F-theory picture. The work thus provides a concrete geometric bridge between M-theory and F-theory descriptions of heterotic compactifications, clarifying how 5-brane dynamics on the M-theory side emerge from F-theory stable degenerations.

Abstract

If one begins with the assertion that the type IIA string compactified on a K3 surface is equivalent to the heterotic string on a four-torus one may try to find a statement about duality in ten dimensions by decompactifying the four-torus. Such a decompactification renders the K3 surface highly singular. The resultant K3 surface may be analyzed in two quite different ways - one of which is natural from the point of view of differential geometry and the other from the point of view of algebraic geometry. We see how the former leads to a "squashed K3 surface" and reproduces the Horava-Witten picture of the heterotic string in M-theory. The latter produces a "stable degeneration" and is tied more closely to F-theory. We use the relationship between these degenerations to obtain the M-theory picture of a point-like E8-instanton directly from the F-theory picture of the same object.

Paper Structure

This paper contains 5 sections, 28 equations, 5 figures.

Table of Contents

  1. Introduction
  2. The Two-torus
  3. M-Theory
  4. F-Theory
  5. M-theory from F-theory

Figures (5)

  • Figure 1: One-chains producing transcendental 2-cycles.
  • Figure 2: Building a transcendental 2-sphere.
  • Figure 3: The stable degeneration of an elliptic K3.
  • Figure 4: The F-theory picture of moving point-like $E_8$-instantons.
  • Figure 5: Squashing $f$-lines into $\mathcal{I}$-lines.