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Lie Groups, Calabi-Yau Threefolds, and F-Theory

Paul S. Aspinwall, Sheldon Katz, David R. Morrison

TL;DR

This paper develops a geometric framework to determine the gauge algebra and hypermultiplet spectrum of six-dimensional vacua in F-theory, starting from an elliptic Calabi–Yau threefold with a section and yielding a six-dimensional $N=(1,0)$ theory. It shows that the gauge content and massless matter follow from intersection theory, and that non-simply-laced algebras arise via monodromy on Kodaira fibers, with dual descriptions in M-theory and IIA via RR-field tuning. It provides explicit rules for counting hypermultiplets, including non-local contributions from monodromy and collisions, and illustrates with cases such as $sp(k)$, $f_4$, and $so(2k-1)$. Anomaly-cancellation constraints supply sharp numerical relations linking geometry to spectra and guide deformations/Higgsing that change the gauge algebra, resolving ambiguities in ${\mathfrak{su}}(2k+1)$ with ${\mathbb Z}_2$ monodromy.

Abstract

The F-theory vacuum constructed from an elliptic Calabi-Yau threefold with section yields an effective six-dimensional theory. The Lie algebra of the gauge sector of this theory and its representation on the space of massless hypermultiplets are shown to be determined by the intersection theory of the homology of the Calabi-Yau threefold. (Similar statements hold for M-theory and the type IIA string compactified on the threefold, where there is also a dependence on the expectation values of the Ramond-Ramond fields.) We describe general rules for computing the hypermultiplet spectrum of any F-theory vacuum, including vacua with non-simply-laced gauge groups. The case of monodromy acting on a curve of A_even singularities is shown to be particularly interesting and leads to some unexpected rules for how 2-branes are allowed to wrap certain 2-cycles. We also review the peculiar numerical predictions for the geometry of elliptic Calabi-Yau threefolds with section which arise from anomaly cancellation in six dimensions.

Lie Groups, Calabi-Yau Threefolds, and F-Theory

TL;DR

This paper develops a geometric framework to determine the gauge algebra and hypermultiplet spectrum of six-dimensional vacua in F-theory, starting from an elliptic Calabi–Yau threefold with a section and yielding a six-dimensional theory. It shows that the gauge content and massless matter follow from intersection theory, and that non-simply-laced algebras arise via monodromy on Kodaira fibers, with dual descriptions in M-theory and IIA via RR-field tuning. It provides explicit rules for counting hypermultiplets, including non-local contributions from monodromy and collisions, and illustrates with cases such as , , and . Anomaly-cancellation constraints supply sharp numerical relations linking geometry to spectra and guide deformations/Higgsing that change the gauge algebra, resolving ambiguities in with monodromy.

Abstract

The F-theory vacuum constructed from an elliptic Calabi-Yau threefold with section yields an effective six-dimensional theory. The Lie algebra of the gauge sector of this theory and its representation on the space of massless hypermultiplets are shown to be determined by the intersection theory of the homology of the Calabi-Yau threefold. (Similar statements hold for M-theory and the type IIA string compactified on the threefold, where there is also a dependence on the expectation values of the Ramond-Ramond fields.) We describe general rules for computing the hypermultiplet spectrum of any F-theory vacuum, including vacua with non-simply-laced gauge groups. The case of monodromy acting on a curve of A_even singularities is shown to be particularly interesting and leads to some unexpected rules for how 2-branes are allowed to wrap certain 2-cycles. We also review the peculiar numerical predictions for the geometry of elliptic Calabi-Yau threefolds with section which arise from anomaly cancellation in six dimensions.

Paper Structure

This paper contains 7 sections, 2 theorems, 25 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Lie algebras and Calabi--Yau threefolds
  3. Counting hypermultiplets
  4. The case of monodromy on $\operatorname{\mathfrak{su}}(\textrm{odd})$.
  5. The case of $\operatorname{\mathfrak{su}}(3)\to\operatorname{\mathfrak{sp}}(1)$.
  6. Deformation to the $\operatorname{\mathfrak{su}}(\textrm{even})$ case.
  7. Numerical Oddities

Key Result

Theorem 1

Let ${\mathbb Z}_2$ monodromy act on an I$_{2k}$ fiber in F-theory so that the central component of the fiber has moduli space $M_1$ and the outer components have moduli space $M_2$. Thus $M_2\to M_1$ is a double cover. Then the resulting gauge algebra is $\operatorname{\mathfrak{sp}}(k)$ and we hav

Figures (6)

  • Figure 1: Ruled surfaces producing $\operatorname{\mathfrak{sp}}(2)$.
  • Figure 2: The adjoint of $\operatorname{\mathfrak{sp}}(2)$.
  • Figure 3: The generic case of Spin(9) gauge symmetry.
  • Figure 4: The singular surface for monodromy in $\operatorname{\mathfrak{su}}(3)$.
  • Figure 5: The singular surface for monodromy in $\operatorname{\mathfrak{su}}(5)$.
  • ...and 1 more figures

Theorems & Definitions (2)

  • Theorem 1
  • Lemma 1