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Mordell-Weil Torsion, Anomalies, and Phase Transitions

Mboyo Esole, Monica Jinwoo Kang, Shing-Tung Yau

TL;DR

<3-5 sentences>We investigate how Mordell-Weil torsion, particularly MW = $\mathbb{Z}_2$, shapes the Coulomb branches of five-dimensional $\mathcal N=1$ gauge theories engineered by M-theory on elliptically fibered Calabi–Yau threefolds with an $I_2$–$I_4$ collision, yielding gauge algebras $\mathfrak{su}(2)\oplus\mathfrak{sp}(4)$ or $\mathfrak{su}(2)\oplus\mathfrak{su}(4)$. The authors develop a geometry-to-physics dictionary using crepant resolutions, compute triple intersection numbers and weights of fibral curves to fix matter content, and determine the 5d prepotential via the IMS framework in each Coulomb chamber. They demonstrate how the presence or absence of torsion constrains representations, and they verify anomaly cancellation via a Green–Schwarz mechanism in the 6d uplifts across all models studied. The work thus connects Mordell–Weil torsion to precise spectrum data and anomaly consistency in string-derived gauge theories, providing a systematic method to extract 5d/6d physics from elliptic fibrations with collisions.

Abstract

We explore how introducing a non-trivial Mordell-Weil group changes the structure of the Coulomb phases of a five-dimensional gauge theory from an M-theory compactified on an elliptically fibered Calabi-Yau threefolds with a I$_2$+I$_4$ collision of singularities. The resulting gauge theory has a semi-simple Lie algebra $\mathfrak{su}(2)\oplus \mathfrak{sp}(4)$ or $\mathfrak{su}(2)\oplus \mathfrak{su}(4)$. We compute topological invariants relevant for the physics, such as the Euler characteristic, Hodge numbers, and triple intersection numbers. We determine the matter representation geometrically by computing weights via intersection of curves and fibral divisors. We fix the number of charged hypermultiplets transforming in each representations by comparing the triple intersection numbers and the one-loop prepotential. This condition is enough to fix the number of representation when the Mordell-Weil group is $\mathbb{Z}_2$ but not when it is trivial. The vanishing of the fourth power of the curvature forms in the anomaly polynomial is enough to fix the number of representations. We discuss anomaly cancellations of the six-dimensional uplifted. In particular, the gravitational anomaly is also considered as the Hodge numbers are computed explicitly without counting the degrees of freedom of the Weierstrass equation.

Mordell-Weil Torsion, Anomalies, and Phase Transitions

TL;DR

<3-5 sentences>We investigate how Mordell-Weil torsion, particularly MW = , shapes the Coulomb branches of five-dimensional gauge theories engineered by M-theory on elliptically fibered Calabi–Yau threefolds with an – collision, yielding gauge algebras or . The authors develop a geometry-to-physics dictionary using crepant resolutions, compute triple intersection numbers and weights of fibral curves to fix matter content, and determine the 5d prepotential via the IMS framework in each Coulomb chamber. They demonstrate how the presence or absence of torsion constrains representations, and they verify anomaly cancellation via a Green–Schwarz mechanism in the 6d uplifts across all models studied. The work thus connects Mordell–Weil torsion to precise spectrum data and anomaly consistency in string-derived gauge theories, providing a systematic method to extract 5d/6d physics from elliptic fibrations with collisions.

Abstract

We explore how introducing a non-trivial Mordell-Weil group changes the structure of the Coulomb phases of a five-dimensional gauge theory from an M-theory compactified on an elliptically fibered Calabi-Yau threefolds with a I+I collision of singularities. The resulting gauge theory has a semi-simple Lie algebra or . We compute topological invariants relevant for the physics, such as the Euler characteristic, Hodge numbers, and triple intersection numbers. We determine the matter representation geometrically by computing weights via intersection of curves and fibral divisors. We fix the number of charged hypermultiplets transforming in each representations by comparing the triple intersection numbers and the one-loop prepotential. This condition is enough to fix the number of representation when the Mordell-Weil group is but not when it is trivial. The vanishing of the fourth power of the curvature forms in the anomaly polynomial is enough to fix the number of representations. We discuss anomaly cancellations of the six-dimensional uplifted. In particular, the gravitational anomaly is also considered as the Hodge numbers are computed explicitly without counting the degrees of freedom of the Weierstrass equation.

Paper Structure

This paper contains 36 sections, 11 theorems, 195 equations, 11 figures, 15 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Weierstrass models and Deligne's formulaire
  4. Elliptic fibrations with Mordell-Weil group $\mathbb{Z}_2$
  5. $G$-models and representations
  6. Intersection theory
  7. Anomaly Cancellation
  8. Summary of results
  9. Crepant resolutions
  10. Euler characteristics
  11. Hodge numbers for Calabi-Yau elliptic threefolds
  12. Hyperplane arrangements
  13. Triple intersection numbers
  14. The prepotential of the five-dimensional theories
  15. Number of charged hypermultiplets.
  16. ...and 21 more sections

Key Result

Theorem 2.2

An elliptic fibration over a smooth base $B$ and with Mordell-Weil group $\mathbb{Z}_2$ is birational to the following (singular) Weierstrass model:

Figures (11)

  • Figure 1: This is the fiber structure of $G=(\text{SU($2$)}\times \text{Sp($4$)})/\mathbb{Z}_2$ uptil codimension two for the Calabi-Yau threefolds.
  • Figure 3: This is the fiber structure of $G=(\text{SU($2$)}\times \text{SU($4$)})/\mathbb{Z}_2$ uptil codimension two for the Calabi-Yau threefolds.
  • Figure 5: There are three chambers in I$_2^{\text{ns}}+$I$_4^{\text{ns}}$-model with a Mordell-Weil group $\mathbb{Z}_2$. Each chamber is noted as the signs of $[\varpi_1,\varpi_2]$ where $\varpi_1=\phi _1-\psi _1$ and $\varpi_2=\psi _1+\phi _1-\phi _2$. For chamber $1$, $\varpi_1<0 ,\ \varpi_2>0$; for chamber $2$, $\varpi_1>0 ,\ \varpi_2>0$; and for chamber $3$, $\varpi_1>0 ,\ \varpi_2<0$.
  • Figure 6: Chambers of the hyperplane arrangement I($A_1\oplus A_2,\mathbf{R}$) with $\mathbf{R}=(\bf{3},\bf{1})\oplus(\bf{1},\bf{15})\oplus(\mathbf{1},\mathbf{6})\oplus(\mathbf{2},\mathbf{4})\oplus(\mathbf{2},\mathbf{\bar{4}})\oplus(\mathbf{1},\mathbf{4})\oplus(\mathbf{1},\mathbf{\bar{4}})$. Each circle corresponds to a chamber. The label on the edge connecting two chambers is the wall separating them. In the sign vector, an entry $s$ means a sign $(-1)^{s+1}$ for the corresponding form, that is, $s=0$ (resp. $s=1$) means that the corresponding linear form is negative (resp. positive). For example, the chamber $1a^-$ corresponds to $(010001001)$, which gives the sign vector $(-1,1,-1,-1,-1,1,-1,-1)$.
  • Figure 7: Chambers of the hyperplane arrangement I($A_1\oplus A_2,\mathbf{R}$) with $\mathbf{R}=(\bf{3},\bf{1})\oplus(\bf{1},\bf{15})\oplus (\bf{2},\bf{4})\oplus (\bf{2},\bf{\bar{4}})\oplus(\bf{1},\bf{6})$. Each circle corresponds to a chamber. The label on the edge connecting two chambers is the wall separating them. In the sign vector, an entry $s$ means a sign $(-1)^{s+1}$ for the corresponding form, that is, $s=0$ (resp. $s=1$) means that the corresponding linear form is negative (resp. positive). For example, the chamber $1a^-$ corresponds to $(010001001)$, which gives the sign vector $(-1,1,-1,-1,-1,1,-1,-1)$.
  • ...and 6 more figures

Theorems & Definitions (22)

  • Definition 2.1
  • Theorem 2.2
  • Definition 2.3: $G$-model
  • Definition 2.4: $\mathcal{K}_1+\cdots+\mathcal{K}_n$-model
  • Definition 2.5: Weight vector of a vertical curve
  • Definition 2.6: Saturated set of weights
  • Definition 2.7: Saturation of a subset
  • Proposition 2.8: Humphreys
  • Theorem 2.9: Bourbaki, Bourbaki.GLA79
  • Definition 2.10: Representation of a $G$-model
  • ...and 12 more