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D$_4$-flops of the E$_7$-model

Mboyo Esole, Sabrina Pasterski

TL;DR

This work advances the birational understanding of $E_7$-models by explicitly constructing four crepant resolutions $Y_4,Y_5,Y_6,Y_8$ of a Weierstrass model with Kodaira type $III^*$ and showing their flop relations realize a $D_4$ subdiagram inside the conjectured $E_8$ flop graph. The flops are interpreted through suspended pinch-point geometry, linking 3-fold birational transitions to SSP resolutions, and the authors compute triple-intersection numbers to derive the five-dimensional prepotential and hypermultiplet content. They further connect these geometric data to six-dimensional anomaly cancellations, obtaining consistent counts of adjoint and fundamental matter that match both 5d Coulomb-branch analysis and 6d Green–Schwarz constraints. Overall, the paper provides explicit minimal models, clarifies the chamber structure of the hyperplane arrangement $I(E_7,\mathbf{56})$, and integrates geometric, field-theoretic, and anomaly considerations for $E_7$-models.

Abstract

We study the geography of crepant resolutions of E$_7$-models. An E$_7$-model is a Weierstrass model corresponding to the output of Step 9 of Tate's algorithm characterizing the Kodaira fiber of type III$^*$ over the generic point of a smooth prime divisor. The dual graph of the Kodaira fiber of type III$^*$ is the affine Dynkin diagram of type E$_7$. A Weierstrass model of type E$_7$ is conjectured to have eight distinct crepant resolutions whose flop diagram is a Dynkin diagram of type E$_8$. We construct explicitly four of these eight crepant resolutions forming a sub-diagram of type D$_4$. We explain how the flops between these four crepant resolutions can be understood using the flops between the crepant resolutions of two well-chosen suspended pinch points.

D$_4$-flops of the E$_7$-model

TL;DR

This work advances the birational understanding of -models by explicitly constructing four crepant resolutions of a Weierstrass model with Kodaira type and showing their flop relations realize a subdiagram inside the conjectured flop graph. The flops are interpreted through suspended pinch-point geometry, linking 3-fold birational transitions to SSP resolutions, and the authors compute triple-intersection numbers to derive the five-dimensional prepotential and hypermultiplet content. They further connect these geometric data to six-dimensional anomaly cancellations, obtaining consistent counts of adjoint and fundamental matter that match both 5d Coulomb-branch analysis and 6d Green–Schwarz constraints. Overall, the paper provides explicit minimal models, clarifies the chamber structure of the hyperplane arrangement , and integrates geometric, field-theoretic, and anomaly considerations for -models.

Abstract

We study the geography of crepant resolutions of E-models. An E-model is a Weierstrass model corresponding to the output of Step 9 of Tate's algorithm characterizing the Kodaira fiber of type III over the generic point of a smooth prime divisor. The dual graph of the Kodaira fiber of type III is the affine Dynkin diagram of type E. A Weierstrass model of type E is conjectured to have eight distinct crepant resolutions whose flop diagram is a Dynkin diagram of type E. We construct explicitly four of these eight crepant resolutions forming a sub-diagram of type D. We explain how the flops between these four crepant resolutions can be understood using the flops between the crepant resolutions of two well-chosen suspended pinch points.

Paper Structure

This paper contains 26 sections, 2 theorems, 86 equations, 13 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Defining the E$_7$-model
  3. E$_7$ facts and conjectures
  4. Summary of results
  5. Preliminaries
  6. $G$-models and Coulomb phases
  7. Geography of minimal models: decomposition of the relative movable cone into relative nef-cones.
  8. The hyperplane arrangement I($\text{E}_7, \mathbf{56}$)
  9. E$_7$ root system and Dynkin diagram
  10. Root system of types E$_7$ and E$_8$, and representation $\mathbf{56}$ of E$_7$.
  11. Chamber structure of the hyperplane I(E$_7$,$\mathbf{56}$)
  12. Minimal models $Y_4$, $Y_5$, $Y_6$, and $Y_8$
  13. Overview of the sequence of blowups defining the resolutions
  14. The geometry of $Y_4$
  15. The geometry of $Y_5$
  16. ...and 11 more sections

Key Result

Theorem 3.1

The hyperplane arrangement I$(E_7, \mathbf{56})$ has eight chambers. Each chamber is simplicial. The adjacency graph of the chambers is isomorphic to the Dynkin diagram of type E$_8$.

Figures (13)

  • Figure 1.1: Adjacency graph of the chambers of the hyperplane arrangement I$(E_7, \mathbf{56})$. Each chamber is simplicial. A weight $\varpi$ between two nodes indicates that the corresponding chambers are separated by the hyperplane $\varpi^\bot$. For example, one goes from Ch$_1$ to Ch$_2$ by crossing the hyperplane $\varpi_{19}^\bot$. The colored chambers forming a subgraph of type D$_4$ are those corresponding to the nef-cone of the crepant resolutions constructed in this paper.
  • Figure 3.1: Affine Dynkin diagram of type $\widetilde{\text{E}}_7$. Removing the black node gives the Dynkin diagram of type E$_7$. The numbers inside the nodes are the multiplicities of the Kodaira fiber of type III$^*$ and the Dynkin labels of the highest root. Thus, the Coxeter number of E$_7$ is $18$. The root $\alpha_1$ is the highest weight of the adjoint representation while $\alpha_6$ is the highest weight of the fundamental represenation $\mathbf{56}$.
  • Figure 3.2: Affine Dynkin diagram of type $\widetilde{\text{E}}_8$. Removing the black node gives the Dynkin diagram of type E$_8$. The number in the nodes are the multiplicities of the Kodaira fiber of type II$^*$.
  • Figure 3.3: Hasse diagram for the weights of the representation $\bf{56}$ of E$_7$. The node $i$ represents the weight $\varpi_i$ given above and written in the basis of fundamental weights. A root $\alpha$ between two nodes $i$ and $j$ (with $i<j$) indicates that $\varpi_j=\varpi_i -\alpha$. The white nodes are those corresponding to the weights used to define the sign vector that characterizes the chambers of the hyperplane arrangement I($\bf{56}$, E$_7$). In a given chamber of I($\bf{56}$, E$_7$), each white node takes a specific sign (see section \ref{['sec:IE756']}). The red (resp. blue) nodes correspond to weights in the positive (resp. negative) conical hull of positive roots. In particular, a white node corresponds to a weight $\varpi$ such that the hyperplane $\varpi^\bot$ intersects the interior of the dual fundamental Weyl chamber while that is not the case for a weight corresponding to a blue or red node.
  • Figure 3.4: Up to a sign, there are seven weights of the representation $\mathbf{56}$ of E$_7$ such that each of them has a kernel intersecting the interior of the dual fundamental Weyl chamber. The partial order of weights provide the Hasse diagram presented above and corresponding to a decorated Dynkin diagram of type E$_7$. We write $\varpi_i \xrightarrow{-\alpha_\ell} \varpi_j$ to indicate that $\varpi_i-\alpha_\ell= \varpi_j$. The linear forms defined by these weights give the sign vector for the hyperplane arrangement I$(\text{E}_7, \mathbf{56})$.
  • ...and 8 more figures

Theorems & Definitions (12)

  • Theorem 3.1
  • proof
  • Remark 4.1
  • Remark 4.2
  • Remark 6.1
  • Definition A.1
  • Definition A.2
  • Definition A.3: $\mathbb{Q}$-Gorenstein, $\mathbb{Q}$-factorial, nef-divisors
  • Definition A.4: Minimal models
  • Definition A.5: Modifications, (crepant) resolutions of singularities
  • ...and 2 more