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The frozen phase of F-theory

Lakshya Bhardwaj, David R. Morrison, Yuji Tachikawa, Alessandro Tomasiello

TL;DR

This work extends F-theory to include frozen O7_+ planes, clarifying how gauge algebras and matter arise on seven-branes and their intersections in six dimensions. By introducing frozen divisors and a generalized Green–Schwarz framework, the authors realize 6d SCFTs that were previously inaccessible, including theories with hypermultiplets in the symmetric representation of $ rak{su}$. They develop perturbative and duality tools to read off spectra, analyze anomaly cancellation with frozen singularities, and construct both noncompact and compact models that exhibit novel tensor and conformal-matter sectors. The results show that frozen branes modify canonical gauge assignments, enable new tensor branches, and connect to perturbative string realizations through dual frames, with implications for the 6d SCFT landscape and its heterotic/IIA connections.

Abstract

We study the interpretation of O7+ planes in F-theory, mainly in the context of the six-dimensional models. In particular, we study how to assign gauge algebras and matter content to seven-branes and their intersections, and the implication of anomaly cancellation in our construction, generalizing earlier analyses without any O7+ planes. By including O7+ planes we can realize 6d superconformal field theories hitherto unobtainable in F-theory, such as those with hypermultiplets in the symmetric representation of su. We also examine a couple of compact models. These reproduce some famous perturbative models, and in some cases enhance their gauge symmetries non-perturbatively.

The frozen phase of F-theory

TL;DR

This work extends F-theory to include frozen O7_+ planes, clarifying how gauge algebras and matter arise on seven-branes and their intersections in six dimensions. By introducing frozen divisors and a generalized Green–Schwarz framework, the authors realize 6d SCFTs that were previously inaccessible, including theories with hypermultiplets in the symmetric representation of . They develop perturbative and duality tools to read off spectra, analyze anomaly cancellation with frozen singularities, and construct both noncompact and compact models that exhibit novel tensor and conformal-matter sectors. The results show that frozen branes modify canonical gauge assignments, enable new tensor branches, and connect to perturbative string realizations through dual frames, with implications for the 6d SCFT landscape and its heterotic/IIA connections.

Abstract

We study the interpretation of O7+ planes in F-theory, mainly in the context of the six-dimensional models. In particular, we study how to assign gauge algebras and matter content to seven-branes and their intersections, and the implication of anomaly cancellation in our construction, generalizing earlier analyses without any O7+ planes. By including O7+ planes we can realize 6d superconformal field theories hitherto unobtainable in F-theory, such as those with hypermultiplets in the symmetric representation of su. We also examine a couple of compact models. These reproduce some famous perturbative models, and in some cases enhance their gauge symmetries non-perturbatively.

Paper Structure

This paper contains 51 sections, 67 equations, 15 figures, 1 table.

Table of Contents

  1. Introduction
  2. Frozen seven-branes and their properties
  3. Basics of orientifold planes
  4. Action on the closed strings:
  5. Action on the open strings:
  6. The RR-charge:
  7. The gauge group:
  8. D$q$-branes intersecting O$p$-planes:
  9. T-duality:
  10. Other types of orientifolds:
  11. Frozen divisors in F-theory
  12. Intersections: perturbative analysis
  13. $\widehat{I}^*$--$I$ intersection
  14. $I^*$--$I^*$, $I^*$--$\widehat{I}^*$, $\widehat{I}^*$--$\widehat{I}^*$ intersections
  15. $I^*$--$I^*$ intersection:
  16. ...and 36 more sections

Figures (15)

  • Figure 2.1: A model with two O$p$-planes with opposite sign is turned by T-duality.
  • Figure 2.2: An O7--D7 intersection, interpreted in F-theory as an intersection between an $\widehat{I}^*_{n+4}$ and an $I^{\rm ns}_{2m}$.
  • Figure 2.3: NS5-branes, D6-branes, and T-duality. The compact and noncompact directions of the cylinder are called respectively directions 4 and 3 in the text.
  • Figure 2.4: On the IIA side, we can move $m$ of the D6s off the NS5s and make them recombine. On the IIB side, this corresponds to a gauge algebra $\mathfrak{su}_m$ that is shared between two curves meeting at a point. We denote this with a double-sided arrow.
  • Figure 2.5: Two configurations with O7$_\pm$-planes, and their T-duals. The dots now represent half-NS5s
  • ...and 10 more figures