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Strings of Minimal 6d SCFTs

Babak Haghighat, Albrecht Klemm, Guglielmo Lockhart, Cumrun Vafa

TL;DR

This work demonstrates a concrete quiver description for strings of the n=4 minimal 6d SCFT, derived via Sen's limit of F-theory, and validates the elliptic genus against M-theory/topological string dualities. By exploiting localization and JK-residue techniques, the authors obtain explicit elliptic genera for one and two strings and connect these to refined BPS invariants of the associated elliptic Calabi-Yau geometries. The paper then systematically builds the corresponding toric Calabi–Yau manifolds for all n=1…8,12, solving topological strings to extract genus-zero BPS data and illustrating modular properties via E2 completions. The results provide a coherent bridge between worldsheet quivers of 6d SCFT strings, F/M-theory dualities, and the refined topological string/BPS spectrum, offering predictions for elliptic genera across the entire class of minimal 6d SCFTs.

Abstract

We study strings associated with minimal 6d SCFTs, which by definition have only one string charge and no Higgs branch. These theories are labelled by a number n with 1 <= n <= 8 or n = 12. Quiver theories have previously been proposed which describe strings of SCFTs for n = 1, 2. For n > 2 the strings interact with the bulk gauge symmetry. In this paper we find a quiver description for the n = 4 string using Sen's limit of F-theory and calculate its elliptic genus with localization techniques. This result is checked using the duality of F-theory with M-theory and topological string theory whose refined BPS partition function captures the elliptic genus of the SCFT strings. We use the topological string theory to gain insight into the elliptic genus for other values of n.

Strings of Minimal 6d SCFTs

TL;DR

This work demonstrates a concrete quiver description for strings of the n=4 minimal 6d SCFT, derived via Sen's limit of F-theory, and validates the elliptic genus against M-theory/topological string dualities. By exploiting localization and JK-residue techniques, the authors obtain explicit elliptic genera for one and two strings and connect these to refined BPS invariants of the associated elliptic Calabi-Yau geometries. The paper then systematically builds the corresponding toric Calabi–Yau manifolds for all n=1…8,12, solving topological strings to extract genus-zero BPS data and illustrating modular properties via E2 completions. The results provide a coherent bridge between worldsheet quivers of 6d SCFT strings, F/M-theory dualities, and the refined topological string/BPS spectrum, offering predictions for elliptic genera across the entire class of minimal 6d SCFTs.

Abstract

We study strings associated with minimal 6d SCFTs, which by definition have only one string charge and no Higgs branch. These theories are labelled by a number n with 1 <= n <= 8 or n = 12. Quiver theories have previously been proposed which describe strings of SCFTs for n = 1, 2. For n > 2 the strings interact with the bulk gauge symmetry. In this paper we find a quiver description for the n = 4 string using Sen's limit of F-theory and calculate its elliptic genus with localization techniques. This result is checked using the duality of F-theory with M-theory and topological string theory whose refined BPS partition function captures the elliptic genus of the SCFT strings. We use the topological string theory to gain insight into the elliptic genus for other values of n.

Paper Structure

This paper contains 19 sections, 127 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Minimal SCFTs in six-dimensions
  3. Strings of the $\mathcal{O}(-1)\rightarrow \mathbb{P}^1$ and $\mathcal{O}(-2)\rightarrow \mathbb{P}^1$ models
  4. From strings of 6d SCFTs to topological strings
  5. Quiver for the $\mathcal{O}(-4) \rightarrow \mathbb{P}^1$ model
  6. Localization computation
  7. Modular anomaly
  8. Refined BPS invariants
  9. The Calabi-Yau geometries with elliptic singularities
  10. The local geometries
  11. Solution of the topological string on the toric hypersurface Calabi-Yau spaces
  12. ${\cal O}(-n)\rightarrow \mathbb{P}^1$ geometries with $n=1,2$
  13. ${\cal O}(-3)\rightarrow \mathbb{P}^1$ geometry with $\hat{A}_2$ resolution
  14. ${\cal O}(-4)\rightarrow \mathbb{P}^1$ geometry with $\hat{D}_4$ resolution
  15. ${\cal O}(-5)\rightarrow \mathbb{P}^1$ geometry with $\hat{F}_4$ resolution
  16. ...and 4 more sections

Figures (8)

  • Figure 1: The quiver describing the $\mathcal{O}(-2) \rightarrow \mathbb{P}^1$ strings.
  • Figure 2: Configuration of $(-2)$ curves that gives rise to the $O(-2)\to \mathbb{P}^1$ local geometry. We have also indicated the degeneration of the elliptic fiber over each curve that gives rise to the M-string geometry.
  • Figure 3: The quiver for $\mathcal{O}(-1) \rightarrow \mathbb{P}^1$ strings.
  • Figure 4: The local geometry that gives rise to the $SO(8+2p)$ 6d SCFT.
  • Figure 5: The $\mathbb{C}^2/\mathbb{Z}_4$ quiver.
  • ...and 3 more figures