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On Exceptional Instanton Strings

Michele Del Zotto, Guglielmo Lockhart

Abstract

According to a recent classification of 6d (1,0) theories within F-theory there are only six "pure" 6d gauge theories which have a UV superconformal fixed point. The corresponding gauge groups are $SU(3),SO(8),F_4,E_6,E_7$, and $E_8$. These exceptional models have BPS strings which are also instantons for the corresponding gauge groups. For $G$ simply-laced, we determine the 2d $\mathcal{N}=(0,4)$ worldsheet theories of such BPS instanton strings by a simple geometric engineering argument. These are given by a twisted $S^2$ compactification of the 4d $\mathcal{N}=2$ theories of type $H_2, D_4, E_6, E_7$ and $E_8$ (and their higher rank generalizations), where the 6d instanton number is mapped to the rank of the corresponding 4d SCFT. This determines their anomaly polynomials and, via topological strings, establishes an interesting relation among the corresponding $T^2 \times S^2$ partition functions and the Hilbert series for moduli spaces of $G$ instantons. Such relations allow to bootstrap the corresponding elliptic genera by modularity. As an example of such procedure, the elliptic genera for a single instanton string are determined. The same method also fixes the elliptic genus for case of one $ F_4 $ instanton. These results unveil a rather surprising relation with the Schur index of the corresponding 4d $\mathcal{N}=2$ models.

On Exceptional Instanton Strings

Abstract

According to a recent classification of 6d (1,0) theories within F-theory there are only six "pure" 6d gauge theories which have a UV superconformal fixed point. The corresponding gauge groups are , and . These exceptional models have BPS strings which are also instantons for the corresponding gauge groups. For simply-laced, we determine the 2d worldsheet theories of such BPS instanton strings by a simple geometric engineering argument. These are given by a twisted compactification of the 4d theories of type and (and their higher rank generalizations), where the 6d instanton number is mapped to the rank of the corresponding 4d SCFT. This determines their anomaly polynomials and, via topological strings, establishes an interesting relation among the corresponding partition functions and the Hilbert series for moduli spaces of instantons. Such relations allow to bootstrap the corresponding elliptic genera by modularity. As an example of such procedure, the elliptic genera for a single instanton string are determined. The same method also fixes the elliptic genus for case of one instanton. These results unveil a rather surprising relation with the Schur index of the corresponding 4d models.

Paper Structure

This paper contains 32 sections, 159 equations, 5 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Minimal 6d (1,0) SCFTs
  3. F-theory engineering of 6d SCFTs in a nutshell
  4. Minimal 6d (1,0) SCFTs from F-theory orbifolds
  5. Instanton strings and $\widetilde{H_G}^{(k)}$ theories
  6. A lightning review of $\widetilde{H}_G^{(k)}$ models
  7. The $\beta$-twisted $\widetilde{H}_G^{(k)}$ models and 6d instanton strings
  8. Some generalities about the 2d $(0,4)$ $\widetilde{h}_G^{(k)}$ SCFTs
  9. Central Charges $(c_L,c_R)$
  10. Anomaly polynomial
  11. Elliptic genus
  12. Strings of the $SO(8)$ 6d $(1,0)$ minimal SCFT revisited
  13. The $E_6$ case: $k=1$
  14. Topological strings and elliptic genera
  15. 6d BPS strings and topological strings
  16. ...and 17 more sections

Figures (5)

  • Figure 1: IIB brane engineering of the $\widetilde{H}^{(k)}_G$ models.
  • Figure 2: IIB description of the tensor branch of the 6d (1,0) theory.
  • Figure 3: 2d quiver corresponding to the $H_{SO(8)}^{(k)}$ theory.
  • Figure 4: Gaiotto $T_3$ theory from degeneration of the $SU(3), N_f = 6$ curve.
  • Figure 5: Schematic IIB description of the 6d (1,0) $\Omega$ background.