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S-duality in $\mathcal{N} = 1$ orientifold SCFTs

Iñaki García-Etxebarria, Ben Heidenreich

TL;DR

The paper develops a systematic framework for S-duality among four dimensional N=1 SCFTs engineered by D3 branes at toric Calabi-Yau orientifold singularities. By combining toric geometry, brane tilings, and a discrete torsion dictionary for NSNS and RR sectors, it maps geometric data to dual gauge theories and constructs TO_k CFTs via deconfinement. It provides explicit classifications and $SL(2,Z)$ multiplets for a broad set of toric diagrams up to five sides, including del Pezzo 1, F0, Y^{4,0}, and dP2, with nontrivial checks using superconformal indices. The results yield a scalable method to generate infinite families of N=1 S-duals and reveal deep links between geometry, discrete fluxes, and 4d SCFT dynamics, with potential extensions to non-isolated and flavored setups.

Abstract

We present a general solution to the problem of determining all S-dual descriptions for a specific (but very rich) class of $\mathcal{N} = 1$ SCFTs. These SCFTs are indexed by decorated toric diagrams, and can be engineered in string theory by probing orientifolds of isolated toric singularities with D3 branes. The S-dual phases are described by quiver gauge theories coupled to specific types of conformal matter which we describe explicitly. We illustrate our construction with many examples, including S-dualities in previously unknown SCFTs.

S-duality in $\mathcal{N} = 1$ orientifold SCFTs

TL;DR

The paper develops a systematic framework for S-duality among four dimensional N=1 SCFTs engineered by D3 branes at toric Calabi-Yau orientifold singularities. By combining toric geometry, brane tilings, and a discrete torsion dictionary for NSNS and RR sectors, it maps geometric data to dual gauge theories and constructs TO_k CFTs via deconfinement. It provides explicit classifications and multiplets for a broad set of toric diagrams up to five sides, including del Pezzo 1, F0, Y^{4,0}, and dP2, with nontrivial checks using superconformal indices. The results yield a scalable method to generate infinite families of N=1 S-duals and reveal deep links between geometry, discrete fluxes, and 4d SCFT dynamics, with potential extensions to non-isolated and flavored setups.

Abstract

We present a general solution to the problem of determining all S-dual descriptions for a specific (but very rich) class of SCFTs. These SCFTs are indexed by decorated toric diagrams, and can be engineered in string theory by probing orientifolds of isolated toric singularities with D3 branes. The S-dual phases are described by quiver gauge theories coupled to specific types of conformal matter which we describe explicitly. We illustrate our construction with many examples, including S-dualities in previously unknown SCFTs.

Paper Structure

This paper contains 35 sections, 163 equations, 61 figures, 5 tables.

Table of Contents

  1. Introduction
  2. The geometry of toric orientifolds
  3. Toric Calabi-Yau singularities
  4. Orientifolds of toric Calabi-Yau singularities
  5. Discrete torsion and AdS/CFT
  6. Discrete torsion for toric orientifolds
  7. The divisor basis and intersection form
  8. Brane tilings, orientifolds, and NSNS torsion
  9. Review of brane tilings
  10. Orientifolds and NSNS torsion
  11. Constructing the orientifold gauge theory
  12. $\TO_k$TOk CFTs and RR torsion
  13. Constructing $\TO_k$ CFTs using deconfinement
  14. RR torsion
  15. Details of $\TO_2$TO2 CFTs
  16. ...and 20 more sections

Figures (61)

  • Figure 1: \ref{['sfig:toricDiags']} Toric diagrams for (clockwise from upper left) the complex cone over $dP_1$, the conifold, the complex cone over $\bF_0$, and $\bC^2/\bZ_2\times\bC$. \ref{['sfig:equivTDs']} Toric diagrams related by rotations and shears are equivalent. These all represent $\bC^3/\bZ_3$.
  • Figure 2: \ref{['sfig:partialRes']} Toric diagrams for some partial and complete resolutions of the $dP_1$ singularity. \ref{['sfig:dP1web']} The web diagram for the affine $dP_1$ singularity. \ref{['sfig:dP1resolvedweb']} Web diagrams for the resolutions of the $dP_1$ singularity shown in \ref{['sfig:partialRes']}.
  • Figure 3: \ref{['sfig:nonToricInvol']} Examples of non-toric involutions for (clockwise from upper left) the $dP_2$ singularity, $Y^{4,0}$, and the conifold. \ref{['sfig:toricInvol']} A toric involution. The red stars indicate the even sublattice. \ref{['sfig:toricInvolRes']} A complete resolution of the resulting toric orientifold. Each starred vertex represents an O7 plane, which wraps a compact (non-compact) divisor if the vertex lies in the interior (on the boundary) of the toric diagram. Each shaded triangle represents an O3 plane arising after complete resolution of the singularity.
  • Figure 4: The four types of O3 plane correspond to different choices of RR and NSNS discrete torsion ($[F]$ and $[H]$, respectively) in the gravity dual, where the action of $\SL(2,\bZ)$ on the O3 planes can be inferred from the known action on the RR and NSNS two form connections.
  • Figure 5: The action of $\SL(2,\bZ)$ on $[H], [F]$ in the case where $H^3(X_5,\tbZ) \cong \bZ_2^2$. The sixteen choices of discrete torsion fall into a singlet, three triplets, and a sextet under $\SL(2,\bZ)$, colored black, red/green/blue, and purple, respectively.
  • ...and 56 more figures