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Higher S-dualities and Shephard-Todd groups

Sergio Cecotti, Michele Del Zotto

TL;DR

This work extends Seiberg–Witten S-duality from ${\mathcal N}=2$ SQCD to non-Lagrangian ${SU(2)}$-coupled SCFTs whose matter sectors carry weights of ${E_6,E_7,E_8}$. Using a homological framework, the authors identify S-duality with auto-equivalences of the derived category of coherent sheaves on weighted projective lines of tubular type and show that the action on matter charges factors through exceptional Shephard–Todd complex reflection groups ${G_4}$ and ${G_8}$. The analysis unifies the p=1,2 (Lagrangian) cases with p=3,4,6 (non-Lagrangian) by replacing rational reflection groups with complex CM-field reflections, thereby generalizing Spin(8) triality to the exceptional lattices ${\mathfrak g}=E_6,E_7,E_8$. The key mechanism is the pair of telescopic functors $T$ and $L$ generating a ${\mathcal B}_3$ braid action that, on the matter lattice, reduces to finite complex reflection groups whose representations encode the S-duality action on ${E^{(1,1)}_r}$ models. The results provide a precise algebraic characterization of S-duality for these theories and connect 4d dualities with the McKay correspondence and the geometry of cluster categories.

Abstract

Seiberg and Witten have shown that in N=2 SQCD with $N_f=2N_c=4$ the S-duality group PSL(2,Z) acts on the flavor charges, which are weights of Spin(8), by triality. There are other N=2 SCFTs in which SU(2) SYM is coupled to strongly-interacting non-Lagrangian matter: their matter charges are weights of $E_6$, $E_7$ and $E_8$ instead of Spin(8). The S-duality group PSL(2,Z) acts on these weights: what replaces Spin(8) triality for the $E_6,E_7,E_8$ root lattices? In this paper we answer the question. The action on the matter charges of (a finite central extension of) PSL(2,Z) factorizes trough the action of the exceptional Shephard--Todd groups $G_4$ and $G_8$ which should be seen as complex analogs of the usual triality group $\mathfrak{S}_3\simeq \mathrm{Weyl}(A_2)$. Our analysis is based on the identification of S-duality for SU(2) gauge SCFTs with the group of automorphisms of the cluster category of weighted projective lines of tubular type.

Higher S-dualities and Shephard-Todd groups

TL;DR

This work extends Seiberg–Witten S-duality from SQCD to non-Lagrangian -coupled SCFTs whose matter sectors carry weights of . Using a homological framework, the authors identify S-duality with auto-equivalences of the derived category of coherent sheaves on weighted projective lines of tubular type and show that the action on matter charges factors through exceptional Shephard–Todd complex reflection groups and . The analysis unifies the p=1,2 (Lagrangian) cases with p=3,4,6 (non-Lagrangian) by replacing rational reflection groups with complex CM-field reflections, thereby generalizing Spin(8) triality to the exceptional lattices . The key mechanism is the pair of telescopic functors and generating a braid action that, on the matter lattice, reduces to finite complex reflection groups whose representations encode the S-duality action on models. The results provide a precise algebraic characterization of S-duality for these theories and connect 4d dualities with the McKay correspondence and the geometry of cluster categories.

Abstract

Seiberg and Witten have shown that in N=2 SQCD with the S-duality group PSL(2,Z) acts on the flavor charges, which are weights of Spin(8), by triality. There are other N=2 SCFTs in which SU(2) SYM is coupled to strongly-interacting non-Lagrangian matter: their matter charges are weights of , and instead of Spin(8). The S-duality group PSL(2,Z) acts on these weights: what replaces Spin(8) triality for the root lattices? In this paper we answer the question. The action on the matter charges of (a finite central extension of) PSL(2,Z) factorizes trough the action of the exceptional Shephard--Todd groups and which should be seen as complex analogs of the usual triality group . Our analysis is based on the identification of S-duality for SU(2) gauge SCFTs with the group of automorphisms of the cluster category of weighted projective lines of tubular type.

Paper Structure

This paper contains 40 sections, 217 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction and Summary
  2. Homological approach to $S$--duality
  3. Remark.
  4. Overview
  5. Charge lattices.
  6. Serre duality and Coxeter transformation.
  7. Auto--equivalences of the derived category.
  8. Remark.
  9. Coherent sheaves on weighted projective lines
  10. Weighted projective lines GL1lenzing1lenzing2lenzingmeltzermeltzerdirkrevLINE.
  11. Explicit formulae in the canonical basis.
  12. Specializing to the $\chi(\boldsymbol{p})=0$ case.
  13. Remark.
  14. Structure of the derived category $\mathsf{D}^b\,\mathsf{coh}\,\mathbb{X}(\boldsymbol{p})$.
  15. The $\chi(\boldsymbol{p})=0$ case.
  16. ...and 25 more sections

Figures (1)

  • Figure 1: The assignments of integral variables $(\boldsymbol{y})$ to the nodes of the $E_8$ graph. For $D_4,E_6,E_7$ just restrict to the corresponding Dynkin subgraph.