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Homological S-Duality in 4d N=2 QFTs

Matteo Caorsi, Sergio Cecotti

TL;DR

Caorsi and Cecotti develop a comprehensive homological framework to study S-duality in 4d ${\rm N}=2$ QFTs by identifying the S-duality group with triangle auto-equivalences of cluster categories modulo physically trivial automorphisms. The duality group acquires a generalized braid-group structure and embeds, up to commensurability, into the Siegel modular group ${Sp}(2g,\mathbb{Z})$, tying electromagnetic and flavor lattices via the 4d/2d correspondence. They analyze four major families, $(G,G')$, $D_p(G)$, $E_r^{(1,1)}(G)$, and $(G,\widehat H)$, showing notable enhancements of S-duality in many models and explicit matrix realizations through telescopic functors and Thomas–Seidel twists. The approach unifies categorical constructions with explicit quiver and root-category data, providing concrete tools for computing dualities in both conformal and asymptotically-free theories and offering a general framework for extending these results to broader DZVX-type models. The work thus provides a powerful, structurally explicit route to understanding non-perturbative dualities in strongly coupled supersymmetric quantum field theories, with clear connections between algebraic categories, quiver mutations, and physical charge lattices.

Abstract

The $S$-duality group $\mathbb{S}(\mathcal{F})$ of a 4d $\mathcal{N}=2$ supersymmetric theory $\mathcal{F}$ is identified with the group of triangle auto-equivalences of its cluster category $\mathscr{C}(\mathcal{F})$ modulo the subgroup acting trivially on the physical quantities. $\mathbb{S}(\mathcal{F})$ is a discrete group commensurable to a subgroup of the Siegel modular group $Sp(2g,\mathbb{Z})$ ($g$ being the dimension of the Coulomb branch). This identification reduces the determination of the $S$-duality group of a given $\mathcal{N}=2$ theory to a problem in homological algebra. In this paper we describe the techniques which make the computation straightforward for a large class of $\mathcal{N}=2$ QFTs. The group $\mathbb{S}(\mathcal{F})$ is naturally presented as a generalized braid group. The $S$-duality groups are often larger than expected. In some models the enhancement of $S$-duality is quite spectacular. For instance, a QFT with a huge $S$-duality group is the Lagrangian SCFT with gauge group $SO(8)\times SO(5)^3\times SO(3)^6$ and half-hypermultiplets in the bi- and tri-spinor representations. We focus on four families of examples: the $\mathcal{N}=2$ SCFTs of the form $(G,G^\prime)$, $D_p(G)$, and $E_r^{(1,1)}(G)$, as well as the asymptotically-free theories $(G,\widehat{H})$ (which contain $\mathcal{N}=2$ SQCD as a special case). For the $E_r^{(1,1)}(G)$ models we confirm the presence of the $PSL(2,\mathbb{Z})$ $S$-duality group predicted by Del Zotto, Vafa and Xie, but for most models in this class $S$-duality gets enhanced to a much larger group.

Homological S-Duality in 4d N=2 QFTs

TL;DR

Caorsi and Cecotti develop a comprehensive homological framework to study S-duality in 4d QFTs by identifying the S-duality group with triangle auto-equivalences of cluster categories modulo physically trivial automorphisms. The duality group acquires a generalized braid-group structure and embeds, up to commensurability, into the Siegel modular group , tying electromagnetic and flavor lattices via the 4d/2d correspondence. They analyze four major families, , , , and , showing notable enhancements of S-duality in many models and explicit matrix realizations through telescopic functors and Thomas–Seidel twists. The approach unifies categorical constructions with explicit quiver and root-category data, providing concrete tools for computing dualities in both conformal and asymptotically-free theories and offering a general framework for extending these results to broader DZVX-type models. The work thus provides a powerful, structurally explicit route to understanding non-perturbative dualities in strongly coupled supersymmetric quantum field theories, with clear connections between algebraic categories, quiver mutations, and physical charge lattices.

Abstract

The -duality group of a 4d supersymmetric theory is identified with the group of triangle auto-equivalences of its cluster category modulo the subgroup acting trivially on the physical quantities. is a discrete group commensurable to a subgroup of the Siegel modular group ( being the dimension of the Coulomb branch). This identification reduces the determination of the -duality group of a given theory to a problem in homological algebra. In this paper we describe the techniques which make the computation straightforward for a large class of QFTs. The group is naturally presented as a generalized braid group. The -duality groups are often larger than expected. In some models the enhancement of -duality is quite spectacular. For instance, a QFT with a huge -duality group is the Lagrangian SCFT with gauge group and half-hypermultiplets in the bi- and tri-spinor representations. We focus on four families of examples: the SCFTs of the form , , and , as well as the asymptotically-free theories (which contain SQCD as a special case). For the models we confirm the presence of the -duality group predicted by Del Zotto, Vafa and Xie, but for most models in this class -duality gets enhanced to a much larger group.

Paper Structure

This paper contains 93 sections, 7 theorems, 372 equations, 12 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Notations/definitions/conventions
  3. Organization of the paper.
  4. Cluster categories and $S$-duality groups
  5. BPS objects vs. cluster categories
  6. Step 1.
  7. Step 2.
  8. Step 3.
  9. Step 4.
  10. $\mathrm{Aut}(\mathscr{C}({\cal F}))$ and the $S$-duality group $\mathbb{S}({\cal F})$
  11. Categorical 4d/2d correspondence and branes
  12. The correspondence $\tilde{{\cal F}}\rightarrow {\cal F}$
  13. The root category $\mathscr{R}(\tilde{{\cal F}})$
  14. The 2d and 4d quantum monodromies
  15. Comparing $\mathrm{Aut}(\mathscr{R}(\tilde{{\cal F}}))$ and $\mathrm{Aut}(\mathscr{C}({\cal F}))$
  16. ...and 78 more sections

Key Result

Lemma 1

The form (linear on the first argument, anti-linear in the second one) is Hermitian and $\boldsymbol{H}_d$ invariant. It decomposes over $\mathbb{Q}[\zeta]$ into a direct sum of Hermitian forms $H_{d,\ell}$ on each space $W_{d,\ell}$. The dimension of the radical of $H_{d,\ell}$ and its signature are independent of $\ell\in({\mathbb Z}/d{\mathbb Z})^\times$.

Figures (12)

  • Figure 1: The $D^{(1,1)}(A_{2N-1})$ models as quiver gauge theories. A circle with a $N$ (resp. $2N$) stands for a $SU(N)$ (resp. $SU(2N)$) ${\cal N}=2$ SYM sector and an edge connecting two circles to a $SU(2N)\times SU(N)$ bifundamental hypermultiplet.
  • Figure 2: Reference orientations of Dynkin quivers. The node numbers are chosen so that for all arrows $\psi$ one has $t(\psi)>s(\psi)$.
  • Figure 3: The AR quiver of $\mathscr{R}(A_3)$. Objects are labelled by their Grothendieck class in the root lattice of $G$. The objects on the right are cyclically identified with the corresponding ones on the left. $\tau$ acts by horizontal translation to the left. Examples of mesh relations are $\psi\sigma(\psi)=0$ and $\alpha\sigma(\alpha)+\beta\sigma(\beta)=0$. The $\tau$ orbit of $P_2=011$ is actually twice a half-orbit.
  • Figure 4: The first half of the AR quiver of $\mathscr{R}(D_4)$; the second half repeats up to a shift (i.e. an overall minus, in terms of Grothendieck classes). The double arrows represent a non-trivial path from $P_s$ to $\tau^{-2}P_s\equiv \tau^{1-h(D_4)/2}P_s$ as described in the text.
  • Figure 5: The $(A_{pm-1},\widehat{A}(p,1))$ theory as a linear quiver gauge theory. Nodes represent $SU((p-k)m)$ gauge sectors coupled through bifundamental hypermultiplets represented by the edges. The $\beta$--functions of all gauge coupling vanish, except for the leftmost node which has a negative $\beta$-function with coefficient $(p+1)m$.
  • ...and 7 more figures

Theorems & Definitions (49)

  • Definition 1
  • Definition 2
  • Remark 1
  • Remark 2
  • Remark 3
  • Definition 3
  • Example 1
  • Definition 4
  • Definition 5
  • Remark 4
  • ...and 39 more