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$Y$ systems, $Q$ systems, and 4D $\mathcal{N}=2$ supersymmetric QFT

Sergio Cecotti, Michele Del Zotto

TL;DR

The paper develops a unified framework in which $Y$-systems and $Q$-systems encode the BPS spectra of 4D ${ m mathcal{N}=2}$ QFTs. It shows that finite BPS chambers of UV SCFTs yield periodic $Y$-systems governed by fractional and full quantum monodromies, while asymptotically free theories yield $Q$-systems whose iterates satisfy constant-coefficient linear recurrences, connected to frieze patterns and cluster mutations. The authors construct and solve new $Y$-systems for Minahan–Nemeschanski theories with $E_6,E_7,E_8$ and for $D_2(G)$ models, and present new $Q$-systems for SYM theories with carefully chosen matter content to keep the YM beta function negative. Through detailed examples and a thorough review of the TBA, wall-crossing, and cluster-algebra machinery, the work ties Seiberg–Witten geometry to integrable recurrences, offering a versatile toolkit for generating and solving new $Y$- and $Q$-systems and for probing BPS spectra and RG flows in 4D $ m mathcal{N}=2$ theories.

Abstract

We review the connection of $Y$- and $Q$-systems with the BPS spectra of $4D$ $\mathcal{N}=2$ supersymmetric QFTs. For each finite BPS chamber of a $\mathcal{N}=2$ model which is UV superconformal, one gets a periodic $Y$-system, while for each finite BPS chamber of an asymptotically-free $\mathcal{N}=2$ QFT one gets a $Q$-system i.e. a rational recursion all whose solutions satisfy a linear recursion with constant coefficients (depending on the initial conditions). For instance, the classical $ADE$ $Y$-systems of Zamolodchikov correspond to the $ADE$ Argyres-Douglas $\mathcal{N}=2$ SCFTs, while the usual $ADE$ $Q$-systems to pure $\mathcal{N}=2$ SYM. After having motivated the correspondence both from the QFT and the TBA sides, and having introduced the basic tricks of the trade, we exploit the connection to construct and SOLVE new $Y$- and $Q$-systems. In particular, we present the new $Y$-systems associated to the $E_6,E_7,E_8$ Minahan-Nemeshanski SCFTs and to the $D_2(G)$ SCFTs. We also present new $Q$-system corresponding to SYM coupled to specific matter systems such that the YM $β$-function remains negative.

$Y$ systems, $Q$ systems, and 4D $\mathcal{N}=2$ supersymmetric QFT

TL;DR

The paper develops a unified framework in which -systems and -systems encode the BPS spectra of 4D QFTs. It shows that finite BPS chambers of UV SCFTs yield periodic -systems governed by fractional and full quantum monodromies, while asymptotically free theories yield -systems whose iterates satisfy constant-coefficient linear recurrences, connected to frieze patterns and cluster mutations. The authors construct and solve new -systems for Minahan–Nemeschanski theories with and for models, and present new -systems for SYM theories with carefully chosen matter content to keep the YM beta function negative. Through detailed examples and a thorough review of the TBA, wall-crossing, and cluster-algebra machinery, the work ties Seiberg–Witten geometry to integrable recurrences, offering a versatile toolkit for generating and solving new - and -systems and for probing BPS spectra and RG flows in 4D theories.

Abstract

We review the connection of - and -systems with the BPS spectra of supersymmetric QFTs. For each finite BPS chamber of a model which is UV superconformal, one gets a periodic -system, while for each finite BPS chamber of an asymptotically-free QFT one gets a -system i.e. a rational recursion all whose solutions satisfy a linear recursion with constant coefficients (depending on the initial conditions). For instance, the classical -systems of Zamolodchikov correspond to the Argyres-Douglas SCFTs, while the usual -systems to pure SYM. After having motivated the correspondence both from the QFT and the TBA sides, and having introduced the basic tricks of the trade, we exploit the connection to construct and SOLVE new - and -systems. In particular, we present the new -systems associated to the Minahan-Nemeshanski SCFTs and to the SCFTs. We also present new -system corresponding to SYM coupled to specific matter systems such that the YM -function remains negative.

Paper Structure

This paper contains 17 sections, 182 equations, 5 figures, 8 tables.

Table of Contents

  1. Introduction and Overview
  2. TBA, ${\cal N}=2$ supersymmetry, $Y$-- and $Q$--systems
  3. BPS spectral problem and cluster algebras
  4. A lightning review of the BPS quiver approach
  5. Wall crossing and quantum cluster algebras
  6. 2d/4D and properties of the quantum monodromy
  7. The $Y$ system of a 4D ${\cal N}=2$ model
  8. From $\mathscr{Y}_q$ to $Y_{i,s}$
  9. SCFT and periodic $Y$ systems.
  10. A detailed example: Minahan--Nemeschansky theories
  11. $Y$ systems of $D_2(G)$ theories.
  12. Asymptotically free models, friezes, and Q systems
  13. Fractional monodromy and frieze pattern
  14. The case of asymptotically free $SU(2)$ gauge theories
  15. The case of $SU(N+1)$ SYM
  16. ...and 2 more sections

Figures (5)

  • Figure 1: UP: The quiver $\vec{A}_3 \, \boxtimes \, \vec{E}_6$. DOWN: The quiver $A_3 \, \square \, E_6$. The Coxeter factorized sequence of mutation of type $E_6 \coprod E_6 \coprod E_6$ is simply obtained mutating first all $\circ$'s then all $\bullet$'s and iterating. The one of type $A_3\coprod\dots\coprod A_3$ mutating first all $\bullet$'s then all $\circ$'s and iterating.
  • Figure 2: The quivers in the family $Q(r,s)$.
  • Figure 3: The DWZ--reduced quivers ${\cal D}(G)$ for the $D_2(G)$ SCFT's.
  • Figure 4: Our conventions on the nodes of the affine quivers. For $\widehat{A}(p,q)$ each node is an extending one, for the other affine quivers above the extending nodes are circled.
  • Figure 5: The quiver of the system $A(3,3)\boxtimes A_3$.