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Classification of complete N=2 supersymmetric theories in 4 dimensions

Sergio Cecotti, Cumrun Vafa

TL;DR

This work establishes that complete ${ m N}=2$ theories in four dimensions—those with freely deformable BPS central charges—are precisely described by BPS quivers of finite mutation type. By leveraging the 4d/2d correspondence, Cecotti and Vafa map these theories to quivers arising from ideal triangulations of bordered surfaces (generalized Gaiotto theories) and 11 exceptional classes, including the Derksen–Owen quivers. The classification hinges on quiver mutation-finiteness and its topological realization via surface triangulations, yielding a clean dictionary between UV-complete 4d theories and finite mutation quivers. They further analyze conformal cases, surface surgeries, and gauging of ${ m N}=2$ subsystems, providing new dualities and detailed BPS-spectra descriptions that reinforce the deep link between 4d gauge dynamics and 2d defect theories. The results illuminate how to construct and decompose complete theories from simpler building blocks and suggest avenues for extending the program to non-complete ${ m N}=2$ theories and broader 4d/2d correspondences.

Abstract

We define the notion of a complete N=2 supersymmetric theory in 4 dimensions as a UV complete theory for which all the BPS central charges can be arbitrarily varied as we vary its Coulomb branch parameters, masses, and coupling constants. We classify all such theories whose BPS spectrum can be obtained via a quiver diagram. This is done using the 4d/2d correspondence and by showing that such complete N=2 theories map to quivers of finite mutation type. The list of such theories is given by the (generalized) Gaiotto theories consisting of two 5-branes wrapping Riemann surfaces with punctures, as well as 11 additional exceptional cases, which we identify.

Classification of complete N=2 supersymmetric theories in 4 dimensions

TL;DR

This work establishes that complete theories in four dimensions—those with freely deformable BPS central charges—are precisely described by BPS quivers of finite mutation type. By leveraging the 4d/2d correspondence, Cecotti and Vafa map these theories to quivers arising from ideal triangulations of bordered surfaces (generalized Gaiotto theories) and 11 exceptional classes, including the Derksen–Owen quivers. The classification hinges on quiver mutation-finiteness and its topological realization via surface triangulations, yielding a clean dictionary between UV-complete 4d theories and finite mutation quivers. They further analyze conformal cases, surface surgeries, and gauging of subsystems, providing new dualities and detailed BPS-spectra descriptions that reinforce the deep link between 4d gauge dynamics and 2d defect theories. The results illuminate how to construct and decompose complete theories from simpler building blocks and suggest avenues for extending the program to non-complete theories and broader 4d/2d correspondences.

Abstract

We define the notion of a complete N=2 supersymmetric theory in 4 dimensions as a UV complete theory for which all the BPS central charges can be arbitrarily varied as we vary its Coulomb branch parameters, masses, and coupling constants. We classify all such theories whose BPS spectrum can be obtained via a quiver diagram. This is done using the 4d/2d correspondence and by showing that such complete N=2 theories map to quivers of finite mutation type. The list of such theories is given by the (generalized) Gaiotto theories consisting of two 5-branes wrapping Riemann surfaces with punctures, as well as 11 additional exceptional cases, which we identify.

Paper Structure

This paper contains 51 sections, 207 equations, 12 figures, 1 table.

Table of Contents

  1. Introduction
  2. BPS Quivers
  3. Generalities of quivers
  4. Examples of BPS quivers
  5. BPS spectra and representation theory
  6. Definition of Complete ${\cal N}=2$ theories
  7. 4d-2d Correspondence Reviewed
  8. Complete ${\cal N}=2$ theories and quivers of finite mutation type
  9. Completeness and finiteness of mutation type
  10. Quivers of finite mutation type
  11. Quivers from ideal triangulations of bordered surfaces
  12. Some basic features of mutation-finite quivers
  13. Identification of a large class of quivers of finite mutation type as generalized Gaiotto theories
  14. $4d/2d$ correspondence and ideal triangulations
  15. Lagrangian $A$--branes as ideal triangulations
  16. ...and 36 more sections

Figures (12)

  • Figure 1: The three elliptic $E$--type Dynkin diagrams oriented as to give finite mutation quivers, and the two Derksen--Owen quivers.
  • Figure 2: The quiver blocks of Type I--V fomin.
  • Figure 3: A $g=2$$n=3$ 'snake' ideal triangulation.
  • Figure 4: The adjacency quiver corresponding to the ideal triangulation \ref{['pipio']} of a $g=2$ surface with three punctures. The numeration of the nodes corresponds to the numeration of arcs in \ref{['pipio']}. In the left side of the quiver we see the 'segment of a quiver $SU(2)$ theory' associated to the three punctures.
  • Figure 5: Inequivalent 'snake' triangulation of the same surface.
  • ...and 7 more figures