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Tri-vertices and SU(2)'s

Amihay Hanany, Noppadol Mekareeya

Abstract

We examine a class of N=2 supersymmetric gauge theories in (3+1) dimensions whose Lagrangians are determined by graphs consisting of two building blocks, namely a tri-vertex and a line. A line represents an SU(2) gauge group and a tri-vertex represents a matter field in the trifundamental representation of SU(2)^3. These graphs can be topologically classified by the genus and the number of external legs. This paper focuses on the hypermultiplet moduli spaces of the aforementioned theories. We compute the Hilbert series which count all chiral operators on the hypermultiplet moduli space. Several examples show that theories corresponding to different graphs with the same genus and the same number of external legs possess the same Hilbert series. This is in agreement with the conjecture that such theories are related to each other by S-duality. We also give a general expression for the Hilbert series for the graph with any genus and any number of external legs.

Tri-vertices and SU(2)'s

Abstract

We examine a class of N=2 supersymmetric gauge theories in (3+1) dimensions whose Lagrangians are determined by graphs consisting of two building blocks, namely a tri-vertex and a line. A line represents an SU(2) gauge group and a tri-vertex represents a matter field in the trifundamental representation of SU(2)^3. These graphs can be topologically classified by the genus and the number of external legs. This paper focuses on the hypermultiplet moduli spaces of the aforementioned theories. We compute the Hilbert series which count all chiral operators on the hypermultiplet moduli space. Several examples show that theories corresponding to different graphs with the same genus and the same number of external legs possess the same Hilbert series. This is in agreement with the conjecture that such theories are related to each other by S-duality. We also give a general expression for the Hilbert series for the graph with any genus and any number of external legs.

Paper Structure

This paper contains 44 sections, 116 equations, 19 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Skeleton diagrams of ${\cal N}=2$ gauge theories
  3. The theory with a free trifundamental of $SU(2)^3$
  4. The $SU(2)$ ${\cal N}=4$ gauge theory with two free singlets
  5. $SU(2)$ gauge theory with 4 flavours
  6. The Kibble branch of the moduli space
  7. Topology of the skeleton diagram.
  8. Theories with genus zero
  9. The $T_2$ theory $(g=0, e=3)$
  10. $SU(2)$ gauge theory with $4$ flavours $(g=0, e=4)$
  11. A branching rule of $SO(8)$ to $SU(2)^4$.
  12. Gluing two $T_2$ theories.
  13. Three phases of $SU(2)$ theory with 4 flavours
  14. Theories with genus one
  15. The tadpole theory $(g=1, e=1)$
  16. ...and 29 more sections

Figures (19)

  • Figure 1: The theory with a free trifundamental field of $SU(2)^3$.
  • Figure 2: (The tadpole) The $SU(2)$${\cal N}=4$ gauge theory with two singlets.
  • Figure 3: The skeleton diagram of the $SU(2)$ gauge theory with 4 flavours.
  • Figure 4: The quiver diagram of the $SU(2)$ gauge theory with 4 flavours.
  • Figure 5: The three weak coupling limits of an $SU(2)$ gauge theory with 4 flavours.
  • ...and 14 more figures