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Rank-3 antisymmetric matter on 5-brane webs

Hirotaka Hayashi, Sung-Soo Kim, Kimyeong Lee, Futoshi Yagi

TL;DR

The work develops Type IIB 5-brane web realizations for 5d $SU(6)$ and $Sp(3)$ gauge theories with rank-3 antisymmetric matter and UV fixed points, validating the constructions via the one-loop prepotential and monopole-string tensions. It uses the topological vertex to compute Nekrasov partition functions for selected theories, including a first explicit result for $SU(6)_{\frac{5}{2}}$ with a half TAS. The study extends to marginal theories with various TAS content, identifies dualities, and uncovers 6d uplifts linked to Tao diagrams and orientifold configurations. The results illuminate the 5d/6d connections of these theories, offering new UV completions and a framework for exploring broader classes of higher-rank antisymmetric matter in 5d SCFTs.

Abstract

We discuss Type IIB 5-brane configurations for 5d $\mathcal{N}=1$ gauge theories with hypermultiplets in the rank-3 antisymmetric representation and with various other hypermultiplets, which flow to a UV fixed point at the infinite coupling. We propose 5-brane web diagrams for the theories of $SU(6)$ and $Sp(3)$ gauge groups with rank-3 antisymmetric matter and check our proposed 5-brane webs against several consistency conditions implied from the one-loop corrected prepotential. Using the obtained 5-brane webs for rank-3 antisymmetric matter, we apply the topological vertex method to compute the partition function for one of these $SU(6)$ gauge theories.

Rank-3 antisymmetric matter on 5-brane webs

TL;DR

The work develops Type IIB 5-brane web realizations for 5d and gauge theories with rank-3 antisymmetric matter and UV fixed points, validating the constructions via the one-loop prepotential and monopole-string tensions. It uses the topological vertex to compute Nekrasov partition functions for selected theories, including a first explicit result for with a half TAS. The study extends to marginal theories with various TAS content, identifies dualities, and uncovers 6d uplifts linked to Tao diagrams and orientifold configurations. The results illuminate the 5d/6d connections of these theories, offering new UV completions and a framework for exploring broader classes of higher-rank antisymmetric matter in 5d SCFTs.

Abstract

We discuss Type IIB 5-brane configurations for 5d gauge theories with hypermultiplets in the rank-3 antisymmetric representation and with various other hypermultiplets, which flow to a UV fixed point at the infinite coupling. We propose 5-brane web diagrams for the theories of and gauge groups with rank-3 antisymmetric matter and check our proposed 5-brane webs against several consistency conditions implied from the one-loop corrected prepotential. Using the obtained 5-brane webs for rank-3 antisymmetric matter, we apply the topological vertex method to compute the partition function for one of these gauge theories.

Paper Structure

This paper contains 31 sections, 56 equations, 39 figures, 2 tables.

Table of Contents

  1. Introduction
  2. $SU(6)$ gauge theories with rank-3 antisymmetric matter
  3. Decoupling from $SO(12)$ gauge theory with conjugate spinor matter
  4. Monopole string tension
  5. Nekrasov partition function
  6. $SU(6)$ gauge theories with $N_{\text{\bf TAS}} = \frac{3}{2}$ and $2$
  7. Marginal $SU(6)$ gauge theory with rank-3 antisymmetric matter
  8. 5-brane web for $SU(6)_0+2\,\mathbf{TAS}$
  9. 5-brane webs for $SU(6)+1\,\mathbf{TAS}$ with various hypermultiplets
  10. $SU(6)_0+1{\bf TAS}+10\mathbf{F}$.
  11. $SU(6)_\frac{3}{2}+1{\bf TAS}+9\mathbf{F}$.
  12. $SU(6)_0+1{\bf TAS}+1{\bf AS}+4\mathbf{F}$.
  13. 5-brane webs for $SU(6)+1/2\,\mathbf{TAS}$ with various hypermultiplets
  14. $SU(6)_0+\frac{1}{2}{\bf TAS}+13\mathbf{F}$.
  15. $SU(6)_3+\frac{1}{2}{\bf TAS}+9\mathbf{F}$
  16. ...and 16 more sections

Figures (39)

  • Figure 1: A 5-brane diagram which realizes the $SO(12)$ gauge theory with a half-hypermultiplet in the conjugate spinor representation. In the left upper conner, two 5-branes of the charge $(-3,1)$ should be understood as they are bound by a single 7-brane of the same charge $(-3,1)$, respecting the S-rule.
  • Figure 2: A 5-brane diagram realizing the $SO(12)$ gauge theory with a conjugate spinor which is obtained after performing a generalized flop transition to the diagram in Figure \ref{['fig:so12wcs']}.
  • Figure 3: A 5-brane diagram realizing an $SU(6)$ gauge theory with a half-hypermultiplet in the rank-3 antisymmetric representation. It will turn out that the Chern-Simons level of this theory is $\kappa = \frac{5}{2}$ in section \ref{['sec:su6wtsatension']}.
  • Figure 4: (a): A 5-brane diagram for the $SO(12)$ gauge theory with a massless hypermultiplet in the conjugate spinor representation. (b): A 5-brane diagram for an $SU(6)$ gauge theory with a massless hypermultiplet in the rank-3 antisymmetric representation, obtained by applying generalized flop transitions and decoupling to the diagram in Figure \ref{['fig:so12wcs2']}. It will turn out that the Chern-Simons level of this theory is $\kappa = 3$ in section \ref{['sec:su6wtsatension']}.
  • Figure 5: (a): A parameterization of Coulomb branch moduli for the diagram in Figure \ref{['fig:su6whtsa']}. (b): A labeling for the area of faces in the diagram in Figure \ref{['fig:su6whtsa']}. As the external 5-branes are bound by 7-branes such that they satisfy the S-rule, some of the faces are in fact connected. For instance, $\textcircled{\scriptsize 1}$ and $\textcircled{\scriptsize 2}$ are connected and so are $\textcircled{\scriptsize 6}$ and $\textcircled{\scriptsize 7}$.
  • ...and 34 more figures