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Hanany-Witten effect and SL(2,Z) dualities in matrix models

Benjamin Assel

TL;DR

The paper develops a comprehensive framework to test and map dualities in 3d $\mathcal{N}=4$ quiver SCFTs using exact $S^3$ partition functions computed from matrix models. By organizing the partition function into 5-brane factors and implementing local $SL(2,\mathbb{Z})$ actions and Hanany-Witten move identities, the authors prove that partition functions of dual theories coincide up to unphysical phases, and establish mirror symmetry for YM quivers with arbitrary node ranks. They extend the duality web to include YM–CS level-rank dualities and GW-type interpolating couplings, derive and prove the NTY formula for linear quivers, and provide explicit abelian checks. The work tightens the brane/gauge correspondence by showing a direct, calculable map between brane moves and matrix-model manipulations, with implications for exact results and AdS$_4$ holography. The methods offer a powerful toolkit for exploring dualities across a broad class of 3d $\mathcal{N}=4$ theories and their gravity duals.

Abstract

We provide tests of dualities for three-dimensional N=4 quiver SCFTs with brane realizations in IIB string theory, by matching their exact partition functions on $S^3$. The dualities are generated by SL(2,Z) transformations and Hanany-Witten 5-brane moves. These contain mirror symmetry as well as dualities identifiying fixed points of Yang-Mills quivers and Chern-Simons theories. The partition function is given by a matrix model, that can be nicely rearranged into a sequence of factors mimicking the brane realization. Identities obeyed by these elementary factors can be used to match the partition functions of dual theories, providing tests for the full web of dualities. In particular we are able to check mirror symmetry for linear and circular quivers with gauge nodes of arbitrary ranks. Our analysis also leads to a proof of a conjectured formula evaluating the matrix models of linear quiver theories.

Hanany-Witten effect and SL(2,Z) dualities in matrix models

TL;DR

The paper develops a comprehensive framework to test and map dualities in 3d quiver SCFTs using exact partition functions computed from matrix models. By organizing the partition function into 5-brane factors and implementing local actions and Hanany-Witten move identities, the authors prove that partition functions of dual theories coincide up to unphysical phases, and establish mirror symmetry for YM quivers with arbitrary node ranks. They extend the duality web to include YM–CS level-rank dualities and GW-type interpolating couplings, derive and prove the NTY formula for linear quivers, and provide explicit abelian checks. The work tightens the brane/gauge correspondence by showing a direct, calculable map between brane moves and matrix-model manipulations, with implications for exact results and AdS holography. The methods offer a powerful toolkit for exploring dualities across a broad class of 3d theories and their gravity duals.

Abstract

We provide tests of dualities for three-dimensional N=4 quiver SCFTs with brane realizations in IIB string theory, by matching their exact partition functions on . The dualities are generated by SL(2,Z) transformations and Hanany-Witten 5-brane moves. These contain mirror symmetry as well as dualities identifiying fixed points of Yang-Mills quivers and Chern-Simons theories. The partition function is given by a matrix model, that can be nicely rearranged into a sequence of factors mimicking the brane realization. Identities obeyed by these elementary factors can be used to match the partition functions of dual theories, providing tests for the full web of dualities. In particular we are able to check mirror symmetry for linear and circular quivers with gauge nodes of arbitrary ranks. Our analysis also leads to a proof of a conjectured formula evaluating the matrix models of linear quiver theories.

Paper Structure

This paper contains 26 sections, 118 equations, 23 figures, 1 table.

Table of Contents

  1. Introduction
  2. $\mathcal{N}=4$ theories with brane realizations
  3. Yang-Mills quiver SCFTs
  4. $T(U(N))$ theory
  5. Chern-Simons SCFTs
  6. SCFTs realized with $(p,q)$ 5-branes (GW quivers):
  7. $SL(2,\mathbb{Z})$ dualities
  8. Partition function on $S^3$
  9. Repackaging of matrix models
  10. Local SL(2,$\mathbb{Z}$) action on matrix models
  11. $(p,q)$-5brane matrix factor
  12. D5-factor with $N\neq \widetilde{N}$
  13. Tests of SL(2,$\mathbb{Z}$) dualities
  14. Mirror symmetry and other dualities involving HW moves
  15. Hanany-Witten moves
  16. ...and 11 more sections

Figures (23)

  • Figure 1: Brane realization of linear quivers. $N_j$ refers to the number of coincident D3-branes (parallel black lines) in a D3-segment and $M_j$ to the number of D5-branes crossing it. The horizontal direction can be thought as $x^3$. The vertical direction can be $x^{456}$ for the D5s and $x^{789}$ for the NS5s.
  • Figure 2: Brane realization of circular quivers. The D3-segments on the left and on the right are identified.
  • Figure 3: Graphs associated to linear and circular quivers.
  • Figure 4: $T(SU(N))$ quiver diagram.
  • Figure 5: Chern-Simons quiver and its associated graph. The label below the nodes denotes the Chern-Simons level, zero meaning a node with auxiliary vector multiplet. The elements of the graph are white dots for NS5-branes and grey dots for $(1,k)$-5branes. $Q$ is the total number of 5-branes, or dots, in the brane picture, or graph.
  • ...and 18 more figures