[Paper Review] 3d $Spin(N)$ Seiberg dualities

Keita Nii|arXiv (Cornell University)|Feb 12, 2020

Black Holes and Theoretical Physics66 references4 citations

TL;DR

This paper proposes 3d $Ν=2$ $Spin(N)$ Seiberg dualities for gauge theories with vector and spinor matter fields, extending 4d $Spin(N)$ dualities by modifying the magnetic superpotential to include dressed monopole operators instead of determinants. The key result is a consistent duality framework in 3d, validated through superconformal index matching, RG flows, and consistency with $Ν=4$ dualities upon deformation.

ABSTRACT

We study low-energy aspects of 3d $\mathcal{N}=2$ $Spin(N)$ gauge theories with matters in vector and (conjugate) spinor representations. Extending the construction of the 4d $\mathcal{N}=1$ $Spin(N)$ Seiberg duality, we find 3d magnetic dual descriptions with tree-level superpotentials slightly different from the 4d ones. We test various consistency checks including RG flows to known 3d dualities and supersymmetry enhancement deformation which leads to a 3d $\mathcal{N}=4$ duality between $SU(2)$ with three hypermultiplets and $U(1)$ with four hypermultiplets.

Motivation & Objective

• To extend 4d $Spin(N)$ Seiberg dualities to 3d $Ν=2$ supersymmetric gauge theories with vector and spinor matter representations.
• To resolve the breakdown of naive 4d duality in 3d due to restored anomalous $U(1)$ symmetries by modifying the magnetic superpotential.
• To establish consistent dual descriptions for $Spin(7)$, $Spin(8)$, $Spin(9)$, and $Spin(10)$ gauge groups with one or two spinor fields.
• To verify duality through consistency checks including superconformal index matching, complex mass deformations, and flows to known dualities.
• To explore the role of dressed monopole operators and their relation to massive gaugino modes in the Coulomb branch.

Proposed method

• Construct electric and magnetic descriptions of 3d $Ν=2$ $Spin(N)$ gauge theories with vector and spinor matter fields, using the same matter content as in 4d $Spin(N)$ Seiberg duality.
• Modify the magnetic theory's superpotential to include dressed monopole operators instead of $∇−\bar{s}$, accounting for the absence of chiral anomalies in 3d.
• Analyze the Higgs and Coulomb branches using gauge-invariant operators such as $M_{QQ} = Q^2$, $M_{SS} = S^2$, $P_3 = SQ^3S$, and $B = Q^7$.
• Perform complex mass deformations to flow from higher-rank $Spin(N)$ theories to lower-rank dualities, including $Spin(7)$ and $Spin(6)$.
• Compute superconformal indices for $Spin(7)$ theories and confirm matching between electric and magnetic sides.
• Use Hilbert series and moduli space analysis to verify that $Spin(8)$ duality flows to the known $Ν=4$ $SU(2)$–$U(1)$ duality upon supersymmetry enhancement.

Experimental results

Research questions

• RQ1 How can 4d $Spin(N)$ Seiberg dualities be consistently extended to 3d $Ν=2$ theories with spinor matter?
• RQ2 Why does the naive 4d duality fail in 3d, and what modification to the magnetic superpotential restores duality?
• RQ3 How do dressed monopole operators in the magnetic theory account for the restored $U(1)$ symmetries in 3d?
• RQ4 What is the role of the superconformal index in verifying duality for $Spin(7)$ theories?
• RQ5 How do complex mass deformations and Higgsing procedures connect the proposed dualities across different $Spin(N)$ groups?

Key findings

• The 3d $Spin(7)$ duality is established with a modified magnetic superpotential that includes dressed monopole operators instead of $∇\bar{s}$, ensuring consistency with 3d anomaly structure.
• Superconformal indices for the $Spin(7)$ duality match exactly between electric and magnetic sides, confirming operator matching in the IR.
• The $Spin(8)$ duality flows to the known 3d $Ν=4$ duality between $SU(2)$ with three hypermultiplets and $U(1)$ with four hypermultiplets upon supersymmetry enhancement.
• Complex mass deformations of the $Spin(10)$ theory reproduce the $Spin(9)$ and $Spin(7)$ dualities, demonstrating a consistent duality web.
• Higgsing the $Spin(7)$ theory via $\langle M_{QQ} \rangle$ vev leads to a $Spin(9)$ duality, while $\langle P_1 \rangle$ vev yields the $Spin(7)$ duality with $F-1$ spinors.
• The duality framework reveals that certain gauge-invariant operators with anti-symmetrized flavor indices map to exotic dressed monopole operators involving massive gaugino fields $\tilde{W}_\alpha$ in the monopole background.