Line Operators in 4d Chern-Simons Theory and Cherkis Bows

Abstract

We show that the phase spaces of a large family of line operators in 4d Chern-Simons theory with $\text{GL}_n$ gauge group are given by Cherkis bow varieties with $n$ crosses. These line operators are characterized by Hanany-Witten type brane constructions involving D3, D5, and NS5 branes in an $Ω$-background. Linking numbers of the five-branes and mass parameters for the D3 brane theories determine the phase spaces and in special cases they correspond to vacuum moduli spaces of 3d $\mathcal{N}=4$ quiver theories. Examples include line operators that conjecturally create T, Q, and L-operators in integrable spin chains.

Figures (17)

• Figure 1: Brane configuration for the line operator $\mathbb{L}_{\bm\varrho}(\bm K, \bm L)$ in $\text{GL}_n$ 4d CS theory labeled by linking numbers $\bm K=(K_1, \cdots, K_n)$ and $\bm L=(L_1, \cdots, L_p)$ satisfying $\sum_{i=1}^n K_i = \sum_{j=1}^p L_j = N$. $\hbar\varrho = \hbar(\varrho_1^\mathbb{C}, \cdots, \varrho_{p-1}^\mathbb{C})$ are complex FI parameters, they are not visible in the classical brane picture. We denote the 3d ${\mathcal{N}}=4$ theory describing the low energy dynamics of the D3 branes by $T^\vee_{\bm\varrho}[\text{U}(N)]_{\bm K}^{\bm L}$.
• Figure 2: A brane configuration found after applying some Hanany-Witten transitions to the configuration in Fig. \ref{['fig:LOKL']}, assuming the constraints (\ref{['cobalanced']}). The two brane configurations lead to the same IR description of the D3 brane world-volume theory in terms of the same 3d ${\mathcal{N}}=4$ gauge theory.
• Figure 3: Quiver for the 3d ${\mathcal{N}}=4$ theory $T^\vee_{\bm\varrho}[\text{U}(N)]_{\bm K}^{\bm L}$ with $N = \sum_{i=1}^n K_i = \sum_{j=1}^p L_j$. The ranks of the gauge and flavor groups are defined form $\bm K$ and $\bm L$ via (\ref{['cobalanced']}). The complex FI parameter associated to the abelian factor of the $j$th gauge node is $\hbar\varrho_j^\mathbb{C}$ and $\bm\varrho = (\varrho_1^\mathbb{C}, \cdots, \varrho_{p-1}^\mathbb{C})$.
• Figure 4: A generic D5 type boundary determined by $\bm K = (K_1, \cdots, K_n)$ and $K_i$ is the linking number of the $i$th D5 brane. We denote the total number of D3 branes by $N = \sum_{i=1}^n K_i$.
• Figure 5: The bow diagram for the D5 type boundary whose brane diagram is given in Fig. \ref{['fig:D5boundary']}. Here $\bm K=(K_1, \cdots, K_n)$ is an $n$-tuple of integers satisfying (\ref{['NK']}).

Theorems & Definitions (7)

• Remark 4.1: Further Comments on Twisted Masses
• Proposition 5.1
• Lemma 5.2
• proof
• Remark 5.3: Comparing the Higgs and Coulomb Branches
• proof : Proof of Proposition \ref{['prop:Tweight']}
• Remark 5.4: All the Q-operators from Bazhanov:2010jq