Coulomb Branches of Noncotangent Type: a Physics Perspective

TL;DR

The paper develops a complete framework for the Coulomb-branch algebra ${\mathcal A}_C$ of 3D ${\rm N}=4$ gauge theories with half-hypermultiplets (noncotangent theories) by extending hemisphere localization to cancel the ${\mathbb Z}_2$ parity anomaly and to accommodate incompatible $(2,2)$ boundary conditions. It constructs shift/difference operators that implement monopole insertions on ${HS^3}$ and uses the ${HS^3}$ partition function as a module to represent ${\mathcal A}_C$, enabling explicit generators and relations for SU(2) theories with general matter, analysis of the ${D_2}$ Coulomb branch, and computation of the Coulomb branch for $A_n$-type quivers with spin-${3\over 2}$ half-hypers. The work carefully treats monopole bubbling, the polynomiality conjecture, and operator mixing to ensure the final algebra has polynomial (denominator-free) relations, independent of the Lagrangian splitting ${\mathbb L}$. It also provides a suite of detailed examples—ranging from general SU(2) with half-hypers to quiver theories—demonstrating how the noncotangent Coulomb branch can be solved and how the protected 1D sector encodes correlators in the Coulomb branch. Overall, the results extend localization-based Coulomb-branch analysis to a broad class of noncotangent theories, offering explicit computational tools for protected sector data and new insights into bubbling phenomena and IR singularities on the Coulomb side.

Abstract

We study the Coulomb-branch sector of 3D $\mathcal{N}=4$ gauge theories with half-hypermultiplets in general pseudoreal representations $\mathbf{R}$ ("noncotangent" theories). This yields (short) quantization of the Coulomb branch and correlators of the Coulomb branch operators captured by the 1d topological sector. This is done by extending the hemisphere partition function approach to noncotangent matter. In this setting one must first cancel the parity anomaly, and overcome an obstacle that $(2,2)$ boundary conditions for half-hypers are generically incompatible with gauge symmetry. Using the Dirichlet boundary conditions for the gauge fields and a careful treatment of half-hypermultiplet boundary data, we describe the resulting shift/difference operators implementing monopole insertions (including bubbling effects) on $HS^3$, and use the $HS^3$ partition function as a natural module on which the Coulomb-branch operator algebra $\mathcal{A}_C$ is represented. As applications we derive generators and relations of $\mathcal{A}_C$ for $SU(2)$ theories with general matter (including half-integer spin representations), analyze theories with Coulomb branch $y^2=z(x^2-1)$, compute the Coulomb branch of an $A_n$ quiver with spin-$\frac32$ half-hypers, and check consistency of a general monopole-antimonopole two-point function.

Figures (4)

• Figure 1: Along the fixed circle $S^1_\varphi \subset S^3$, one can insert the twisted-translated Higgs or Coulomb branch operators to compute the protected correlation functions.
• Figure 2: Each hemisphere partition function is viewed as a "wavefunction", and the gluing operation -- as an inner product \ref{['eq:inner-prod']}. The insertions of twisted-translated Coulomb branch operators are represented via certain shift operators acting on the hemisphere wavefunction.
• Figure 3: A twisted-translated Coulomb branch operator $\mathcal{O}(\varphi)$ entering the hemisphere through the N endpoint of $HS^1$, that is $\mathcal{O}(0)$, is represented by an operator $\mathcal{O}_N$ acting on the hemisphere partition function, while $\mathcal{O}(\pi)$ is represented by $\mathcal{O}_S$.
• Figure 4: Quiver of $SU(2)$ groups with half-hyper flavors connected by the dashed lines.