Janus Configurations, Chern-Simons Couplings, And The Theta-Angle in N=4 Super Yang-Mills Theory

TL;DR

This work extends the half-BPS Janus construction of ${\cal N}=4$ super Yang–Mills to allow a position-dependent theta-angle $\theta(y)$ in addition to a varying gauge coupling, linking this setup to novel three-dimensional ${\cal N}=4$ Chern–Simons theories and to D3–NS5 configurations with nonzero $\theta$. A central mechanism is the inequivalent embedding of the 3d ${\cal N}=4$ SUSY algebra into the 4d algebra via an outer automorphism, enabling $y$-dependent SUSY realizations; the authors derive explicit relations between the running couplings, the angle parameter $\psi$, and the complex coupling $\tau=\theta/(2\pi)+2\pi i/e^2$, finding that $\tau(y)$ traces SL$(2,\mathbb{R})$ geodesics in the upper half-plane. They develop a comprehensive 3d superfield approach to construct ${\cal N}=4$ Chern–Simons theories from ${\cal N}=1$ data, classify admissible gauge/matter content using supergroups like $U(N|M)$ and $OSp(N|M)$, and connect these structures to 4d Janus configurations and brane realizations, including bifundamental defects and D3–NS5 systems with nonzero $\theta$. The conformal limit is carefully analyzed, revealing a natural field redefinition that yields a well-defined discontinuity in $\tau$ while preserving ${\cal N}=4$ supersymmetry. Overall, the paper provides a unified framework linking generalized Janus domain walls, 3d ${\cal N}=4$ CS theories, and brane constructions, with precise conditions on couplings and boundary data for preserving extended supersymmetry.

Abstract

We generalize the half-BPS Janus configuration of four-dimensional N=4 super Yang-Mills theory to allow the theta-angle, as well as the gauge coupling, to vary with position. We show that the existence of this generalization is closely related to the existence of novel three-dimensional Chern-Simons theories with N=4 supersymmetry. Another closely related problem, which we also elucidate, is the D3-NS5 system in the presence of a four-dimensional theta-angle.

Figures (3)

• Figure 1: A configuration with $N$ D3-branes ending on an NS5-brane from the left, while $M$ D3-branes end from the right. The D3-brane worldvolumes span the 0123 directions, and those of the NS5-branes span the 012456 directions. The horizontal direction in the figure represents spacetime direction $x^3$, and the vertical direction represents spacetime directions 456.
• Figure 2: A system of $N$ parallel D3-branes intersecting successive NS5-branes.
• Figure 3: (a) A quiver associated with the first non-trivial example of a hyper-Kahler manifold obeying the fundamental identity. (b) More general linear quivers leading to solutions of the fundamental identity.