Supersymmetric Boundary Conditions in N=4 Super Yang-Mills Theory

TL;DR

Gaiotto and Witten develop a comprehensive framework for half-BPS boundary conditions in four-dimensional N=4 SYM, describing how Dirichlet/Neumann choices can be extended via Nahm’s equations, boundary poles, and couplings to boundary CFTs or hypermultiplets. They show that boundary data can be organized by a ρ SU(2) embedding, a boundary subgroup H, and a boundary theory B, with the moduli spaces of Nahm solutions realized as hyper-Kähler quotients or Slodowy slices, and they connect these structures to D3-brane configurations (D5, NS5, orientifolds) and to S-duality. The work also explores domain walls, shift parameters (FI terms and mass parameters), and the possibility of deformations that break Lorentz invariance while preserving SUSY, outlining how duality acts on these boundary data. Overall, the paper provides a detailed, brane-informed toolkit for classifying boundary conditions, understanding boundary vacua via Nahm’s equations, and probing dualities in N=4 SYM with potential links to geometric Langlands via principal SU(2) embeddings. The results establish a rich dictionary between field-theoretic boundary conditions, hyper-Kähler geometry, and brane constructions, with implications for boundary CFTs and the structure of moduli spaces of vacua.

Abstract

We study boundary conditions in N=4 super Yang-Mills theory that preserve one-half the supersymmetry. The obvious Dirichlet boundary conditions can be modified to allow some of the scalar fields to have a ``pole'' at the boundary. The obvious Neumann boundary conditions can be modified by coupling to additional fields supported at the boundary. The obvious boundary conditions associated with orientifolds can also be generalized. In preparation for a separate study of how electric-magnetic duality acts on these boundary conditions, we explore moduli spaces of solutions of Nahm's equations that appear in the presence of a boundary. Though our main interest is in boundary conditions that are Lorentz-invariant (to the extent possible in the presence of a boundary), we also explore non-Lorentz-invariant but half-BPS deformations of Neumann boundary conditions. We make preliminary comments on the action of electric-magnetic duality, deferring a more serious study to a later paper.

Figures (12)

• Figure 1: A brane configuration whose purpose is to illustrate the general half-BPS boundary condition. A collection of semi-infinite D3-branes with worldvolume in the 0123 directions (portrayed by horizontal solid lines) ends on a collection of D5-branes that run in the 012456 directions (portrayed by vertical dotted lines) and one or more coincident NS5-branes that run in the 012789 directions (portrayed by the symbol $\bigotimes$). In this and subsequent pictures, the horizontal direction parametrizes $x^3$ and the vertical direction represents the 456 directions in spacetime.
• Figure 2: Here and later, horizontal solid lines denote D3-branes whose world-volume is parametrized by $x^0,x^1,x^2,x^3$. Vertical dotted lines denote D5-branes supported at $x^3=x^7=x^8=x^9=0.$
• Figure 3: A system of parallel D3-branes interacting with two parallel D5-branes. The D3-branes can "break" in crossing the D5-branes. The position of a D3-brane that connects two D5-branes is parametrized by the value of a hypermultiplet.
• Figure 4: In this example, the number of D3-branes jumps by 1 in crossing a D5-brane.
• Figure 5: By displacing a D3-brane that is on the right of the D5-brane very far from the others in the $x^4-x^5-x^6$ directions, we can reduce to a case with equals numbers of D3-branes on both sides.