Boundaries and Defects of N=4 SYM with 4 Supercharges, Part I: Boundary/Junction Conditions

TL;DR

Hashimoto, Ouyang, and Yamazaki extend Gaiotto-Witten’s 1/2 BPS boundary framework to 1/4 BPS in ${\cal N}=4$ SYM by systematically constructing boundary/junction data from D3-branes ending on NS5 and D5 branes and by deriving generalized Nahm equations. They develop explicit boundary and junction conditions, including localized degrees of freedom, mass and FI deformations, and the role of the complexified gauge group in describing moduli spaces, revealing nontrivial lifting of bulk degrees in non-Abelian sectors. The work connects bulk/defect data to 3d boundary theories (notably the $T[SU(N)]$ family and its 1/4 BPS generalizations) and emphasizes the utility of S-duality as a tool to probe quantum-corrected structures. These constructions provide building blocks for engineering 3d ${\cal N}=2$ theories in 2+1 dimensions from 4d ${\cal N}=4$ membranes and offer a framework to study moduli spaces and dual descriptions across boundaries and defects, with detailed analysis reserved for Part II.

Abstract

We consider ${\cal N}=4$ supersymmetric Yang Mills theory on a space with supersymmetry preserving boundary conditions. The boundaries preserving half of the 16 supercharges were analyzed and classified in an earlier work by Gaiotto and Witten. We extend that analysis to the case with fewer supersymmetries, concentrating mainly on the case preserving one quarter. We develop tools necessary to explicitly construct boundary conditions which can be viewed as taking the zero slope limit of a system of D3 branes intersecting and ending on a collection of NS5 and D5 branes oriented to preserve the appropriate number of supersymmetries. We analyze how these boundary conditions constrain the bulk degrees of freedom and enumerate the unconstrained degrees of freedom from the boundary/defect field theory point of view. The key ingredients used in the analysis are a generalized version of Nahm's equations and the explicit boundary/interface conditions for the NS5-like and D5-like impurities and boundaries, which we construct and describe in detail. Some bulk degrees of freedom suggested by the naive brane diagram considerations are lifted.

Figures (21)

• Figure 1: By collapsing the boundary conditions we can construct more complicated boundary conditions. For this analysis it is necessary to keep track of the bulk degrees of freedom between the two defects, which is constrained by \ref{['bulk_half']}.
• Figure 2: 1/2 BPS boundary for a $U(3)$ theory preserving $U(2)$ gauge symmetry.
• Figure 3: An example of $1/2$ BPS boundary for the ${\cal N}=4$$U(10)$ SYM with $H=U(6)$, $\mathfrak{B}$ is $U(1) \times U(3)$ gauge theory with 1 and 6 flavors of quarks, respectively. The data $\rho$ characterizes the sequence non-decreasing linking numbers which for this example is $\{1,3\}$ and encodes the pattern of symmetry breaking $10 \rightarrow 7 \rightarrow 6$. This data can also be represented by the Young diagram as shown.
• Figure 4: The generalization of $\rho$ when D5 and D5$^{\prime}$ branes are present. We have restricted our attention to the case where the linking number is non-decreasing so that a generalized Young diagram can be drawn as illustrated, although there are no a priori reason to only consider this case.
• Figure 5: Brane construction of ${\cal N}=2$$U(1)$$N_f=3$ theory in two different brane orderings.