S-Duality of Boundary Conditions In N=4 Super Yang-Mills Theory

TL;DR

The paper develops a comprehensive framework for S-duality of boundary conditions in N=4 super Yang–Mills theory by encoding boundary data in triples $( ho,H,rak B)$ and realizing them via brane constructions. It introduces the 3D boundary theories T(SU(n)) and its generalizations T(G), including Nahm-pole data T^ρ(SU(n)), and shows how S-duality acts by coupling to these boundary SCFTs and performing ungauging procedures, often yielding domain walls and composite 3D theories that mirror the 4D duals. A central achievement is a general duality recipe B_H ×_H T^ρ(G) that constructs the S-dual boundary theory when the dual gauge symmetry remains unbroken, with a detailed treatment of unitary, orthogonal, and symplectic groups via branes, orientifolds, and orbifolds. The work also develops a robust IR analysis through 3D monopole operators, classifies linear quivers as Good/Bad/Ugly, and extends the duality framework to domain walls and Janus configurations, thereby connecting 4D S-duality to 3D mirror symmetry and to a broad class of quiver gauge theories. Finally, it generalizes to accommodate the full SL(2,Z) duality group, including theta-angle deformations and fractional Chern-Simons couplings, with extensive treatment of orientifold/orbifold realizations and their nonperturbative dualities. This framework provides a powerful, geometrically grounded method to compute dual boundary conditions across a wide range of gauge groups and boundary matter content, with implications for IR fixed points, monopole operator spectra, and boundary domain-wall phenomena.

Abstract

By analyzing brane configurations in detail, and extracting general lessons, we develop methods for analyzing S-duality of supersymmetric boundary conditions in N=4 super Yang-Mills theory. In the process, we find that S-duality of boundary conditions is closely related to mirror symmetry of three-dimensional gauge theories, and we analyze the IR behavior of large classes of quiver gauge theories.

Figures (75)

• Figure 1: A brane configuration that determines a half-BPS boundary condition in ${\mathcal{N}}=4$ super Yang-Mills theory. Here and later, horizontal solid lines designate D3-branes spanning directions 0123; vertical dotted lines designate D5-branes spanning directions 012456. In this example, there are eight D3-branes and the gauge group is $U(8)$. The symbol $\bigoplus$ denotes a further fivebrane system, of which some possible examples are sketched in fig. \ref{['Fig2']}.
• Figure 2: Some brane configurations, any one of which can correspond to the symbol $\bigoplus$ on the left of fig. \ref{['Fig1']}. Here and later, the symbol $\bigotimes$ represents an NS5-brane spanning directions 012789. In (a), three D3-branes end on a single NS5-brane. This leads to Neumann boundary conditions in $U(3)$ gauge theory. In (b), the D3-branes intersect a D5-brane before terminating on a single NS5-brane. This leads (in the limit that all fivebrane separations in the $y=x^3$ direction are taken to zero) to Neumann boundary conditions with a fundamental hypermultiplet supported on the boundary. The hypermultiplet comes from the brane intersection. In (c), (d), and (e), there is more than one NS5-brane. This leads to Neumann boundary conditions modified by coupling to a non-trivial boundary SCFT, as described in the text.
• Figure 3: A quiver such as this one gives a convenient way to summarize the construction of a gauge theory with suitable gauge group and matter representation. A circle containing an integer $n$ represents a $U(n)$ factor in the gauge group. The gauge group is the product of such factors, one for each circle. A line joining two circles labeled by $n$ and $m$ represents a bifundamental hypermultiplet, that is a collection of hypermultiplets transforming under $U(n)\times U(m)$ as $({\bf n},\overline{\bf m})\oplus (\overline{\bf n},{\bf m})$. Finally, if a circle labeled by $n$ is linked to square labeled $p$, this means that there are $p$ fundamental hypermultiplets of $U(n)$. For every square labeled by $p$, there is a $U(p)$ global symmetry acting on the corresponding hypermultiplets. The specific quiver drawn here represents the boundary SCFT that arises from the brane configuration of fig. \ref{['Fig2']}(e).
• Figure 4: (a) A configuration of 3 D5-branes and 3 NS5-branes in $U(7)$ gauge theory. Each fivebrane has a linking number, defined as the number of fivebranes of the opposite kind that are to the left of the given fivebrane, plus the net number of threebranes ending on the right of the given fivebrane. In the figure, the linking number of a D5-brane (or an NS5-brane) is given by the integer that is written just above (or below) the brane in question. This configuration has been chosen so that the linking numbers of fivebranes of a given type are non-decreasing if one reads the figure from left to right. (b) In the boundary condition derived from (a), a $U(5)$ subgroup of the gauge group is coupled at the boundary to an SCFT with $U(5)$ symmetry. This SCFT can be obtained as the infrared limit of the three-dimensional gauge theory associated with the quiver indicated here (together with a free fundamental hypermultiplet from interaction with the D5-brane of linking number 3).
• Figure 5: Two NS5-branes with $k$ D5-branes between them. To the left of the NS5-branes, between them, and to their right, there are respectively $n'$, $n$, and $n"$ D3-branes. Here $n',n,n"$ equal 2,3, and 1, respectively.