One-point functions for doubly-holographic BCFTs and backreacting defects

TL;DR

This work addresses the computation of defect one-point functions for ambient chiral primaries in 4d $\mathcal{N}=4$ SYM with 1/2-BPS boundaries, defects, and interfaces engineered by backreacting D3/D5/NS5 brane systems that realize double holography. The authors combine supersymmetric localization (yielding multi-matrix models on $S^4$ or hemispheres) with fully backreacted supergravity duals to derive exact large-$N$ one-point functions for ambient operators $\mathcal{O}_J=\mathrm{tr}(u\cdot\Phi)^J$, including intricate mixing with lower-dimension and multi-trace operators. They provide explicit saddle-point densities and matrix-model insertions, derive auxiliary $\langle\hat{\mathcal{O}}_J\rangle$ and physical $\langle\mathcal{O}_J\rangle$ correlators for D3/D5 defects, D3/D5 interfaces, D3/NS5 BCFTs, and D3/D5/NS5 interfaces, and verify nontrivial S-duality relations across these setups. The results extend probe-limit findings to fully backreacted (large 5-brane) regimes and establish concrete links between localization data, holographic densities, and dual 3d boundary theories, with potential implications for double holography and boundary bootstrap analyses. Overall, the work provides a comprehensive, cross-checked framework for exact ambient operator one-point functions in a broad class of holographic BCFTs and ICFTs.

Abstract

We derive one-point functions in 4d $\mathcal N=4$ SYM with $\tfrac{1}{2}$-BPS boundaries, defects and interfaces which host large numbers of defect degrees of freedom. The theories are engineered by Gaiotto-Witten D3/D5/NS5 brane setups with large numbers of D5 and NS5 branes, and have holographic duals with fully backreacted 5-branes. They include BCFTs with 3d SCFTs on the boundary which allow for the notion of double holography, as well as D3/D5 defects and interfaces with large numbers of D5-branes. Through a combination of supersymmetric localization and holography we derive one-point functions for 4d chiral primary operators.

Figures (5)

• Figure 1: Left: Brane construction for an interface between $U(6)$ and $U(18)$$\mathcal{N}=4$ SYM with Nahm pole boundary conditions. D3-branes are shown as as horizontal lines, D5-branes as vertical lines. 12 D3-branes on the left end on two groups of D5-branes to reduce the rank of the gauge group. The first 5-brane group contains 3 D5-branes with 2 D3-branes ending on each, the second contains 2 D5-branes with 3 D3-branes ending on each. Right: Defect in 4d $\mathcal{N}=4$ SYM realized by D5-branes intersecting D3-branes.
• Figure 2: Eigenvalue densities for the D3/D5 defect with $k=0$ (yellow) and interface with $k\neq0$ (green) compared to unmodified $\mathcal{N}=4$ SYM (blue) at the same $N$ and $g$.
• Figure 3: Brane constructions for BCFTs, with D3-branes as horizontal lines, NS5-branes as ellipses and D5-branes as vertical lines, for the D3/NS5 BCFT with the quiver in (\ref{['eq:D3NS5-quiver']}), for $N_5=4$, $K=3$.
• Figure 4: Left: the D3/D5/NS5 BCFT engineered by $N_{\rm D3}$ semi-infinite D3-branes ending on a combination of $N_{\rm D5}$ D5-branes and $N_5$ NS5-branes, as described in the text. Right after Hanany-Witten transitions to make the quiver in (\ref{['eq:D5NS5K-quiver']}) manifest, for $N_5=5$, $N_{\rm D5}=3$, $R=2$, $S=-2$.
• Figure 5: $\Sigma$ as half disc in the $u$ coordinate with Wilson loop D5$^\prime$ embeddings as blue curves. Those emerging from the NS5 source describe Wilson loops associated with 3d gauge nodes ($0<z<1$). The embeddings emerging from the points marked F1 describe Wilson loops associated with the two 4d gauge nodes ($z=0$ and $z=1$). The eigenvalue densities $\varrho(z,x)$ can be obtained from the corresponding D5$^\prime$ branes. The points marked F1 are solutions to $\frac{dv}{du}=0$; they host fundamental strings describing Wilson loops (for details we refer to Coccia:2021lpp). These are also branch points in the map between the $u$ and $v$ coordinates. On each side the line extending between F1 and the D3, and the line between F1 and the NS5, are mapped to the same branch cut in the $v$ plane.