Monodromy Defects in Maximally Supersymmetric Yang-Mills Theories from Holography

We study three Type II supergravity solutions holographically dual to codimension-2 supersymmetric defects in $(p+1)$-dimensional SU($N$) maximally supersymmetric Yang-Mills theory ($p=2,3,4$). In all of these cases, the defects have a non-trivial monodromy for the maximal abelian subgroup for the SO($9-p$) R-symmetry. Such solutions are obtained by considering branes wrapping spindle configurations, changing the parameters (which alters the coordinate domain), and imposing suitable boundary conditions. We provide a prescription to compute the entanglement entropy of the effective theory on the defect. We find the resulting quantity to be proportional to the free energy of the ambient theory. A similar analysis is performed for the D5-brane wrapping a spindle, but we find that changing the coordinate domain does not lead to a defect solution, but rather to a circle compactification.

💡 Research Summary

The paper investigates holographic realizations of codimension‑2 supersymmetric monodromy defects in maximally supersymmetric SU(N) Yang‑Mills (SYM) theories in dimensions p + 1 with p = 2, 3, 4. The authors start from the near‑horizon geometries of stacks of Dp‑branes (p ≤ 4) in Type II supergravity, expressed in the “brane frame” (also called the dual frame) where the metric takes a conformally AdS_{p+2} × S^{8‑p} form. By performing a change of variables (u, ỹ) → (ρ, y) they rewrite the background so that the asymptotic region (UV) appears as AdS_{p} × S^{1}, while the internal two‑dimensional space is a spindle (a topological sphere with conical deficits at its poles).

To construct defect solutions, the authors modify the integration constants of the known spindle solutions, extending the coordinate range of one spindle direction from a finite interval to a semi‑infinite one. This produces a geometry that asymptotes to the vacuum AdS_{p} × S^{1} foliation but carries a non‑trivial gauge field holonomy around the S^{1}. The gauge fields belong to the maximal abelian subgroup of the SO(9‑p) R‑symmetry, i.e. a U(1)_R together with (r‑1) flavour U(1)’s (r = ⌊(9‑p)/2⌋). The holonomy implements a background gauge field A_R = μ_R dz, A_F^I = μ_I dz, which induces a monodromy for fields charged under these symmetries when circling the defect (ρ = 0). The authors show that supersymmetry forces n = 1 (no conical excess) for the U(1)_R defect, in agreement with earlier conformal analyses.

The paper treats each value of p separately using the appropriate gauged supergravity (4‑D for p = 2, 5‑D for p = 3, 6‑D for p = 4). For each case they write down the BPS equations, impose regularity at the spindle pole (y = 1) and the required asymptotics at y → ∞, and verify that the solutions preserve at least two supercharges.

The D5‑brane case (p = 5) is examined in Section 4. Because the D5‑brane background does not admit an AdS frame, the same coordinate manipulation does not generate a defect but rather a compactification on a circle. Hence no codimension‑2 monodromy defect exists for p = 5 within this construction.

A central part of the work is the computation of the defect entanglement entropy (EE). The authors adapt the Ryu‑Takayanagi prescription to the warped, dilaton‑dependent backgrounds, defining a UV cut‑off at y → ∞ and adding appropriate holographic counterterms. After renormalization, the defect EE is found to be proportional to the holographic free energy (or central charge) of the ambient SYM theory:
S_{EE}^{defect} = α_p c_{hol}^{(p)}.
The proportionality constant α_p depends only on p and matches known results for the conformal p = 3 case, thereby extending the EE–free‑energy relation to non‑conformal SYM theories (p = 2, 4).

In the conclusion the authors summarize their findings, emphasize that the spindle‑based construction provides a unified holographic description of monodromy defects across dimensions, and outline future directions such as studying defect operator spectra, possible enhancements of supersymmetry, and extensions to M‑brane setups. An appendix collects the explicit gauged supergravity Lagrangians and uplift formulas used throughout the analysis.

Overall, the paper presents a systematic method to embed codimension‑2 monodromy defects in holographic duals of maximally supersymmetric Yang‑Mills theories, demonstrates that the defect entanglement entropy scales with the ambient free energy even in non‑conformal settings, and clarifies why the D5‑brane case does not admit such a defect.