Monodromy Defects in Maximally Supersymmetric Yang-Mills Theories from Holography

Abstract: We study three Type II supergravity solutions which are holographically dual to codimension-2 supersymmetric defects in $(p+1)$-dimensional SU($N$) maximally supersymmetric Yang-Mills ($p=2,3,4$). 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. 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 a circle compactification.

• The paper introduces holographic duals for codimension-2 monodromy defects in maximally supersymmetric SU(N) Yang-Mills theories, extending approaches to non-conformal cases.
• It constructs explicit supergravity solutions using wrapped branes and modified spindle geometries, matching field theory monodromy parameters with bulk gauge field holonomies.
• The study reveals that the renormalized defect entanglement entropy is proportional to the ambient theory’s free energy, offering a new probe of generalized conformal symmetry.

Holographic Analysis of Codimension-2 Monodromy Defects in Maximally Supersymmetric Yang-Mills Theories

Introduction and Motivation

The paper "Monodromy Defects in Maximally Supersymmetric Yang-Mills Theories from Holography" (2512.10767) presents a systematic study of codimension-2 supersymmetric monodromy defects in maximally supersymmetric SU($N$) Yang-Mills theories (SYM) in $(p+1)$ dimensions for $p=2,3,4$ via holographic duality. The analysis combines nontrivial supergravity solutions—specifically branes wrapping spindle geometries with modified coordinate domains and boundary conditions—with a detailed investigation of the field theory interpretation in terms of monodromy defects. The work generalizes methods previously used for conformal theories (e.g., 4d $\mathcal{N}=4$ SYM, ABJM, and the $(2,0)$ SCFT) to the non-conformal case, thus filling a notable gap in the AdS/CFT literature.

Construction of Holographic Duals to Monodromy Defects

The construction begins with the Type II supergravity backgrounds describing the near-horizon geometry of D$p$-branes, which are holographically dual to maximally supersymmetric SU($N$) SYM theories in $(p+1)$ dimensions. For $p\neq 3$ these theories are not conformal, yet the supergravity backgrounds can still be written, in brane frame, as a fibration that is conformally $AdS_{p+2} \times S^{8-p}$. This allows the extension of techniques initially developed for conformal monodromy defects (i.e., those preserving subgroups of the superconformal algebra).

The duals to monodromy defects are constructed by starting from existing brane solutions wrapping spindles, and then analytically continuing or modifying the range of the spindle coordinates such that one becomes semi-infinite. Imposing appropriate boundary conditions ensures the geometry shrinks smoothly at one end and acquires the required asymptotics at infinity. Crucially, the background admits nontrivial Wilson lines (monodromies) for the maximal abelian subgroup of the $SO(9-p)$ R-symmetry, corresponding to turning on constant values for specific U(1) gauge fields in the bulk.

In detail:

• For $p=2,3,4$, explicit solutions are obtained as consistent truncations of higher-dimensional gauged supergravity theories, typically involving several $U(1)$ gauge fields and scalars parameterizing the moduli associated to the R-symmetry's maximal torus.
• For $p=3$ (the conformal case), previously known solutions are recovered and reviewed for completeness. For $p=2,4$ (non-conformal), the setup extends these ideas to non-conformal brane backgrounds, using the structure of the "brane frame" to mimic AdS behavior.

Field Theoretic Interpretation and Implementation of Monodromy Defects

On the field theory side, the codimension-2 defect is located at the origin in a transverse $\mathbb{R}^{2}$, with the metric expressed in polar coordinates and a possible conical defect parameter $n$. The defect is implemented by coupling SYM to background gauge fields for the maximal abelian subgroup of the R-symmetry, generating nontrivial holonomy (monodromy) around the defect. This holonomy is precisely matched with the nontrivial bulk expectation values for the corresponding supergravity gauge fields in the holographic dual.

A key observation is that supersymmetric monodromy defects for the R-symmetry can coexist only with the presence of conical singularities in the transverse space ($n \neq 1$), which agrees with previous results in the conformal cases. For $p\neq 3$, the SYM theories and defects are not conformal, but possess a generalized conformal structure in which the (dimensionful) gauge coupling transforms as a spurion under Weyl rescalings.

Detailed Analysis of the Solutions

For each case ($p=2$ D2-branes, $p=3$ D3-branes, $p=4$ D4-branes), the authors analyze the spectrum of allowed solutions, the dependence on the monodromy parameters, and the possible enhancement of supersymmetry for special choices of background gauge fields. The parameterization uses the integration constants of the lower-dimensional gauged supergravity and their field theory interpretation via the holonomies.

• For $p=2$ (D2-brane), the 4d gauged supergravity truncation with $U(1)^3$ gauge symmetry is employed, matching three independent monodromies. The conditions for supersymmetry, smoothness, and regularity are presented explicitly. A rich structure of sub-sectors with enhanced supersymmetry is detailed for particular background field configurations.
• For $p=3$ (D3-brane), the conformal case is revisited, exhibiting full agreement with earlier results and explicit relations between geometric parameters and field theory data (monodromies, conical parameters).
• For $p=4$ (D4-brane), the extension to the non-conformal case leverages 6d $U(1)^2$ gauged supergravity. The solutions exhibit similar qualitative features to the conformal case but differ due to the running of the effective gauge coupling.

The case $p=5$ (D5-brane) is distinct, since the D5 background is not asymptotically AdS in any frame. Instead, it corresponds to compactification on a circle, and the defect interpretation in this case is qualitatively different and cannot be mapped directly to a codimension-2 surface defect.

Holographic Entanglement Entropy and Renormalization

A significant portion of the work involves providing a prescription—and actual computation—for the defect entanglement entropy (dEE) associated to the effective theory localized on the defect.

The entanglement entropy is computed by adapting the free energy prescription of [Macpherson et al., Bea et al.] as a shortcut for entanglement entropy, treating the supergravity background as the dual of a $(p-1)$-dimensional QFT. The computation reveals an unavoidable UV divergence associated with the non-compactness of one internal coordinate. The authors establish a renormalization scheme, introducing a UV cutoff defined in terms of the Fefferman-Graham coordinate and subtracting the contribution from the vacuum (defect-free) background, weighted appropriately by the conical parameter $n$. This procedure, justified by the universal structure of the asymptotics, yields a finite, physical result.

A notable and explicit claim is that, for $p=2,3,4$, the renormalized defect entanglement entropy is proportional to the free energy (Weyl anomaly, or central charge) of the ambient SYM theory, up to explicit $n$ and monodromy-dependent factors. In the conformal case ($p=3$), it further decomposes into a linear combination of the defect Weyl anomaly and its conformal weight, a relation that mirrors and extends earlier results.

Implications and Outlook

The demonstrated proportionality between the defect entanglement entropy and the intrinsic free energy of the host theory has both conceptual and computational implications:

• Holographic Characterization: The results provide a nontrivial check of holography beyond the conformal setting, exhibiting the operational meaning of generalized conformal symmetry via the dependence on the energy scale (through the brane frame coordinate) in non-conformal field theories.
• Defect Quantities: For conformal defects, entanglement entropy can be related to the defect conformal weight and defect anomalies. The results suggest extending this perspective to generalized conformal structures, possibly yielding holographically computable quantities analogous to the central charge for defects in non-conformal theories.
• Applications: The analysis enables the identification and computation of universal data associated with codimension-2 defects (e.g., surface operators, monodromy defects) in strongly coupled SYM theories in various dimensions, with applications to precision holography, quantization of solitonic configurations, and RG flows across defects.

The work suggests further lines of inquiry, such as extending the computation to non-supersymmetric defects, studying the interplay with modified Ward identities induced by the generalized conformal structure, and exploring analogous constructions for defects of higher codimension or for D5/D6 brane backgrounds where the conventional AdS asymptotics break down.

Conclusion

This paper fills a significant gap in the study of holographic duals for codimension-2 monodromy defects beyond the conformal case, providing explicit supergravity constructions dual to such defects in maximally supersymmetric SU($N$) Yang-Mills theories for $p=2,3,4$. The authors establish that the renormalized defect entanglement entropy, constructed from the holographic background, is universally proportional to the free energy of the ambient theory, independent of specific details of the monodromy or the conical parameters. These results robustly extend the analytic control of defect observables in holographic field theories, open up new directions for the study of non-conformal defects, and suggest deeper connections between bulk geometry and defect CFT data.