Search for a command to run...
This work addresses a growing structural problem in modern holography: the increasing reliance on ad hoc cutoff prescriptions and higher-dimensional boundary counterterms in situations where the standard AdS_{d+1} holographic renormalization framework no longer applies. Recent progress in higher-dimensional holography—particularly involving bubbling geometries, defects, and heavy operators—has made it clear that many physically relevant observables lie outside the scope of any finite-field lower-dimensional truncation. In these regimes, renormalization must be performed directly in ten or eleven dimensions, and finite observables are typically obtained only after introducing carefully chosen cutoff surfaces and higher-dimensional boundary terms. While such constructions can be technically successful and reproduce known field-theoretic results, they are intrinsically prescription-based: their justification relies on a posteriori agreement rather than on an internal consistency principle. The central claim of this paper is that this situation reflects a missing global consistency structure, not a technical shortcoming of holographic renormalization. The work provides that missing structure. The paper introduces the notion of a context to organize higher-dimensional holographic renormalization. A context packages all data required to define a regulated holographic observable, including the choice of cutoff surface, asymptotic presentation, subtraction and normalization conventions, admissible boundary terms, and boundary conditions. Different regulator prescriptions are therefore not treated as gauge redundancies, but as distinct contexts that must be related by explicit transport rules. The collection of contexts and admissible changes between them naturally forms a groupoid, which becomes the minimal mathematical structure needed to make statements about scheme dependence precise. Within this framework, the role of counterterms is reinterpreted. Rather than being viewed solely as devices for cancelling divergences in a single scheme, counterterms are identified as coherence (descent) data: they provide the finite gluing information required to compare renormalized quantities across different contexts. This shift in perspective reveals that the real problem in higher-dimensional holography is no longer ultraviolet divergence, but the comparison of finite answers obtained in different regulator settings. The main result of the paper is a global consistency criterion that cleanly separates three logically distinct situations. First, if no coherent transport law exists between contexts, the comparison problem itself is ill-posed. Second, if coherent transport exists but cannot be trivialized by object-wise redefinitions, one obtains a well-defined global object with an intrinsic obstruction. Third, only in the special case where the transport data can be trivialized does one obtain a strictly context-independent observable. The paper proves that anomalies correspond precisely to the second situation: they are obstruction classes associated with nontrivial transport around loops in context space. Importantly, the work emphasizes that strictification is not required for global consistency; coherent but non-strictifiable assignments are perfectly well-posed and are the correct mathematical avatars of anomalous observables. A central part of the paper is a detailed, surgical analysis of a recent ten-dimensional holographic computation of defect anomalies. That computation successfully reproduces a known field-theory result by choosing a specific higher-dimensional cutoff and adding a specific boundary counterterm. This work shows that, while the computation is correct, it cannot by itself answer several global questions: which counterterms are admissible, whether the prescription is unique, how different cutoff choices should be compared, or what invariant is actually being measured. By mapping each element of the computation into the context–groupoid framework, the paper demonstrates that the higher-dimensional boundary counterterm functions as coherence data and that the extracted anomaly is the strictification-invariant residue of the comparison problem. In this way, the framework explains both why such counterterms are required and why the resulting anomaly is robust, without appealing to calibration against known results. To make the framework practically useful, the paper includes a concrete, step-by-step recipe for holographers. The recipe instructs the reader to identify admissible contexts, specify admissible changes between them, compute the induced finite transition terms, test coherence under composition, and then either strictify the result or extract the anomaly as a loop invariant. This procedure upgrades prescription-based computations into controlled analyses and clarifies when scheme dependence is removable and when it is intrinsically physical. The scope and limitations of the framework are discussed explicitly. The approach does not replace explicit holographic calculations, nor does it compute anomaly coefficients on its own. Instead, it governs what can be made canonical and what cannot, and it predicts when additional counterterms must appear, when anomalies are inevitable, and when scheme dependence carries physical meaning. Brief connections to defect holography, higher-form symmetries, and non-invertible structures are outlined to indicate how the framework naturally interfaces with current developments. Overall, this work reframes higher-dimensional holographic renormalization from a collection of successful but prescription-dependent computations into a principled global comparison problem with a sharp consistency criterion. Its core message is simple and precise: higher-dimensional holography does not need more prescriptions; it needs a coherence principle.