DRSN VII: Drifted Holographic Geometry and Spectral Gauge–Gravity Duality (De Rerum Spectrale Natura, Report VII, Version 1.0)

DRSN VII: DRIFTED HOLOGRAPHIC GEOMETRY AND SPECTRAL GAUGE–GRAVITY DUALITY De Rerum Spectrale Natura series REPORT VII (Version 1.0) GDs J. Pinho-da-Cruz Department of Mechanical Engineering University of Aveiro September 2025 •Holography reformulated as a spectral correspondence. •Boundary theories arise as projected Dirac sectors. •Radial direction interpreted as spectral drift / RG flow. •Bulk–boundary duality encoded in a single spectral action. Spectral Holography from Drift Geometry: Gauge–Gravity Duality, RG Flow and Operator Correspondence J. Pinho-da-Cruz 1, ∗ 1 Department of Mechanical Engineering, University of Aveiro, Portugal We formulate holography and gauge–gravity correspondence within the framework of drifted spectral geometry. Bulk and boundary theories are described as distinct spectral sectors of a single Dirac operator, related by projection and controlled drift deformation. The radial holographic direction is interpreted as a spectral flow, providing an operatorial realisation of renormalisation group evolution. This approach yields a unified, background-independent formulation of holography, bridging AdS/CFT, brane constructions, and noncommutative geometry. Keywords: Spectral Action; Holography; Gauge–Gravity Correspondence; Renormalisation Group Flow; Drift Geometry; AdS/CFT; Noncommutative Geometry. CONTENTS I. Introduction 3 II. Spectral Holography Framework 4 A. Bulk and Boundary as Spectral Sectors 4 B. Radial Direction as Spectral Flow 4 C. Consistency Conditions 4 III. Spectral Renormalisation Group Flow and Boundary Dynamics 5 A. Spectral Flow as RG Evolution 5 B. Beta Functions from Spectral Variations 6 C. Energy Scales and Spectral Cutoff 6 D. Boundary Dynamics and Operatorial Running 6 E. Structural Summary 6 IV. Spectral Bulk–Boundary Dictionary and AdS/CFT Correspondence 7 A. Bulk Operators and Boundary Observables 7 ∗jp[email protected] 3 B. Sources, States, and Generating Functionals 7 C. AdS/CFT as a Spectral Equivalence 8 D. Matching Conditions and Anomalies 8 E. Dictionary Summary 8 F. Scope and Implications 8 V. Examples, Consistency Checks, and Applications 9 A. AdS–Like Geometries and Spectral Boundaries 9 B. Consistency Check: Matching of Degrees of Freedom 9 C. Entanglement and Area Laws 10 D. Application to Brane-Induced Holography 10 E. Structural Summary 10 F. Outlook on Applications 10 VI. Conclusions, Outlook, and Future Directions 11 A. Summary of the Spectral Holography Framework 11 B. Conceptual Advances 11 C. Relation to the DSRN Programme 12 D. Future Directions 12 A. Spectral Projections and Boundary Stability 12 B. Spectral Interpretation of Entanglement 13 References 13 I. INTRODUCTION Holography has emerged as a central paradigm in modern theoretical physics, asserting an equivalence between gravitational dynamics in a bulk spacetime and quantum field theories living on its boundary. The most prominent realisation of this idea is the AdS/CFT correspondence, where bulk gravity in anti–de Sitter space is dual to a conformal field theory defined on its boundary. Despite its success, conventional holography relies heavily on background-specific constructions, asymptotic boundary conditions, and string-theoretic embeddings. In particular, the dictionary between bulk and boundary degrees of freedom is often introduced in a model-dependent manner. 4 In the present report we propose a reformulation of holography as a purely spectral phenomenon. Building on the drifted Dirac framework developed in the previous reports, we show that bulk and boundary theories arise naturally as different spectral sectors of a single operatorial structure. Projections define boundary degrees of freedom, while the drift parameter generates a controlled spectral flow interpretable as renormalisation group evolution. This perspective unifies holography, brane dynamics, and compactification within a single spectral action, eliminating the need for separate bulk and boundary postulates. II. SPECTRAL HOLOGRAPHY FRAMEWORK A. Bulk and Boundary as Spectral Sectors Let Ds be a drifted Dirac operator acting on a bulk Hilbert space Hbulk . A boundary theory is defined by a spectral projection P∂onto a subspace H∂⊂ Hbulk. The induced boundary operator D∂:= P∂DsP∂(II.1) encodes the dynamics of boundary degrees of freedom. Definition 1 (Spectral Boundary).A holographic boundary is defined by a projection P∂ such that H∂=P∂Hbulk, with dynamics governed by D∂. B. Radial Direction as Spectral Flow The drift parameter s generates a family of operators Ds . We interpret this family as defining a spectral flow connecting bulk and boundary descriptions. In this picture, the holographic radial coordinate is identified with the drift parameter, and evolution in s corresponds to renormalisation group flow in the boundary theory. C. Consistency Conditions Consistency of spectral holography requires approximate commutativity [D2 s, P∂]≈0,(II.2) ensuring that boundary dynamics is stable under bulk evolution. This condition plays the role of holographic matching and renormalisation conditions in conventional formulations. Remark 2. The holographic dictionary is encoded operatorially rather than postulated. 5 Bulk Dirac Ds Spectral Flow (s) Boundary Dirac D∂ FIG. 1. Spectral interpretation of holography: bulk and boundary are related by projection and drift-induced spectral flow. III. SPECTRAL RENORMALISATION GROUP FLOW AND BOUNDARY DYNAMICS A central advantage of the spectral holography framework is that renormalisation group (RG) flow acquires a precise operatorial interpretation. Rather than being introduced as a scale-dependent deformation of a boundary action, RG evolution arises naturally as spectral flow generated by the drifted Dirac operator. A. Spectral Flow as RG Evolution Let Ds denote the drifted bulk Dirac operator. The family {Ds}s∈R defines a continuous spectral flow, with d dsDs= [Ds, φ],(III.1) where φ is the drift generator. When restricted to the boundary sector, this induces an evolution equation for the boundary operator d dsD∂=P∂[Ds, φ]P∂.(III.2) This equation provides an intrinsic definition of RG flow for the boundary theory. B. Beta Functions from Spectral Variations The spectral action associated with the boundary operator, S∂(s) = TrH∂(f(D∂/Λ)) ,(III.3) 6 inherits s–dependence from the bulk drift. Differentiating with respect to syields d dsS∂(s) = TrH∂f′(D∂/Λ) 1 Λ dD∂ ds ,(III.4) which plays the role of a spectral beta function. Fixed points of the RG flow correspond to stationary spectral configurations satisfying d dsS∂(s)=0.(III.5) Remark 3. Conformal boundary theories correspond to spectral fixed points of the drift flow. C. Energy Scales and Spectral Cutoff In the spectral action formalism, the cutoff scale Λplays the role of an ultraviolet regulator. The drift parameter s acts as an independent scale variable, controlling how spectral modes are redistributed between bulk and boundary. The combined dependence on ( s, Λ) naturally encodes multi-scale RG behaviour. D. Boundary Dynamics and Operatorial Running Boundary coupling constants are encoded in the lower-order terms of the boundary Dirac operator. As s varies, these terms evolve according to the induced spectral flow, yielding running couplings without introducing separate renormalisation prescriptions. This operatorial running unifies geometric and quantum aspects of RG evolution. E. Structural Summary Spectral Object RG Interpretation Drift parameter sRG scale Spectral flow Ds(s)RG trajectory Stationary spectrum Fixed point Boundary projection Effective field theory TABLE I. Spectral interpretation of renormalisation group flow. The spectral RG framework thus provides a background-independent and operatorially exact formulation of boundary dynamics, forming the backbone of spectral holography. 7 IV. SPECTRAL BULK–BOUNDARY DICTIONARY AND ADS/CFT CORRESPONDENCE We now formulate the bulk–boundary dictionary in purely spectral terms. In contrast with conventional holographic constructions, where the dictionary is introduced via asymptotic expansions and boundary conditions, the present framework encodes the correspondence operatorially through projections, commutators, and spectral invariants. A. Bulk Operators and Boundary Observables Let Ds be the drifted bulk Dirac operator acting on Hbulk , and P∂ the projection defining the boundary sector. Bulk observables correspond to spectral functionals of Ds , while boundary observables are obtained by restriction, O∂=P∂Obulk(Ds)P∂.(IV.1) Correlation functions on the boundary are therefore traces over H∂, ⟨O∂1· · · O∂n⟩= TrH∂(O∂1· · · O∂n),(IV.2) providing a direct operatorial definition of boundary correlators. B. Sources, States, and Generating Functionals In conventional AdS/CFT, boundary sources couple to bulk fields evaluated at the boundary. In the spectral framework, sources correspond to perturbations of the boundary operator, D∂−→ D∂+J∂,(IV.3) where J∂ is a bounded operator acting on H∂ . The generating functional of boundary correlators is then defined spectrally by Z∂[J∂] = exp−TrH∂fD∂+J∂ Λ.(IV.4) Functional derivatives with respect to J∂ reproduce boundary correlation functions, establishing the spectral analogue of the GKPW prescription. 8 C. AdS/CFT as a Spectral Equivalence Within this framework, AdS/CFT correspondence is interpreted as a spectral equivalence between two descriptions of the same operatorial data: •a bulk description in terms of Dsacting on Hbulk, •a boundary description in terms of D∂acting on H∂. The equivalence is mediated by the projection P∂ and controlled by the drift parameter s , which governs the redistribution of spectral weight between bulk and boundary. D. Matching Conditions and Anomalies Spectral consistency requires that bulk and boundary variations of the spectral action agree up to boundary-localised terms, δTr(f(Ds/Λ)) = δTrH∂(f(D∂/Λ)) + δSanom.(IV.5) The term Sanom captures boundary anomalies and topological contributions. In the spectral framework, anomalies arise from index-theoretic data and are fixed unambiguously by the operator structure, rather than by regularisation choices. E. Dictionary Summary Bulk Quantity Boundary Interpretation DsRG-dependent Hamiltonian Spectral action Trf(Ds)Generating functional Projection P∂Boundary limit Drift parameter sEnergy scale / radial coordinate Index density Anomaly term TABLE II. Spectral bulk–boundary dictionary. F. Scope and Implications The spectral bulk–boundary dictionary is background independent and does not rely on asymptotic expansions. It applies equally to AdS-like geometries, compactifications with boundaries, and 9 brane-induced holographic scenarios. In this sense, AdS/CFT emerges as a special case of a more general spectral holography principle. V. EXAMPLES, CONSISTENCY CHECKS, AND APPLICATIONS To illustrate the spectral holography framework and to test its internal consistency, we discuss representative examples and applications. The emphasis is on structural checks rather than modelspecific computations, highlighting how known holographic scenarios emerge naturally from the operatorial formalism. A. AdS–Like Geometries and Spectral Boundaries Consider a bulk geometry whose spectral properties approximate those of an anti–de Sitter spacetime. In the spectral formulation, no explicit coordinate realisation of AdS is required. Instead, AdS–like behaviour is characterised by: •a continuous spectral flow of Dswith respect to the drift parameter s, •approximate scale invariance at spectral fixed points, •a well-defined boundary projection P∂. At such fixed points, the boundary spectral action becomes invariant under drift, corresponding to a conformal boundary theory. B. Consistency Check: Matching of Degrees of Freedom A basic consistency requirement of holography is the matching of effective degrees of freedom between bulk and boundary descriptions. In the spectral framework, this requirement translates into the relation TrHbulk (f(Ds/Λ)) ≃TrH∂(f(D∂/Λ)) + Oboundary,(V.1) where Oboundary denotes controlled boundary-localised contributions. This equality ensures that the bulk spectral density reorganises into boundary degrees of freedom without loss of information.