DRSN VI: Drifted Brane Geometry and Spectral Worldvolume Dynamics (De Rerum Spectrale Natura, Report VI, Version 2.0)

DRSN VI: DRIFTED BRANE GEOMETRY AND SPECTRAL WORLDVOLUME DYNAMICS De Rerum Spectrale Natura series REPORT VI (Version 2.0) GDs J. Pinho-da-Cruz Department of Mechanical Engineering University of Aveiro September 2025 • Branes arise as spectral localisations of the bulk Dirac operator, rather than as independently postulated worldvolume objects. • Worldvolume geometry and dynamics are derived from the spectral action through operatorial restriction to the brane sector. • Dirac–Born–Infeld and Wess–Zumino structures emerge universally from heat-kernel invariants and indextheoretic contributions. • The drift deformation controls bulk–brane coupling and stability, providing a tunable and analytically controlled interaction mechanism. Drifted Brane Spectral Geometry: Worldvolume Dynamics, Spectral Embeddings and the DBI–WZ Action J. Pinho-da-Cruz 1, ∗ 1 Department of Mechanical Engineering, University of Aveiro, Portugal We extend the drifted spectral framework to the geometry and dynamics of branes. Branes are characterised as spectrally localised sectors of the bulk Dirac operator, and their worldvolume geometry arises from operatorial restriction rather than independent postulates. Standard brane actions emerge naturally from the spectral action under drift deformation. Keywords: Spectral Action; Drift Geometry; Branes; Dirac–Born–Infeld; Wess–Zumino Terms; Bulk–Brane Systems. ∗jp[email protected] 3 CONTENTS I. Introduction 5 II. Spectral Characterisation of Branes and Embedding Geometry 5 III. Spectral Localisation of Branes 6 A. Induced Dirac Operator 6 B. Extrinsic Geometry from Commutators 6 C. Summary of Geometric Data 7 IV. Worldvolume Dirac Operator and Induced Geometry 7 A. Induced Worldvolume Operator 7 B. Intrinsic Geometry 7 C. Extrinsic Geometry from Commutators 8 D. Summary of Geometric Data 8 V. Spectral Derivation of Dirac–Born–Infeld and Wess–Zumino Actions 8 A. Restriction of the Spectral Action 8 B. Emergence of the DBI Structure 9 C. Wess–Zumino Couplings 9 D. Structural Summary 9 VI. Bulk–Brane Consistency, Stability, and Drift Dynamics 10 A. Spectral Consistency Conditions 10 B. Backreaction and Bulk–Brane Coupling 10 C. Stability Under Drift 11 D. Spectral Balance Between Bulk and Brane 11 E. Dynamical Interpretation 11 F. Summary 11 VII. Conclusions, Outlook, and Relation to Holography 12 A. Summary of Results 12 B. Conceptual Implications 12 C. Relation to Holography and Gauge–Gravity Correspondence 13 D. Outlook 13 4 Appendices 14 A. Spectral Projections and Brane Localisation 14 B. Heat Kernel Expansion on the Brane 14 C. Drift Dependence of Brane Couplings 14 D. Summary of the Brane Spectral Framework 15 References 16 5 I. INTRODUCTION Branes play a central role in modern high–energy theory. Within the drifted spectral framework, they arise naturally as localised sectors of the bulk Dirac operator, removing the conceptual separation between bulk and worldvolume dynamics. The operator-theoretic framework of drifted spectral geometry developed in DRSN I–V establishes the analytic and structural foundations on which the present drifted brane construction is built, including bounded similarity deformations, exact heat-kernel factorisation, compactification control and the spectral moduli layer [1–5]. About this report. This work constitutes Report VI of the DRSN series (De Rerum Spectrale Natura), a sequence of independent but thematically unified studies on spectral drift geometry. The DRSN series is developed within an open research community on spectral geometry and fundamental physics; related materials, preprints and versioned updates are archived at https://zenodo.org/communities/dsrn/. The present report develops the brane and worldvolume layer of drifted spectral geometry. Branes are formulated as spectrally localised sectors of the bulk Hilbert space via bounded projections, and worldvolume dynamics is derived from the restriction of the spectral action to the projected sector, yielding Dirac–Born–Infeld and Wess–Zumino type structures in a purely operatorial manner. The emphasis is structural and analytic: we formulate the spectral consistency conditions governing bulk–brane coupling, backreaction control and stability under drift. No new phenomenological claims are made. Subsequent reports in the DRSN series will build on this brane layer to develop holographic and gauge–gravity correspondences within the drifted spectral programme. II. SPECTRAL CHARACTERISATION OF BRANES AND EMBEDDING GEOMETRY A brane is defined spectrally by a projection onto a localised subspace of the bulk Hilbert space. Definition 1 (Spectral Brane).Let Ds act on Hbulk . A brane Σis defined by a projection PΣ such that HΣ=PΣHbulk, DΣ=PΣDsPΣ. The induced geometry and extrinsic data arise from commutators between Dsand PΣ. 6 Bulk Dirac Ds PΣ DΣ FIG. 1. Spectral construction of a brane via projection. III. SPECTRAL LOCALISATION OF BRANES The restriction of the bulk Dirac operator to a spectral brane defines an effective worldvolume Dirac operator encoding intrinsic and extrinsic geometry. A. Induced Dirac Operator The worldvolume operator DΣ=PΣDsPΣ acts on HΣand inherits ellipticity and self-adjointness from Ds. B. Extrinsic Geometry from Commutators Extrinsic curvature terms arise from [Ds, PΣ], which encode how the brane is embedded in the bulk geometry. Remark 2. This provides an operatorial origin for the second fundamental form, without introducing embedding fields by hand. 7 Spectral Object Geometric Meaning PΣBrane localisation DΣIntrinsic geometry [Ds, PΣ]Extrinsic curvature TABLE I. Operatorial encoding of brane geometry. C. Summary of Geometric Data IV. WORLDVOLUME DIRAC OPERATOR AND INDUCED GEOMETRY The spectral characterisation of a brane introduced in the previous section naturally induces an effective Dirac operator on the brane worldvolume. This operator encodes both intrinsic and extrinsic geometric data, and its properties follow directly from those of the bulk drifted Dirac operator. A. Induced Worldvolume Operator Let Ds be the drifted bulk Dirac operator acting on the Hilbert space Hbulk , and let PΣ be the spectral projector associated with a brane Σ. The induced worldvolume Dirac operator is defined as DΣ:= PΣDsPΣ,(IV.1) acting on the projected Hilbert space HΣ:= PΣHbulk. Since PΣ is a bounded projection and Ds is self-adjoint for real drift, the operator DΣ inherits ellipticity and self-adjointness on its natural domain. These properties follow from standard results on bounded perturbations and holomorphic families of self-adjoint operators [14,15]. B. Intrinsic Geometry The square of the induced operator, D2 Σ=PΣD2 sPΣ+O([Ds, PΣ]),(IV.2) defines a Laplace-type operator on the brane worldvolume. Its principal symbol coincides with the induced metric on Σ, while lower-order terms encode curvature and torsion contributions inherited from the bulk. Thus, intrinsic worldvolume geometry arises purely from the operatorial restriction. 8 C. Extrinsic Geometry from Commutators Extrinsic curvature data is encoded in the commutator KΣ:= [Ds, PΣ].(IV.3) This operator measures the failure of Ds to preserve the subspace HΣ and plays the role of a spectral second fundamental form. Quadratic combinations of KΣ generate the analogue of extrinsic curvature squared terms familiar from geometric brane actions. Remark 3. No explicit embedding map or extrinsic curvature tensor is introduced. All geometric information is encoded in operator commutators. D. Summary of Geometric Data Spectral Quantity Geometric Interpretation PΣBrane localisation DΣIntrinsic Dirac operator [Ds, PΣ]Extrinsic curvature D2 ΣInduced Laplacian TABLE II. Operatorial encoding of intrinsic and extrinsic brane geometry. The induced worldvolume operator thus provides a complete spectral description of brane geometry, setting the stage for the derivation of effective brane actions. V. SPECTRAL DERIVATION OF DIRAC–BORN–INFELD AND WESS–ZUMINO ACTIONS We now derive the effective brane action from the spectral action associated with the induced worldvolume operator. Remarkably, standard Dirac–Born–Infeld (DBI) and Wess–Zumino (WZ) terms emerge naturally from the spectral framework, without being postulated independently. A. Restriction of the Spectral Action Consider the bulk spectral action Sbulk = Tr(f(Ds/Λ)) .(V.1) 9 Restricting this action to the brane sector amounts to inserting the projector PΣ, SΣ= Tr(PΣf(Ds/Λ) PΣ) = TrHΣ(f(DΣ/Λ)) .(V.2) The heat kernel expansion of this trace yields local invariants constructed from D2 Σ and the commutators [Ds, PΣ]. B. Emergence of the DBI Structure At leading order in derivatives, the spectral action produces a volume term proportional to the induced metric determinant on Σ. Higher-order contributions involve quadratic and quartic combinations of KΣ= [Ds, PΣ], yielding a non-linear structure of the form SDBI ∼ZΣq−det(gind +F),(V.3) where F denotes effective worldvolume gauge and curvature contributions. This reproduces the characteristic Dirac–Born–Infeld action [13,16]. C. Wess–Zumino Couplings Topological terms in the heat kernel expansion give rise to Wess–Zumino type couplings. These arise from mixed traces involving bulk flux operators and the projection PΣ , leading schematically to SWZ ∼ZΣ C∧eF,(V.4) where C denotes bulk form potentials encoded spectrally. The coupling is fixed by index-theoretic data and requires no additional normalisation, in agreement with the standard dielectric brane couplings [17]. D. Structural Summary The appearance of DBI and WZ structures is therefore not an assumption but a consequence of the spectral framework. Remark 4. The drift deformation controls the relative strength of bulk–brane couplings and plays a stabilising role for brane dynamics. 16 REFERENCES [1] J. Pinho-da-Cruz, DRSN I: A Unified Geometric Framework for Matter, Torsion and Dark Energy, August 2025. Version 3.0, doi: 10.5281/zenodo.17981231. [2] J. Pinho-da-Cruz, DRSN II: Drift Geometry and the Unified Spectral Operator, September 2025. Version 2.0, doi: 10.5281/zenodo.17982131. [3] J. 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