DRSN IX: Spectral Standard Model and Drifted Strings (De Rerum Spectrale Natura, Report IX, Version 2.0)

DRSN IX: SPECTRAL STANDARD MODEL AND DRIFTED STRINGS De Rerum Spectrale Natura series REPORT IX (Version 2.0) GDs J. Pinho-da-Cruz Department of Mechanical Engineering University of Aveiro September 2025 •The Standard Model arises as a localised spectral fixed point of the drifted Dirac operator. • String-like objects appear as extended spectral excitations, characterised by continuous spectral bands and Regge-like behaviour. • Particles and strings are unified as distinct regimes of a single spectral operator, rather than as fundamentally different entities. • Phenomenological transitions between particle and string regimes are controlled by spectral drift and curvature in spectral flow. Spectral Standard Model and Drifted Strings: Particles and Strings as Regimes of a Single Drifted Dirac Operator J. Pinho-da-Cruz 1, ∗ 1 Department of Mechanical Engineering, University of Aveiro, Portugal We develop a unified spectral formulation of the Standard Model of particle physics and its extension to drifted string degrees of freedom. The Standard Model arises as a spectral fixed point of the drifted Dirac operator associated with an almost–commutative geometry, while Yukawa couplings, Higgs dynamics, and gauge interactions are encoded in internal spectral data. We further interpret strings as extended spectral excitations generated by non–local drift modes, providing a bridge between particle physics and string theory within a single spectral action framework. Keywords: Spectral Standard Model; Almost–Commutative Geometry; Yukawa Couplings; Higgs Sector; Drift Geometry; Strings; Noncommutative Geometry. ∗jp[email protected] 3 CONTENTS I. Introduction 5 II. Spectral Standard Model under Drift 6 A. Almost–Commutative Geometry and the Standard Model 6 B. Drifted Internal Dirac Operator 6 C. Spectral Fixed Point and Low–Energy Physics 6 D. Structural Summary 6 III. Drifted Strings as Extended Spectral Excitations 6 A. From Pointlike to Extended Spectral Modes 7 B. Worldsheet Emergence from Spectral Flow 7 C. Effective String Tension 7 D. Coupling to Gauge and Higgs Sectors 8 E. Relation to Fundamental Strings 8 F. Structural Summary 8 G. Remarks 9 IV. Spectral Unification of Particles and Strings 9 A. Unified Spectral Spectrum 9 B. Energy Regimes and Crossover 9 C. Unified Effective Action 10 D. Gauge and Gravitational Universality 10 E. Relation to Dualities 10 F. Structural Summary 10 G. Implications 10 V. Phenomenological Consequences and Predictions 11 A. Running Couplings and Unification 11 B. Higgs Sector and Vacuum Stability 11 C. Particle–String Transition Scale 11 D. Signatures of Drifted Strings 12 E. Cosmological Implications 12 F. Structural Summary 12 4 G. Remarks 13 VI. Conclusions, Outlook, and Final Remarks 13 A. Summary of Results 13 B. Conceptual Implications 13 C. Outlook 14 Appendices 15 A. Almost–Commutative Geometry and Internal Spectral Data 15 B. Spectral Origin of Extended Excitations 15 References 16 5 I. INTRODUCTION The spectral formulation of the Standard Model, based on almost–commutative geometry, provides a remarkable unification of gravity, gauge interactions, and scalar dynamics. In this framework, fermions, gauge bosons, and the Higgs field emerge from the spectral properties of a single Dirac operator acting on a product geometry. The drifted spectral paradigm developed in the previous reports extends this construction by introducing a controlled deformation of the Dirac operator that preserves its essential analytic properties while generating new dynamical sectors. In particular, drift induces universal potentials, stabilises moduli, and provides a geometric origin for cosmological and quantum effects. The aim of the present report is twofold. First, we show that the Standard Model arises naturally as a spectral fixed point of the drifted internal Dirac operator [ 1 – 3 ]. Second, we extend the framework to incorporate string–like excitations as non–local spectral modes generated by drift, thereby connecting particle physics and string theory within a unified spectral action. The operator-theoretic framework of drifted spectral geometry developed in DRSN I–VIII establishes the analytic and structural foundations on which the present particle–string unification analysis is built, including bounded similarity deformations, spectral action dynamics, compactification control, brane localisation, holographic correspondence, and quantum spectral flow [1–8]. About this report. This work constitutes Report IX 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 particle–string unification layer of drifted spectral geometry. Elementary particles and strings are formulated as different spectral regimes of a single underlying Dirac operator, corresponding to localised and extended spectral excitations, respectively. The emphasis is structural and analytic: we analyse how Standard Model degrees of freedom, stringlike excitations, and Regge behaviour arise from the same spectral data under drift. No new phenomenological claims are made. Subsequent reports will build on this unification layer to address arithmetic, information-theoretic, and cosmological aspects of drifted spectral geometry. The spectral formulation employed here follows the standard framework of noncommutative geometry and the spectral action principle [9–11]. 6 II. SPECTRAL STANDARD MODEL UNDER DRIFT A. Almost–Commutative Geometry and the Standard Model The spectral triple describing the Standard Model is given by the product (A,H, D) = (C∞(M)⊗ AF, L2(M, S)⊗ HF, DM⊗⊮+γ5⊗DF), where AF and DF encode internal gauge and Yukawa data [ 9 , 10 , 12 ]. Gauge bosons and the Higgs field arise from inner fluctuations of D. B. Drifted Internal Dirac Operator We introduce a drift deformation of the internal Dirac operator, (DF)s=esφFDFe−sφF,(II.1) where φF is a bounded internal multiplier. This deformation preserves the spectrum of DF while modifying its lower–order structure. As a consequence, Yukawa couplings and scalar potentials acquire controlled s–dependence. C. Spectral Fixed Point and Low–Energy Physics At the spectral fixed point s = s∗ , the effective potential for the drift parameter is minimised and the internal spectral data stabilise. The resulting configuration reproduces the observed pattern of fermion masses, gauge couplings, and Higgs vacuum expectation value [ 10 – 12 ]. In this sense, the Standard Model appears as an infrared spectral fixed point of the drifted geometry. D. Structural Summary III. DRIFTED STRINGS AS EXTENDED SPECTRAL EXCITATIONS In this section we introduce a spectral interpretation of string degrees of freedom within the drifted geometry framework. Rather than postulating strings as fundamental one–dimensional objects embedded in spacetime, we show that string–like excitations arise naturally as extended, non–local modes of the drifted Dirac operator. The interpretation of extended spectral excitations and Regge-like behaviour is aligned with the standard string-theoretic framework [13]. 7 Spectral Ingredient Physical Sector DMGravity DFYukawas and Higgs Inner fluctuations Gauge bosons Drift parameter sDynamical scale / stabilisation Spectral fixed point Standard Model vacuum TABLE I. Spectral encoding of the Standard Model under drift. A. From Pointlike to Extended Spectral Modes In almost–commutative geometry, particle states correspond to localised spectral modes of the Dirac operator. When drift is introduced, the operator acquires an additional scale–dependent structure that allows for delocalised eigenmodes. These modes extend along one effective direction and behave dynamically as string–like excitations [13]. Concretely, consider families of eigenstates ψn ( s )of Ds whose support spreads over extended regions of the underlying manifold as s varies. Such families define coherent spectral excitations with effective one–dimensional worldsheets. B. Worldsheet Emergence from Spectral Flow The drift parameter s generates a continuous spectral flow. For extended modes, the dependence on s defines an additional internal coordinate, which can be interpreted as a worldsheet direction. The pair ( τ, s ), where τ denotes physical time, parametrises an effective two–dimensional surface swept by the excitation [2,13]. In this picture, the string worldsheet is not fundamental but emerges from the operatorial structure of the drifted Dirac operator. No independent worldsheet fields are introduced; all degrees of freedom are inherited from the bulk spectral data. C. Effective String Tension The effective tension of a drifted string is determined spectrally by the variation of eigenvalues under drift. Schematically, one finds Teff ∼∂2λeff n(s) ∂s2    s=s∗ ,(III.1) 8 where λeff n ( s )denote effective spectral data associated with extended or projected spectral sectors of the drifted operator, rather than global spectral invariants. [ 1 , 3 ]. Because the drift potential stabilises at s = s∗ , the resulting tension is finite and dynamically generated. This provides a geometric origin for string tension within the spectral framework. D. Coupling to Gauge and Higgs Sectors Drifted string excitations couple naturally to gauge and Higgs fields through the internal Dirac operator. Since gauge bosons and scalars arise from inner fluctuations of DF , extended spectral modes automatically carry gauge quantum numbers and interact with the Standard Model sector. This coupling is fixed by the spectral data and does not require additional interaction terms. E. Relation to Fundamental Strings The spectral strings described here share structural similarities with fundamental strings: •they possess an effective worldsheet description, •they exhibit a dynamically generated tension, •they couple universally to gauge and gravitational sectors. They differ conceptually in that they are emergent, operatorial spectral excitations rather than fundamental degrees of freedom. F. Structural Summary Spectral Feature String Interpretation Extended eigenmodes String states Spectral flow in sWorldsheet coordinate Eigenvalue curvature String tension Inner fluctuations Gauge and Higgs couplings Spectral action Effective string action TABLE II. Spectral interpretation of drifted string excitations. 9 G. Remarks Drifted strings emerge inevitably once non–local spectral modes are allowed. They provide a bridge between particle physics and string theory without introducing new fundamental postulates. IV. SPECTRAL UNIFICATION OF PARTICLES AND STRINGS We now present the unifying principle underlying the coexistence of particle and string degrees of freedom in the drifted spectral framework. The central claim is that particles and strings are not distinct fundamental entities, but rather different regimes of excitation of a single spectral operator. A. Unified Spectral Spectrum Let Ds denote the drifted Dirac operator associated with an almost–commutative geometry. Its spectrum decomposes naturally into: • localised modes, sharply peaked in spacetime and internal space, corresponding to particle states; • extended modes, delocalised along one effective direction generated by drift, corresponding to string–like excitations. Both sectors are eigenstates of the same operator and are governed by the same spectral action. B. Energy Regimes and Crossover The distinction between particle and string behaviour is controlled by energy and scale. At energies well below the drift scale, only localised modes contribute, and the effective theory reduces to the Standard Model. As energy increases, extended spectral modes become dynamically accessible, and string–like behaviour emerges. The crossover between these regimes is governed by the drift parameter and its stabilised value s∗, which sets the characteristic scale separating particle and string physics. 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. Pinho-da-Cruz, DRSN VIII: Quantum Drift Geometry, September 2025. Version 2.0, doi: 10.5281/zenodo.17997240. [4] J. Pinho-da-Cruz, DRSN III: Supersymmetric Drift Geometry and the Spectral Master Operator, September 2025. Version 2.0, doi: 10.5281/zenodo.17986205. [5] J. Pinho-da-Cruz, DRSN IV: Drifted M-Theory Spectral Geometry, September 2025. Version 2.0, doi: 10.5281/zenodo.17987598. [6] J. Pinho-da-Cruz, DRSN V: Drifted Compactifications and Spectral Moduli Dynamics, September 2025. Version 2.0, doi: 10.5281/zenodo.17990670. [7] J. Pinho-da-Cruz, DRSN VI: Drifted Brane Geometry and Spectral Worldvolume Dynamics, September 2025. Version 2.0, doi: 10.5281/zenodo.17992350. [8] J. Pinho-da-Cruz, DRSN VII: Drifted Holographic Geometry and Spectral Gauge–Gravity Duality, September 2025. Version 2.0, doi: 10.5281/zenodo.17992722. [9] A. Connes, Noncommutative Geometry, Academic Press, San Diego (1994). [10] A. H. Chamseddine and A. Connes, “The Spectral Action Principle,” Commun. Math. Phys. 186, 731–750 (1997). [11] A. H. Chamseddine and A. Connes, “The Uncanny Precision of the Spectral Action,” Commun. Math. Phys. 293, 867–897 (2010). [12] W. D. van Suijlekom, Noncommutative Geometry and Particle Physics, Springer, Dordrecht (2015). [13] J. Polchinski, String Theory, Vols. I–II, Cambridge University Press (1998).