DRSN IX: Spectral Standard Model and Drifted Strings (De Rerum Spectrale Natura, Report IX, Version 1.0)

DSRN IX: SPECTRAL STANDARD MODEL AND DRIFTED STRINGS De Rerum Spectrale Natura series REPORT IX (Version 1.0) GDs J. Pinho-da-Cruz Department of Mechanical Engineering University of Aveiro September 2025 •Standard Model as a spectral fixed point. •Yukawas and Higgs from drifted internal Dirac operators. •Gauge couplings unified spectrally. •Strings as extended spectral excitations. •Matter, forces and strings from a single spectral action. 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 (Dated: December 13, 2025) 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. CONTENTS I. Introduction 3 II. Spectral Standard Model under Drift 4 A. Almost–Commutative Geometry and the Standard Model 4 B. Drifted Internal Dirac Operator 4 C. Spectral Fixed Point and Low–Energy Physics 4 D. Structural Summary 4 III. Drifted Strings as Extended Spectral Excitations 4 A. From Pointlike to Extended Spectral Modes 5 B. Worldsheet Emergence from Spectral Flow 5 C. Effective String Tension 5 D. Coupling to Gauge and Higgs Sectors 6 E. Relation to Fundamental Strings 6 F. Structural Summary 6 G. Remarks 6 IV. Spectral Unification of Particles and Strings 7 A. Unified Spectral Spectrum 7 3 B. Energy Regimes and Crossover 7 C. Unified Effective Action 7 D. Gauge and Gravitational Universality 8 E. Relation to Dualities 8 F. Structural Summary 8 G. Implications 8 V. Phenomenological Consequences and Predictions 8 A. Running Couplings and Unification 9 B. Higgs Sector and Vacuum Stability 9 C. Particle–String Transition Scale 9 D. Signatures of Drifted Strings 9 E. Cosmological Implications 10 F. Structural Summary 10 G. Remarks 10 VI. Conclusions, Outlook, and Final Remarks 10 A. Summary of Results 10 B. Conceptual Implications 11 C. Outlook 11 A. Almost–Commutative Geometry and Internal Spectral Data 11 B. Spectral Origin of Extended Excitations 11 References 12 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. Second, we extend the framework to incorporate 4 string–like excitations as non–local spectral modes generated by drift, thereby connecting particle physics and string theory within a unified spectral action. 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 [ 1 , 2 , 4 ]. 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. 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. 5 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. 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 τdenotes physical time, parametrises an effective two–dimensional surface swept by the excitation. 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 ∼∂2λn(s) ∂s2    s=s∗ ,(III.1) where λn ( s )are eigenvalues of Ds . 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. 6 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. However, they differ conceptually in that they are emergent, operatorial objects rather than fundamental degrees of freedom. In appropriate limits, the spectral string sector may reproduce effective actions resembling those of perturbative string theory. 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. 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. In this sense, strings appear as collective excitations of the same spectral geometry that gives rise to the Standard Model. 7 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. C. Unified Effective Action The full spectral action admits a decomposition of the schematic form Sspectral =SSM +Sstring +Smix,(IV.1) where: •SSM reproduces the Standard Model action; •Sstring governs the dynamics of extended spectral modes; •Smix encodes interactions between particle and string sectors. All three terms arise from the same trace Tr(f(Ds/Λ)) and are therefore not independent ingredients. 8 D. Gauge and Gravitational Universality A crucial feature of the unified framework is universality of couplings. Since both particle and string excitations originate from the same Dirac operator, they couple universally to gravity and gauge fields. This provides a natural explanation for the universality of gravitational coupling and the absence of arbitrary relative normalisations between particle and string sectors. E. Relation to Dualities The spectral unification framework accommodates familiar dualities of string theory in a new light. Dual descriptions correspond to different spectral decompositions of the same operator. In particular, particle–string duality appears as a reorganisation of spectral modes rather than a change of fundamental degrees of freedom. F. Structural Summary Spectral Regime Physical Interpretation Localised modes Particles (Standard Model) Extended modes Strings Drift scale s∗Particle–string crossover Single Dirac operator Unified dynamics Spectral action Universal effective action TABLE III. Spectral unification of particles and strings. G. Implications The spectral unification of particles and strings suggests that string theory may be viewed as an emergent sector of a more fundamental spectral geometry. Conversely, particle physics appears as the low–energy, localised limit of the same operatorial structure. This perspective provides a conceptually economical and mathematically controlled route to unification. V. PHENOMENOLOGICAL CONSEQUENCES AND PREDICTIONS The spectral unification of particles and strings yields concrete phenomenological implications. Because both sectors originate from the same drifted Dirac operator, low–energy physics, high–energy excitations, 9 and possible string signatures are tightly correlated. In this section we outline robust consequences and testable predictions of the framework. A. Running Couplings and Unification In the spectral Standard Model, gauge couplings are determined by spectral coefficients of the internal Dirac operator. Under drift, these coefficients undergo controlled spectral flow, leading to modified renormalisation group trajectories. A generic prediction is a mild deformation of standard unification patterns, with threshold corrections induced by the onset of extended spectral modes. B. Higgs Sector and Vacuum Stability The Higgs potential arises spectrally from inner fluctuations and is therefore sensitive to drift corrections. Quantum drift effects stabilise the Higgs vacuum through correlated shifts in quartic couplings. This mechanism alleviates metastability issues of the Standard Model Higgs sector without introducing new scalar fields. C. Particle–String Transition Scale The drift condensate value s∗ defines a characteristic scale separating particle–like and string–like regimes. Below this scale, physics is well described by the Standard Model. Above it, extended spectral excitations contribute, leading to deviations from pointlike behaviour in scattering amplitudes. This predicts a smooth crossover rather than an abrupt transition to string physics. D. Signatures of Drifted Strings Drifted strings couple universally to gauge and gravitational sectors. Observable consequences may include: •softening of high–energy scattering amplitudes, •deviations from point–particle form factors, •additional Regge–like trajectories emerging from spectral modes, •correlated effects across different interaction channels. Unlike conventional string theory, these effects are controlled by a single spectral parameter and are therefore highly constrained.