DRSN V: Drifted Compactifications and Spectral Moduli Dynamics (De Rerum Spectrale Natura, Report V, Version 1.0)

DSRN V: DRIFTED COMPACTIFICATIONS AND SPECTRAL MODULI DYNAMICS De Rerum Spectrale Natura series REPORT V (Version 1.0) GDs J. Pinho-da-Cruz Department of Mechanical Engineering University of Aveiro September 2025 •Compactification arises from spectral factorisation of a drifted Dirac operator, without geometric truncation. •Internal moduli appear as zero–order spectral sectors generated by drifted Laplace–type operators. •Drift produces a universal quartic spectral potential governing moduli dynamics and stability. •G2,SU(3) and Spin(7) compactifications are treated within a unified drifted spectral framework. •Effective four–dimensional gravity and scalar dynamics emerge directly from the reduced spectral action. Spectral Compactifications from Eleven Dimensions: Drift, Moduli and Effective Four-Dimensional Physics J. Pinho-da-Cruz 1, ∗ 1 Department of Mechanical Engineering, University of Aveiro, Portugal We develop a fully spectral framework for compactifications from eleven to four dimensions based on drifted Dirac operators. Starting from an eleven–dimensional drifted spectral geometry, we show that dimensional reduction, moduli dynamics, and effective four–dimensional physics emerge from the spectral action without the introduction of auxiliary fields or phenomenological truncation ansätze. The drift deformation induces a universal quartic potential, providing a spectral mechanism for moduli stabilisation, supersymmetry breaking patterns, and cosmological dynamics. We analyse compactifications based on G2 , SU (3)–structure, and Spin (7) geometries, and derive the corresponding effective four–dimensional theories in a unified operatorial language. The resulting framework offers a conceptually economical and analytically controlled route from eleven–dimensional geometry to four–dimensional physics, preparing the ground for brane dynamics, holography, and phenomenological applications within the DSRN programme. Keywords: Spectral Action; Drift Geometry; M–theory Compactifications; Special Holonomy; Moduli Stabilisation; Noncommutative Geometry; Effective Four–Dimensional Physics; Cosmology. CONTENTS I. Introduction 4 II. Drifted Compactification Framework 5 A. Geometric Setup and Spectral Decomposition 5 B. Heat Kernel Factorisation and Mode Expansion 5 C. Effective Four–Dimensional Spectral Action 6 D. Spectral Origin of Moduli and Stabilisation 6 E. Remarks on Consistency and Scope 7 III. Drifted G2Compactifications 7 A. G2Geometry and Spectral Data 7 ∗jp[email protected] 3 B. Zero Modes, Supersymmetry, and Drift 8 C. Effective Potential and Moduli Stabilisation 8 D. Remarks and Comparison 8 IV. Drifted SU(3)-Structure Compactifications 9 A. SU(3)-Structure Geometry and Spectral Data 9 B. Half-Flat Structures, Torsion, and Drift 9 C. Effective Four–Dimensional Theory 10 D. Remarks and Relation to Mirror Symmetry 10 V. Drifted Spin(7) Compactifications 11 A. Spin(7) Geometry and Spectral Encoding 11 B. Supersymmetry Breaking and Zero Modes 11 C. Effective Potential and Stability 12 D. Remarks and Comparison 12 VI. Effective Four–Dimensional Cosmology from Drifted Spectral Geometry 13 A. Spectral Reduction to Homogeneous Cosmology 13 B. Friedmann Equations and Drift Dynamics 13 C. Inflationary and Dark–Energy Regimes 14 D. Spectral Interpretation of Vacuum Energy 14 E. Remarks and Outlook 14 VII. Conclusions and Outlook 15 A. Summary of Results 15 B. Comparison of Compactification Classes 15 C. Conceptual Implications 16 D. Relation to the DSRN Programme 16 E. Outlook 17 A. Spectral Decomposition under Compactification 17 B. Reduction of the Heat Kernel and Effective Action 17 C. Semigroup and Resolvent Control 18 4 References 18 I. INTRODUCTION Compactification has traditionally been treated as a separate dynamical mechanism, implemented through dimensional truncations, flux backgrounds, or effective field–theoretic assumptions. While phenomenologically successful, such approaches often obscure the geometric origin of low–energy degrees of freedom and require substantial model–dependent input. Within spectral geometry, by contrast, physical information is encoded directly in the spectrum of a Dirac–type operator, suggesting that compactification itself should admit a purely spectral formulation. In the preceding reports of the DSRN series, we introduced drifted Dirac operators as controlled similarity deformations preserving spectral and analytic structure, and showed that gravitational, matter, and cosmological sectors arise from the spectral action. In particular, Report IV established an eleven–dimensional drifted spectral framework capturing the bosonic sector of M–theory in operatorial form. The present report builds directly on that result, addressing the question of how four–dimensional physics emerges spectrally from eleven dimensions. The central idea pursued here is that compactification is not an additional dynamical ingredient, but a manifestation of spectral factorisation. By analysing the heat kernel expansion of drifted Dirac operators on product geometries, we show that internal geometry, moduli fields, and effective potentials are encoded in the Seeley–DeWitt coefficients of a single operator. This perspective eliminates the need for ad hoc truncations and provides a unified treatment of different classes of special holonomy. We systematically study drifted compactifications based on G2 , SU (3)–structure, and Spin (7) geometries, highlighting both their distinctive features and their shared universal drift sector. A key outcome is the appearance of a universal quartic drift potential, which stabilises moduli and governs cosmological dynamics independently of the detailed topology of the internal space. As a result, inflationary and dark–energy–like regimes arise as calculable consequences of spectral geometry. The structure of this report is as follows. Section 2 introduces the general drifted compactification framework and the spectral reduction mechanism. Sections 3–5 analyse compactifications based on G2 , SU (3), and Spin (7) geometries, respectively. Section 6 derives the effective four–dimensional cosmology. Section 7 summarises the results and situates them within the broader DSRN programme. 5 Spectral decomposition of the drifted Dirac operator D(11) s→D(4) s⊕D(7) s FIG. 1. Spectral splitting of the drifted Dirac operator under compactification M11 →M4×X7. II. DRIFTED COMPACTIFICATION FRAMEWORK A. Geometric Setup and Spectral Decomposition We consider an eleven–dimensional drifted spectral geometry ( A11,H11, D(11) s )as established in DSRN Report IV, and assume a product (or warped product) decomposition of the underlying manifold, M11 ≃M4×X7.(II.1) At the spectral level, this induces a canonical decomposition of the Hilbert space, H11 ≃ H4⊗H7,(II.2) and of the drifted Dirac operator, D(11) s=D(4) s⊗ 1 + Γ4⊗D(7) s,(II.3) where Γ4denotes the four–dimensional chirality operator. The drift deformation acts by conjugation, D(11) s=esΦD(11)e−sΦ,(II.4) with Φa smooth scalar function on M11, allowing a decomposition Φ(x, y) = ϕ4(x)+ϕ7(y),(II.5) up to zero–order corrections. B. Heat Kernel Factorisation and Mode Expansion Since the drift deformation preserves the principal symbol and ellipticity, the square of the operator in (II.3) factorises up to bounded zero–order terms, (D(11) s)2= (D(4) s)2⊗ 1 + 1 ⊗(D(7) s)2+O0.(II.6) 6 Consequently, the heat kernel admits an asymptotic factorisation, TrH11 e−t(D(11) s)2≃TrH4e−t(D(4) s)2TrH7e−t(D(7) s)2,(II.7) up to exponentially suppressed corrections, in the sense of standard heat kernel theory [3,4]. Let {λ(7) n}be the eigenvalues of (D(7) s)2. The spectral trace can then be written as TrH11 e−t(D(11) s)2=X n e−tλ(7) nTrH4e−t(D(4) s)2,(II.8) making explicit the separation between zero modes ( λ(7) n = 0) and massive Kaluza–Klein contributions. C. Effective Four–Dimensional Spectral Action The spectral action principle yields S11 = Tr f (D(11) s)2 Λ2!!,(II.9) with fa smooth cutoff function and Λthe spectral scale [1,2]. Using the heat kernel expansion and integrating over X7 , one obtains an effective four–dimensional action of the form S(4) eff =ZM4 √−g4M2 P 2R4−1 2KIJ (φ)∂µφI∂µφJ−Veff(s, φ)+··· .(II.10) Crucially, the drift parameter sgenerates a universal contribution Veff(s)=α s2+β s4, β > 0,(II.11) independent of the detailed topology of X7 , in direct analogy with the lower–dimensional analysis of DSRN Report I. D. Spectral Origin of Moduli and Stabilisation The moduli fields φIarise spectrally as parameters controlling •the spectrum of D(7) s, •the internal curvature and flux contributions, •the zero–order endomorphism terms induced by the drift. Their dynamics is therefore fully encoded in the Seeley–DeWitt coefficients of ( D(11) s ) 2 , without introducing additional ad hoc scalar sectors [6]. 7 4D Term 11D Origin Drift Dependence Interpretation R4a(11) 2none Gravity ∂φ ∂φ a(11) 4indirect Moduli kinetics s2C2 1quadratic Mass term s4(C2 1)2quartic Stabilisation TABLE I. Spectral origin of the main four–dimensional terms after compactification. E. Remarks on Consistency and Scope This framework is: •self–contained: all ingredients are spectral and operatorial, •background–independent: no specific choice of X7is required at this stage, •stable under drift: ellipticity and spectral properties are preserved, •non–redundant: no duplication of the 11D field content is introduced. The present section establishes the universal backbone of drifted compactifications. Specific internal geometries and phenomenological regimes will be analysed in the following sections. III. DRIFTED G2COMPACTIFICATIONS A. G2Geometry and Spectral Data We specialise the internal space to a seven–dimensional manifold X7 endowed with a G2 –structure, characterised by a stable three–form φ and its Hodge dual ψ = ⋆7φ . In the torsion–free case, dφ = dψ = 0, the Levi–Civita connection has holonomy contained in G2 , yielding a single covariantly constant spinor. For drifted compactifications, the internal Dirac operator is deformed as D(7) s=esϕ7D(7)e−sϕ7,(III.1) where ϕ7∈C∞ ( X7 ). This preserves the principal symbol and ellipticity, while inducing zero–order endomorphism terms encoding torsion–like contributions at the spectral level. 8 G2spectral data under drift φ, ψ −→ D(7) drift −−−−→ D(7) s FIG. 2. G2geometry encoded spectrally and its drift deformation. Mode Spectral Origin Drift Effect 4D Role G2zero spinor ker D(7) none N= 1 SUSY Metric moduli a(7) 4quadratic Scalar fields Drift scalar sBCH tower quartic Stabilisation KK modes λ(7) nshifted Massive tower TABLE II. G2modes and their spectral fate under drift. B. Zero Modes, Supersymmetry, and Drift In the absence of drift, the kernel of D(7) is one–dimensional, corresponding to the unique covariantly constant spinor. Under drift, the kernel is preserved as a vector space, but the associated zero mode acquires an effective four–dimensional profile through the factorisation Ψ(x, y) = ψ4(x)⊗e−sϕ7(y)η(y),(III.2) where ηis the G2–invariant spinor. Supersymmetry in four dimensions is therefore preserved at the level of zero modes, while massive Kaluza–Klein excitations are shifted by drift–dependent terms. This realises a controlled partial lifting of degeneracies without breaking the underlying G2structure. C. Effective Potential and Moduli Stabilisation The spectral action induces a four–dimensional effective potential of the schematic form Veff(s, φG2)=α s2+β s4+X I γI(φG2)s2+··· ,(III.3) where φG2 denotes geometric moduli of the G2 structure. The universal quartic term stabilises the drift direction, while the mixed terms provide masses for a subset of G2moduli. D. Remarks and Comparison Drifted G2compactifications: •preserve N= 1 supersymmetry at low energies, 9 SU(3)-structure spectral reduction X7≃X6×S1−→ (J, Ω) −→ D(6) s FIG. 3. Spectral encoding of SU(3)–structure data and drift deformation. •stabilise the universal drift direction without fluxes, •induce moduli masses spectrally, avoiding ad hoc superpotentials, •remain compatible with the eleven–dimensional spectral framework. These features distinguish the present construction from standard flux–based G2 compactifications, while remaining fully geometric and operatorial [1,2,6]. IV. DRIFTED SU(3)-STRUCTURE COMPACTIFICATIONS A. SU(3)-Structure Geometry and Spectral Data We now consider internal geometries admitting an SU (3)–structure, realised either as X7≃ X6×S1 or, more generally, as seven–manifolds endowed with a globally defined SU (3)–invariant spinor. The structure is characterised by a real two–form J and a complex three–form Ωon X6 , satisfying the standard compatibility conditions. At the spectral level, the internal Dirac operator decomposes as D(7) ≃D(6) +γ7∂y,(IV.1) where y denotes the coordinate along S1 (or an effective circle direction). The drift deformation acts by D(7) s=esϕ7D(7)e−sϕ7,(IV.2) with ϕ7=ϕ6(x)+ϕS1(y)up to bounded zero–order corrections. B. Half-Flat Structures, Torsion, and Drift Unlike torsion–free Calabi–Yau compactifications, generic SU (3)–structures admit intrinsic torsion, organised into five torsion classes Wi . In the present framework, these torsion contributions appear as zero–order endomorphisms in (D(7) s)2, while preserving ellipticity and spectral control. 16 a flexible yet spectrally controlled setting. Spin (7) compactifications yield rigid geometries with intrinsic supersymmetry breaking. Despite their differences, all three cases share the same universal drift sector, whose quartic potential governs stability and cosmological behaviour. This universality is a direct consequence of the operatorial nature of the drift deformation and is independent of the detailed topology of the internal space. C. Conceptual Implications From a conceptual standpoint, the results of this report support the view that compactification is not a separate dynamical mechanism but an intrinsic aspect of spectral geometry. The distinction between geometry, matter, and cosmology becomes blurred: all are encoded in the spectrum of a single operator and its controlled deformations. In particular, the drift parameter plays a dual role: it acts both as a geometric deformation at high energies and as an effective scalar degree of freedom at low energies. This provides a natural bridge between ultraviolet spectral data and infrared physics, without invoking additional effective field theory assumptions. D. Relation to the DSRN Programme Within the broader DSRN programme, the present report occupies a pivotal position. It translates the eleven–dimensional spectral framework of Report IV into concrete four–dimensional physics, thereby closing the dimensional reduction sector of the theory. The results obtained here serve as direct input for brane dynamics, holography, and phenomenological applications. In particular: •Report VI will extend the spectral formalism to branes and worldvolume dynamics, •Report VII will address holographic and gauge–gravity correspondences, •subsequent reports will explore quantum corrections and particle phenomenology. The compactification framework developed here provides the common ground for all these extensions. 17 E. Outlook Several directions merit further investigation. First, a detailed analysis of perturbations around the spectral vacuum may yield testable cosmological signatures. Second, the interplay between drifted compactifications and brane embeddings is expected to shed light on non–perturbative sectors of the theory. Finally, the operatorial nature of the construction suggests that quantum corrections should be incorporated at the spectral level, rather than through ad hoc loop expansions. This points towards a genuinely non–perturbative formulation of quantum gravity within the drifted spectral paradigm. In conclusion, drifted spectral compactifications provide a unified, analytically controlled, and conceptually economical route from eleven–dimensional geometry to four–dimensional physics, completing the objectives of the present report. [1–3,5,6] Appendix A: Spectral Decomposition under Compactification We collect here the technical results underlying the spectral factorisation used throughout the report. Let D(11) s be a drifted Dirac operator on a product geometry M11 = M4×X7 . Under the assumptions stated in Section 2, the operator decomposes as D(11) s=D(4) s⊗ 1 + Γ4⊗D(7) s+O0,(A.1) where O0denotes bounded zero–order endomorphisms. Squaring the operator yields a Laplace–type operator, (D(11) s)2= (D(4) s)2⊗ 1 + 1 ⊗(D(7) s)2+O0,(A.2) ensuring that standard heat kernel techniques apply. The validity of the asymptotic expansion follows from general results on elliptic operators with bounded perturbations [3,4]. Appendix B: Reduction of the Heat Kernel and Effective Action Let K11 ( t )denote the heat kernel of ( D(11) s ) 2 . Up to exponentially suppressed terms, one has the factorisation TrK11(t)≃TrK4(t) TrK7(t),(B.1) where K4 and K7 are the heat kernels of ( D(4) s ) 2 and ( D(7) s ) 2 , respectively. This factorisation justifies the separation of four–dimensional and internal contributions in the spectral action. 18 Integrating over the internal manifold X7yields the effective four–dimensional spectral action, S(4) eff =X n f4−nΛ4−nZM4 a(4) n(x)√−g4d4x, (B.2) where the coefficients a(4) n depend parametrically on the internal spectral data. This is the precise origin of moduli fields and drift–induced potentials in the effective theory. Appendix C: Semigroup and Resolvent Control The drift deformation is implemented by similarity transformations of Dirac–type operators. As reviewed in the methodological appendix of Report IV, such deformations preserve self–adjointness and spectral properties. The resolvent admits the Laplace representation (λI −iDs)−1=Z∞ 0 e−λteitDsdt, ℜλ > 0,(C.1) which is valid under the standard hypotheses of semigroup theory [5]. This representation underlies the control of higher–order spectral corrections and ensures that all drift–induced terms remain bounded and well defined. No additional analytic assumptions are required beyond those already imposed in the eleven–dimensional framework. [1] A. Connes, Noncommutative Geometry, Academic Press, San Diego (1994). [2] A. H. Chamseddine and A. Connes, “The Spectral Action Principle”, Communications in Mathematical Physics 186, 731–750 (1997). [3] E. B. Davies, Heat Kernels and Spectral Theory, Cambridge University Press, Cambridge (1989). [4] P. B. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah–Singer Index Theorem, 2nd ed., CRC Press, Boca Raton (1995). [5] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York (1983). [6] W. D. van Suijlekom, Noncommutative Geometry and Particle Physics, Springer, Dordrecht (2015).