DRSN III: Supersymmetric Drift Geometry and the Spectral Master Operator (De Rerum Spectrale Natura, Report III, Version 1.0)

DRSN III: SUPERSYMMETRIC DRIFT GEOMETRY AND THE SPECTRAL MASTER OPERATOR De Rerum Spectrale Natura series REPORT III (Version 1.0) GDs J. Pinho-da-Cruz Department of Mechanical Engineering University of Aveiro September 2005 • Supersymmetric drift generates a controlled deformation Qα,s = e−sxQαesx preserving the worldsheet SUSY algebra and BRST cohomology. • The drifted supersymmetric worldsheet operator and the drifted ten-dimensional Dirac operator combine into the supersymmetric spectral master operator D(susy) s , which admits full analytic control including domain stability, self-adjointness, holomorphicity, and principal symbol invariance. • Exact factorisation ( D(susy) s ) 2 = H(susy) s⊗ 1+1 ⊗ ( D(10) s ) 2 yields heat-kernel factorisation and a convolution formula for Seeley–DeWitt coefficients. • Supersymmetric drift encodes internal fluxes, torsion and non-geometry through a supersymmetric BCH tower compatible with supergravity. • Worldsheet supersymmetric conformal invariance, target-space spectral field equations and unified drift stationarity are equivalent to the operator equation [(D(susy) s)2, X] = 0. Supersymmetric Drift Geometry and the Spectral Master Operator J. Pinho-da-Cruz 1, ∗ 1 Department of Mechanical Engineering, University of Aveiro, Portugal (Dated: December 11, 2025) The drift deformation of Dirac operators introduced in the first report of this series (DSRN I, DOI: 10.5281/zenodo.17873909) and extended to worldsheet–target coupling in DSRN II (DOI: 10.5281/zenodo.17887608) provides a robust analytic framework for modifying lowerorder geometric terms of Dirac-type operators while preserving domain, principal symbol and spectrum. In this third report we construct the supersymmetric drift geometry, introducing a drift deformation of worldsheet supercharges and coupling it to a drifted ten-dimensional supersymmetric Dirac operator. A bounded, self-adjoint generator X=x⊗1+1⊗φ, acting simultaneously on the worldsheet and target Hilbert spaces, produces the supersymmetric spectral master operator D(susy) s=D(susy) ws,s ⊗1 + Γws ⊗D(10) s. We prove that this operator satisfies domain stability, self-adjointness, principal symbol invariance and holomorphicity, and we establish an exact factorisation formula for its square, leading to heat-kernel and Seeley–DeWitt coefficient factorisation. We further construct the supersymmetric unified Spectral Action and show that supersymmetric conformal invariance on the worldsheet, ten-dimensional spectral field equations and drift-stationarity of the master operator are all equivalent to the single operator identity [(D(susy) s)2, X]=0. This establishes a unified operator-theoretic foundation for supersymmetric drift geometry and prepares the ground for the drifted eleven-dimensional geometry developed in Report IV. Keywords: Spectral Action; Drift Geometry; Supersymmetric Dirac Operators; Worldsheet Supercharges; BRST–SUSY Cohomology; Heat Kernel Factorisation; Fluxes and Torsion; BCH Expansion; Supergravity; M-Theory. ∗jp[email protected] 3 CONTENTS I. Introduction 4 II. Supersymmetric Drift Geometry on the Worldsheet 6 A. Supersymmetric Algebra and BRST Framework 6 B. Supersymmetric Drifted Charges 6 C. Drifted BRST Operator 7 D. Drifted Supersymmetric Worldsheet Dirac Operator 7 E. Spectral Renormalisation Group Flow 7 F. Supersymmetric BCH Expansion 7 G. Diagram of Worldsheet Drift Flow 8 H. Summary Table 9 III. Supersymmetric Drift Geometry in Ten Dimensions 9 A. Diagram of the 10D Drift 9 B. Summary Table 9 IV. The Supersymmetric Spectral Master Operator 9 V. Heat-Kernel Factorisation and the Supersymmetric Unified Spectral Action 11 A. Exact Factorisation of the Square 11 B. Heat-Kernel Factorisation 12 C. Diagram of Supersymmetric Heat-Kernel Flow 12 D. Seeley–DeWitt Convolution Coefficients 12 E. Summary Table 13 F. The Supersymmetric Unified Spectral Action 13 VI. The Supersymmetric Drift–Conformal–Spectral Equivalence Theorem 13 A. Worldsheet Supersymmetric Beta Functions 13 B. Target-Space Spectral Stationarity 14 C. Unified Drift Stationarity 14 D. Diagrammatic Summary 14 E. Interpretation 14 VII. Examples, Diagrams and Physical Illustrations 15 4 A. Supersymmetric Drift Vacua 15 B. Supersymmetric Moduli Flow 16 C. Non-Geometric SUSY Flux Layers 16 D. Spectral Reorganisation Under Drift 16 E. Interpretation and Connection to Report IV 17 VIII. Conclusions and Outlook 18 Appendices 19 Appendices 19 A. Operator-Theoretic Foundations of Supersymmetric Drift 19 1. Bounded Conjugation and Domain Stability 19 2. Holomorphic Families of Type (A) 19 3. Drift Flow Identity 20 4. Tensor Product Operators 20 B. Supersymmetric Drifted Lichnerowicz Formula 20 1. Drift Expansion 20 C. Heat-Kernel Factorisation: Proofs 21 D. Supersymmetric BCH Algebra and Non-Geometric Layers 21 References 21 I. INTRODUCTION The drift deformation of Dirac operators, formulated in DSRN Report I [ 1 ] (DOI: 10.5281/zenodo.17873909), established a method for modifying the lower-order structure of a Dirac-type operator through bounded conjugation while preserving its fundamental analytic properties. This mechanism generated a universal quartic drift potential and provided an operator-theoretic viewpoint on geometric deformations compatible with the spectral action framework. DSRN Report II [ 2 ] (DOI: 10.5281/zenodo.17887608) extended the formalism to the worldsheet of string theory by applying drift to the BRST operator and coupling it to the drifted target-space Dirac operator. A unified non-supersymmetric master operator was constructed, and an exact 5 factorisation of its square led to a convolution formula for Seeley–DeWitt coefficients. Most notably, worldsheet conformal invariance, target spectral field equations and unified drift-stationarity were shown to be equivalent to a single operator condition. The present report introduces and develops the supersymmetric extension of this drift geometry. Supersymmetry complicates the drift mechanism in several ways: supercharges must remain nilpotent and preserve their algebra; the supersymmetric BRST operator must drift without altering cohomology; and the drifted target-space operator must maintain compatibility with ten-dimensional supergravity structures, including fluxes, torsion and the Clifford algebra. To address these challenges, we construct: i) a drift deformation of worldsheet supersymmetry, Qα,s =e−sxQαesx, preserving the full SUSY algebra; ii) a drift deformation of the ten-dimensional supersymmetric Dirac operator, D(10) s=esφD(10)e−sφ, compatible with supergravity torsion and fluxes; iii) a supersymmetric spectral master operator, D(susy) s=D(susy) ws,s ⊗1 + Γws ⊗D(10) s, unifying both deformations under a single bounded generator. We prove that this structure satisfies domain invariance, self-adjointness, symbol invariance, holomorphicity and exact heat-kernel factorisation. We then construct the supersymmetric unified Spectral Action and derive the equations of motion. A spectral equivalence theorem is established, identifying supersymmetric worldsheet conformality, target-space spectral dynamics and driftstationarity as manifestations of the same operator identity. The supersymmetric drift geometry developed here provides the mathematical infrastructure necessary for the drifted eleven-dimensional geometry (M-theory) presented in Report IV. This is Report III of the De Rerum Spectrale Natura collection. 6 II. SUPERSYMMETRIC DRIFT GEOMETRY ON THE WORLDSHEET Supersymmetry on the worldsheet is generated by a set of supercharges {Qα}N α=1 satisfying the standard supersymmetric algebra of two-dimensional superconformal field theory. For definiteness, we work within the RNS framework, although the following drift construction is representation–independent and applies equally to heterotic and type II worldsheet formalisms. The purpose of this section is to introduce a supersymmetric drift flow, Qα,s =e−sxQαesx, prove preservation of the worldsheet supersymmetry algebra, construct the drifted supersymmetric Dirac operator D(susy) ws,s =QBRST,s +Q† BRST,s, and establish its analytic properties. Figure 1provides a structural overview, while Table I summarises the main operators. A. Supersymmetric Algebra and BRST Framework Let Qα: Dom(Qα)⊂ Hws → Hws be worldsheet supercharges satisfying {Qα, Qβ}= 2(γµ)αβPµ,[Pµ, Qα] = 0, where Pµ generate worldsheet translations. Let QBRST denote the BRST operator, satisfying Q2 BRST = 0 and [QBRST, Qα] = 0. B. Supersymmetric Drifted Charges Let x∈ B(Hws)be bounded and self-adjoint. Define Qα,s := e−sxQαesx. Proposition 1. For every α, β and all s∈R, {Qα,s, Qβ,s}= 2(γµ)αβPµ. 7 Proof. Bounded similarity transformations preserve algebraic identities: {Qα,s, Qβ,s}=e−sx{Qα, Qβ}esx = 2(γµ)αβe−sxPµesx = 2(γµ)αβPµ. C. Drifted BRST Operator Define QBRST,s =e−sxQBRSTesx. Proposition 2. The drifted BRST operator satisfies: Q2 BRST,s = 0,[Qα,s, QBRST,s]=0. D. Drifted Supersymmetric Worldsheet Dirac Operator Define D(susy) ws := QBRST +Q† BRST, D(susy) ws,s := QBRST,s +Q† BRST,s. Proposition 3. If D(susy) ws is self-adjoint, then D(susy) ws,s is self-adjoint on Dom(D(susy) ws ). E. Spectral Renormalisation Group Flow Differentiation yields: ∂D(susy) ws,s ∂s = [D(susy) ws,s , x]. Proposition 4. {D(susy) ws,s } satisfies a worldsheet supersymmetric spectral renormalisation group equation. F. Supersymmetric BCH Expansion The BCH expansion reads: Qα,s =Qα+sC1,α +s2 2C2,α +s3 6C3,α +· · · , Cn,α = adn x(Qα). Its interpretations: 8 •C1,α: marginal SUSY deformations; •C2,α: torsion and NS–NS flux layers; •C3,α: non-geometric SUSY layers (Q-flux precursors). G. Diagram of Worldsheet Drift Flow Figure 1illustrates the drift flow for the SUSY and BRST operators. QαQα,s QBRST QBRST,s D(susy) ws,s e−sx(·)esx e−sx(·)esx FIG. 1. Supersymmetric drift flow on the worldsheet. 9 Operator Drifted Form Role QαQα,s Worldsheet supercharge QBRST QBRST,s Gauge-fixed SUSY BRST operator D(susy) ws D(susy) ws,s Drifted SUSY Dirac operator TABLE I. Supersymmetric drifted worldsheet structures. H. Summary Table III. SUPERSYMMETRIC DRIFT GEOMETRY IN TEN DIMENSIONS The ten-dimensional supersymmetric Dirac operator on a type II or heterotic supergravity background is D(10) =iΓM∇M+F, where Fencodes the NS–NS and RR flux terms. Let φ∈ B(Htarget)be bounded and self-adjoint. Define D(10) s:= esφD(10)e−sφ. Proposition 5. D(10) sis self-adjoint on Dom(D(10))and σ(D(10) s)=σ(D(10)). The drift flow satisfies ∂D(10) s ∂s = [D(10) s, φ]. The drifted Lichnerowicz identity becomes (D(10) s)2=−gMN ∇(s) M∇(s) N+E0+s∆φ+s2∥∇φ∥2+Fs. A. Diagram of the 10D Drift Figure 2illustrates the supersymmetric drift flow in ten dimensions. B. Summary Table IV. THE SUPERSYMMETRIC SPECTRAL MASTER OPERATOR Define the total Hilbert space Htot =Hws ⊗ Htarget. 16 s V(s) V(s)=αs2+βs4 β > 0 FIG. 6. Supersymmetric universal drift potential. B. Supersymmetric Moduli Flow Drift induces flows of moduli fields M: M(s)=M(0) + s δM+· · · . Figure 7depicts a drift-induced SUSY moduli flow. M(0) M(s) M(s)=M(0) + s δM FIG. 7. Supersymmetric drift-induced moduli flow. C. Non-Geometric SUSY Flux Layers The BCH tower for D(10) s, C(n)= adn φ(D(10)), produces geometric and non-geometric layers (Table V): C(1) →moduli shifts, C(2) →torsion & NS–NS flux, C(3) →non-geometry (Q-flux), . . . D. Spectral Reorganisation Under Drift At a SUSY drift vacuum s∗ , the drift reorganises the spectrum of D(10) s and of H(susy) s . Schematically, eigenvalues λn(0) flow to λn(s∗), shifting the spectral distribution. Figure 8provides a schematic representation of drift-induced eigenvalue shifts. 17 BCH layer Structure Physical Interpretation C(1) adφ(D(10))Linear SUSY deformation, moduli shifts C(2) ad2 φ(D(10))Torsion, NS–NS flux renormalisation C(3) ad3 φ(D(10))Non-geometric SUSY flux (Q-layer) C(n≥4) — Higher non-geometry (R-flux) TABLE V. Supersymmetric BCH layers and corresponding flux structures. n λn λn(0) λn(s∗) FIG. 8. Schematic reorganisation of the supersymmetric spectrum induced by drift at s = s∗ . Blue dots represent the undeformed eigenvalues λn(0) and red dots the drifted eigenvalues λn(s∗). E. Interpretation and Connection to Report IV These examples illustrate how supersymmetric drift geometry affects: •the behaviour of moduli, •the structure of SUSY fluxes, •the formation of supersymmetric condensates, •the spectral properties of worldsheet and target-space Dirac operators. In DSRN Report IV we will extend these constructions to eleven dimensions, showing how: 1. the supersymmetric drift lifts naturally to a drifted Dirac operator in eleven dimensions, 2. BCH layers generate the 3-form C3and 4-form G4of M-theory, 3. non-geometric fluxes are encoded in higher-order commutators of the 11D drift generator, 4. exact factorisation persists in the 2⊗10 ⊗11-dimensional master operator. 18 VIII. CONCLUSIONS AND OUTLOOK In this report we have constructed and analysed the supersymmetric extension of the drift geometry programme initiated in DSRN Reports I and II. Working simultaneously on the worldsheet and in ten-dimensional target space, we introduced supersymmetric drift flows acting on the worldsheet supercharges, on the BRST operator of the gauge-fixed superstring, and on the tendimensional supersymmetric Dirac operator of type II or heterotic supergravity. A central achievement of this report is the construction of the supersymmetric spectral master operator D(susy) s=D(susy) ws,s ⊗1 + Γws ⊗D(10) s, which unifies worldsheet and target-space supersymmetry under a single drift generator. We proved domain stability, self-adjointness, principal symbol invariance and holomorphicity of this operator, together with exact factorisation of its square, leading to product structure for heat kernels and a convolution rule for the Seeley–DeWitt coefficients. We established the Supersymmetric Drift–Conformal–Spectral Equivalence Theorem, showing that worldsheet supersymmetric conformal invariance, ten-dimensional supersymmetric spectral field equations and unified drift-stationarity are equivalent to the operator identity [(D(susy) s)2, X]=0. This identifies drift geometry as an operator-theoretic unification of worldsheet supersymmetry, target-space supergravity and spectral field equations. In Section VII we illustrated supersymmetric drift phenomena through examples including drift vacua, drift-induced moduli flow, non-geometric flux layers arising from the supersymmetric BCH tower, and drift-induced spectral reorganisation. Outlook. In the next instalment of this series (DSRN IV) we shall extend supersymmetric drift geometry to eleven dimensions. This will involve: •constructing the drifted eleven-dimensional Dirac operator D(11) s, •analysing the BCH tower generating the M-theoretic forms C3and G4, •establishing heat-kernel factorisation in the 2⊗10 ⊗11 setting, •deriving the eleven-dimensional unified Spectral Action, 19 •proving drift-stationarity as the M-theoretic analogue of supersymmetric field equations. Supersymmetric drift geometry therefore provides a coherent operator-theoretic framework from which worldsheet supersymmetry, ten-dimensional supergravity and eleven-dimensional M-theory structures emerge naturally. This is Report III of the De Rerum Spectrale Natura collection. APPENDICES Appendix A: Operator-Theoretic Foundations of Supersymmetric Drift This appendix collects the operator-theoretic tools used in the supersymmetric drift framework. The results generalise those of DSRN Reports I and II and are included here to make the present report self-contained. 1. Bounded Conjugation and Domain Stability Let A be a densely defined, closed operator on a Hilbert space H , and let B∈ B ( H )be bounded and self-adjoint. Define As:= e−sBAesB. Proposition 10. For every s∈R: a) Dom(As) = Dom(A); b) if Ais closed, then Asis closed; c) if Ais self-adjoint, then Asis self-adjoint on Dom(A); d) σ(As)=σ(A). 2. Holomorphic Families of Type (A) For As=A+sC with Cbounded, the drift family is holomorphic of type (A): s7→ Asψis analytic for all ψ∈Dom(A). 20 3. Drift Flow Identity Differentiating As=e−sBAesB yields ∂As ∂s = [As, B]. 4. Tensor Product Operators If A and B are self-adjoint on H1 and H2 , then A⊗ 1and 1 ⊗B are self-adjoint on Dom ( A ) ⊗ Dom(B)and commute strongly, enabling heat-kernel factorisation used throughout this report. Appendix B: Supersymmetric Drifted Lichnerowicz Formula The ten-dimensional supersymmetric Dirac operator is of the form D(10) =iΓM∇M+F, where Fdenotes flux contributions. The standard Lichnerowicz identity is (D(10))2=−gMN ∇M∇N+E0, with E0containing curvature, torsion and flux bilinears. 1. Drift Expansion Because [φ, [φ, D(10)]] = 0, the BCH expansion truncates: D(10) s=D(10) +s[D(10), φ]. Squaring gives (D(10) s)2= (D(10))2+s{D(10),[D(10), φ]}+s2[D(10), φ]2. Introduce the drifted spin connection: ∇(s) M:= ∇M+s ∂Mφ. Proposition 11. The supersymmetric drifted Lichnerowicz identity is (D(10) s)2=−gMN ∇(s) M∇(s) N+E0+s∆φ+s2∥∇φ∥2+Fs, where Fscontains drifted flux contributions from BCH layers C(n). 21 Appendix C: Heat-Kernel Factorisation: Proofs Let (D(susy) s)2=H(susy) s⊗1+1⊗(D(10) s)2. Since the two factors commute strongly, exponentiation yields: e−t(D(susy) s)2=e−tH(susy) s⊗e−t(D(10) s)2. Taking the trace gives: Tr(e−t(D(susy) s)2) = Trws(e−tH(susy) s)Trtarget(e−t(D(10) s)2). This establishes the factorisation used in Section V. Appendix D: Supersymmetric BCH Algebra and Non-Geometric Layers The BCH tower of the drifted operator is defined by C(n)= adn φ(D(10)). Its first layers encode: C(1) :moduli and linear SUSY deformations, C(2) :torsion and NS–NS flux corrections, C(3) :non-geometric SUSY layers (Q-flux). Higher layers (n≥4) represent non-geometric R-flux and higher-order operator corrections. [1] J. Pinho-da-Cruz, DSRN Report I: Drift Spectral Relativity and Noncommutative Geometry, September 2005. DOI: 10.5281/zenodo.17873909. [2] J. Pinho-da-Cruz, DSRN Report II: Drift Geometry and the Unified Spectral Operator of Superstring Theory, September 2005. DOI: 10.5281/zenodo.17887608. [3] A. Connes, Noncommutative Geometry, Academic Press, 1994. [4] A. H. Chamseddine and A. Connes, “The Spectral Action Principle”, Commun. Math. Phys. 186 (1997), 731–750. [5] E. B. Davies, Heat Kernels and Spectral Theory, Cambridge Tracts in Mathematics 92, Cambridge University Press, 1989. [6] M. Reed and B. Simon, Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness, Academic Press, 1975. 22 [7] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, 1983. [8] P. B. Gilkey, Invariance Theory, the Heat Equation and the Atiyah–Singer Index Theorem, CRC Press, 1995. [9] M. B. Green, J. H. Schwarz, and E. Witten, Superstring Theory, Vols. 1–2, Cambridge University Press, 1987. [10] J. Polchinski, String Theory, Vols. 1–2, Cambridge University Press, 1998. [11] V. Moretti, Spectral Theory and Quantum Mechanics, UNITEXT, Springer, 2013. [12] M. Lewin, Spectral Theory and Quantum Mechanics, Universitext, Springer, 2024.