DRSN II: Drift Geometry and the Unified Spectral Operator (De Rerum Spectrale Natura, Report II, Version 1.0)

DRSN II: DRIFT GEOMETRY AND THE UNIFIED SPECTRAL OPERATOR De Rerum Spectrale Natura series REPORT II (Version 1.0) GDs J. Pinho-da-Cruz Department of Mechanical Engineering University of Aveiro September 2025 • Worldsheet drift generates a controlled one-parameter deformation Qs = e−sx Qesx preserving nilpotency and BRST cohomology. • The drifted worldsheet operator and the drifted target-space Dirac operator combine into the unified master operator Ds. • The family Ds admits full analytic control: domain stability, self-adjointness, holomorphicity and iterated drift calculus. • Exact factorisation D2 s = Hs⊗ 1+1 ⊗D2 s yields heat-kernel factorisation and a convolution formula for Seeley–DeWitt coefficients. • The deformation parameter s encodes a spectral coupling between worldsheet and target geometry, generating a unified renormalisation flow. • Worldsheet conformal invariance, target-space spectral field equations and unified drift stationarity are equivalent to the operator equation [D2 s, X] = 0. Drift Geometry and the Unified Spectral Operator of Superstring Theory J. Pinho-da-Cruz 1, ∗ 1 Department of Mechanical Engineering, University of Aveiro, Portugal (Dated: December 10, 2025) The drift deformation of Dirac operators introduced in DSRN Report I [ 1 ] provides a method for modifying lower-order geometric terms while preserving domain, principal symbol and spectrum. In this report we extend drift geometry to the worldsheet of superstring theory by introducing a drift deformation of the BRST operator and coupling it to the drifted target-space Dirac operator through a single unified construction. We define a bounded, self-adjoint generator X=x⊗1+1⊗φ, whose conjugation flow produces the master drifted Dirac operator Ds=Dws,s ⊗1 + Γws ⊗Ds. We prove self-adjointness, principal symbol invariance and holomorphicity of the drifted family, as well as exact factorisation of the square and heat kernel. The resulting unified Spectral Action exhibits a convolution formula for Seeley–DeWitt coefficients and contains the universal quartic drift potential discovered in Report I. We further show that worldsheet conformal invariance (vanishing beta functions) is equivalent to stationarity of both the target-space Spectral Action and the unified Spectral Action, and to a single drift-stationarity operator equation [ D2 s, X ] = 0. This establishes an operator-theoretic bridge between BRST worldsheet geometry and spectral target-space dynamics. Keywords: Spectral Action; Drift Geometry; BRST Operator; Superstring Theory; Worldsheet Conformal Invariance; Heat Kernel Expansion; Noncommutative Geometry; Dirac Operator; Spectral Renormalisation Group. CONTENTS I. Introduction 4 II. Worldsheet Drift Geometry 6 A. BRST Framework 6 ∗jp[email protected] 3 B. Drifted BRST Operator 7 C. Nilpotency and Cohomology 7 D. Worldsheet Symmetry Preservation 7 E. Drifted Worldsheet Dirac Operator 7 F. Spectral Renormalisation Group Flow 8 G. BCH Expansion 8 III. Target–Space Drift Geometry (Review of Report I) 8 A. Dirac Operators and Bounded Multipliers 9 B. Drifted Dirac Operator 9 C. Drift Flow and Holomorphicity 9 D. Drifted Lichnerowicz Formula 9 E. Heat Kernel and Drift Potential 10 F. Role in the Unified Geometry 10 IV. The Unified Drift Generator and the Master Operator 11 A. Product Hilbert Space and Grading 11 B. Undeformed Composite Operator 11 C. Unified Drift Generator 11 D. Drifted Master Operator 11 E. Analytic Properties 12 V. Heat Kernel Factorisation and the Unified Spectral Action 12 A. Square Factorisation 12 B. Heat Kernel Factorisation 12 C. Asymptotics 13 D. Unified Spectral Action 13 VI. Beta Functions and Spectral Field Equations 13 A. Worldsheet Beta Functions 13 B. Target-Space Drift Stationarity 14 C. Unified Drift Equation 14 D. Spectral Action Variation 14 E. Main Equivalence Theorem 15 4 F. Interpretation 15 VII. Examples and Illustrations 15 A. Unified Drift Flow 15 B. Heat Kernel Factorisation 15 C. Universal Drift Potential 16 D. Structure of the Master Operator 16 E. Toy Model 16 VIII. Conclusions and Outlook 16 Appendices 17 Appendices 17 A. Operator-Theoretic Foundations of Drift Geometry 17 1. Bounded Conjugation and Domain Stability 18 2. Holomorphic Families of Type (A) 19 3. Drift Flow Identity 19 B. Drifted Lichnerowicz Formula 19 C. Heat Kernel Factorisation 20 D. BRST Drift Algebra 20 1. Nilpotency and Cohomology 20 2. Worldsheet Symmetries 20 3. Drifted Worldsheet Dirac Operator 21 References 21 I. INTRODUCTION The drift deformation of Dirac operators, introduced in DSRN Report I [ 1 ], provides a method for modifying lower-order geometric terms of a Dirac-type operator while preserving its domain, principal symbol and spectrum. The deformation is realised by bounded conjugation, Ds=esφDe−sφ, 5 where φ is a bounded, real multiplication operator. In Riemannian spin geometry and in the spectral action framework [ 2 – 5 ], this drift induces universal analytic structures, including a quartic potential V ( s ) = αs2 + βs4 with β > 0, arising from the Seeley–DeWitt coefficients of the drifted operator. The aim of this report is to extend drift geometry to the worldsheet of superstring theory and to construct a unified drifted operator that simultaneously incorporates: i) the drifted BRST charge and worldsheet Dirac structure, ii) the drifted target-space Dirac operator of spectral geometry, iii) an operator-theoretic coupling between worldsheet and target sectors. We introduce a bounded, self-adjoint operator X=x⊗1+1⊗φ, where x acts on the worldsheet Hilbert space and φ acts on the target-space Hilbert space. The unified drift of the composite Dirac operator D0=Dws ⊗1 + Γws ⊗D is then defined by Ds=e−sX D0esX, and yields the master drifted operator Ds=Dws,s ⊗1 + Γws ⊗Ds. The analytic properties of Ds (self-adjointness, symbol invariance, holomorphicity) follow from functional-analytic results in [6–8,13,14]. A key result is the exact factorisation D2 s=Hs⊗1+1⊗D2 s, where Hs = D2 ws,s , which leads to factorisation of the heat kernel and a convolution formula for the unified Seeley–DeWitt coefficients. Figures 1–4 and Tables 1–2 illustrate the drift flows, heat kernel decomposition, drift potential and unified operator structure. The central conceptual result of this report is the equivalence between worldsheet conformal invariance and target-space spectral field equations: βws(s)=0 ⇐⇒ δStarget δΨ= 0 ⇐⇒ δSunif δΨ= 0 ⇐⇒ [D2 s, X]=0. 6 Here Starget denotes the drifted target-space Spectral Action of Report I, and Sunif is the unified Spectral Action constructed in this work. This identifies drift-stationarity of Ds as the unified operator-theoretic condition encoding both vanishing worldsheet beta functions and target-space spectral dynamics. The rest of the report is organised as follows. Section II develops drift geometry for the worldsheet BRST and Dirac operators. Section III reviews the target-space Dirac drift of Report I. Section 4 introduces the unified drift generator and the master operator. Section 5 proves heat kernel factorisation and constructs the unified Spectral Action. Section 6 establishes the equivalence theorem between beta functions and spectral field equations. Section 7 gives illustrative examples and diagrams. Section 8 contains conclusions and outlook. Appendices A–D provide analytic foundations and supporting computations. II. WORLDSHEET DRIFT GEOMETRY The worldsheet BRST operator plays a central role in the gauge-fixed formulation of string theory [ 11 , 12 ]. In this section we introduce a drift deformation of the BRST operator and establish its analytic properties. These results form the worldsheet half of the unified drift geometry and are essential for constructing the master operator in Section 4. The worldsheet drift flow will later align with the target-space drift flow, as depicted in Figure 1 and summarised in Table 1. A. BRST Framework Let Q: Dom(Q)⊂ Hws → Hws be the BRST operator acting on the worldsheet Hilbert space Hws =Hmatter ⊗ Hghosts. We assume: •Qis closed and densely defined; •Q2= 0 (BRST nilpotency); •the BRST cohomology H•(Q) = ker(Q)/im(Q)defines physical states; •Qcommutes with the Virasoro or super-Virasoro generators. 7 B. Drifted BRST Operator Let x∈ B(Hws)be bounded and self-adjoint. Define the drifted BRST operator Qs:= e−sxQesx, s ∈R. By bounded similarity, Dom(Qs) = Dom(Q), Qsis closed. C. Nilpotency and Cohomology Since conjugation is an algebra automorphism, (Qs)2=e−sxQ2esx = 0. Proposition 1. The drifted BRST operator Qsis nilpotent for all s∈R. Moreover, Qsψ= 0 iff Q(esxψ) = 0, so: Proposition 2. H•(Qs)∼ =H•(Q). Thus drift does not change the physical-state space. D. Worldsheet Symmetry Preservation If [x, Ln]=0,[x, Gr]=0, then [Qs, Ln]=0,[Qs, Gr]=0. Proposition 3. Worldsheet conformal and supersymmetry algebras are preserved under drift. E. Drifted Worldsheet Dirac Operator Define the worldsheet Dirac-type operator Dws := Q+Q†. Its drift deformation is Dws,s := e−sxDwsesx =Qs+Q† s. 8 Proposition 4. If Dws is self-adjoint, then Dws,s is self-adjoint on Dom(Dws). This operator appears in the master operator Dsand is one of the entries in Table 1. F. Spectral Renormalisation Group Flow Differentiating, ∂Dws,s ∂s = [Dws,s, x]. Proposition 5. The drifted worldsheet Dirac operator satisfies the spectral renormalisation group equation ∂Dws,s ∂s = [Dws,s, x]. G. BCH Expansion The BCH expansion gives Qs=Q+sC1+s2 2C2+s3 6C3+··· , Cn= adn x(Q). The operators Cnmay be interpreted as: •C1: marginal or gauge/matter deformations of the worldsheet theory; •C2: torsion-like and flux-like corrections; •C3and higher: non-geometric contributions. These layers will align with target-space BCH corrections in Section III. III. TARGET–SPACE DRIFT GEOMETRY (REVIEW OF REPORT I) The drift geometry of Dirac operators introduced in DSRN Report I [ 1 ] is essential for constructing the unified operator of Section 4. This section provides a concise and self-contained review of those results, drawing on tools from spectral geometry [ 2 – 5 ], heat-kernel theory [ 6 , 9 ], and functional analysis [7,8,13,14]. 9 A. Dirac Operators and Bounded Multipliers Let D: Dom(D)⊂ Htarget → Htarget be the Dirac operator on a four-dimensional Riemannian spin manifold. Let φ∈ B ( Htarget )be a bounded, real multiplier. From [7], [D, φ]∈ B(Htarget), and the commutator is D-relatively bounded with relative bound zero. B. Drifted Dirac Operator Define Ds:= esφDe−sφ. Then: Dom(Ds) = Dom(D), Dsis self-adjoint, σ(Ds)=σ(D). Thus drift modifies only lower-order terms. C. Drift Flow and Holomorphicity Differentiating, ∂Ds ∂s = [Ds, φ]. By Kato theory [7]: {Ds}s∈Ris a holomorphic family of type (A). D. Drifted Lichnerowicz Formula The undeformed operator satisfies [9]: D2=−gµν∇µ∇ν+E0. 16 C. Universal Drift Potential From DSRN Report I [1], V(s)=αs2+βs4, β > 0. Figure 3 shows the typical quartic structure. The presence or absence of a condensate depends on the sign of α. D. Structure of the Master Operator The operator Ds=Dws,s ⊗1 + Γws ⊗Ds is diagrammed in Figure 4 and summarised in Table 1. E. Toy Model Let Dws =   0A A†0   , x =   x10 0x2   . Then [Hs, x]=0⇐⇒ x1=x2. This captures the essence of the unified condition [D2 s, X]=0. VIII. CONCLUSIONS AND OUTLOOK This report extends the drift geometry of DSRN Report I [ 1 ] to the worldsheet of superstring theory and constructs a unified operator Ds=Dws,s ⊗1 + Γws ⊗Ds, which simultaneously incorporates worldsheet BRST geometry and target-space spectral geometry. We established: 17 •preservation of analytic structures under drift; •exact factorisation D2 s=Hs⊗1+1⊗D2 s; •heat kernel factorisation and Seeley–DeWitt convolution; •the universal quartic drift potential reappearing in the unified Spectral Action; •the main equivalence theorem: βws(s)=0⇐⇒ δStarget δΨ=0⇐⇒ [D2 s, X]=0. This identifies drift-stationarity as the common operator-theoretic origin of worldsheet conformal invariance and target-space spectral field equations. Future reports will extend this framework to: 1. supersymmetric drift geometry (Report III), 2. drifted M-theory geometry (Report IV), 3. drifted compactifications and flux towers (Report V), 4. drifted brane dynamics (Report VI). Dws Dws,s DsDsD e−sx(·)esx ⊗1Γws esφ(·)e−sφ FIG. 1. Unified drift flow linking worldsheet and target-space operators. e−tD2 s e−tHse−tD2 s ⊗ FIG. 2. Exact heat-kernel factorisation of the drifted master operator. APPENDICES Appendix A: Operator-Theoretic Foundations of Drift Geometry We recall here some functional-analytic facts used in the main text. Background may be found in [6–8,13,14]. 18 s V(s) V(s)=αs2+βs4 FIG. 3. Universal quartic drift potential with β > 0. Dws,s Γws Ds Ds=Dws,s ⊗1 + Γws ⊗Ds FIG. 4. Tensor structure of the unified drifted master operator. Operator Definition Sector Dws,s e−sxDwsesx Worldsheet DsesφDe−sφ Target-space DsDws,s ⊗1 + Γws ⊗DsUnified TABLE I. Drifted operators appearing in the unified construction. Coefficient Drift Dependence a2(s)a2(0) + αs2 a4(s)a4(0) + βs4 TABLE II. Universal drift dependence of Seeley–DeWitt coefficients. 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. Then: •Dom(As) = Dom(A)for all s∈R; 19 •if Ais closed, then Asis closed; •if Ais self-adjoint, then Asis self-adjoint on Dom(A). These are standard consequences of bounded similarity transformations [7]. 2. Holomorphic Families of Type (A) For As = A + sC with C bounded, the family {As} is holomorphic of type (A): the domain is independent of s and s7→ Asψ is analytic for each ψ∈Dom ( A )[ 7 ]. This applies to Dws,s , Ds and Ds. 3. Drift Flow Identity Differentiating As=e−sBAesB yields ∂As ∂s = [As, B]. This is the basic drift flow equation used throughout Sections 2–6. Appendix B: Drifted Lichnerowicz Formula We sketch the derivation of the drifted Lichnerowicz formula used in Section III, following [ 1 , 2 , 9 ]. Assume the undeformed Dirac operator satisfies D2=−gµν∇µ∇ν+E0. Let φbe a scalar multiplier. Then [φ, [φ, D]] = 0, and the BCH expansion truncates: Ds=esφDe−sφ =D+s[D, φ]. Computing D2 swe find D2 s=D2+s{D, [D, φ]}+s2[D, φ]2. 20 Introduce ∇(s) µ=∇µ+s ∂µφ. Then D2 s=−gµν∇(s) µ∇(s) ν+E0+s∆φ+s2∥∇φ∥2, which is the drifted Lichnerowicz identity quoted in Section III. Appendix C: Heat Kernel Factorisation Let A and B be self-adjoint operators on Hilbert spaces H1 and H2 respectively. Suppose [A⊗1,1⊗B] = 0 on H1⊗ H2. Then the heat semigroup factorises: e−t(A⊗1+1⊗B)=e−tA ⊗e−tB, and so Tr(e−t(A⊗1+1⊗B)) = TrH1(e−tA) TrH2(e−tB). Applying this to A = Hs and B = D2 s yields the factorisation used in Section 5; the convolution formula for aunif k(s)follows directly from the product of the two asymptotic expansions [6,9]. Appendix D: BRST Drift Algebra We recall here the algebraic properties of the drifted BRST operator, as used in Section II. Standard BRST constructions can be found in [11,12]. Let Qs=e−sxQesx with xbounded and self-adjoint. 1. Nilpotency and Cohomology Since ( Qs ) 2 = e−sxQ2esx = 0, BRST nilpotency is preserved. Moreover, Qsψ = 0 iff Q ( esxψ ) = 0, so H•(Qs)∼ =H•(Q). 2. Worldsheet Symmetries If [ x, Ln ]=[ x, Gr ]=0, then [ Qs, Ln ]=[ Qs, Gr ]=0. Thus Virasoro and super-Virasoro symmetries are compatible with drift. 21 3. Drifted Worldsheet Dirac Operator Defining Dws,s =e−sxDwsesx with Dws =Q+Q†, we have ∂Dws,s ∂s = [Dws,s, x], which is the worldsheet SRG equation used in Section II. [1] J. Pinho-da-Cruz, DSRN Report I: Drift Spectral Relativity and Noncommutative Geometry, September 2005. doi: 10.5281/zenodo.17873909. [2] A. Connes, Noncommutative Geometry, Academic Press, 1994. [3] A. H. Chamseddine and A. Connes, “The Spectral Action Principle”, Commun. Math. Phys. 186 (1997), 731–750. [4] M. Eckstein and B. Iochum, Spectral Action in Noncommutative Geometry, SpringerBriefs in Mathematical Physics 27, Springer, 2018. [5] W. D. van Suijlekom, Noncommutative Geometry and Particle Physics, Springer, 2015. [6] E. B. Davies, Heat Kernels and Spectral Theory, Cambridge Tracts in Mathematics 92, Cambridge University Press, 1989. [7] M. Reed and B. Simon, Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness, Academic Press, 1975. [8] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, Springer, 1983. [9] P. B. Gilkey, Invariance Theory, the Heat Equation and the Atiyah–Singer Index Theorem, CRC Press, 1995. [10] R. Haag, Local Quantum Physics: Fields, Particles, Algebras, 2nd ed., Springer, 1996. [11] M. B. Green, J. H. Schwarz, and E. Witten, Superstring Theory, Vols. 1–2, Cambridge University Press, 1987. [12] J. Polchinski, String Theory, Vols. 1–2, Cambridge University Press, 1998. [13] V. Moretti, Spectral Theory and Quantum Mechanics, UNITEXT, Springer, 2013. [14] M. 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