DRSN II: Drift Geometry and the Unified Spectral Operator (De Rerum Spectrale Natura, Report II, Version 2.0)

DRSN II: DRIFT GEOMETRY AND THE UNIFIED SPECTRAL OPERATOR De Rerum Spectrale Natura series REPORT II (Version 2.0) 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 The drift deformation of Dirac operators introduced in DRSN 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. ∗jp[email protected] 3 CONTENTS I. Introduction 6 II. Worldsheet Drift Geometry 7 A. BRST Framework 8 B. Drifted BRST Operator 8 C. Nilpotency and Cohomology 8 D. Worldsheet Symmetry Preservation 9 E. Drifted Worldsheet Dirac Operator 9 F. Spectral Renormalisation Group Flow 9 G. BCH Expansion 10 III. Target–Space Drift Geometry (Review of Report I) 10 A. Dirac Operators and Bounded Multipliers 10 B. Drifted Dirac Operator 10 C. Drift Flow and Holomorphicity 11 D. Drifted Lichnerowicz Formula 11 E. Heat Kernel and Drift Potential 12 F. Role in the Unified Geometry 12 IV. The Unified Drift Generator and the Master Operator 12 A. Product Hilbert Space and Grading 12 B. Undeformed Composite Operator 13 C. Unified Drift Generator 13 D. Drifted Master Operator 13 E. Analytic Properties 13 V. Heat Kernel Factorisation and the Unified Spectral Action 14 A. Square Factorisation 14 B. Heat Kernel Factorisation 14 C. Asymptotics 14 D. Unified Spectral Action 15 VI. Beta Functions and Spectral Field Equations 15 4 A. Worldsheet Beta Functions 15 B. Target-Space Drift Stationarity 15 C. Unified Drift Equation 16 D. Spectral Action Variation 16 E. Main Equivalence Theorem 16 F. Interpretation 16 VII. Examples and Illustrations 17 A. Unified Drift Flow 17 B. Heat Kernel Factorisation 17 C. Universal Drift Potential 17 D. Structure of the Master Operator 18 E. Toy Model 18 VIII. Conclusions and Outlook 18 Appendices 21 A. Operator-Theoretic Foundations of Drift Geometry 21 1. Bounded Conjugation and Domain Stability 21 2. Holomorphic Families of Type (A) 21 3. Drift Flow Identity 21 B. Drifted Lichnerowicz Formula 22 C. Heat Kernel Factorisation 22 D. BRST Drift Algebra 23 1. Nilpotency and Cohomology 23 2. Worldsheet Symmetries 23 3. Drifted Worldsheet Dirac Operator 23 E. Effective Condensate s∗and Constraint Intersection 23 Definition of the effective condensate 24 Independent constraint family 24 Constraint intersection outcome 24 5 References 26 6 I. INTRODUCTION The drift deformation of Dirac operators, introduced in DRSN 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, 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, 7 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. 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. About this report. This work constitutes Report II 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 focuses on the operator-theoretic extension of drift geometry to the worldsheet of superstring theory and on the construction of a unified drifted spectral operator coupling worldsheet and target-space sectors. Its purpose is structural and analytic: no new phenomenological claims are made. Subsequent reports in the DRSN series will build on this unified framework to develop additional geometric, dynamical and physical layers of drift geometry. 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 8 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. 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. 9 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. 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]. 16 C. Unified Drift Equation From Section 4, Ds=Dws,s ⊗1 + Γws ⊗Ds,D2 s=Hs⊗1+1⊗D2 s. Differentiating, ∂D2 s ∂s = [Hs, x]⊗1+1⊗[D2 s, φ]. Let X=x⊗1+1⊗φ. Then [D2 s, X]=0⇐⇒ [Hs, x] = 0 and [D2 s, φ] = 0. D. Spectral Action Variation The unified Spectral Action is Sunif(s) = Tr f(Ds/Λ)∼X k F6−kΛ6−kaunif k(s). Since worldsheet coefficients aws m(s)are independent of target-space fields Ψ, δSunif δΨ=X m+n=k F6−(m+n)Λ6−(m+n)aws m(s)δatarget n(s) δΨ. Thus, δSunif δΨ=0⇐⇒ δStarget δΨ= 0. E. Main Equivalence Theorem Combining all identities: βws(s)=0⇐⇒ δStarget δΨ=0⇐⇒ δSunif δΨ=0⇐⇒ [D2 s, X]=0. F. Interpretation Worldsheet conformality, target-space spectral dynamics, and unified drift-stationarity arise from a single operator equation. The drift parameter s thus plays the role of a common renormalisation coordinate for worldsheet and target sectors. 17 VII. EXAMPLES AND ILLUSTRATIONS This section provides diagrams and examples clarifying the abstract structures developed in previous sections. Figures 1–4 and Tables 1–2 accompany the discussion. A. Unified Drift Flow The drift equations ∂Dws,s ∂s = [Dws,s, x],∂Ds ∂s = [Ds, φ], combine into ∂Ds ∂s = [Ds, X]. Figure 1 illustrates the parallel drift flows in worldsheet and target sectors and their fusion through the operator X. B. Heat Kernel Factorisation Section 5 proved the identity e−tD2 s=e−tHs⊗e−tD2 s. Figure 2 depicts this decomposition. Table 2 records the drift dependence of the Seeley–DeWitt coefficients. C. Universal Drift Potential From DRSN 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. 18 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 DRSN 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: •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. 19 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 Ds Ds D e−sx(·)esx ⊗1 Γws esφ(·)e−sφ FIG. 1. Unified drift flow linking worldsheet and target-space operators. 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. 20 e−tD2 s e−tHse−tD2 s ⊗ FIG. 2. Exact heat-kernel factorisation of the drifted master operator. 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. Coefficient Drift Dependence a2(s)a2(0) + αs2 a4(s)a4(0) + βs4 TABLE II. Universal drift dependence of Seeley–DeWitt coefficients. 21 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]. 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; •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. 22 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. 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]. 23 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. 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. Appendix E: Effective Condensate s∗and Constraint Intersection In this appendix we document the logical status and use of the effective condensate s∗ within the DRSN framework, for the specific spectral model analysed in the present report. No new structural or mathematical results are introduced. The purpose of this appendix is purely methodological: to make explicit how s∗ is defined at the effective level and how its admissibility is assessed through the intersection of independent constraints. Non-claim. The parameter s∗ is not a spectrally rigid invariant, not a fundamental constant, and not a universal quantity. Any numerical benchmark associated with s∗ is effective and schemedependent, depending on truncation order, normalisation, background choice, and internal-sector inputs. No ontological or uniqueness claim is attached to its value. 24 Definition of the effective condensate In the quartic-truncated effective regime, the drift-induced spectral potential takes the universal form V(s)=αs2+βs4, β > 0. Whenever α < 0, the potential admits a non-trivial stationary point defined by s2 ∗=−α 2β. Here α and β are effective coefficients extracted after truncation of the spectral action and specification of the geometric and internal background. Independent constraint family The physical admissibility of s∗is tested against a family of independent constraints: 1. Existence: a non-trivial minimum requires α < 0and β > 0. 2. Spectral stability: absence of runaway behaviour at large |s|. 3. Dynamical viability (when applicable): consistency with the effective equations of motion (e.g. FRW relaxation). 4. Scale consistency (when applicable): the effective vacuum energy V ( s∗ )lies in an admissible regime for the chosen normalisation. 5. Non-fine-tuned regime: exclusion of parametrically extreme values |s∗|≪1or |s∗|≫1. Each constraint defines an admissible subset of parameter space. A physically meaningful condensate exists if and only if the intersection of all relevant constraints is non-empty. Constraint intersection outcome The outcome of the constraint intersection for the model analysed in this report is summarised in Table III. 25 TABLE III. Constraint status for the effective condensate s∗ in the Standard Model internal drifted sector (SDM–I). Constraint Status Note Existence (α < 0,β > 0)Noαint ≥0 Spectral stability Yes βint >0 Dynamical viability N/A No independent cosmology Scale consistency N/A — Non-fine-tuned regime N/A — Intersection Empty No viable s∗= 0