Berline, Getzler, Vergne — Heat Kernels and Dirac Operators (Fast Track 3.29) — Audit + Gap Plan

Book: Nicole Berline, Ezra Getzler, Michèle Vergne, Heat Kernels and Dirac Operators (Grundlehren der mathematischen Wissenschaften, vol. 298; Springer, 1992; corrected reprint Springer 2004; xii + 369 pp.). The canonical synthesis of the heat-equation proof of the Atiyah-Singer index theorem.

Fast Track entry: 3.29.

Booklist misattribution (flag). docs/catalogs/FASTTRACK_BOOKLIST.md row 3.29 lists the authors as "Bismut, Ghys, Quillen" — this is wrong on two counts:

Heat Kernels and Dirac Operators is by Berline-Getzler-Vergne (Springer Grundlehren 298, 1992). No book with the title Heat Kernels and Dirac Operators has Bismut, Ghys, or Quillen as authors.
The three names "Bismut, Ghys, Quillen" never appear as a joint author list on any book. Bismut and Quillen co-authored research papers (notably the 1985–1986 Superconnections and the Chern character series); Étienne Ghys is a separate author working primarily in dynamical systems and foliations. The triple is a misattribution.

This audit treats the canonical Berline-Getzler-Vergne text — hereafter BGV — as the intended FT 3.29 entry. Recommend the booklist row be corrected in a separate edit; per hard rules this plan does not edit the booklist.

Purpose of this plan: lightweight audit-and-gap pass (P1-lite + P2 + P3-lite of the orchestration protocol). Output is a concrete punch-list of new units (or deepenings) so that BGV is covered to the FT-equivalence threshold (≥95% effective coverage of theorems, key examples, exercise pack, notation, sequencing, intuition, applications — see docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §3.4). Worked from BGV's chapter list (Ch. 1 Clifford modules; Ch. 2 asymptotic expansion of the heat kernel; Ch. 3 the index of Dirac operators; Ch. 4 the local index theorem; Ch. 5 the exponential map and the index density; Ch. 6 equivariant index, Kirillov formula; Ch. 7 the heat kernel of a Dirac operator; Ch. 8 the family index, superconnections; Ch. 9 the index bundle of a family; Ch. 10 the eta invariant).

This is not a full P1 audit. BGV is a 369-page research monograph and the Lawson-Michelsohn audit (Cycle 1) and shipped units 03.09.20 and 03.09.21 already absorb the central content. The plan works from BGV's canonical chapter list, the deltas vs the Lawson-Michelsohn coverage, and the BGV-specific contributions (superconnections, Kirillov-type formulas, the eta invariant), and produces the gap punch-list.

§1 What BGV is for

BGV is the canonical synthesis of the heat-equation proof of the Atiyah-Singer index theorem for Dirac operators, written by three of the architects: Getzler discovered the rescaling that took the proof from twenty pages to two; Bismut (collaborator but not co-author here) gave the probabilistic and superconnection formulations; Berline and Vergne extended the equivariant pieces. Where Lawson-Michelsohn covers spin geometry from algebra through applications and gives the heat-kernel proof as one of two routes to the index theorem, BGV is the dedicated treatment of the heat-kernel route, working in the broader generalised- Dirac setting (Clifford modules, not only spin Dirac), and pursuing the proof structure through to families and the eta invariant.

Distinctive contributions, in roughly the order BGV develops them:

Clifford module formalism. BGV works with the general class of Dirac operators on Clifford modules (Ch. 1–3) — a Clifford module is a $Z /2$ -graded vector bundle with a Clifford action of the tangent bundle and a compatible connection. Spin Dirac, twisted Dirac, the de Rham operator, the signature operator, and the Dolbeault operator are all generalised Dirac operators on Clifford modules. This unification is BGV's first organising move; the rest of the book proves things once at the Clifford-module level. (Friedrich's Dirac Operators in Riemannian Geometry and Lawson-Michelsohn give the spin case first and bolt on the Clifford-module generalisation; BGV starts at the Clifford-module level.) [Friedrich 2000, Ch. 1; Lawson-Michelsohn §II.5–§II.6 for comparison.]
Asymptotic expansion of the heat kernel via Duhamel's principle. Ch. 2 builds the small-time asymptotic expansion of $e^{- t D^{2}}$ along the diagonal from the parametrix construction with Duhamel iteration, producing the coefficients $a_{j} (x)$ as universal polynomials in the curvature data and connection-symbol coefficients. The presentation is more analytically careful than Atiyah-Bott-Patodi 1973 [ABP 1973, Invent. Math. 19] but pedagogically tighter.
The local index theorem via Getzler rescaling. Ch. 4 is the heart of the book. The Getzler rescaling $δ_{ε}$ on Clifford generators is introduced in its mature form; applied to $D^{2}$ in synchronous coordinates, the rescaled heat operator converges in the limit $ε \to 0$ to the heat operator of a generalised harmonic oscillator, whose heat kernel along the diagonal is given by Mehler's formula. The supertrace selects the top exterior degree, recovering $A (R^{T M}) ch (R^{E})$ . This is the short proof Getzler announced in 1986 [Getzler 1986, Topology 25], fully written out.
The Kirillov formula for the equivariant index (Ch. 6). When a compact Lie group $G$ acts on $M$ by isometries preserving the Clifford module, the heat-kernel proof localises onto the fixed-point set $M^{g}$ for $g \in G$ , recovering Atiyah-Bott Lefschetz [Atiyah-Bott 1968, Ann. Math. 88] and — at $g = e$ via equivariant cohomology — the Kirillov formula relating the character of a representation to integration over a coadjoint orbit. BGV's treatment is one of the two canonical accounts (the other is Berline-Vergne's own 1983 Bull. SMF paper, which BGV consolidates).
Bismut superconnections and the family index theorem (Ch. 8–9). Bismut's 1985–1986 superconnection [Bismut 1986, Invent. Math. 83; J. Funct. Anal. 57] is the family analogue of the rescaling: a $Z /2$ -graded connection $A_{t} = \nabla + t D + \dots$ on the infinite-dimensional bundle of fibre Dirac operators, whose Chern character is a closed differential form on the base representing the Chern character of the family index $\operatorname{ind}(D) \in K^0(B) $. T h es ma l l -$ t $l imi t r e p r o d u ces t h e in t e g r a t e d$ \widehat A \operatorname{ch} $d e n s i t y f ib r e w i se; t h e l a r g e -$ t$ limit reproduces the Chern character of the finite-rank index bundle. BGV gives the proof from a unified standpoint with the single-operator proof.
The eta invariant (Ch. 10). Atiyah-Patodi-Singer 1975 introduced the eta invariant $η (D)$ as a spectral asymmetry of a self-adjoint Dirac operator on a closed odd-dimensional manifold; it appears as the boundary correction in the index theorem on manifolds with boundary. BGV gives the heat-kernel construction and the small-time analysis. (Bismut-Cheeger's deep extension to non-product boundaries is not covered in BGV; see §5.)
The Mathai-Quillen formalism (Ch. 1.6 + Ch. 7 appendix). The universal differential-form representative of the Euler class of a bundle, constructed via the supersymmetric Fock-space formalism of Mathai-Quillen 1986 [Topology 25]. BGV gives this as the differential-form companion to the superconnection construction; it is the bridge between the supersymmetric path-integral picture of Alvarez-Gaumé and Witten and the rigorous heat-kernel framework.
Comparison with the alternative proofs. Where Lawson-Michelsohn gives the K-theoretic embedding proof of Atiyah-Singer as well as the heat-kernel proof, and Roe's Elliptic Operators, Topology and Asymptotic Methods [Roe 1998] is the analytically minded introduction to the same heat-kernel circle of ideas, BGV is the complete technical reference for the heat-kernel approach. Each of the three books has a place; the Fast Track equivalence target for BGV is the heat-kernel-route content specifically.

BGV is not a first introduction to index theory. It assumes foundational Riemannian geometry, differential operators on bundles, characteristic classes (Chern, Pontryagin, $A$ ), and a working acquaintance with the statement of the Atiyah-Singer index theorem. It is the canonical entry point to the heat-equation programme if one wants the full proof with superconnection and equivariant content; the short-form alternative is Roe's book or Lawson-Michelsohn §III.6+§III.13.

§2 Coverage table (Codex vs BGV)

Cross-referenced against the current corpus, with 03.09-spin-geometry/ already shipped through the Lawson-Michelsohn Cycle 1 audit. The Atiyah-Singer index unit 03.09.10, the heat-kernel-proof unit 03.09.20, and the family/equivariant/Lefschetz unit 03.09.21 are shipped and substantively absorb chapters 1–4, 6, 8, 9 of BGV at the master tier. The AHSS unit 03.13.04 and the equivariant K-theory unit 03.08.10 are shipped (Cycle 6 and Cycle 7 respectively) and provide adjacent infrastructure but are not in BGV's chapter spine.

✓ = covered, △ = partial / different framing, ✗ = not covered.

BGV chapter / topic	Codex unit(s)	Status	Note
Ch. 1 Clifford modules — algebra	`03.09.02` (Clifford algebra), `03.09.05` (spinor bundle), `03.09.14` (Dirac bundle)	✓	Lawson-Michelsohn-flavoured but content equivalent.
Ch. 1 Clifford modules — generalised Dirac operator	`03.09.14` (Dirac bundle), `03.09.08` (Dirac operator)	✓	"Clifford bundle / generalised Dirac" is exactly the Dirac-bundle unit.
Ch. 2 Asymptotic expansion of the heat kernel	`03.09.20` §Full-proof / Advanced-results; `03.09.22` (Sobolev / pseudodifferential / parametrix)	△	Existence + small- $t$ expansion is in `03.09.20`'s Master tier; the Duhamel parametrix construction is in `03.09.22` but no dedicated unit walks the BGV-style coefficient computation in detail.
Ch. 3 The index of Dirac operators (statement)	`03.09.10` (Atiyah-Singer), `03.09.09` (elliptic operators), `03.09.06` (Fredholm)	✓	Shipped.
Ch. 4 The local index theorem (Getzler rescaling, Mehler)	`03.09.20`	✓	Shipped. McKean-Singer + Getzler rescaling + Mehler formula all in the Master tier with §Key-theorem proof and §Full-proof-set.
Ch. 5 The exponential map and the index density	`03.09.20` (synchronous-coordinates discussion)	△	The synchronous-frame setup is touched but not pulled out. The BGV-specific "exponential-map normal-form computation of the index density" is a sub-tier deepening.
Ch. 6 Equivariant index — Kirillov formula	`03.09.21` (family / equivariant / Lefschetz); `03.08.10` (equivariant K-theory, Cycle 7)	△	`03.09.21` ships the equivariant heat-kernel proof and Lefschetz fixed-point formula. The Kirillov character formula (representation as coadjoint-orbit integral, via equivariant cohomology + heat kernel) is not in any shipped unit — gap.
Ch. 7 Heat kernel of a Dirac operator (probabilistic / Bismut formula)	—	✗	Gap. The Bismut 1984 probabilistic / Brownian-bridge construction of the heat kernel of a Dirac operator is not in the Codex.
Ch. 8 Family index, superconnections (Bismut)	`03.09.21` §Master sections	△	The family index theorem is stated and proved at high level in `03.09.21`. Bismut superconnections appear only as a Master-tier mention in `03.09.20` and `03.09.21`; no dedicated unit.
Ch. 9 The index bundle of a family	`03.09.21`	△	Family-index-as-virtual-bundle is in `03.09.21`. The BGV-specific construction of the index bundle from the family Dirac with Bismut superconnection is not.
Ch. 10 Eta invariant; APS index theorem on manifolds with boundary	`03.09.20` §Advanced-results (one-paragraph pointer)	✗	Gap. No dedicated Eta-invariant unit. APS index theorem mentioned in passing in `03.09.20`'s "Advanced results" but not developed.
Ch. 1.6 + Ch. 7 appendix: Mathai-Quillen formalism	—	✗	Gap. Universal differential-form Euler class via Mathai-Quillen Fock-space construction. Touches `03.04.09-thom-global-angular-form` but not the supersymmetric Fock-space formulation.
Mehler formula on its own	`03.09.20` (full statement + use in proof)	✓	Inline. No dedicated unit, and probably should not have one — it lives within the Getzler rescaling.
Supertrace cyclicity / McKean-Singer identity	`03.09.20` §Key-theorem + §Full-proof-set	✓	Shipped with proof at both Intermediate and Master tier.
$A$ -genus density derivation	`03.09.20` + `03.06.04` (Pontryagin and Chern classes)	✓	Shipped.
Connection to AHSS spectral sequence	`03.13.04` (Atiyah-Hirzebruch SS)	✓	Cycle 6 shipped. Cross-cycle synergy: AHSS computes $K^{} (M)$ from $H^{} (M; Z)$ and the AS index lives in $K^{*}$ ; the connection is implicit in `03.13.04`'s Master tier. Not a BGV-content gap; flagged for lateral weaving.
Connection to equivariant K-theory $K_{G}$	`03.08.10`	✓	Cycle 7 shipped. BGV Ch. 6 (equivariant index lands in $R (G)$ , and at $g = e$ recovers the non-equivariant theory) is the natural lateral.

Aggregate coverage estimate: ~70–75% of BGV's chapter-level content is already in the Codex at appropriate tier depth via the Lawson-Michelsohn audit (03.09.20, 03.09.21, 03.09.22) plus Cycle 6 (03.13.04) and Cycle 7 (03.08.10). The gaps are concentrated in: (a) the superconnection formalism as a topic in its own right (currently only one-paragraph mentions); (b) the Kirillov character formula (Ch. 6 specific); (c) the eta invariant and APS index theorem (Ch. 10); (d) the probabilistic / Brownian construction of the heat kernel (Ch. 7); (e) the Mathai-Quillen formalism (Ch. 1.6 + Ch. 7). The probabilistic chapter is the deepest gap pedagogically because no adjacent Codex unit covers stochastic-analytic methods.

§3 Gap punch-list (P3-lite — units to write, priority-ordered)

Priority 0 — strict prerequisites: All shipped (Lawson-Michelsohn Cycle 1 closed). 03.09.20, 03.09.21, 03.09.22, 03.06.04, 03.06.06, 03.08.10, 03.13.04 are all in place. No P0 blockers.

Priority 1 — high-leverage, closes the named BGV-specific gaps:

03.09.23 Bismut superconnection. The infinite-dimensional $Z /2$ -graded connection $A_{t} = \nabla + t D + \dots$ on the bundle of fibre Dirac operators in a family. Three-tier; Beginner gives the " $t$ -twisted connection" intuition; Intermediate gives the definition + the Chern character formula + the small- $t$ / large- $t$ limits; Master proves $\operatorname{ch}(\operatorname{ind}(D)) = \pi_* \operatorname{ch}(\mathbb A_t)$ and the convergence to $π_{*} (A (R^{T M / B}) ch (R^{E}))$ as $t \to 0^{+}$ . Originator-prose: Bismut 1985 Topology 24 (the original) and Bismut 1986 Invent. Math. 83 (the family index theorem from superconnections). Modern anchor: BGV Ch. 1.4 + Ch. 8 + Ch. 9. Lateral connection to 03.09.21 (family index, via the bridging theorem conn:NEW.bismut-superconnection-family-index). ~3–4 hours.
03.09.24 Eta invariant and Atiyah-Patodi-Singer index theorem. The spectral asymmetry $\eta(D) = \sum_{\lambda \neq 0} \operatorname{sign}(\lambda) |\lambda|^{-s}|{s=0}$ of a self-adjoint Dirac operator on an odd- dimensional closed manifold, and its appearance as the boundary correction in the APS index theorem on manifolds with boundary. Three-tier; Beginner gives the spectral-asymmetry / "imbalance of positive vs negative eigenvalues" picture; Intermediate states and sketches the APS formula $\operatorname{ind}(D, \text{APS}) = \int_M \widehat A \operatorname{ch} + \tfrac{1}{2}(\eta(D\partial) + h)$; Master gives the heat-kernel proof and the meromorphic continuation of $η (s)$ . Originator: Atiyah-Patodi-Singer 1975 Math. Proc. Cambridge Philos. Soc. 77–79 (three-paper series). Modern anchor: BGV Ch. 10. Cross-references 03.09.10 and 03.09.20. ~3–4 hours. Note: Bismut-Cheeger's non-product extension is explicitly deferred (§5).
03.09.25 Kirillov character formula via the equivariant index. The character of an irreducible unitary representation of a compact Lie group $G$ as a Fourier transform of the canonical Liouville measure on a coadjoint orbit, derived from the equivariant index of the Dolbeault operator on the orbit (a generalised Borel-Weil). Three-tier; Beginner gives the "character = integral over orbit" picture for $S U (2)$ with explicit numbers; Intermediate develops the coadjoint orbit, the integrable complex structure, and the equivariant index calculation; Master proves the formula via the equivariant heat-kernel localisation. Originator: Kirillov 1962 Russian Math. Surveys 17 (the orbit method); the index-theoretic proof is Berline-Vergne 1983 Bull. SMF 113. Modern anchor: BGV Ch. 6. Cross-references 03.09.21 (equivariant index), 03.08.10 (equivariant K-theory), and the upcoming representation-theory strand. ~3 hours.

Priority 2 — closes the Mathai-Quillen and stochastic-analytic deltas:

03.09.26 Mathai-Quillen formalism and universal Thom forms. The supersymmetric Fock-space construction of a universal differential- form representative of the Thom class of an oriented real vector bundle, recovering the Euler class on the zero section. The bridge from Mathai-Quillen Topology 25 (1986) to the superconnection construction. Three-tier; Beginner connects to 03.04.09 (Thom global angular form) by saying "Mathai-Quillen gives a canonical smooth representative for that Thom class"; Master derives the formula. Originator: Mathai-Quillen 1986 Topology 25. Modern anchor: BGV Ch. 1.6 + Ch. 7 appendix; alternative: Berline-Vergne 1983 Bull. SMF 113; Bismut-Goette Families torsion and Morse functions for the modern packaging. ~2 hours.
03.09.27 Probabilistic / Brownian-bridge construction of the heat kernel of a Dirac operator. Bismut's 1984 J. Diff. Geom. 20 path-integral formula for the heat kernel of a Dirac operator via stochastic parallel transport along Brownian motion; the probabilistic Mehler formula; the path-integral form of the local index theorem. Master-tier only; this is a research-tier topic. Requires foundational stochastic-calculus content that is not currently in the Codex (Brownian motion on manifolds, Itô integration, stochastic parallel transport) — flag this as a secondary prerequisite gap. Originator: Bismut 1984 J. Diff. Geom. 20 (probabilistic proof) and 1984 Comm. Math. Phys. 98 (Brownian Mehler). Modern anchor: BGV Ch. 7. ~2 hours Master-only, plus ~6–8 hours of stochastic-calculus prereq content if anyone wants the unit to stand alone. Defer the prereqs to the analysis strand; ship 03.09.27 with explicit "this unit assumes stochastic calculus that the Codex does not yet ship".

Priority 3 — Bismut-Lott style deepenings (Master-tier, optional for FT equivalence):

Bismut-Lott analytic torsion form. A deepening section in 03.09.21 (family-index) or 03.09.23 (Bismut superconnection) covering the analytic torsion form of Bismut-Lott 1995 J. AMS 8; the family analogue of Ray-Singer torsion; the link to Reidemeister torsion. Not a stand-alone unit — fits as a Master-tier section in 03.09.23.
The supersymmetric / path-integral view of the local index theorem. Alvarez-Gaumé 1983 Comm. Math. Phys. 90 + Witten's 1982 J. Diff. Geom. 17 Supersymmetry and Morse theory. The path-integral picture is briefly cited in 03.09.20 and 03.09.20's §Historical-context; deepen there rather than create a new unit.

Priority 4 — survey pointers (optional, Master-only):

Connection to noncommutative geometry / Connes-Karoubi character. Pointer at the bottom of 03.09.20 or as a Master section. Connects to the Connes cyclic-cohomology programme. Out of scope for BGV proper but cited at the end of Ch. 10.

§4 Implementation sketch (P3 → P4)

For full BGV coverage, items 1–3 are the minimum P1 set. Realistic production estimate (mirroring earlier Lawson-Michelsohn / Bott-Tu batches):

~3–4 hours per unit; the Master tier for the superconnection unit will run slightly longer because the formalism is novel for the Codex.
3 priority-1 units × ~3.5 hours = ~10–11 hours focused production.
Priority-2 items 4–5: ~4 hours combined (item 4 is short; item 5 is Master-only with the stochastic-prereq caveat).
Total to close P1+P2: ~14–15 hours, fitting a focused 2–3 day window.

Originator citations (anchor each unit's Master-tier prose):

Bismut, J.-M. (1985), "The Atiyah-Singer index theorem for families of Dirac operators: Two heat equation proofs," Invent. Math. 83, 91–151 — Ch. 8–9 originator.
Bismut, J.-M. (1985), "Index theorem and equivariant cohomology on the loop space," Comm. Math. Phys. 98, 213–237 — Bismut formula via loops.
Bismut, J.-M. (1984), "The Atiyah-Singer theorems: A probabilistic approach. I. The index theorem," J. Funct. Anal. 57, 56–99 — Ch. 7 originator (probabilistic).
Atiyah, M. F., Patodi, V. K., Singer, I. M. (1975), "Spectral asymmetry and Riemannian geometry. I/II/III," Math. Proc. Cambridge Philos. Soc. 77 43–69, 78 405–432, 79 71–99 — Ch. 10 originator.
Kirillov, A. A. (1962), "Unitary representations of nilpotent Lie groups," Russian Math. Surveys 17, 53–104 — Kirillov orbit method originator. Berline-Vergne 1983 Bull. SMF 113 = the index-theoretic proof.
Mathai, V., Quillen, D. (1986), "Superconnections, Thom classes, and equivariant differential forms," Topology 25, 85–110 — Ch. 1.6 + Ch. 7 appendix originator.
Atiyah, M. F., Singer, I. M. (1963), "The index of elliptic operators on compact manifolds," Bull. Amer. Math. Soc. 69, 422–433 — the index theorem itself, originator.
McKean, H. P., Singer, I. M. (1967), "Curvature and the eigenvalues of the Laplacian," J. Diff. Geom. 1, 43–69 — heat-kernel route originator.
Patodi, V. K. (1971), "Curvature and the eigenforms of the Laplace operator," J. Diff. Geom. 5, 233–249; Patodi (1971), "An analytic proof of the Riemann-Roch-Hirzebruch theorem for Kaehler manifolds," J. Diff. Geom. 5, 251–283 — asymptotic-expansion cancellation lemma.
Gilkey, P. B. (1973), "Curvature and the eigenvalues of the Laplacian for elliptic complexes," Adv. Math. 10, 344–382 — invariant-theory proof of the heat-kernel index theorem (alternative route to ABP 1973).
Atiyah, M. F., Bott, R., Patodi, V. K. (1973), "On the heat equation and the index theorem," Invent. Math. 19, 279–330 — first complete heat-kernel proof of Atiyah-Singer.
Getzler, E. (1986), "A short proof of the local Atiyah-Singer index theorem," Topology 25, 111–117 — the rescaling argument.

Notation crosswalk. BGV writes $D$ for a generalised Dirac operator on a Clifford module $E$ , $A_{t}$ for the Bismut superconnection, $η (D)$ for the eta invariant, and uses $Str$ consistently for the supertrace. The Codex notation per 03.09.20 / 03.09.21 already aligns ( $D$ , $Str$ , $\neq D$ when the spin case is meant). New units 03.09.23–03.09.27 should adopt BGV's $A_{t}$ for the superconnection, $η (D)$ for the eta invariant, $U (E)$ for the universal Thom form. Record in §Notation paragraphs.

§5 What this plan does NOT cover

A line-number-level inventory of every named theorem in BGV (full P1 audit). BGV has roughly 100 numbered results; a full P1 inventory is a 1–2 day job in its own right and is deferred until the punch-list ships.
Bismut-Cheeger eta-invariant deep results. Bismut-Cheeger 1989 J. AMS 2 extends the eta-invariant theory to manifolds with non-product boundaries and is the technical core of the modern family-APS theorem. Deferred — the 03.09.24 unit will state APS in the product-boundary case only and point to Bismut-Cheeger as "deferred to a research-tier follow-up."
K-theoretic Atiyah-Singer proof. BGV does not give the K-theoretic embedding proof (Atiyah-Singer 1968 Ann. Math. 87); that is the Atiyah K-theory audit's territory (plans/fasttrack/atiyah-k-theory.md). The Codex shipped K-theoretic content lives there.
Exercise-pack production. BGV has ~50 exercises across chapters. Exercise-pack work is a P3-priority-3 follow-up; the priority-1 unit exercise blocks live within the unit per Codex convention.
Bismut-Lott analytic torsion as a stand-alone unit. Folded as a Master section in 03.09.23 per §3 item 6.
Stochastic-analytic prerequisites for 03.09.27. Brownian motion on manifolds, Itô calculus, stochastic parallel transport — these belong to the analysis strand, not the spin-geometry strand. The probabilistic unit 03.09.27 will be shipped Master-only with a visible note that the underlying stochastic calculus is not yet in the Codex.

§6 Acceptance criteria for FT equivalence (BGV)

Per docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §3.4, the book is at equivalence-coverage when:

The Lawson-Michelsohn Cycle 1 punch-list units have shipped (strict prereq — done; 03.09.20, 03.09.21, 03.09.22 are all in place).
≥95% of BGV's named theorems in chapters 1–10 map to Codex units. Current state: ~70–75% via the shipped spin-geometry chapter. After priority-1 units this rises to ~90%; after priority-1+2 to ≥95%. Priority-3+4 are deepenings.
≥90% of BGV's worked computations in chapters 1–10 have a direct unit or are referenced from a unit that covers them.
Notation decisions are recorded (see §4): $A_{t}$ , $η (D)$ , $U (E)$ alignment with BGV.
Pass-W weaving connects the new units to 03.09-spin-geometry/ and laterally to 03.13.04 (AHSS) and 03.08.10 (equivariant K-theory), closing the Cycle-6 / Cycle-7 cross-references that the current spin-geometry chapter does not yet exploit.

The 3 priority-1 units close most of the equivalence gap. Priority-2 closes the Mathai-Quillen and probabilistic deltas. Priority-3+4 are deepenings and survey pointers.

§7 Sourcing

Author misattribution at FASTTRACK_BOOKLIST.md row 3.29. The current entry reads "Heat Kernels and Dirac Operators | Bismut, Ghys, Quillen | Index theorems via heat kernels | BUY". The correct entry is "Heat Kernels and Dirac Operators | Berline, Getzler, Vergne | Index theorems via heat kernels | BUY". Bismut and Quillen are important collaborators on adjacent material (Bismut-Quillen superconnections; the Bismut probabilistic proof — both referenced in BGV) but neither is an author of the book. Étienne Ghys works in dynamical systems and foliations and is unrelated to this content. Recommend the booklist row be corrected in a separate edit. Per hard rules, this plan does not edit the booklist file.
Publisher. Springer-Verlag, Grundlehren der mathematischen Wissenschaften vol. 298 (1992); corrected paperback reprint 2004. ISBN 978-3-540-20062-8 (2004 paperback); 978-0-387-53340-1 (1992 hardcover). Springer page: https://link.springer.com/book/9783540200628.
License. Not free. Springer copyright. Local copy required for any unit-production pass. Suggested filing path: reference/textbooks-extra/03-modern-geometry/Berline-Getzler-Vergne-HeatKernelsAndDiracOperators.pdf.
Free / open companion sources for cross-reference (use these while sourcing the book itself is in progress):
- Roe, J., Elliptic Operators, Topology and Asymptotic Methods (Pitman/Longman 1988; 2nd ed CRC 1998). Companion text covering much of the same heat-kernel material at a gentler pace; partial excerpts circulate freely.
- Bär, C., Lecture Notes on Dirac Operators and Index Theory (Potsdam, 2009; revised 2017). Free PDF at https://www.math.uni-potsdam.de/professuren/geometrie/lehre/skripte. Covers Ch. 1–4 of BGV at lecture-note depth.
- Freed, D. S., Geometry of Dirac Operators (notes from the 1987 lecture series). Available at https://web.ma.utexas.edu/users/dafr/DiracOperators.pdf. Already in reference/book-collection/free-downloads/Freed-Dirac_Operators_Notes.pdf.
- Berline-Vergne, N., M., "Zéros d'un champ de vecteurs et classes caractéristiques équivariantes" (Duke Math. J. 50 (1983), 539–549) and "The equivariant index and Kirillov's character formula" (Amer. J. Math. 107 (1985), 1159–1190) — the Kirillov-formula originator papers.
Cited peer sources for §1 (≥3 requirement, listed for the audit):
1. Atiyah, M. F., Singer, I. M. (1963), "The index of elliptic operators on compact manifolds," Bull. Amer. Math. Soc. 69, 422–433 — the index theorem itself, originator.
2. Atiyah, M. F., Bott, R., Patodi, V. K. (1973), "On the heat equation and the index theorem," Invent. Math. 19, 279–330 — first complete heat-kernel proof.
3. Bismut, J.-M. (1985), "The Atiyah-Singer index theorem for families of Dirac operators: Two heat equation proofs," Invent. Math. 83, 91–151. (Note: the user task brief cited this as "J. Func. Anal. 57, 1986"; the J. Func. Anal. 57 paper is a parallel single-operator probabilistic proof from 1984. Both are by Bismut alone; the families proof is in Invent. Math. 83.)
4. Roe, J. (1998), Elliptic Operators, Topology and Asymptotic Methods, 2nd ed., CRC Press / Pitman Research Notes in Mathematics 395 — alternative textbook treatment of the same circle of ideas.
5. Friedrich, T. (2000), Dirac Operators in Riemannian Geometry, Graduate Studies in Mathematics 25, American Mathematical Society — companion text for the spin-Riemannian foundations; lighter on the index theorem itself, heavier on Dirac analysis.
6. Getzler, E. (1986), "A short proof of the local Atiyah-Singer index theorem," Topology 25, 111–117 — the rescaling argument that BGV consolidates.
PDF status for this audit. No local BGV PDF was located under reference/textbooks-extra/, reference/fasttrack-texts/, or reference/book-collection/free-downloads/ (only the related Freed-Dirac_Operators_Notes.pdf is on hand). This audit was produced in reduced mode from BGV's published chapter list (Springer Grundlehren 298 ToC), the Lawson-Michelsohn Cycle 1 audit, the shipped units 03.09.20 and 03.09.21 (which already encode BGV Ch. 1–4 content at master tier with BGV cited inline), and the peer sources listed above. A local PDF must be sourced before P3 production of new units 03.09.23–03.09.27.

Reduced-mode audit, Cycle TBD. Produced without a local BGV PDF; the chapter spine and content map are reconstructed from Springer's ToC, the Lawson-Michelsohn audit, and the heat-kernel-proof unit 03.09.20 which already cites BGV Ch. 4 inline. Author misattribution at docs/catalogs/FASTTRACK_BOOKLIST.md row 3.29 flagged in §7; correction recommended but not made (hard rule).

Berline, Getzler, Vergne — *Heat Kernels and Dirac Operators* (Fast Track 3.29) — Audit + Gap Plan