Atiyah — *Geometry of Yang-Mills Fields* (Fast Track 3.20) — Audit + Gap Plan

Book: Michael F. Atiyah, Geometry of Yang-Mills Fields (Lezioni Fermiane). Accademia Nazionale dei Lincei / Scuola Normale Superiore, Pisa, 1979. iv + 99 pp. (Edizioni della Normale; reprinted as Publications of the Scuola Normale Superiore, ISBN 978-88-7642-303-1.)

Fast Track entry: 3.20. Modern-Geometry strand (B), gauge-theory slot. Sits between Kobayashi-Nomizu Vol. 2 (FT 3.19, sibling audit running this cycle) and Donaldson's Floer Homology Groups in Yang-Mills Theory (FT 3.06). It is the canonical concise launchpad text for the geometric / algebraic-bundle side of four-dimensional gauge theory and the historical entry point to Donaldson theory.

Source PDF: Not present in reference/textbooks-extra/. Cover image is archived at reference/fast-track/images/Atiyah-Yang-Mills-Fields-742x1024__05da96f1d2.jpg (already cited from 03.07.05-yang-mills-action.md). The book itself is sold by Edizioni della Normale (Scuola Normale Superiore Pisa); a scanned copy circulates on academic mirrors (Ranicki / Edinburgh maths) but is not bundled with the Codex. WebFetch of the canonical Ranicki mirror returned a 404 in this audit pass; the chapter structure used below is reconstructed from the table-of-contents excerpts surfaced by WebSearch (researchgate, semanticscholar, scribd, inspirehep) and cross-checked against Donaldson-Kronheimer (1990) §3 + Freed-Uhlenbeck (1991) §3 + Atiyah-Hitchin-Drinfeld-Manin 1978 Phys. Lett. A 65 (the ADHM paper Atiyah's Pisa lectures are designed to present).

Audit type: P1-lite audit + P2 gap pass + P3-lite punch-list, mirroring brown-higgins-sivera-nonabelian-algebraic-topology.md and milnor-stasheff-characteristic-classes.md.

Plan status: reduced — no direct PDF access in this pass. Audit works from the book's published table of contents and the well-documented ADHM construction that the book exists to present. Sufficient to drive the production punch-list to FT-equivalence. Full P1 line-number inventory deferred pending direct PDF access.

---

§1 What Atiyah's Geometry of Yang-Mills Fields is for

Atiyah's Pisa lectures are the canonical concise monograph presenting the 1977–1978 geometric reformulation of four-dimensional SU(2) instantons. Where Yang-Mills theory had begun in 1954 as a physics theory of non-abelian gauge fields, by 1975 Belavin, Polyakov, Schwartz and Tyupkin had discovered the first finite-action solution (the BPST instanton on $S^4$), and by 1978 Atiyah, Drinfeld, Hitchin and Manin had reduced the problem of constructing all instantons on $S^4$ to a problem in linear algebra (the ADHM construction). The Pisa lectures crystallise this entire programme in roughly 100 pages, written by Atiyah while the dust was still settling.

The book is the launchpad text for Donaldson theory: Donaldson's 1983 thesis and the 1990 Donaldson-Kronheimer book both build directly on the moduli-of-instantons picture Atiyah presents here.

Distinctive contributions, in roughly the order the seven chapters develop them:

1. Physics background and the field equations (Ch. 1). From Yang-Mills 1954 to the self-dual / anti-self-dual decomposition in four dimensions. The Yang-Mills Lagrangian $\frac{1}{2}\Vert F_A{\Vert }^2$, the action minimisers in a fixed topological sector $c_2=k\in \mathbb{Z}$, and the Bogomolny / Belavin-Polyakov- Schwartz-Tyupkin 1975 lower bound $\mathrm{YM}(A)\ge 8{\pi }^2|k|$ with equality precisely on (anti-)self-dual connections. Asymptotic conditions justifying the one-point conformal compactification ${\mathbb{R}}^4\to S^4$.
2. Instantons on $S^4$ (Ch. 2). Quaternionic description of the basic BPST instanton, its $\mathrm{SU}(2)$-bundle on $S^4$ with $c_2=1$, and the conformal-invariance / quaternionic-multiplication structure that makes the instanton equation tractable. Geometrical interpretation via ${\mathbb{HP}}^1\cong S^4$.
3. Penrose twistor space (Ch. 3). The Penrose correspondence $S^4\leftarrow {\mathbb{P}}^3(\mathbb{C})$ — twistor space of $S^4$ is ${\mathbb{CP}}^3$ — and its real structure. Twistor lines as the fibres of ${\mathbb{CP}}^3\to S^4$. This is the bridge to algebraic geometry: ASD connections on a bundle over $S^4$ pull back to holomorphic structures on bundles over ${\mathbb{CP}}^3$.
4. Holomorphic-bundle reformulation / Ward correspondence (Ch. 4). The Atiyah-Ward theorem: ASD $\mathrm{SU}(2)$-connections on $S^4$ correspond bijectively to rank-2 holomorphic bundles on ${\mathbb{CP}}^3$ that are trivial on twistor lines and carry a real structure. Complex coordinates on ${\mathbb{R}}^4$, holomorphic versus unitary gauges, the twistor interpretation of instantons. Bundles over ${\mathbb{P}}^1(\mathbb{C})$ as the local model.
5. Construction of algebraic bundles / Horrocks–ADHM (Ch. 5). The signature theorem of the book. The Horrocks monad construction for algebraic bundles on ${\mathbb{CP}}^3$; the linear-algebraic data (a quadruple of matrices satisfying a quadratic ADHM constraint) from which every $k$-instanton is reconstructed. Quaternionic reformulae. This is the Atiyah-Drinfeld-Hitchin-Manin construction (1978).
6. Linear field equations in a Yang-Mills background (Ch. 6). The Penrose transform at linear level: spinor / Dirac fields in an instanton background. Bundles and sheaf cohomology ($H^1({\mathbb{CP}}^3,\mathcal{O}(-2))\to$ massless fields on $S^4$). The 't Hooft ansatz as a special case. Relation with the Radon transform.
7. Theorems on algebraic bundles (Ch. 7). Cohomology of the Horrocks construction; Barth's theorem on rank-2 bundles on ${\mathbb{CP}}^3$; the reality constraints that pick out $\mathrm{SU}(2)$- versus $\mathrm{SL}(2,\mathbb{C})$-bundles. Closes the loop: every ASD $\mathrm{SU}(2)$-instanton on $S^4$ is ADHM, and every ADHM datum gives an instanton.

The book is not a first introduction to bundles or Yang-Mills. It assumes Kobayashi-Nomizu-level differential geometry (principal bundles, connections, curvature, Chern classes), basic algebraic geometry over ${\mathbb{CP}}^n$ (sheaf cohomology, line bundles $\mathcal{O}(n)$), and physicist-level Yang-Mills vocabulary. It is the canonical bridge from the differential-geometric and physics sides into the algebraic-geometric / twistor reformulation.

Peer cross-references (the four cited reference texts and where they overlap Atiyah's Pisa lectures):

• Donaldson, Kronheimer, The Geometry of Four-Manifolds (Oxford, 1990). §2 (Yang-Mills, ASD equations), §3 (the moduli space of instantons), §6 (ADHM construction). Donaldson-Kronheimer is the definitive modern treatment; Atiyah's Pisa lectures are the original launchpad that Donaldson-Kronheimer §3 and §6 expand and rigorise. Codex anchor: 03.07.05 cites Donaldson-Kronheimer §2 as a Master-tier source.
• Freed, Uhlenbeck, Instantons and Four-Manifolds (MSRI Publications 1, Springer, 1984; 2nd ed. 1991). §3 (the ADHM construction), §4 (the moduli space). Freed-Uhlenbeck is the most pedagogical complement to Atiyah's Pisa lectures: the bare minimum prerequisites are spelled out and ADHM is constructed with full proofs.
• Bleecker, Gauge Theory and Variational Principles (Addison-Wesley 1981; Dover reprint 2005). Ch. 3–4. Bleecker treats the differential-geometric foundations of Yang-Mills (principal bundles with connection, curvature, the action) in detail; complements Atiyah by filling in the bundle-theoretic prerequisites Atiyah assumes.
• Atiyah, Hitchin, Drinfeld, Manin, "Construction of instantons," Physics Letters A 65 (1978) 185–187. The four-page paper that the Pisa lectures expand into a book. The originating ADHM citation.

Two further critical originator citations:

• Yang, C. N. and Mills, R. L., "Conservation of isotopic spin and isotopic gauge invariance," Physical Review 96 (1954) 191–195. The action functional Atiyah's book studies.
• Belavin, A. A., Polyakov, A. M., Schwartz, A. S., Tyupkin, Yu. S., "Pseudoparticle solutions of the Yang-Mills equations," Physics Letters B 59 (1975) 85–87. The original BPST one-instanton.

---

§2 Coverage table (Codex vs Atiyah Pisa lectures)

Cross-referenced against the current corpus. The 03-modern-geometry/07-gauge-theory/ chapter has exactly one shipped unit: 03.07.05-yang-mills-action.md. The 05-bundles/ chapter covers principal bundles, connections, curvature (03.05.01, 03.05.07, 03.05.09). The 06-characteristic-classes/ chapter covers Pontryagin / Chern classes (03.06.04) and Chern-Weil (03.06.06). The Riemann-surfaces chapter (Ch. 6) has the Atiyah-Bott Riemann-surface picture sketched in 03.07.05's Master tier but no dedicated unit. The four-manifold / instanton / ADHM machinery is essentially absent from the Codex.

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

Atiyah topic	Chapter	Codex unit(s)	Status	Note
Yang-Mills Lagrangian $\frac{1}{2}|F_A{|}^2$	1	03.07.05	✓	Shipped at all three tiers; gauge invariance proved.
Euler-Lagrange equation $d_A^{\ast }{F}_A=0$	1	03.07.05	✓	Derived in Master-tier proof set.
Bianchi identity $d_A{F}_A=0$	1	03.05.09	✓	Master tier of curvature unit.
Self-dual / anti-self-dual decomposition ${\Omega }^2={\Omega }_{+}^2\oplus {\Omega }_{-}^2$ on 4-manifolds	1	03.07.05 (Master)	△	Stated and used; no dedicated unit on the Hodge-* eigenspace decomposition in dimension 4.
ASD equation $F_A^{+}=0$	1	03.07.05 (Master, Exercise 6)	△	Proved to be Yang-Mills; not its own unit.
Topological charge $c_2(E)=\frac{1}{8{\pi }^2}\int \mathrm{tr}(F\wedge F)$	1	03.05.09 + 03.06.04	△	Closedness of $\mathrm{tr}(F\wedge F)$ shown; quantisation $\in \mathbb{Z}$ and normalisation $8{\pi }^2$ stated in 03.07.05 Master Synthesis but not derived.
Bogomolny / BPST bound $\mathrm{YM}(A)\ge 8{\pi }^2|c_2(E)|$	1	03.07.05 (Master)	△	Stated in Synthesis; no proof unit.
Asymptotic conditions, conformal compactification ${\mathbb{R}}^4\to S^4$	1	—	✗	Gap. Foundational for the instanton moduli space (finite-action solutions extend).
BPST one-instanton (1975)	2	—	✗	Gap. No worked unit on the explicit $k=1$ instanton; mentioned only in 03.07.05 Historical context.
Quaternionic description of $S^4\cong {\mathbb{HP}}^1$	2	—	✗	Gap. ${\mathbb{HP}}^1$ is mentioned in 09-spin-geometry triality unit but not as the conformal compactification of ${\mathbb{R}}^4$.
Conformal invariance of Yang-Mills in dim 4	1–2	—	✗	Gap. Foundational: explains why finite action on ${\mathbb{R}}^4$ ⇔ extension to $S^4$.
Penrose twistor space ${\mathbb{CP}}^3\to S^4$	3	—	✗	Gap. Twistor theory is essentially absent from the Codex.
Real structure on ${\mathbb{CP}}^3$ (twistor)	3	—	✗	Gap.
Ward correspondence (ASD bundles on $S^4$ ↔ holomorphic bundles on ${\mathbb{CP}}^3$ trivial on lines)	4	—	✗	Gap (high priority — the central theorem of Ch. 3–4).
Holomorphic vs unitary gauge	4	—	✗	Gap.
Holomorphic bundles on ${\mathbb{CP}}^1$ (Grothendieck splitting)	4	04.05.03 (line bundle, alg-geom side)	△	Line bundles covered; Grothendieck's splitting theorem $E=⨁\mathcal{O}(a_i)$ not in a dedicated unit.
Horrocks monad construction on ${\mathbb{CP}}^3$	5	—	✗	Gap. Pure algebraic-geometry input to ADHM.
ADHM construction (Atiyah-Drinfeld-Hitchin-Manin 1978)	5	—	✗	Gap (high priority — the apex theorem of the book).
ADHM quadratic constraint	5	—	✗	Gap.
Moduli space ${\mathcal{M}}_k(S^4)$ of $k$-instantons	5–7	—	✗	Gap (high priority). Dimension formula $\dim {\mathcal{M}}_k=8k-3$ for $\mathrm{SU}(2)$.
Penrose transform at linear level	6	—	✗	Gap. Master-tier survey.
Sheaf cohomology $H^1({\mathbb{CP}}^3,\mathcal{O}(-2))$ → massless fields	6	04.06.02 + adjacent	△	Sheaf cohomology machinery present in alg-geom chapter; the Penrose-transform application is absent.
't Hooft ansatz	6	—	✗	Gap. Worked instanton family.
Radon transform relation	6	—	✗	Gap (low priority — survey pointer only).
Barth's theorem on rank-2 bundles on ${\mathbb{CP}}^3$	7	—	✗	Gap.
Reality constraints (real structure picks out $\mathrm{SU}(2)$-instantons)	7	—	✗	Gap.
Atiyah-Bott Yang-Mills over Riemann surfaces	—	03.07.05 (Master)	△	Referenced in 03.07.05 Master Historical context; not in Atiyah's Pisa book (later 1983 paper) and not in this audit's scope.

Aggregate coverage estimate. Counting the ~25 top-level Atiyah topics in the table:

• ✓ covered: 3
• △ partial: 6
• ✗ gap: 16

Coverage is roughly ~15–20% weighted by load-bearing, ~12% by raw topic count. The Yang-Mills action side is in good shape thanks to the single shipped 03.07.05 unit; the instanton side, the twistor / Ward side, and the ADHM apex are essentially absent. This is consistent with the chapter having only one shipped unit overall.

---

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

Priority 0 — strict prerequisites in the corpus or in sibling audits. The principal-bundle / connection / curvature / Chern-class backbone (03.05.01, 03.05.07, 03.05.09, 03.06.04, 03.06.06) is already shipped; the Kobayashi-Nomizu Vol. 2 sibling audit (FT 3.19, running this cycle) will refine the curvature-form and Chern-Weil treatment at master tier, which several P1 units below depend on for the topological-charge derivation. Soft blocker: the Kobayashi-Nomizu punch-list must not regress the 03.05.09 / 03.06.04 / 03.06.06 API surface.

Priority 1 — high-leverage, captures Atiyah's central content:

1. 03.07.06 Anti-self-dual (ASD) equation $F_A^{+}=0$ on a 4-manifold. Hodge-* eigenspace decomposition ${\Omega }^2(M;\mathfrak{g})={\Omega }_{+}^2\oplus {\Omega }_{-}^2$ in dimension 4. Statement of the ASD equation; proof that ASD connections are Yang-Mills (already a Master exercise in 03.07.05, promoted to standalone unit). Three-tier, ~1500 words. Atiyah Ch. 1 anchor; Donaldson-Kronheimer §2.1 anchor; Freed-Uhlenbeck §3 anchor.

2. 03.07.07 BPST instanton and the Bogomolny bound. Explicit $k=1$ instanton on ${\mathbb{R}}^4$: $A=\mathrm{Im}(\bar{q}dq)/(1+|q{|}^2)$ in quaternionic coordinates; finite-action extension to $S^4$; computation ${\int }_{S^4}\frac{1}{8{\pi }^2}\mathrm{tr}(F\wedge F)=1$; Bogomolny lower bound $\mathrm{YM}(A)\ge 8{\pi }^2|c_2(E)|$ with equality on (anti-)self-dual connections. Three-tier, ~1500 words. Atiyah Ch. 1–2 anchor; Belavin-Polyakov-Schwartz-Tyupkin 1975 originator citation.

3. 03.07.08 Conformal compactification $\mathbb{R}^4 \to S^4 \cong \mathbb{HP}^1$ and finite-action solutions. Stereographic projection, conformal invariance of Yang-Mills in dimension 4, ${\mathbb{HP}}^1$ as quaternionic projective line, and the theorem: finite-action ASD connections on ${\mathbb{R}}^4$ extend (after a gauge transformation) to smooth connections on $S^4$ (Uhlenbeck removable-singularities, 1982). Intermediate + Master. Atiyah Ch. 1 anchor; Uhlenbeck 1982 cited.

4. 03.07.09 Moduli space of ASD connections ${\mathcal{M}}_k(S^4)$. Definition (gauge-equivalence classes of irreducible ASD connections of charge $k$); slice theorem; deformation complex $0 \to \Omega^0(\mathfrak{g}_E) \to \Omega^1(\mathfrak{g}E) \to \Omega^2+(\mathfrak{g}_E) \to 0$; the dimension formula $\dim {\mathcal{M}}_k(S^4,\mathrm{SU}(2))=8k-3$ via Atiyah-Singer index theorem applied to this complex; reducibility and singular strata. Master-tier; Intermediate sketch only. Atiyah Ch. 5–7 anchor; Donaldson-Kronheimer §3 anchor; Freed-Uhlenbeck §4 anchor. Cross-link to 03.09.10 (Atiyah-Singer) — the dimension formula is its first gauge-theoretic application.

5. 03.07.10 ADHM construction (Atiyah-Drinfeld-Hitchin-Manin). The apex unit. Statement of the ADHM data: a quadruple of matrices $(B_1,B_2,I,J)$ with $B_i\in \mathrm{End}({\mathbb{C}}^k)$, $I\in \mathrm{Hom}({\mathbb{C}}^2,{\mathbb{C}}^k)$, $J\in \mathrm{Hom}({\mathbb{C}}^k,{\mathbb{C}}^2)$, satisfying the real and complex ADHM equations; explicit construction of the instanton from the ADHM datum via the Dirac operator on the trivial ${\mathbb{C}}^{2k+2}$-bundle; the Atiyah-Drinfeld-Hitchin- Manin theorem (1978): every $\mathrm{SU}(2)$ $k$-instanton on $S^4$ arises from an ADHM datum, unique up to $\mathrm{U}(k)$ action. Master-tier full proof sketch; Intermediate tier states the result and exhibits the $k=1$ specialisation (recovers BPST). Atiyah Ch. 5–7 anchor; Atiyah-Hitchin-Drinfeld-Manin 1978 Phys. Lett. A 65 originator citation; Donaldson-Kronheimer §3 and §6 cross-references.

Priority 2 — twistor / Ward bridge (essential to Atiyah's framing):

6. 03.07.11 Penrose twistor space and the Ward correspondence. Twistor space ${\mathbb{CP}}^3\to S^4$ with real structure; twistor lines; the Atiyah-Ward theorem: ASD $\mathrm{SU}(2)$-bundles on $S^4$ correspond bijectively to rank-2 holomorphic bundles on ${\mathbb{CP}}^3$ trivial on twistor lines and carrying a real structure. Master-tier survey unit; Intermediate tier states the correspondence and gives one worked direction. Atiyah Ch. 3–4 anchor; Atiyah-Ward 1977 originator citation.

7. 03.07.12 Horrocks monad and rank-2 bundles on ${\mathbb{CP}}^3$. The algebraic-geometry input: every rank-2 bundle on ${\mathbb{CP}}^3$ with $c_1=0$ arises from a monad $\mathcal{O}(-1{)}^k\to {\mathcal{O}}^{2k+2}\to \mathcal{O}(1{)}^k$ modulo the monad-equivalence relation; ties to Barth's theorem. Connects 03.07.10 (ADHM) to standard algebraic geometry of ${\mathbb{CP}}^n$. Master-only, ~1500 words. Atiyah Ch. 5 + 7 anchor.

Priority 3 — Master-tier deepenings:

8. 03.07.13 Grothendieck's splitting theorem on ${\mathbb{CP}}^1$. Every holomorphic vector bundle on ${\mathbb{CP}}^1$ splits as $⨁\mathcal{O}(a_i)$. Used by Atiyah Ch. 4 ("Bundles over ${\mathbb{P}}^1(\mathbb{C})$") as the local twistor-line model. Standalone short unit (~1200 words); could also be sited in 04-algebraic-geometry/05-divisors/ if the editorial choice favours that chapter. Grothendieck 1957 originator citation.

9. 03.07.14 Penrose transform at linear level. Massless field equations on $S^4$ ↔ $H^1({\mathbb{CP}}^3,\mathcal{O}(-2))$ etc. Atiyah Ch. 6 anchor. Master-only pointer unit. Includes the 't Hooft ansatz as the worked example.

Priority 4 — survey pointers (Master-only, not load-bearing):

10. 03.07.15 Barth's theorem and reality constraints on ${\mathbb{CP}}^3$. Pointer unit closing the algebraic-geometry loop. Atiyah Ch. 7 anchor. Optional; not required for FT-equivalence.

11. Master section to 03.07.05 on the Atiyah-Bott Riemann-surface picture as the dimension-2 analogue. Already partially shipped (Synthesis paragraph); upgrade to a dedicated Master subsection rather than a new unit.

---

§4 Implementation sketch (P3 → P4)

For Atiyah-Pisa coverage to reach FT-equivalence, items 1–5 are the minimum set. Realistic production estimate (mirroring Lawson-Michelsohn and Milnor-Stasheff batches):

• ~3.5–4 hours per unit. Atiyah-Pisa units skew higher than the corpus average because the master tier requires careful coordination across differential geometry (curvature, Hodge-*), algebraic geometry (sheaf cohomology, monads), and index theory (deformation-complex dimension formula).
• Priority 1: 5 units × ~3.75 hours = ~19 hours.
• Priority 2: 2 units × ~3.5 hours = ~7 hours.
• Priority 3: 2 units × ~3 hours = ~6 hours.
• Priority 4: ~2 hours total.
• Total: ~34 hours of focused production. Fits a focused 5-day window.

Originator-prose target. Atiyah is himself an originator of the key results in chapters 4–7 (Ward correspondence with Ward; ADHM with Drinfeld, Hitchin, Manin). Units 5 and 6 should carry originator-prose treatment per docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §10, citing:

• C. N. Yang, R. L. Mills, "Conservation of isotopic spin and isotopic gauge invariance," Physical Review 96 (1954) 191–195 — originating the Yang-Mills action; already cited from 03.07.05.
• A. A. Belavin, A. M. Polyakov, A. S. Schwartz, Yu. S. Tyupkin, "Pseudoparticle solutions of the Yang-Mills equations," Physics Letters B 59 (1975) 85–87 — originating the BPST instanton.
• M. F. Atiyah, R. S. Ward, "Instantons and algebraic geometry," Comm. Math. Phys. 55 (1977) 117–124 — originating the twistor / Ward reformulation.
• M. F. Atiyah, V. G. Drinfeld, N. J. Hitchin, Yu. I. Manin, "Construction of instantons," Physics Letters A 65 (1978) 185–187 — originating the ADHM construction.
• M. F. Atiyah, Geometry of Yang-Mills Fields, Lezioni Fermiane, Scuola Normale Superiore Pisa, 1979 — the consolidated monograph.

Notation crosswalk. Atiyah uses $A$ for the connection and $F$ for the curvature (matching the Codex's 03.05.07–03.05.09 convention). Twistor-side uses ${\mathbb{P}}^3={\mathbb{CP}}^3$ over $\mathbb{C}$ with the standard real structure $\sigma : [z_0, z_1, z_2, z_3] \mapsto [\bar{z}_1, -\bar{z}_0, \bar{z}_3, -\bar{z}_2]$ (quaternionic conjugation); ADHM data typically written $(B_1,B_2,I,J)$ in the modern (Nakajima) convention, or as $(\alpha ,\beta ,a,b)$ in Atiyah's notation — record both in the §Notation paragraph of 03.07.10. Quaternionic coordinates $q=x_1+x_{2}i+x_{3}j+x_{4}k\in \mathbb{H}$ identify ${\mathbb{R}}^4$ with $\mathbb{H}$; the BPST instanton is $A=\mathrm{Im}(\bar{q}dq)/(1+|q{|}^2)$ in this convention.

---

§5 What this plan does NOT cover

• Donaldson invariants of smooth 4-manifolds. The moduli-of-instantons machinery feeds them, but the invariants themselves and the orientability / compactification arguments are deferred to FT 3.06 Donaldson Floer audit (separate plan stub).
• Seiberg-Witten theory. Deferred to a later wave; not in the Atiyah-Pisa book at all (post-dates it by ~15 years).
• Floer homology in gauge theory. FT 3.06 territory.
• A line-number-level exercise inventory of Atiyah's lectures. The book has no formal exercise sections; this audit treats the worked examples in chapters 2 + 5 as the implicit "exercise pack" and reproduces them in the master tier of the punch-list units.
• The Atiyah-Bott Yang-Mills-over-Riemann-surfaces story (1983 paper, post-dates Pisa). Touched in 03.07.05 Master tier; a dedicated Atiyah-Bott audit is appropriate at FT entry to be determined.
• Hyperkähler / quiver-variety modern reformulation of ADHM (Nakajima, 1990s). Pointer in 03.07.10 Master Connections only.

---

§6 Acceptance criteria for FT equivalence (Atiyah Pisa)

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

• The Kobayashi-Nomizu Vol. 2 sibling audit has not regressed 03.05.09 / 03.06.04 / 03.06.06 (soft prereq).
• ≥95% of Atiyah's named theorems in chapters 1–7 map to Codex units (currently ~15–20%; after priority-1 units rises to ~65%; after priority-1+2 to ~85%; full ≥95% requires priority-1+2+3 + selective priority-4).
• ≥90% of Atiyah's worked computations (BPST one-instanton in Ch. 2, Horrocks monad in Ch. 5, ADHM construction in Ch. 5–7, 't Hooft ansatz in Ch. 6) appear directly in a unit or are referenced by a unit covering them.
• Notation decisions are recorded (see §4) in 03.07.10 and 03.07.11.
• Pass-W weaving connects the new units to 03.07.05 (Yang-Mills action), 03.05.09 (curvature, Bianchi), 03.06.04 (Chern classes), and 03.09.10 (Atiyah-Singer, for the moduli dimension formula).

The 5 priority-1 units close the load-bearing equivalence gap on the differential-geometric and ADHM sides. Priority-2 closes the twistor bridge. Priority-3+4 are deepenings.

---

§7 Sourcing

• Print. Edizioni della Normale (Scuola Normale Superiore Pisa), ISBN 978-88-7642-303-1. Available from the publisher and the usual resellers. Status in docs/catalogs/FASTTRACK_BOOKLIST.md: BUY.
• PDF mirror. A scanned copy is widely linked from academic mirrors (Edinburgh / Ranicki archive; researchgate; inspirehep #150867). Not bundled with the Codex; not licensed for redistribution. Add to reference/textbooks-extra/ as Atiyah-GeometryOfYangMillsFields-Pisa1979.pdf if a clean copy is obtained, mirroring the pattern of Lawson-Michelsohn and similar monographs in reference/textbooks-extra/.
• Companion texts (free or already in the corpus). Tong's gauge theory lectures (reference/tong/md/pages/gaugetheory.md, already cited from 03.07.05) for physics-style background; Donaldson-Kronheimer The Geometry of Four-Manifolds (1990) and Freed-Uhlenbeck Instantons and Four-Manifolds (1991) as the standard modern companions — both flagged for the corpus and cross-referenced from 03.07.05.
• License. Atiyah-Pisa is a commercial publication; cite by ISBN. Free academic copies on Edinburgh and researchgate mirrors are for individual reading only.

---

Reduced audit pass (no direct PDF access in this cycle). Chapter structure reconstructed from the published table of contents and the canonical companion texts. Full P1 line-number inventory deferred pending direct PDF access. Soft prerequisite: Kobayashi-Nomizu Vol. 2 sibling audit running this cycle must not regress the curvature / Chern-Weil API.