Brylinski — Loop Spaces, Characteristic Classes and Geometric Quantization (Fast Track 3.47) — Audit + Gap Plan

Book: Jean-Luc Brylinski, Loop Spaces, Characteristic Classes and Geometric Quantization (Progress in Mathematics Vol. 107, Birkhäuser Boston 1993, xvi + 300 pp.; reprinted in Modern Birkhäuser Classics, Birkhäuser/Springer 2008, ISBN 978-0-8176-4730-8). Not openly hosted by the author or publisher; paywalled via SpringerLink (DOI 10.1007/978-0-8176-4731-5).

Fast Track entry: 3.47 (Modern Geometry block; flagged BUY in docs/catalogs/FASTTRACK_BOOKLIST.md with topic tags "Higher cohomology, stacks").

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 to write so that Brylinski is covered to the Fast Track equivalence threshold (≥95% effective coverage of theorems, key examples, exercise pack, notation, sequencing, intuition, applications — see docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §3.4).

This pass is intentionally not a full line-number P1 audit. The book is a 300-page research monograph synthesising sheaf hypercohomology, gerbe theory and loop-group representation theory; a complete inventory at line-number granularity is a multi-week job and is deferred to a dedicated audit pass after the priority-1 punch-list ships. This plan works from the book's canonical 7-chapter outline and Brylinski's distinctive editorial choices, produces the gap punch-list, and stops there.

Reduced-source note. No local PDF was available at the three project paths checked (reference/fasttrack-texts/, ~/Downloads/, ~/Documents/Code Projects/B.I.B.L.E/). WebFetch attempts against SpringerLink and Google Books were either redirected to authentication walls or returned 404. The Birkhäuser/Springer publisher page, ncatlab.org/nlab entries on Deligne cohomology, gerbe, Bundle gerbe and Jean-Luc Brylinski, and the canonical citation pattern of the book in later literature (Murray 1996, Hitchin 1999, Bunke-Schick 2012) were used to reconstruct the chapter outline. This plan is therefore marked REDUCED: the topic taxonomy below is reliable (the book's outline is well-known to specialists) but per-chapter section numbers and specific named-theorem inventories cannot be cited at line-number granularity until a copy is acquired.

§1 What Brylinski is for

Brylinski 1993 is the canonical bridge between Deligne's sheaf hypercohomology and the geometric (bundle / gerbe / line-bundle-on-loop- space) realisations that string theory and higher gauge theory later needed. Where Pressley–Segal Loop Groups (1986) is the canonical text on loop groups as infinite-dimensional Lie groups and their positive-energy representations, and Kac Infinite-Dimensional Lie Algebras is the canonical text on the affine-algebra side, Brylinski is the canonical text that introduces smooth Deligne cohomology, develops gerbes with band $U (1)$ as a geometric model for $H^{3} (X, Z)$ , and shows how line bundles over the free loop space $L M$ are controlled by gerbes on $M$ — the geometric core of the WZW central extension.

Distinctive contributions, in roughly the order Brylinski develops them across the 7 chapters:

Complexes of sheaves and hypercohomology (Ch. 1). A self-contained development of sheaf hypercohomology, the Čech-de-Rham double complex, and the smooth Deligne complex $\mathbb{Z}(p)_D^\infty = [\underline{\mathbb{Z}} \to \underline{\Omega}^0 \to \underline{\Omega}^1 \to \cdots \to \underline{\Omega}^{p-1}]$ as the differential-cohomology refinement of integral cohomology. This is the foundation the rest of the book builds on. Originating reference: Deligne, Théorie de Hodge II, Pub. IHES 40 (1971) — but Brylinski's smooth-manifold reformulation is the reference standard.
Line bundles and geometric quantization (Ch. 2). Kostant–Weil isomorphism $H^2(M, \mathbb{Z}) \cong {\text{iso classes of Hermitian line bundles} / M} $, t h es m oo t h D e l i g n eco m pl e x$ \mathbb{Z}(2)_D^\infty$ classifying line bundles with connection, and the Kostant–Souriau prequantum line bundle. This recasts geometric quantization as a Deligne-cohomology theorem, not a hand-built construction.
Kähler geometry of the space of loops / knots (Ch. 3). The free loop space $L M$ and (Brylinski's variant) the space of unparameterised smooth knots in $R^{3}$ are infinite-dimensional Kähler manifolds, with the Kähler form constructed from a transgression of a 3-form on $M$ . This sets up the geometric side of the loop-group story.
Degree-3 cohomology, Dixmier–Douady theory (Ch. 4). $H^3(M, \mathbb{Z})$ classifies infinite-dimensional Hilbert-bundle / continuous- trace- $C^{*}$ -algebra structures (Dixmier–Douady class). Brylinski rebuilds the classification using the smooth Deligne complex $Z (3)_{D}^{\infty}$ . This is the "what $H^{3}$ classifies" chapter and is the conceptual entry into gerbes.
Sheaves of groupoids / gerbes with band $U (1)$ (Ch. 5). The centrepiece. A gerbe with band $U (1)$ on $M$ is a sheaf of groupoids satisfying local-non-emptiness and local-connectedness, whose isomorphism classes are in bijection with $H^{3} (M, Z)$ (the Brylinski–Giraud theorem). A gerbe with connective structure and curving corresponds to smooth Deligne cohomology $H^{3} (M, Z (3)_{D}^{\infty})$ . This is the canonical geometric model that Murray (1996) later reformulated as bundle gerbes — Brylinski's stacky / sheaf-of-groupoids form and Murray's submersion-with-line-bundle form are equivalent but differ in computability; both descend from this chapter.
Line bundles over loop spaces (Ch. 6). A gerbe on $M$ with band $U (1)$ canonically transgresses to a line bundle on $L M$ ; conversely line bundles on $L M$ pull back from gerbes on $M$ when sufficiently $S^{1}$ -equivariant. This is the transgression theorem, geometrising the cohomological transgression $H^{3} (M) \to H^{2} (L M)$ .
Gerbes and central extensions of loop groups (Ch. 7). The level- $k$ central extension $L G \to L G$ of a simple compact Lie group $G$ is the transgression of the level- $k$ gerbe on $G$ (equivalently, the gerbe with Dixmier–Douady class $k$ times the generator of $H^{3} (G, Z) ≅ Z$ ). Brylinski gives this construction directly from the Deligne / gerbe machinery, recovering Pressley–Segal's central extension geometrically.

Brylinski is not a first introduction to anything it covers. It assumes sheaf cohomology (Hartshorne / Iversen), de Rham theory (Bott–Tu), Lie groups and Lie algebras, basic line-bundle geometry (Kobayashi–Nomizu), and the existence of the free-loop-space formalism (Pressley–Segal, Kac). It is the canonical text for the specific bridge: Deligne cohomology ⇔ gerbes ⇔ loop-space line bundles ⇔ loop-group central extensions.

Peer sources confirming this framing (≥3 required):

Pressley & Segal, Loop Groups (Oxford University Press 1986). The canonical text on $L G$ , positive-energy representations and central extensions from the Lie-theoretic side — Brylinski Ch. 7 is the geometric / sheaf-cohomological complement.
Bunke & Schick, "Smooth K-theory" (Astérisque 328, 2010) and follow-ups; these treat differential cohomology / smooth Deligne cohomology as the universal example, citing Brylinski Ch. 1 as the reference standard for the smooth Deligne complex on a manifold.
Hitchin, "Lectures on special Lagrangian submanifolds" (in Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds, AMS/IP Studies in Advanced Math. 23, 2001) — the first widely-circulated notes that pushed gerbes (in Brylinski's sense) into the string-theory / mirror-symmetry mainstream; cites Brylinski Ch. 5 as the geometric reference for the $B$ -field.
Murray, "Bundle gerbes" (J. London Math. Soc. 54 (1996) 403–416). The geometric reformulation: a submersion $Y \to M$ together with a line bundle on $Y \times_{M} Y$ satisfying a cocycle condition. Murray opens by citing Brylinski as the originating treatment of $H^{3}$ -classified geometric objects and frames bundle gerbes as a computable alternative.

§2 Coverage table (Codex vs Brylinski)

Cross-referenced against the current Codex corpus (per file scan of content/03-modern-geometry/{04,05,06,07,08,11,12,13}-*/ and content/05-symplectic/). ✓ = covered; △ = partial / different framing; ✗ = not covered.

Brylinski topic	Chapter	Codex unit(s)	Status	Note
Sheaf hypercohomology, Čech–de-Rham double complex	1	`03.04.11-cech-de-rham.md`	△	Single-complex Čech-de-Rham only; gap: hypercohomology of a complex of sheaves; spectral sequence of the Čech-hypercohomology bicomplex.
Smooth Deligne complex $Z (p)_{D}^{\infty}$	1	—	✗	Gap (high priority — Brylinski's foundational tool; load-bearing for everything downstream).
Deligne / differential cohomology $H^{p} (M, Z (p)_{D}^{\infty})$	1	—	✗	Gap (high priority).
Hermitian line bundles classified by $H^{2} (M, Z)$ (Kostant–Weil)	2	`03.05.08-complex-vector-bundle.md`, `03.06.04-pontryagin-chern-classes.md`	△	First Chern class as obstruction is implicit; the Kostant–Weil iso (line bundles $≅ H^{2} (\cdot, Z)$ ) is not stated.
Line bundles with connection $\leftrightarrow$ $H^{2} (M, Z (2)_{D}^{\infty})$	2	—	✗	Gap.
Prequantum line bundle / Kostant–Souriau geometric quantization	2	`05.03.01-coadjoint-orbit.md` (KKS form only)	△	KKS symplectic form on coadjoint orbits is named but the prequantization condition $[ω /2 π] \in H^{2} (M, Z)$ and the construction of the prequantum line bundle are not units.
Kähler structure on $L M$ / space of knots	3	—	✗	Gap. Loop-space geometry is absent from `03.11-infinite-dim-lie/` (which covers algebraic central extensions only).
Transgression $H^{p + 1} (M) \to H^{p} (L M)$	3, 6	—	✗	Gap.
$H^{3} (M, Z)$ as classifying group (Dixmier–Douady)	4	—	✗	Gap (high priority).
Continuous-trace $C^{*}$ -algebras, infinite-dim Hilbert bundles	4	—	✗	*Gap (low priority — operator-algebra side, defer to a $C^{}$ track).**
Gerbes with band $U (1)$ / sheaves of groupoids	5	—	✗	Gap (highest priority — book's centrepiece).
Connective structure + curving on a gerbe	5	—	✗	Gap (high priority).
Bundle gerbes (Murray 1996 reformulation)	5 (cf.)	—	✗	Gap (high priority — the computable form; cite Murray as originator).
$B$ -field / Kalb-Ramond field as a gerbe connection	5–6	—	✗	Gap. Codex has no string-theory $B$ -field unit.
Transgression: gerbe on $M$ $\to$ line bundle on $L M$	6	—	✗	Gap (high priority — the geometric content of Brylinski).
Central extension $L G \to L G$ (algebraic 2-cocycle form)	7	`03.11.01-central-extension.md`, `03.11.02-infinite-dim-lie-reps.md`, `03.11.03-virasoro-algebra.md`	△	Central extension is defined at the Lie-algebra cocycle level only (Kac / Pressley–Segal §4). Brylinski's geometric construction from a gerbe on $G$ is not present.
Level- $k$ gerbe on a simple compact Lie group	7	—	✗	Gap. Connects WZW level to $H^{3} (G, Z) ≅ Z$ .
WZW action and the level- $k$ central extension	7 (cf.)	`03.10.02-cft-basics.md`	△	CFT basics unit exists but does not connect to the gerbe / Deligne picture.
Twisted K-theory pointer (Freed–Hopkins–Teleman)	— (post-Brylinski)	—	✗	Gap (low priority — Master-tier pointer; defer to FHT-specific audit).

Aggregate coverage estimate: ~5% of Brylinski has corresponding Codex units, and most of that 5% is the algebraic / Lie-cocycle side of Ch. 7 (via the three units in 03.11-infinite-dim-lie/). The entire Deligne-cohomology and gerbe apparatus (Chs. 1, 4, 5, 6) is absent. The gap is essentially total for the book's central content.

Prerequisite-blocker note: the Codex has solid bundle/connection foundations (03.05-bundles/, 03.06-characteristic-classes/) and Čech cohomology (03.04.11-cech-de-rham.md, 06.04.02-cech-cohomology-line-bundles.md), which is the minimum substrate. No new prereq stub is needed before starting the Brylinski punch-list — the gap is in the upper-storey construction (hypercohomology of complexes of sheaves, smooth Deligne, gerbes), not in the foundations.

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

Priority 0 — no external blockers. The Codex's existing 03.04.11-cech-de-rham.md, 03.05.*-bundles/, 03.06.*-characteristic-classes/ and 03.11.01-central-extension.md are sufficient prereq substrate.

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

03.04.14 Hypercohomology of a complex of sheaves. Definition via Čech bicomplex; the two spectral sequences. Brylinski Ch. 1 anchor. Standalone unit (~1500 words). Foundational for everything downstream.
03.06.07 Smooth Deligne complex $Z (p)_{D}^{\infty}$ and differential cohomology $\hat{H}^{p} (M, Z)$ . Definition, Bockstein-style exact sequences relating $\hat{H}^{p}$ to ordinary $H^{p}$ and to differential forms with integral periods. Brylinski Ch. 1 anchor; Bunke–Schick for the modern reformulation. Master-tier required (~2000 words). Load-bearing for items 3, 5, 6, 7.
03.06.08 Kostant–Weil isomorphism and prequantum line bundle. $H^{2} (M, Z) ≅ {Hermitian line bundles} / \sim$ ; line bundles with connection $≅ H^{2} (M, Z (2)_{D}^{\infty})$ ; prequantization condition $[ω /2 π] \in H^{2} (M, Z)$ and construction of the prequantum line bundle on a symplectic manifold. Brylinski Ch. 2; Woodhouse Geometric Quantization as the standard pedagogical reference. Three-tier (~2000 words).
03.11.04 Free loop space $L M$ and the transgression $H^{p + 1} (M) \to H^{p} (L M)$ . Smooth structure on $L M$ (Frechet manifold), $S^{1}$ -action, transgression at the level of de Rham complexes. Brylinski Ch. 3, 6. Three-tier (~1500 words).
03.06.09 Dixmier–Douady class and $H^{3} (M, Z)$ . What $H^{3}$ classifies: continuous-trace $C^{*}$ -algebra bundles (statement only — operator-algebra proofs deferred), and (the modern reading) $U (1)$ -gerbes. Brylinski Ch. 4 anchor; Murray 1996 for the bundle-gerbe reformulation cited in §4. Master-tier (~1500 words).
03.06.10 Bundle gerbe (Murray) and gerbe with band $U (1)$ (Brylinski). Two parallel definitions, equivalence statement, classification by $H^{3} (M, Z)$ . Connective structure (curving) and the lift to $H^{3} (M, Z (3)_{D}^{\infty})$ for gerbes with connection. Brylinski Ch. 5 + Murray 1996 as joint anchors. Master-tier (~2500 words; the central unit of this punch-list).
03.06.11 Transgression of a gerbe to a line bundle on the loop space. Statement and construction: a $U (1)$ -gerbe on $M$ with connective structure gives a line bundle with connection on $L M$ ; curvature of the line bundle is the transgressed 3-curvature of the gerbe. Brylinski Ch. 6. Master-tier (~1500 words).
03.11.05 Geometric construction of $L G \to L G$ from the level- $k$ gerbe on $G$ . The Brylinski Ch. 7 construction: level- $k$ generator of $H^{3} (G, Z) ≅ Z$ transgresses to the level- $k$ line bundle on $L G$ , whose total space (with the obvious group structure) is the central extension. Connects to existing 03.11.01-central-extension.md, 03.11.02-infinite-dim-lie-reps.md. Master-tier (~2000 words).

Priority 2 — physics-side connection units:

03.07.16 $B$ -field as a gerbe connection. Pointer unit connecting 03.06.10 to string-theory / sigma-model literature. Hitchin 1999 / 2001 anchor. Intermediate + Master (~1200 words).
03.10.03 WZW action and the level- $k$ extension via Brylinski's geometric construction. Builds on existing 03.10.02-cft-basics.md. Cross-links to 03.11.05. Master-only (~1500 words).

Priority 3 — pointer / Master-tier deepening:

03.06.12 Twisted K-theory and the Freed–Hopkins–Teleman theorem (pointer unit). $K_{G}^{*} (G)$ at level $k$ equals the Verlinde ring; builds on 03.06.10 (gerbes / twists) and 03.08.* (K-theory). Freed–Hopkins–Teleman 2003/2007/2011 papers as originators. Pointer only; defers proof to a dedicated Freed audit. Master-only (~1000 words).

Priority 4 — deferred / explicitly NOT in scope here:

Higher gerbes ( $n$ -gerbes for $n \geq 2$ ) — defer until 2-gerbes / $\infty$ -stack track is opened.
TQFT-specific gerbe applications (Dijkgraaf–Witten, Chern–Simons at the prequantum level) — defer to Freed FT 3.05 audit per the task's explicit non-goal.
The cubical / simplicial-sheaf reformulation of Deligne cohomology — defer; not load-bearing for Brylinski.
Continuous-trace $C^{*}$ -algebra side of Dixmier–Douady — defer to a $C^{*}$ / operator-algebra track.

§4 Implementation sketch (hour estimates, originator citations)

For full Brylinski coverage at the FT-equivalence threshold, the minimum set is items 1–8 (8 priority-1 units). Realistic production estimate (mirroring earlier Bott–Tu / Lawson–Michelsohn / Berline–Getzler–Vergne batches in plans/fasttrack/):

~3–4 hours per unit for definitional / Beginner+Intermediate-tier units (items 1, 4, 9).
~5–6 hours per unit for Master-tier units carrying nontrivial constructions (items 2, 3, 5, 6, 7, 8, 10).
Priority-1 total: 1 × 4h + 1 × 6h + 1 × 5h + 1 × 4h + 1 × 5h + 1 × 6h + 1 × 5h + 1 × 6h = ~41 hours focused production.
Priority-2 total: ~4h + ~5h = ~9 hours.
Priority-3 pointer: ~2 hours.
Grand total: ~52 hours of focused production for FT equivalence
- Master-tier extras. Fits a 7-day production window with reserve.

Originator-prose target. Brylinski is the originator of the smooth Deligne / gerbe synthesis. The Master tier of items 1, 2, 5, 6, 7, 8 must carry originator-prose treatment per docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §10, citing:

P. Deligne, "Théorie de Hodge II," Publications mathématiques de l'I.H.É.S. 40 (1971) 5–58 — originating Deligne cohomology (analytic / algebraic-variety setting).
P. Deligne, "Théorie de Hodge III," Publications mathématiques de l'I.H.É.S. 44 (1974) 5–77 — continuation; mixed Hodge structures.
J.-L. Brylinski, Loop Spaces, Characteristic Classes and Geometric Quantization, Progress in Math. 107, Birkhäuser 1993 — the book itself; originating reference for smooth Deligne cohomology, for gerbes with band $U (1)$ in the differential-geometric setting, and for the gerbe → loop-space-line-bundle → central-extension chain.
M. K. Murray, "Bundle gerbes," Journal of the London Mathematical Society 54 (1996) 403–416 — originating the bundle-gerbe (submersion + line bundle on the fibre product) reformulation; the computable form used in physics.
N. Hitchin, "Lectures on special Lagrangian submanifolds," in Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds (Cumrun Vafa & S.-T. Yau eds.), AMS/IP Studies in Advanced Math. 23 (2001) — first widely-cited treatment putting gerbes (Brylinski's sense) into the string-theory mainstream; originating the modern $B$ -field / gerbe dictionary.
D. S. Freed, M. J. Hopkins & C. Teleman, "Loop groups and twisted K-theory" series, Journal of Topology 4 (2011) 737–798 and earlier parts — originating the FHT theorem identifying twisted equivariant K-theory of $G$ at level $k$ with the Verlinde ring of $L G$ positive- energy representations at level $k$ . Pointer-only in priority-3 item 11.

Notation crosswalk. Brylinski writes $Z (p)_{D}^{\infty}$ for the smooth Deligne complex (with $\infty$ marking the smooth, not analytic / algebraic, version) and uses the truncation $[\underline{\mathbb{Z}} \to \underline{\Omega}^0 \to \cdots \to \underline{\Omega}^{p-1}]$. Bunke–Schick and the differential-cohomology literature write $\hat{H}^{p} (M, Z)$ for $H^{p} (M, Z (p)_{D}^{\infty})$ . Murray writes $G \to M$ for a bundle gerbe; Brylinski writes $C \to M$ for a sheaf of groupoids (gerbe). The Codex notation decision (per docs/specs/UNIT_SPEC.md §11) should: adopt Brylinski's $Z (p)_{D}^{\infty}$ in 03.06.07 for the complex, adopt the differential-cohomology $\hat{H}^{p}$ in unit bodies as the headline name, and use bundle gerbe as the headline term in 03.06.10 (citing Brylinski's gerbe with band $U (1)$ as the equivalent sheaf-of-groupoids form). Record this in a §Notation paragraph of 03.06.07 and 03.06.10.

§5 What this plan does NOT cover

A line-number-level inventory of every named proposition / theorem in Brylinski (full P1 audit; deferred — book is 300 pp. and no local PDF was available in this pass).
The exercise pack. Brylinski has worked-example-style exposition rather than an exercise pack per chapter; exercises for the punch-list units will need to be generated during P3 production, not transcribed. Deferred to a follow-up exercise sweep after the priority-1 units ship.
Higher gerbes ( $n$ -gerbes for $n \geq 2$ ). Brylinski touches these in passing in Ch. 5 but does not develop the theory. Defer to a dedicated $\infty$ -stack / higher-gauge-theory audit.
TQFT-specific applications of the gerbe / Deligne machinery (Dijkgraaf–Witten as a gerbe-cohomology theory, Chern–Simons at the prequantum level, anomaly inflow). These belong to the Freed FT 3.05 audit (plans/fasttrack/freed-lectures-field-theory-topology.md) and are explicitly out of scope for this plan per the task brief.
The operator-algebra side of Dixmier–Douady (continuous-trace $C^{*}$ -algebras). Defer to a $C^{*}$ / non-commutative-geometry track.
The full Freed–Hopkins–Teleman theorem proof; only a pointer unit (item 11) is in scope.

§6 Acceptance criteria for FT equivalence (Brylinski)

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

The 8 priority-1 units ship to the standard tier-rubric. Item 6 (03.06.10 bundle gerbe / gerbe with band $U (1)$ ) is the single highest-impact unit and must include Brylinski-Murray equivalence as a stated theorem plus at least one worked computation (the level- $k$ gerbe on $SU (2)$ is the canonical choice).
≥95% of Brylinski's named theorems in Chs. 1, 2, 4, 5, 6, 7 map to Codex units (currently ~5%; after priority-1 this rises to ~75%; after priority-1 + priority-2 to ~90%; ≥95% additionally needs the exercise-pack sweep noted in §5 and the priority-3 pointer for FHT).
≥90% of Brylinski's signature worked examples map: the prequantum line bundle on a symplectic manifold (Ch. 2), the Kähler form on the space of knots (Ch. 3 — Master-tier example only), the level- $k$ gerbe on $SU (2)$ / $SU (n)$ (Ch. 7), and the resulting central extension of $L SU (n)$ at level $k$ .
Notation decisions in §4 are recorded in the relevant units' §Notation sections.
Pass-W weaving connects the new units laterally to: (i) 03.04.11-cech-de-rham.md ← items 1, 2; (ii) 03.05.07-principal-bundle-connection.md ← items 2, 6; (iii) 03.06.04-pontryagin-chern-classes.md ← items 3, 5; (iv) 03.11.01-central-extension.md, 03.11.02-infinite-dim-lie-reps.md ← item 8; (v) 05.03.01-coadjoint-orbit.md ← item 3; (vi) 03.10.02-cft-basics.md ← item 10.

Priority-1 alone closes the bulk of the equivalence gap. Priority-2 closes the physics-side connection gap. Priority-3 + exercise sweep close to ≥95%.

§7 Sourcing

NOT FREE. Brylinski has not openly hosted the book; the publisher (Birkhäuser/Springer) paywalls the 2008 Modern Birkhäuser Classics reprint via SpringerLink (DOI 10.1007/978-0-8176-4731-5; ISBN 978-0-8176-4730-8). Print is in stock at the publisher (~$70 USD list, often less via institutional access).
Acquire before P3 production. A local PDF (or institutional access via the user's library) is required before writing items 1, 2, 5, 6, 7 at Master tier — these units depend on Brylinski's notation and proof organisation, and the reduced-source reconstruction in this plan cannot substitute for the canonical text at line-number granularity.
Local copy target. When acquired, place at reference/fasttrack-texts/03-modern-geometry/Brylinski-LoopSpacesCharacteristicClassesGeometricQuantization.pdf to mirror the pattern of other 3.x FT texts (Bott, Freed, May, Sternberg already present in that directory).
Companion sources (used to triangulate Brylinski's content in this reduced pass; all should be cited in the units below per docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §10):
- Pressley & Segal, Loop Groups (OUP 1986) — already referenced in 03.11.0[1-3]; cross-reference from item 8.
- Murray, "Bundle gerbes," J. London Math. Soc. 54 (1996) — primary secondary source for item 6.
- Bunke & Schick, "Smooth K-theory" (Astérisque 328, 2010) — primary secondary source for item 2.
- Hitchin, "Lectures on special Lagrangian submanifolds" (2001) — primary secondary source for item 9.
- Freed, Hopkins & Teleman, "Loop groups and twisted K-theory" series (2003 / 2007 / 2011) — primary secondary source for item 11 (pointer-only).
Reduced-pass marker. This plan was produced WITHOUT a local PDF copy of Brylinski. The chapter-by-chapter outline used here is the canonical 7-chapter structure of the book as cited in the secondary literature above; it is reliable at the topic-taxonomy level but not at the level of specific theorem numbers, page references, or proof ordering. The first action on acquiring the PDF is to upgrade §2's coverage table with Brylinski's per-theorem numbering and revise item-by-item word-counts in §4 accordingly.

Brylinski — *Loop Spaces, Characteristic Classes and Geometric Quantization* (Fast Track 3.47) — Audit + Gap Plan