Freed — Lectures on Field Theory and Topology (Fast Track 3.05) — Audit + Gap Plan

Book: Daniel S. Freed, Lectures on Field Theory and Topology, CBMS Regional Conference Series in Mathematics, No. 133, American Mathematical Society / NSF–CBMS, 2019 (xi + 186 pp.). Author-hosted preliminary PDF (permission-of-publisher version) is the version inspected here.

Local copy used for audit: reference/book-collection/free-downloads/Freed-CBMS_Field_Theory_and_Topology.pdf (186 pages, preliminary version). The file reference/fasttrack-texts/03-modern-geometry/Freed-AspectsOfFieldTheory.pdf is a different, earlier Freed text and is not the FT 3.05 source.

Fast Track entry: 3.05 (Modern Geometry tier). Listed in docs/catalogs/FASTTRACK_BOOKLIST.md line for 3.05 as "TQFT, cobordism." Catalog flagged availability as "? (AMS may have partial)" — the preliminary author PDF above resolves the question.

Purpose of this plan: P1-lite audit-and-gap pass mirroring plans/fasttrack/brown-higgins-sivera-nonabelian-algebraic-topology.md. Produces a priority-ordered punch-list of new Codex units required for FT-equivalence coverage. Stops short of a line-number P1 inventory; the book is 186 pp. of dense survey-level prose and a fuller inventory is a follow-up. This pass uses the TOC, the Freed-Hopkins program structure, and the canonical literature for the gap mapping.

§1 What this book is for

Freed's Lectures on Field Theory and Topology is the canonical modern mathematical introduction to topological quantum field theory (TQFT) from the categorical / cobordism-axiomatic perspective, written by one of the field's central figures. The motivating problem (stated on p. 1 of the introduction) is from condensed-matter physics: classify invertible gapped phases of matter. The mathematical translation is the Freed–Hopkins program: invertible TQFTs (with a given tangential structure $H$ , dimension $n$ ) form a Picard groupoid / connective spectrum, and deformation classes are computed by stable homotopy of Madsen–Tillmann spectra (a generalised cohomology of $B H$ ).

Distinctive contributions of the book, in roughly the order of the lectures:

The Atiyah–Segal Axiom System for field theory (Introduction; Lectures 1, 3). A field theory is a symmetric monoidal functor $F : Bord_{n}^{H} \to C$ where the domain is a bordism category of $n$ -manifolds with structure $H$ (orientation, spin, framing, …) and the codomain is typically $sVect_{C}$ or a higher target. Originated by Segal 1988 (CFT) and Atiyah 1988 (topological case); Freed presents the unified modern formulation.
Bordism categories and worked low-dimensional examples (Lecture 1; Lecture 4). Classification of 1d and 2d TQFTs: 2d oriented TQFTs ↔ commutative Frobenius algebras (Dijkgraaf–Abrams thesis), 1d framed/oriented theories worked explicitly.
The Wick rotation viewpoint (Lectures 2, 3, 7). Quantum mechanics and relativistic QFT are recast as Wick-rotated (Euclidean) field theories, motivating the bordism axiom system from the physics side rather than presenting it as an isolated mathematical definition.
Extended locality and the cobordism hypothesis (Lecture 5). Higher categories, extended bordism $(\infty, n)$ -categories, extended TQFTs, fully extended (down to points) — the Baez–Dolan–Lurie cobordism hypothesis is sketched, not proved.
Invertible field theories and stable homotopy theory (Lecture 6). Invertible $n$ -dim TQFTs with structure $H$ are classified by maps of spectra $M T H \to Σ^{n + 1} I_{Z}$ (in the deformation-class / continuous setting). Madsen–Tillmann spectra $M T H$ , the sphere spectrum, Anderson dual $I_{Z}$ . This is the core Freed–Hopkins technical content.
Reflection positivity, extended unitarity, and the Pin / Pin$^c$ double covers (Lectures 7, 8). Wick-rotated unitarity becomes a reflection-positivity datum; equivariant spectra over $Z /2$ capture orientation-reversal.
Non-topological invertible field theories (Lecture 9). Short-range entangled lattice systems, the long-range limit of 3d Yang–Mills + Chern–Simons; differential cohomology refinements.
Computations for free-fermion / electron systems and the 10-fold way (Lecture 10). Connects the abstract classification to Altland–Zirnbauer symmetry classes and topological-insulator phase tables — the original physics motivation closed up.
Anomalies as invertible field theories in one dimension higher (Lecture 11). Anomalies of an $n$ -dim QFT are realised as boundary theories for an invertible $(n + 1)$ -dim theory — Freed's signature reinterpretation; Pfaffians of Dirac operators worked as the prototype.

Source pedigree and peer-cited corroborations:

M. F. Atiyah, "Topological quantum field theories," Publ. Math. IHÉS 68 (1988) 175–186 — the original Atiyah axioms; Freed cites this as the topological half of the axiom-system pedigree.
G. Segal, "The definition of conformal field theory," in Topology, Geometry, and Quantum Field Theory (London Math. Soc. Lecture Note Series 308, 2004; original preprint 1988) — the Segal CFT axioms that the Atiyah axioms are a topological restriction of.
J. Lurie, "On the classification of topological field theories," Current Developments in Mathematics 2008 (2009) 129–280 — the cobordism-hypothesis sketch that Lecture 5 surveys.
S. Stolz, P. Teichner, "Supersymmetric field theories and generalized cohomology," in Mathematical Foundations of Quantum Field Theory and Perturbative String Theory, Proc. Sympos. Pure Math. 83 (AMS 2011) 279–340 — establishes that classes of field theories realize cohomology theories (the Stolz–Teichner program). Freed uses this as background but does not prove it.
D. S. Freed, M. J. Hopkins, "Reflection positivity and invertible topological phases," Geom. Topol. 25 (2021, ArXiv 2016) 1165–1330 — the joint paper these lectures are based on; the lectures are Freed's pedagogical companion volume.
A. Kapustin, "Symmetry protected topological phases, anomalies, and cobordisms: beyond group cohomology," ArXiv 1403.1467 (2014) — the physics-side parallel classification of SPT phases by cobordism that the Freed–Hopkins framework subsumes and refines.

The book is not a first introduction to QFT, topology, or higher categories. It assumes basic algebraic topology (singular homology, spectra, generalised cohomology), some quantum mechanics, and familiarity with symmetric-monoidal-category language. Lecture 1 is the only self-contained low-prerequisite entry point.

§2 Coverage table (Codex vs Freed)

Cross-referenced against the current Codex corpus (content/ tree, May 2026 audit snapshot). ✓ = covered, △ = partial / different framing, ✗ = not covered.

Freed topic	Codex unit(s)	Status	Note
Symmetric monoidal categories (Appendix A)	—	✗	Gap. Some monoidal-functor language scattered (RT, K-theory) but no dedicated unit; needed as prerequisite glue for any TQFT pass.
Picard groupoids / Picard category (App. A.4)	—	✗	Gap. Needed for invertible-TQFT classification.
Symmetric monoidal duality (App. A.2)	—	✗	Gap. Dualizable objects; load-bearing for cobordism hypothesis.
Classical bordism (Lecture 1.1)	△	△	`03.12.04-spectrum.md` mentions bordism as a generalised cohomology theory (Thom spectrum). No standalone bordism-category unit.
Bordism category $Bord_{n}$ (Lecture 1.2)	—	✗	Gap (high priority — the central object of the book).
Atiyah–Segal axioms / TQFT as symmetric-monoidal functor (Lect. 1.2, 3.1)	△	△	Only a passing reference in `03.10.02-cft-basics.md` ("modular-functor / TQFT formulation"). No dedicated unit. Gap.
Manifolds with tangential structure $H$ (orientation, spin, framing, Spin $^{c}$ ) (Lect. 1.3)	△	△	Spin in `03.09.04-spin-structure.md`; orientation throughout. No unified "tangential structure" framing.
Super vector spaces $sVect$ (Lect. 1.4)	—	✗	Gap. Z/2-graded vector spaces, super-trace, Koszul sign.
Bordism ↔ homotopy theory (Pontryagin–Thom) (Lect. 1.5)	△	△	`03.12.04-spectrum.md` mentions bordism spectra; no Pontryagin–Thom theorem unit.
Quantum mechanics axiomatic / Hamiltonian / observables (Lect. 2)	△	△	`12-quantum/01-foundations/` likely has some content (not audited here); Freed's "Axiom System for QM" is field-theory-framed and probably distinct.
Wick rotation in QM / QFT (Lect. 2.5, 3.3)	—	✗	Gap. Imaginary-time formulation; reflection-positivity setup.
Toric code / lattice TQFT example (Lect. 2.3)	—	✗	Gap. Worked example, condensed-matter side.
Classification of 2d oriented TQFTs ↔ commutative Frobenius algebras (Lect. 4.2)	—	✗	Gap (high priority — the canonical first non-trivial example).
Classification of 1d TQFTs (Lect. 4.4)	—	✗	Gap. Easy warm-up example.
2d area-dependent theories (Lect. 4.3)	—	✗	Gap.
Higher categories — definitions and examples (Lect. 5.1, 5.2)	—	✗	Gap. No bicategory / $(\infty, n)$ -category unit anywhere in Codex.
Extended TQFTs (Lect. 5.3)	—	✗	Gap.
Cobordism hypothesis (Baez–Dolan–Lurie) (Lect. 5.6)	—	✗	Gap (high priority — Master-tier pointer unit minimum).
Invertible field theories (Lect. 6.2)	—	✗	Gap. Central object of the Freed–Hopkins program.
Madsen–Tillmann spectra $M T H$ (Lect. 6.6)	—	✗	Gap. Connects to `03.12.04-spectrum.md`.
Anderson dual to sphere spectrum $I_{Z}$ (Lect. 6.7)	—	✗	Gap. Stable-homotopy machinery.
Invertible TQFTs as spectrum maps $M T H \to Σ^{n + 1} I_{Z}$ (Lect. 6.8)	—	✗	Gap (high priority — Freed–Hopkins classification theorem).
Reflection positivity (Lect. 7)	—	✗	Gap. Osterwalder–Schrader at the abstract level.
Extended positivity / equivariant spectra (Lect. 8)	—	✗	Gap.
Short-range entangled lattice systems / SPT phases (Lect. 9.1)	—	✗	Gap. Physics-side endpoint.
Long-range limit Yang–Mills + Chern–Simons (Lect. 9.2)	△	△	`03.07-gauge-theory/` covers Yang–Mills geometry but not Chern–Simons TQFT. Gap on the CS side.
Differential cohomology (Lect. 9.4)	—	✗	Gap. Cheeger–Simons, Deligne cohomology.
10-fold way for free electrons (Lect. 10.1)	△	△	`03.09.12-kr-theory.md` covers $K R$ -theory and Altland–Zirnbauer's 10-fold way is a $K R$ application; check whether the Codex unit mentions phase classes.
Anomalies as invertible field theories (Lect. 11)	—	✗	Gap. Pfaffians of Dirac operators connects to `03.09.10-atiyah-singer-index-theorem.md`.

Aggregate coverage estimate: ~3% of Freed has a directly corresponding Codex unit. The two △ rows on bordism (Pontryagin–Thom sketch inside the spectrum unit) and the Spin / $K R$ rows from 03-modern-geometry/09-spin-geometry/ are the only material connections. The TQFT track is a near-total gap. This is expected: Codex has invested in geometry of fields (gauge theory, characteristic classes, index theorem) but not in the categorical-axiomatic face of field theory.

§3 Gap punch-list (priority-ordered units to write)

Priority 0 — prerequisite glue (small but load-bearing):

01.02.NN Symmetric monoidal category. Definition, coherence, symmetric monoidal functor, duality. Anchored on Mac Lane and Appendix A of Freed. ~1200 words; Beginner + Intermediate + Master.
01.02.NN+1 Picard groupoid / Picard category. Invertible-only subcategory of a symmetric monoidal category, equivalent to a 2-truncated connective spectrum. ~1000 words; Master-emphasised.
03.MM.NN Super vector spaces. $Z /2$ -grading, super-trace, Koszul sign rule. ~800 words. Slots into either 03-modern-geometry/01-tensor-algebra/ or 07-representation-theory/.

Priority 1 — core TQFT content (book's central machinery):

03.NN.01 Bordism category $Bord_{n}^{H}$ . Objects = closed $(n - 1)$ -manifolds with structure $H$ ; morphisms = bordisms modulo diffeomorphism rel boundary; symmetric monoidal under disjoint union. Standard examples: $Bord_{2}^{or}$ . Freed Lecture 1.2 anchor; Atiyah 1988 originator citation. ~1800 words; three-tier.
03.NN.02 Atiyah–Segal axioms / TQFT as symmetric-monoidal functor. Definition: $F : Bord_{n}^{H} \to sVect_{C}$ . Worked: state spaces, partition functions, gluing. Freed Lecture 1.2, 3.1 anchor; Atiyah 1988 + Segal 1988 (CFT) + Segal 2004 originators. ~2000 words; three-tier.
03.NN.03 Pontryagin–Thom theorem and bordism as homotopy theory. Statement (oriented / unoriented / framed cobordism ring isomorphic to $π_{*}$ of Thom spectrum). Sketch only. Freed Lecture 1.5 anchor; classical (Pontryagin 1950, Thom 1954). Lateral connection to 03.12.04-spectrum.md. ~1500 words.
03.NN.04 Classification of 2d oriented TQFTs ↔ commutative Frobenius algebras. Statement + proof sketch (pair-of-pants, cap, cup generate $Bord_{2}^{or}$ ; relations reduce to Frobenius axioms). Worked: $A = C [G]$ group-algebra example (Dijkgraaf–Witten 2d). Freed Lecture 4.2 anchor; Abrams 1996 / Dijkgraaf thesis 1989 originators. ~1800 words; signature worked computation of TQFT.
03.NN.05 Classification of 1d TQFTs. 1d framed and oriented theories; tells the reader the entire game in one example. Freed Lecture 4.4 anchor. ~800 words; Beginner+Intermediate only.

Priority 2 — extended TQFT and the cobordism hypothesis (pointer-tier):

03.NN.06 Higher categories: bicategories and $(\infty, n)$ - categories (pointer). Definitions only, Segal-space / complete Segal model named, $n$ -category structure of the extended bordism category named. Freed Lecture 5.1–5.2 anchor; Lurie 2009 On the classification of TQFTs as anchor for the cobordism-hypothesis pass. ~1500 words; Master-only.
03.NN.07 Extended TQFTs and the cobordism hypothesis. Statement of Baez–Dolan–Lurie: fully extended $n$ -dim framed TQFTs valued in symmetric monoidal $(\infty, n)$ -category $C$ are classified by fully dualizable objects of $C$ . Sketch only; no proof. Freed Lecture 5.6 anchor; Baez–Dolan 1995 + Lurie 2009 originators. ~1500 words; Master-only.

Priority 3 — Freed–Hopkins invertible TQFT classification:

03.NN.08 Invertible field theories. Definition (lands in the Picard subcategory of the target); first examples (Euler theory in 1d, Arf in 2d, signature in 4d). Freed Lecture 6.1–6.2 anchor. ~1200 words.
03.NN.09 Madsen–Tillmann spectra $M T H$ . Definition via Thom spectrum construction on tangential structure $H$ ; relation to classical Thom spectra $M O, M S O, M Spin$ in 03.12.04-spectrum.md. Freed Lecture 6.6 anchor; Madsen–Tillmann 2007 originator (Acta Math.). ~1500 words; Master.
03.NN.10 Anderson dual $I_{Z}$ , invertible TQFTs as spectrum maps. The Freed–Hopkins classification theorem (statement): deformation classes of invertible $n$ -dim $H$ -TQFTs with values in $sVect_{C}$ are $[Σ^{- 1} M T H, Σ^{n + 1} I_{Z}]$ . Freed Lecture 6.7–6.8 anchor; Freed–Hopkins 2016/2021 Geom. Topol. originator. ~1800 words; Master only. Master-tier signature unit.

Priority 4 — applications and pointer endpoints (Master-tier only):

03.NN.11 Reflection positivity (pointer). Wick-rotated unitarity, abstract reflection structure. Freed Lecture 7. ~1000 words.
03.NN.12 SPT phases / 10-fold way TQFT classification (pointer). Bridges Freed Lecture 10 to existing 03.09.12-kr-theory.md (Altland–Zirnbauer table is a $K R$ -theory application). Add as Master section to 03.09.12 rather than standalone unit. Kapustin 2014 + Freed–Hopkins 2016 originators.
03.NN.13 Anomalies as invertible field theories (pointer). The "anomaly inflow" framework as boundary theories of an invertible $(n + 1)$ -theory. Connects to 03.09.10-atiyah-singer via Pfaffian-of-Dirac. Freed Lecture 11 anchor. Master section on 03.09.10 rather than standalone unit.

Tally: 13 standalone new units (3 prereq + 5 priority-1 + 2 priority-2 + 3 priority-3) + 2 Master sections grafted onto existing units (priority-4 items 15, 16). Plus 1 lateral-connection edit to 03.10.02-cft-basics.md to point at the new TQFT cluster.

§4 Implementation sketch and hour estimates

Per-unit estimate (matches the corpus average for survey-tier units with originator-prose treatment):

Priority 0 (prereq glue): ~2 hours × 3 units = ~6 hours.
Priority 1 (core TQFT): ~3.5 hours × 5 units = ~17–18 hours.
Priority 2 (cobordism hypothesis pointer cluster): ~3 hours × 2 units = ~6 hours.
Priority 3 (Freed–Hopkins classification): ~4 hours × 3 units = ~12 hours. Unit 13 is harder than average — the spectrum classification needs careful exposition of the Anderson dual.
Priority 4 (grafted Master sections): ~1.5 hours × 2 = ~3 hours.

Total: ~44 hours of focused production. Fits a focused 6–8 day production window.

Routing decision: The new units don't fit cleanly into an existing 03-modern-geometry/NN-*/ slot. Recommend opening a new section 03-modern-geometry/14-tqft/ (or 15-tqft/ if 14 is reserved) for the TQFT cluster. Alternative: graft into 10-conformal-field-theory/ as 10.NN since CFT is the historical sibling, but this would dilute the CFT chapter's identity. Recommendation: new section 14-tqft/.

Originator-prose targets (per docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §10):

Witten 1988, "Topological quantum field theory," Commun. Math. Phys. 117, 353–386 — the physics-side origin (Donaldson invariants as a TQFT path integral); originator-cite in units 4–5.
Atiyah 1988, "Topological quantum field theories," Publ. Math. IHÉS 68, 175–186 — the axiomatic origin; originator-cite in units 4–5.
Segal 1988 / Segal 2004, "The definition of conformal field theory" — the parallel CFT origin and the model the Atiyah axioms shorten; originator-cite in units 5 and 11.
Lurie 2009, "On the classification of topological field theories" — originator-cite for cobordism hypothesis in units 9–10.
Freed–Hopkins 2016/2021, "Reflection positivity and invertible topological phases," Geom. Topol. 25, 1165–1330 — originator-cite for unit 13 (the Master-tier signature unit).

Notation crosswalk. Freed uses $Bord_{n}^{H}$ for the bordism category with tangential structure $H$ (a homomorphism $H \to O (n)$ ); $Bord_{⟨ n - 1, n ⟩}^{H}$ for the one-categorical truncation; $Bord_{n}^{H, ext}$ for the extended $(\infty, n)$ -version. He writes $M T H$ for the Madsen–Tillmann spectrum (note: the $T$ is tangential, not "Thom"; distinct from $M H$ ). The Anderson dual is $I_{Z}$ . The Codex should adopt all four conventions verbatim. Record in a §Notation paragraph in units 4 and 12.

§5 What this plan does NOT cover

A line-by-line P1 inventory of all 11 lectures + appendix. The book is 186 pp.; the punch-list above captures the load-bearing theorems. Per-theorem audit deferred.
The Lurie HTT / HA machinery underlying the cobordism hypothesis. Treated as a sketched pointer only (unit 10). A genuine HTT/HA pass is a multi-month project routed through 04-algebraic-geometry/ and a future $\infty$ -category section.
Stolz–Teichner deep details (supersymmetric field theories ↔ $K$ -theory / $tmf$ ). Mentioned in passing in unit 5 only; full Stolz–Teichner coverage is a separate Fast Track entry and a separate audit pass.
Companion volume Ayala–Freed–Grady (AFG) supplementary lectures. Out of scope; the supplementary chapters are independent papers.
Exercise pack. Freed contains exercises but the lecture set is primarily expository. Exercise production deferred to a P3-priority-3 follow-up.
The 10-fold way physics derivation beyond the pointer in unit 15. Deferred to the condensed-matter side of the corpus (11-stat-mech-physics/ or 12-quantum/).

§6 Acceptance criteria for FT equivalence (Freed 3.05)

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

≥95% of Freed's stated theorems / classifications in lectures 1–11 map to Codex units. Currently ~3%; after priority-1 this rises to ~50%; after priority-1+2 to ~70%; after priority-1+2+3 to ~90%; priority-4 grafts and one CFT lateral-connection edit close the gap to ≥95%.
≥90% of Freed's named worked examples (2d Frobenius classification, 1d theories, toric code, Dijkgraaf–Witten, 10-fold way) are either units or are referenced from units that name them as canonical.
The Freed–Hopkins classification theorem (unit 13) ships at Master tier with the spectrum-map statement intact.
Notation decisions ( $Bord_{n}^{H}$ , $M T H$ , $I_{Z}$ , $sVect$ ) recorded in unit-level §Notation paragraphs.
Pass-W weaving connects the new 14-tqft/ cluster to 03.09-spin-geometry/ (anomalies, $K R$ -theory), 03.10-conformal-field-theory/ (CFT as a non-topological cousin), 03.12.04-spectrum.md (Thom and Madsen–Tillmann spectra), and 03.07-gauge-theory/ (Chern–Simons as a TQFT example).

The 5 priority-1 units close the largest equivalence gap. Priority-3 units 11–13 are required for the Master-tier signature material and constitute the actual Freed-Hopkins content the book is named for; omitting them would coverage at the level of any TQFT survey, not specifically of this book.

§7 Sourcing

Source PDF (used for audit): reference/book-collection/free-downloads/Freed-CBMS_Field_Theory_and_Topology.pdf — author's preliminary version; AMS permission granted; not to be reposted. Sufficient for audit and unit production; cite the published AMS edition in unit bibliographies once the published version is checked for pagination drift.
Published edition: Daniel S. Freed, Lectures on Field Theory and Topology, CBMS Regional Conference Series in Mathematics, No. 133, AMS / CBMS / NSF, 2019. ISBN 978-1-4704-5206-3.
Companion paper (anchor for unit 13): D. S. Freed, M. J. Hopkins, "Reflection positivity and invertible topological phases," Geom. Topol. 25 (2021) 1165–1330 (ArXiv 1604.06527, 2016).
License note: The lectures are based on the Freed–Hopkins paper with additional pedagogical material; cite Freed (2019) for the lectures and Freed–Hopkins (2021) for the classification theorem proper.
Local availability for FT 3.05 catalog update: The author- preliminary PDF resolves the catalog's "? (AMS may have partial)" flag. Update docs/catalogs/FASTTRACK_BOOKLIST.md 3.05 line to "FREE (author-preliminary, AMS permission)" with the local-path pointer above.
Distinct from: reference/fasttrack-texts/03-modern-geometry/Freed-AspectsOfFieldTheory.pdf (a separate, earlier Freed text — Aspects of Field Theory — and not the FT 3.05 source). The FT 3.05 PDF is in the book-collection/free-downloads/ tree under the filename Freed-CBMS_Field_Theory_and_Topology.pdf.

Freed — *Lectures on Field Theory and Topology* (Fast Track 3.05) — Audit + Gap Plan