Extended TQFT, Bordisms, and the Cobordism Hypothesis

Extended TQFT turns geometric cutting into composition at every codimension. A classification claim is determined only after one fixes the spacetime dimension, tangential structure, extension depth, bordism higher category, symmetric monoidal target, required duals and adjoints, homotopy fixed-point data, invertibility, positivity, and equivalence relation. The cobordism hypothesis then classifies fully extended framed theories in a specified target by its fully dualizable objects. It does not, by itself, classify oriented or reflection-positive theories, unextended functors, microscopic phases, or arbitrary physical QFTs.

Helpful background. Bordism and tangential structures supply the geometric equivalence relation. What is a topological field theory? supplies the physical distinction between metric-independent and metric-dependent observables. State spaces, cobordisms, and gluing supply the elementary sewing picture.

Enter this chapter

The central object is a symmetric monoidal functor

$$Z:\operatorname{Bord}^{\xi}_{n,\le k}\longrightarrow\mathcal C.$$

Here $n$ is spacetime dimension, $\xi$ is a framing, orientation, spin, pin, or other tangential structure, $k$ records how far the theory is extended, and $\mathcal C$ is the target higher category. When $k=1$ in the traditional convention, closed $(n-1)$-manifolds receive state spaces and $n$-bordisms receive maps. Full extension reaches points and all intermediate strata, so the target must have morphisms through level $n$. Turning bordisms around forces duals and adjoints at corresponding levels.

This topological domain has no Lorentzian metric signature: its functorial data are invariant under the structured diffeomorphisms declared in the domain. The site’s $(+---)$ convention resumes when a physical Lorentzian theory is compared with this topological truncation. Reflection positivity is instead a Euclidean reflection condition and must be stated separately. Likewise, “unitary,” “invertible,” and “fully dualizable” name different structures.

The classification mechanism is directional. Bordism generators—points, cups, caps, handles, and their higher analogues—map to objects, evaluation and coevaluation maps, units and counits, and coherences. Morse and Cerf moves become algebraic relations. Full dualizability licenses the framed fully extended functor. A $G$-structured theory additionally needs a coherent homotopy fixed point for the induced $G$ action. Picard factorization licenses invertibility; a dagger and positive reflected pairings license a positive interpretation. Lurie states the framed and structured equivalences in Lurie 2009, Theorems 2.4.6 and 2.4.26, printed pp. 43–47.

The dependency map should be read from left to right. Each stage adds a condition; no reverse arrow is implied. The branch records structures that refine rather than follow automatically from the framed theorem.

Fixing $n$, $\xi$, extension depth, and the target precedes every theorem. Duals and adjoints through the required categorical level produce a fully dualizable point object and hence, in the framed fully extended case, a functor up to equivalence. Structured, invertible, relative, and reflection-positive theories add independent fixed-point, Picard, boundary, or positivity data. Physical realization is a further construction. The diagram is schematic and not to scale. Structured description and source data (JSON)

The chapter sequence

Read the pages in this order.

1. Bordism categories and symmetric monoidal TQFTs define the unextended domain, target, duality, and surface sewing relations.
2. Atiyah–Segal functorial TQFT and gluing turn cuts into compositions, tensor products, traces, and pairings.
3. Fully extended TQFTs and higher categories assign data down to points and type corners as higher morphisms.
4. Higher-categorical targets and levels of dualizability explain how successive adjoints limit extension depth.
5. Dualizability and the cobordism hypothesis state the framed fully extended classification and its exact equivalence.
6. Tangential structures: oriented, spin, and framed theories add structure-group actions and homotopy fixed points.
7. Invertible field theories and generalized cohomology pass from Picard targets to spectra and bordism classifications.
8. Anomalies as relative and invertible field theories make anomalous partition functions line-valued boundary data for an invertible bulk.
9. Reflection positivity, unitarity, and Hermitian TQFTs derive reflected state-space pairings and test their signs.
10. Boundaries, defects, and extended operators in TQFT encode walls by modules or morphisms and junctions by higher maps.
11. Extended-TQFT classification: scope and counterexamples compares theorem domains and rejects missing-target, missing-structure, and missing-positivity upgrades.

The first two pages establish ordinary functoriality. The next four add extension depth, higher targets, full dualizability, and tangential structure. The final five treat special structures and stress-test the classification boundary. This order is pedagogical: a reader doing a focused lookup may enter later after checking the actual hard prerequisites on that page.

Hypotheses and licensed conclusions

Domains, hypotheses, directional conclusions, excluded upgrades, and failure tests for extended TQFT classifications

Object and domain	Required hypotheses	Licensed result	Excluded converse or upgrade	Adversarial check
Unextended oriented bordism functor	Fixed dimension and oriented bordism category; finite-dimensional target; monoidality, diffeomorphism invariance, nondegenerate duality, and sewing	State spaces and bordism maps compatible with disjoint union and gluing	Closed partition functions alone do not define the functor, and no point value follows	Assign different maps to two Frobenius-related decompositions of one surface
Partially or fully extended functor	Bordisms with corners through the declared depth; symmetric monoidal higher target; duals and adjoints at every visible level	Consistent values on points, strata, junctions, and their compositions through that depth	Object duality alone does not provide adjoints for lower morphisms	Use the dual-number algebra or a nonadjointable bimodule and demand a point-level extension
Framed fully extended classification	Dimension $n$; framed fully extended bordism category; specified symmetric monoidal $(\infty,n)$-category; fully dualizable object	Equivalence of the functor groupoid with the maximal groupoid of fully dualizable objects	No oriented, positive, unextended, or physical classification follows automatically	Change the target or omit one adjoint level and compare the resulting functor groupoids
Oriented, spin, pin, or $G$-structured theory	Declared homomorphism $G\to O(n)$ and coherent homotopy fixed-point data for the induced action	A structured refinement of the framed theory	An invariant isomorphism without higher coherence is not a homotopy fixed point	Forget the spin structure in Arf theory and compare the two torus signs
Invertible extended theory	Factorization through the Picard target; exact symmetry type; discrete or continuous target; declared deformation and positivity conditions	A spectrum-map or generalized-cohomology classification in the stated regime	Noninvertible topological order and microscopic realizability are not included	Use a multidimensional state space and attempt to tensor it to the one-dimensional unit
Relative anomalous theory	Invertible bulk anomaly functor; compatible boundary truncation; line-valued partition function; coherent trivialization for cancellation	Functorial anomaly line, inflow transformation, and a precise cancellation condition	A local counterterm in one chart does not remove global holonomy	Transport around a large-gauge loop and compute determinant-line or eta holonomy
Hermitian or reflection-positive TQFT with boundaries	Compatible dagger and reflection structure; positive reflected pairings; module or morphism actions; all sewing and Cardy-type relations required by the model	Positive state-space forms and functorial boundary, wall, and junction data	Dualizability does not imply positivity, and a bare vector-space label is not a boundary condition	Choose a negative Frobenius trace weight or omit the module action

Structured table data (JSON) preserves the caption, scoped headers, rows, and reading order.

The table is a claim filter. It distinguishes a theorem’s input category from properties of a physical realization. It also distinguishes equivalence in a higher target—Morita equivalence in a standard algebraic example—from literal equality or algebra isomorphism.

Where classification attempts fail

The most dangerous errors preserve plausible formulas while changing their type. A commutative Frobenius algebra can define an unextended surface theory without being separable enough for a fully extended Morita-valued theory. A fully dualizable object can lack the coherent fixed point required by orientation. A valid algebraic functor can have a negative reflected norm. A local anomaly can vanish while determinant-line holonomy remains nontrivial. A generalized-cohomology class can exist without a microscopic gapped realization.

The failure map pairs each omitted hypothesis with the first invalid conclusion and the strongest statement that remains.

Each dashed branch removes one necessary condition. The surviving result may be data on a chosen decomposition, a partially extended theory, a framed rather than oriented theory, a noninvertible TQFT, a relative rather than absolute theory, or a Hermitian but indefinite model. None of these downgrades is a classification of arbitrary physical QFT. The diagram is schematic and not to scale. Structured description and source data (JSON)

Scope boundaries

This chapter owns functorial and extended topological field theories as mathematical objects, including their bordism domains, higher targets, dualizability, tangential refinements, invertibility, relative anomaly formulation, positivity structures, and equivalence notions. Low-dimensional Frobenius and Morita models provide exact checks because every generator and relation can be displayed.

Physical topological observables, line-operator dynamics, and symmetry TFTs remain in their physical volumes. General category and bordism prerequisites remain in Mathematical Methods. BV–BFV gluing owns perturbative path-integral pushforwards with boundary residual fields. A functorial TQFT may describe a topological sector or effective limit, but that comparison must identify what was integrated out, which observables survive, and what positivity or realizability theorem applies.

Review the chapter

Before accepting a classification, answer:

1. What are $n$, $\xi$, and the extension depth?
2. Which bordism higher category and collar or corner conventions form the domain?
3. What is the target, and what do equivalences in it mean?
4. Which object duals and which morphism adjoints have actually been constructed?
5. Is the theorem framed, or has the required homotopy fixed point been supplied?
6. Is the theory invertible at every categorical level, or merely dualizable?
7. Does “unitary” mean a dagger structure, nondegenerate pairing, or verified reflection positivity?
8. For an anomaly, is the partition function a number, a section of a line, or a relative boundary datum?
9. For a boundary, what action or morphism makes gluing well defined?
10. Which result constructs a physical realization rather than only the abstract functor?

Synthesis exercise

A finite-dimensional commutative Frobenius algebra has a nondegenerate trace, and its point object has an algebraic dual. May one conclude that it defines an oriented, reflection-positive, fully extended two-dimensional TQFT?

Solution

No. The Frobenius algebra gives an unextended oriented surface theory. Full extension in the Morita target additionally requires separability, equivalently the adjointability conditions for full $2$-dualizability. An oriented point-level theory needs a coherent $SO(2)$ homotopy fixed point, represented by appropriate Calabi–Yau trace data. Reflection positivity further requires a compatible involution and positive trace weights. Each condition must be checked in the specified target and equivalence relation.

References

• Atiyah, Michael F. “Topological Quantum Field Theory.” Publications Mathématiques de l’IHÉS 68 (1988): 175–186. DOI; Open PDF.
• Freed, Daniel S. “Anomalies and Invertible Field Theories.” Proceedings of Symposia in Pure Mathematics 88 (2014): 25–45. DOI; Open PDF.
• Freed, Daniel S., and Michael J. Hopkins. “Reflection Positivity and Invertible Topological Phases.” Geometry & Topology 25 (2021): 1165–1330. DOI; Open PDF.
• Lurie, Jacob. “On the Classification of Topological Field Theories.” In Current Developments in Mathematics 2008, 129–280. Somerville, MA: International Press, 2009. Open PDF.
• Schommer-Pries, Christopher J. The Classification of Two-Dimensional Extended Topological Field Theories. PhD thesis, University of California, Berkeley, 2009; expanded version 2014. Open PDF.