Extended TQFT, Bordisms, and the Cobordism Hypothesis

Extended TQFT turns cutting and gluing at every codimension into higher-categorical composition. The cobordism hypothesis makes this a classification theorem once the dimension, tangential structure, target, and extension depth are fixed.

Enter this chapter

Begin with bordism functors and ordinary gluing. Then track which duals and adjoints become necessary as the theory extends down to points. 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.

Topics

1. Bordism Categories and Symmetric Monoidal TQFTs — Defines geometric composition and disjoint-union multiplication.
2. Atiyah–Segal Functorial TQFT and Gluing — Relates the traditional state-space axioms to sewing.
3. Fully Extended TQFTs and Higher Categories — Adds lower-dimensional strata and their higher morphisms.
4. Higher-Categorical Targets and Levels of Dualizability — Matches extension depth to required duality data.
5. Dualizability and the Cobordism Hypothesis — Classifies fully extended framed theories in a fixed target.
6. Tangential Structures: Oriented, Spin, and Framed Theories — Adds the coherent structure needed to change the tangential category.
7. Invertible Field Theories and Generalized Cohomology — Restricts to theories whose values are tensor-invertible.
8. Anomalies as Relative and Invertible Field Theories — Expresses anomalous theories relative to an invertible bulk.
9. Reflection Positivity, Unitarity, and Hermitian TQFTs — Separates duality and Hermitian structure from positivity.
10. Boundaries, Defects, and Extended Operators in TQFT — Extends the functorial data to labeled boundaries and defects.
11. Extended-TQFT Classification: Scope and Counterexamples — Identifies what the classification does and does not include.

How the ideas fit together

Full dualizability supplies evaluations, adjoints, and coherences matching framed bordisms. The framed theorem classifies theories by their value at a point; a structured theory additionally requires coherent homotopy fixed-point data Lurie 2009, Theorems 2.4.6 and 2.4.26 (PDF). Reflection positivity is further structure Freed and Hopkins 2021, §§ 4.3 and 8.1 (PDF). These results classify functors in specified targets, not arbitrary microscopic systems or every metric-dependent QFT.

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).

Review the chapter

Does a fully dualizable object automatically define an oriented theory?

Solution

It defines the corresponding framed fully extended theory. Removing the framing requires coherent fixed-point data for the relevant rotation-group action. Existence of the object and its duals does not provide that additional structure.

Where to go next

Continue to defects and categorical QFT to study how labeled strata, interfaces, and generalized symmetries fit into this composition picture.

References

• 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.