Extended-TQFT Classification: Scope and Counterexamples

An extended-TQFT classification is meaningful only as an equivalence between a precisely declared functor groupoid and a precisely declared sub-groupoid of a target higher category. The dimension, tangential structure, extension depth, target, dualizability level, fixed-point data, invertibility, positivity, and equivalence notion are theorem hypotheses, not optional annotations. The framed cobordism hypothesis supplies one powerful equivalence; oriented, spin, reflection-positive, unextended, and physically realizable theories require additional or different data. No version classifies arbitrary physical QFTs.

Required background. The cobordism hypothesis supplies the framed fully extended theorem; invertible field theories supply the spectral specialization; and reflection-positive TQFTs supply the positivity condition that dualizability does not imply.

Helpful background. Conformal-net sectors and modular categories provide a comparison with operator-algebraic classification, while what is a topological field theory? keeps physical observables distinct from abstract functor data.

Classification claims as typed equivalences

Before using a cobordism classification, record the tuple

$$(n,\xi,k,\operatorname{Bord}^{\xi}_{n,\le k}, \mathcal C,\mathcal C^{(k)\text{-dual}}, \mathcal P,\simeq),$$

where $n$ is spacetime dimension, $\xi$ is the tangential structure, $k$ is extension depth, $\mathcal C$ is the symmetric monoidal target, $\mathcal P$ denotes any fixed-point, invertibility, Hermitian, or positivity data, and $\simeq$ is the chosen equivalence. The framed fully extended theorem is

$$\operatorname{Fun}^{\otimes} \bigl(\operatorname{Bord}^{\mathrm{fr}}_n,\mathcal C\bigr)^\sim \simeq (\mathcal C^{\mathrm{fd}})^\sim,$$

via evaluation at the positively framed point. Lurie’s statement, including the fully dualizable subcategory and functor-groupoid conclusion, is Lurie 2009, Theorem 2.4.6 and Remarks 2.4.7–2.4.9, printed pp. 43–44.

Changing any entry changes the theorem:

Claimed theory class	Extra datum beyond a framed point object	Correct comparison
Framed, fully extended, Morita-valued 2D TQFT	Finite-dimensional separable algebra	Morita equivalence of fully dualizable objects
Oriented, fully extended 2D TQFT	Coherent SO(2) homotopy fixed point, represented by Calabi–Yau trace data in the standard target	Equivalence preserving the fixed-point structure
Unextended oriented 2D TQFT	Finite-dimensional commutative Frobenius algebra	Monoidal natural equivalence of bordism functors
Reflection-positive Hermitian TQFT	Compatible involution and positive reflected pairings	Hermitian monoidal equivalence preserving positivity
Invertible deformation class	Picard target, symmetry type, stable or discrete convention, and often reflection positivity	Homotopy or deformation class of spectrum maps

The rows are related but not interchangeable. A truncation may forget enough information that non-equivalent extended theories become equivalent on closed hypersurfaces. A change of target can replace algebra isomorphism by Morita equivalence. A positivity structure is additional even when the object is fully dualizable.

The two-dimensional comparison

In the Morita $2$-category over $\mathbb C$, finite-dimensional separable algebras are fully dualizable. Thus framed fully extended two-dimensional theories are classified by such algebras up to Morita equivalence. Matrix algebras illustrate the equivalence relation:

$$M_r(\mathbb C)\simeq_{\mathrm{Morita}}\mathbb C$$

although they are not isomorphic for $r>1$. An oriented refinement requires a coherent $SO(2)$ homotopy fixed point; in this target it is encoded by suitable Calabi–Yau or symmetric Frobenius trace data. Reflection positivity further constrains the induced Hermitian trace weights to be positive.

This is the exact first application returned to topological order and invertible phases: compare framed fully extended Morita-valued 2D theories with oriented semisimple unitary 2D TQFTs and list the added fixed-point and positivity structures. Schommer-Pries proves the two-dimensional extended generators-and-relations classification in Schommer-Pries 2014, Chapter 3, especially §§3.4–3.5. The physical owner decides whether a given functor is realized by a microscopic phase.

Counterexamples to common upgrades

Unextended does not imply fully extended. The dual numbers

$$D=\mathbb C[\epsilon]/(\epsilon^2)$$

form a commutative Frobenius algebra and hence an unextended oriented 2D TQFT. They are not separable, so the evaluation bimodule lacks the adjoints required for full extension in the Morita target.

Framed does not imply oriented. A fully dualizable object need not come with a coherent trivialization of the $SO(n)$ action. Even an isomorphism from the Serre automorphism to the identity is only the first layer of fixed-point data.

Oriented does not imply spin-independent, nor conversely. Arf theory distinguishes spin structures on the same oriented surface. It cannot descend through the forgetful functor unless its spin dependence is trivial.

Dualizable does not imply positive. A semisimple Frobenius algebra with one negative trace weight satisfies algebraic gluing but produces a negative-norm state under reflection.

Invertible does not mean all topological order. A theory with a multidimensional state space on a closed hypersurface has no tensor inverse and lies outside a Picard-spectrum classification.

Functorial classification does not imply microscopic realization. A spectrum map or fully dualizable object produces the declared mathematical functor. It does not supply a local Hamiltonian, a gapped path, reflection positivity, or a continuum limit unless those are separately constructed.

The chapter’s adversarial overclaim—“the framed cobordism hypothesis classifies oriented reflection-positive theories in an unspecified target”—therefore fails at four independent fields: tangential structure, positivity, target, and equivalence. The strongest surviving statement is the framed theorem after one fixes $n$, full extension, a symmetric monoidal $(\infty,n)$-category, full dualizability, and equivalence of functors.

Exercises

Why does the matrix-algebra example rule out “classification up to algebra isomorphism”?

Solution

$M_r(\mathbb C)$ and $\mathbb C$ have different dimensions and are not isomorphic for $r>1$, but the standard column and row bimodules are inverse under relative tensor product. They are equivalent objects in the Morita target, which is the equivalence detected by the functor groupoid.

A paper states only that its point object is dualizable. What is the first question to ask before invoking a three-dimensional fully extended theorem?

Solution

Ask whether the object is fully $3$-dualizable in the specified target: object duality is only the first condition, and evaluation morphisms plus their adjoints must satisfy the next-level requirements. One must also identify the tangential structure and the exact target equivalence.

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.
• 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.