What Is a Topological Field Theory?

A topological field theory (TQFT) is a local quantum field theory whose physical amplitudes, state spaces, and protected operators depend only on the declared topology and tangential data, and whose assignments obey cutting and gluing. In the finite-dimensional unextended setting, the shortest precise model is a symmetric-monoidal assignment from structured bordisms to vector spaces and linear maps. Metric independence of one action term, a metric-independent partition function on closed spacetime, or a finite state space is not enough by itself.

This definition keeps three distinctions visible. An invertible TQFT has a tensor inverse and assigns lines to closed spatial manifolds. A noninvertible TQFT can have higher-dimensional state spaces and nontrivial topological operators. Intrinsic topological order is instead a property of a gapped many-body phase; under suitable locality and unitarity assumptions, its universal infrared sector may be described by a noninvertible TQFT, but the two phrases are not synonyms. The tensor-inverse and line-state criteria are stated precisely in Freed and Hopkins 2021, § 5.2, current arXiv v6, printed pp. 33–34, especially eq. (5.4), Example 5.3, and Definition 5.9, PDF. Here “intrinsic topological order” means the long-range-entangled, noninvertible many-body case; this keeps it separate from the broader terminology that also calls invertible phases “invertible topological orders” Wen 2017, arXiv v3, printed pp. 3, 7, and 9, PDF.

Required background. When Is a Topological Term Well Defined? separates a global exponentiated action from a local density and distinguishes fixed from summed fields. Helpful background. Support, Codimension, and Operator Data supplies the support, isotopy, orientation, and framing data of extended operators.

A TQFT assigns states and composable evolution

Fix a spacetime dimension $d$ and a tangential structure $s$, such as orientation, spin, or framing. Begin with an absolute, anomaly-free, unextended theory over $\mathbb C$. Its operational data are

$$Z:\operatorname{Bord}^{s}_{d}\longrightarrow \operatorname{FinVect}_{\mathbb C}.$$

The notation packages four assignments:

• a closed structured $(d-1)$-manifold $\Sigma$ receives a state space $\mathcal H(\Sigma)$;
• a bordism $M:\Sigma_{\mathrm{in}}\to\Sigma_{\mathrm{out}}$ receives a linear map $Z(M):\mathcal H(\Sigma_{\mathrm{in}})\to \mathcal H(\Sigma_{\mathrm{out}})$;
• a closed $d$-manifold receives a number $Z(M)\in\mathbb C$; and
• disjoint union receives tensor product.

The cylinder is identity evolution,

$$Z(\Sigma\times[0,1]) = \operatorname{id}_{\mathcal H(\Sigma)},$$

and gluing two bordisms along the same structured boundary gives composition:

$$Z(M_2\circ_{\Sigma}M_1) = Z(M_2)\circ Z(M_1).$$

Likewise,

$$\begin{aligned} \mathcal H(\Sigma_1\sqcup\Sigma_2) &\simeq \mathcal H(\Sigma_1)\otimes\mathcal H(\Sigma_2), \\ \mathcal H(\overline\Sigma) &\simeq \mathcal H(\Sigma)^{\vee}, \\ Z(M_1\sqcup M_2) &= Z(M_1)\otimes Z(M_2). \end{aligned}$$

Here $\overline\Sigma$ carries the reversed orientation and compatible reversed $s$-structure whenever the chosen bordism category admits that operation. These are not optional decorations. A collection of manifold invariants that does not supply compatible state spaces and gluing maps is not yet a local field theory. In Atiyah’s unextended axioms, finite generation, disjoint-union monoidality, orientation reversal, and gluing are part of the definition Atiyah 1988, § 2, printed pp. 177–181, axioms (A)–(B) and (1)–(4c), PDF. The theorem-first bordism-category formulation and its precise variants are left to Mathematical QFT.

A useful consequence is the mapping-torus trace. If a structured diffeomorphism $f:\Sigma\to\Sigma$ acts by $Z(f)$, and $M_f$ carries the $s$-structure induced by gluing the structured cylinder, then

$$Z(M_f)=\operatorname{Tr}_{\mathcal H(\Sigma)}Z(f).$$

In the ordinary vector-valued control used on this page, the identity map with the product/glued structure gives

$$Z(\Sigma\times S^1) = \dim\mathcal H(\Sigma).$$

Spin theories valued in super vector spaces can instead produce a trace or a supertrace depending on the spin structure around the circle. With that qualification, the identity turns a closed-manifold path integral into a state-count check. Atiyah 1988, § 2, printed p. 180, PDF derives this trace by gluing. It also exposes a failure immediately: a proposed finite TQFT cannot assign one value to $Z(\Sigma\times S^1)$ and a different dimension to $\mathcal H(\Sigma)$.

Metric independence is necessary but not sufficient

For a conventional local action, metric variation defines the stress tensor,

$$T_{\mu\nu} = -\frac{2}{\sqrt{\lvert g\rvert}} \frac{\delta S}{\delta g^{\mu\nu}}.$$

Vanishing physical metric variation is a strong diagnostic. In a cohomological theory it may hold only on the cohomology of a nilpotent charge, because $T_{\mu\nu}$ is exact there. In a gauge theory the gauge-fixing action can contain a metric even when gauge-invariant observables do not. A quantum framing anomaly can leave dependence on a framing although continuous metric dependence cancels. Spin and orientation dependence are likewise compatible with topological behavior once those structures are declared. The cohomological stress-tensor argument and its quantum-measure qualification are worked out in Witten 1988, § 3, printed pp. 364–365, eqs. (3.1)–(3.9), especially eqs. (3.8)–(3.9) and n. 10, PDF.

The operator test is isotopy invariance. For protected operators $\mathcal O_i(N_i)$ on embedded supports $N_i$, a deformation that preserves labels, framings, incidence data, and the absence of crossings should obey

$$\frac{\mathrm d}{\mathrm dt} \left\langle \mathcal O_1(N_1(t))\cdots\mathcal O_n(N_n(t)) \right\rangle =0.$$

Moving one line through another, passing an endpoint through a wall, or changing a framing is not such a deformation. Those operations can produce braiding, junction, or framing data rather than equality.

Metric-independent numbers still do not establish locality. The decisive extra tests are compatible state spaces, cylinder identity, composition under cutting and gluing, and tensor product under disjoint union. Conversely, finite-dimensional state spaces are not a universal physical definition: noncompact fields, residual zero modes, continuum sectors, or generalized targets can require infinite-dimensional or derived state objects. This page uses $\operatorname{FinVect}_{\mathbb C}$ only for the controlled compact models below.

Five nearby notions answer different questions

The comparison below is the chapter’s compact decision aid. Its rows are not a hierarchy: a microscopic phase can flow to one of the field theories in a different row, and a full QFT can contain an invertible response sector without being invertible as a whole.

Five uses of “topological” compared by object type, metric dependence, state-and-gluing data, and invertibility; the rows are not mutually exclusive classes of theories

Object	Kind of object	Decisive test	States and protected excitations	What the label does not imply
Generic QFT	Whole field theory	Locality, dynamics, and the declared quantum consistency conditions	Usually metric-dependent, with generic propagating states and operators	Any protected topological sector or metric independence
Topological term	One ingredient of an action or exponentiated weight	The factor is invariant under allowed metric or deformation changes and passes its global-definition and coefficient quantization-or-periodicity tests	The rest of the QFT may still have propagating modes and generic state spaces	Neither metric independence of the full theory nor topological order
Invertible field theory or response	Whole field theory or a specified response sector	A gluing-compatible tensor inverse exists	Closed spatial manifolds receive lines; all assigned maps are invertible	Metric independence or a TQFT, a trivial response, or invertibility of every other infrared sector
Noninvertible TQFT	Whole topological field theory	Topological locality and gluing hold, but no stacking inverse exists	Nontrivial topological sectors or higher-dimensional state spaces can occur	Unitarity, semisimplicity, or realization by a microscopic material
Intrinsic topological order	Property of a microscopic gapped phase	A gapped phase has long-range universal structure not removable by a finite-depth local unitary or a gap-preserving local deformation	Topology-dependent degeneracy and anyonic or extended excitations are common diagnostics	That every abstract TQFT is a realizable phase, or that degeneracy alone proves the diagnosis

The second row is illustrated by four-dimensional Maxwell theory with a theta term. The factor $\exp(i\theta Q)$ is topological after its global data are specified, but the Maxwell kinetic term contains a Hodge star and the theory has propagating photons. The full theory is not a TQFT. The third row was developed on the preceding background-response page: line-valued state spaces are necessary, not merely a phase on closed spacetime. The last row requires microscopic and entanglement diagnostics that belong to the Many-Body volume.

First application: three compact topological gauge theories

Work on closed oriented three-manifolds and quantize on a closed oriented surface $\Sigma_g$ of genus $g$. For quantum Chern–Simons theory, include a chosen framing—or an equivalent relative gravitational-counterterm convention—and include spin structure when the level requires it. The BF and untwisted finite-gauge controls below use ordinary oriented data. All gauge fields in the Abelian models are compact, their global sectors are included, and their dynamical path integrals use the appropriate gauge-orbit measure. These assumptions are what turn local topological densities into field theories with finite state spaces.

Compact Abelian Chern–Simons theory

For a dynamical compact $U(1)$ connection $a$ at nonzero level $k$,

$$S_{\mathrm{CS}}[a] = \frac{k}{4\pi}\int_M a\wedge\mathrm da.$$

In the one-component normalization, an ordinary oriented bosonic theory has even $k$, whereas a spin theory permits any integer $k$ and odd $k$ depends on the spin structure. The genus-$g$ state-space dimension is

$$\dim\mathcal H_{U(1)_k}(\Sigma_g) = \lvert k\rvert^g.$$

Thus $\lvert k\rvert>1$ gives a noninvertible TQFT. The standard $k=\pm1$ theory is instead invertible and spin-dependent. The case $k=0$ is degenerate and is not described by this finite-dimensional formula. At the quantum level the Chern–Simons path integral also carries a framing anomaly, so “topological” does not mean “requires no tangential refinement.” To translate Belov–Moore’s convention, their scalar coefficient $k_{\mathrm{BM}}$ uses integral-period curvature and obeys $k_{\mathrm{site}}=2k_{\mathrm{BM}}$, while their integral lattice matrix is the page’s standard $K$-matrix. This is why an integral scalar level there corresponds to an even one-component bosonic level here, while a half-integral scalar level corresponds to an odd spin level. The level lattice and determinant state count are derived in Belov and Moore 2005, §§ 1–2 and § 5.3, arXiv v1, printed pp. 3–4, 7–9, and 26, especially eqs. (1.1)–(1.3) and the prose after eq. (5.17), PDF.

Compact BF theory

For two dynamical compact $U(1)$ connections $a$ and $b$,

$$S_{\mathrm{BF}}[a,b] = \frac{N}{2\pi}\int_M b\wedge\mathrm da, \qquad N\in\mathbb Z_{>0}.$$

The $K$-matrix is

$$K= \begin{pmatrix} 0&N\\ N&0 \end{pmatrix}, \qquad \lvert\det K\rvert=N^2,$$

so

$$\dim\mathcal H_{BF_N}(\Sigma_g)=N^{2g}.$$

This is the general Abelian determinant formula $\dim\mathcal H(\Sigma_g)=\lvert\det K\rvert^g$ Belov and Moore 2005, § 5.3, arXiv v1, printed p. 26, prose after eq. (5.17), PDF. For $N>1$ the torus already has more than one state, and Wilson lines have a nontrivial mutual linking phase. For $N=1$ the finite sector is trivial. The compact global completion, integer level, operator algebra, and finite-gauge interpretation are given in Kapustin and Seiberg 2014, § 3, arXiv v2, printed pp. 9–13, eqs. (3.1)–(3.16), PDF. Treating $a$ and $b$ as ordinary noncompact forms would lose the finite holonomy sectors and invalidate this state count.

Finite gauge theory and the groupoid measure

Let $G$ be finite and let $[\omega]\in H^3(BG;\mathbb R/\mathbb Z)$. A fixed flat bundle $P$ contributes the unit phase

$$\mathcal R_\omega[M;P] = \exp\!\left( 2\pi i\left\langle f_P^*\omega,[M]\right\rangle \right).$$

On a closed manifold this fixed-background factor is a unit phase. It is the closed-spacetime shadow of an invertible background theory only after the compatible local, gluing, and boundary-line data are supplied; the unit phase alone is not a proof of full invertibility. Dijkgraaf–Witten theory instead sums over the finite bundle groupoid,

$$Z_{G,\omega}(M) = \sum_{[P]\in\pi_0\operatorname{Bun}^{\mathrm{flat}}_G(M)} \frac{\mathcal R_\omega[M;P]} {\lvert\operatorname{Aut}(P)\rvert}.$$

The automorphism weight is required by gluing; an unweighted sum is not the same local theory. In the untwisted Abelian case $G=\mathbb Z_N$,

$$\dim\mathcal H(T^2) = \big\lvert H^1(T^2;\mathbb Z_N)\big\rvert =N^2.$$

This matches compact $BF_N$ only after the compact sectors and measure are matched; local flatness equations alone do not prove an equivalence. The finite action and normalized bundle sum appear in Dijkgraaf and Witten 1990, §§ 6.2–6.3, printed pp. 415–417, eqs. (6.8)–(6.17), PDF. Boundary lines, the automorphism-weighted measure, and gluing are constructed in Freed and Quinn 1993, §§ 1–2, current arXiv v3, printed pp. 4–12, especially Lemma 2.4, Theorem 2.13, and eq. (2.17), PDF.

All three theories pass more than a metric test: they have structured state spaces, topological operators, and compatible cutting and gluing. Their state counts also show why “dynamical” and “noninvertible” are different axes: $U(1)_{\pm1}$ is dynamical but invertible, whereas $BF_N$ and untwisted $\mathbb Z_N$ gauge theory are noninvertible for $N>1$.

The mapping-torus trace makes the gluing check numerical. Give $T^3=T^2\times S^1$ the product structure and take $k=N=2$. Then

$$\begin{aligned} Z_{U(1)_2}(T^3) &= \dim\mathcal H_{U(1)_2}(T^2) =2, \\ Z_{BF_2}(T^3) &= \dim\mathcal H_{BF_2}(T^2) =4, \\ Z_{\mathrm{DW},\mathbb Z_2}(T^3) &= \frac{\big\lvert\operatorname{Hom}(\mathbb Z^3,\mathbb Z_2)\big\rvert} {\lvert\mathbb Z_2\rvert} =\frac{8}{2} =4. \end{aligned}$$

The last denominator is the automorphism weight for each Abelian flat bundle. The matching $BF_2$ and untwisted $\mathbb Z_2$ counts are a necessary consistency check for their standard equivalence after global completion; one matching partition function is not by itself a proof of that equivalence.

Ordered interval observables form an E₁ algebra

There is a complementary local-observable formulation. Take a finite-dimensional unital associative algebra $A$ over $\mathbb C$ and view it as an $E_1$ algebra in chain complexes, concentrated in degree zero for this example. For every oriented open interval $I\subset\mathbb R$, set

$$\mathcal F_A(I)=A.$$

If $I_1<\cdots<I_n$ are pairwise disjoint subintervals of a larger interval $J$, define the structure map by ordered multiplication,

$$\mu_{I_1,\ldots,I_n;J}: A^{\otimes n}\longrightarrow A, \qquad a_1\otimes\cdots\otimes a_n \longmapsto a_1\cdots a_n.$$

These formulas first define a locally constant prefactorization assignment; the explicit ordered-interval construction is given in Costello and Gwilliam 2025, § 1.1, current arXiv v2, printed pp. 3–4, PDF. The empty family maps $1\in\mathbb C$ to the unit $1_A$. Inclusions of one interval into another act by the identity under the chosen identifications, so the assignment is locally constant. Composing configurations of subintervals in two stages gives either

$$(a_1a_2)a_3 \qquad\text{or}\qquad a_1(a_2a_3).$$

The factorization compatibility condition is therefore precisely associativity. The orientation of the line orders the inputs, so no commutativity is required. Passing from prefactorization products to a factorization algebra additionally imposes Weiss descent. For the constructible disk assignment used here, the required descent statement is the separate local-to-global input in Karlsson, Scheimbauer, and Walde 2026, Example 5.3.7 and Remark 5.3.8, current arXiv v4, printed pp. 64–65, PDF. With that input, the ordered product is the concrete $E_1$ multiplication encoded by a locally constant factorization algebra on intervals.

This construction is a bounded algebraic model of topological local observables. It is not automatically a full Atiyah-style TQFT: an arbitrary associative algebra does not by itself supply state spaces for every closed spatial manifold, nondegenerate pairings, or bordism maps. Those are the additional functorial data. Reflection positivity and unitarity are further physical hypotheses, not part of the bare Atiyah definition. Their precise variants belong to the theorem-level factorization and extended-TQFT treatments.

Cutting a circle produces Hochschild homology

The interval model has a global invariant that can be computed directly. Write

$$A^e=A\otimes A^{\mathrm{op}}.$$

Cut $S^1$ at two points into two oriented intervals. Each interval carries the regular $A$-bimodule, while the two collars supply the left and right actions. Excision glues the intervals by the derived tensor product,

$$\int_{S^1}A \simeq A\mathop{\otimes}^{\mathbb L}_{A^e}A \simeq CH_\bullet(A),$$

and hence

$$H_\bullet\!\left(\int_{S^1}A\right) \cong HH_\bullet(A).$$

Here $\int_{S^1}A$ is an object of the derived category of chain complexes over $\mathbb C$, defined up to quasi-isomorphism. It is not automatically a number assigned by an Atiyah TQFT or a Hilbert space. This circle calculation and its derived gluing are the one-dimensional case of factorization excision Ayala and Francis 2015, Definition 3.15, Lemma 3.18, and Theorem 3.19, current arXiv v6, printed pp. 18–19, PDF. For an ordinary ungraded algebra, the Hochschild chain group in degree $n$ is $CH_n(A)=A^{\otimes(n+1)}$, with differential

$$\begin{aligned} b(a_0\otimes\cdots\otimes a_n) ={}& \sum_{i=0}^{n-1}(-1)^i a_0\otimes\cdots\otimes a_i a_{i+1} \otimes\cdots\otimes a_n \\ &+(-1)^n a_n a_0\otimes a_1\otimes\cdots\otimes a_{n-1}. \end{aligned}$$

The last term is where the two ends of the cut interval rejoin. Associativity gives $b^2=0$. Refinement-independent derived gluing is a separate consequence of factorization excision; it is not equivalent merely to the chain identity $b^2=0$. The derived symbol matters: replacing it by an underived quotient can erase higher Tor groups.

For the concrete control $A=M_r(\mathbb C)$,

$$HH_0(A)=A/[A,A]\cong\mathbb C, \qquad HH_n(A)=0\quad(n>0).$$

The matrix trace identifies the degree-zero quotient, and separability of the matrix algebra removes higher Hochschild homology. Concretely, the normalized separability idempotent $r^{-1}\sum_{i,j}E_{ij}\otimes E_{ji}^{\mathrm{op}}$ splits $A^e\to A$, so $A$ is projective as an $A^e$-module and the higher $\operatorname{Tor}^{A^e}$ groups vanish Weibel 1994, Chapter 9, § 9.2, Lemma 9.2.10 and Theorem 9.2.11, printed pp. 310–311. This is a useful global calculation, but it still does not turn every associative $A$ into a unitary TQFT. A trace or pairing used to extract numerical amplitudes must be supplied and checked separately.

There is a sharp comparison. If $V=\mathbb C^r$ and $A=\operatorname{End}(V)$, then the factorization-homology calculation gives $\int_{S^1}A\simeq\mathbb C$. The ordinary one-dimensional bordism TQFT that assigns $V$ to a positively oriented point instead gives $Z(S^1)=\operatorname{Tr}(\operatorname{id}_V)=r$. These are different constructions and must not be identified.

The exact interval construction and circle computation are the physical worked examples handed to the forthcoming Mathematical QFT pages on locally constant factorization algebras and $E_n$ algebras and factorization homology and manifold invariants. Those pages will own the homotopy-coherent definitions, general excision theorem, and higher-dimensional classification.

What the definition does not prove

The operational tests above are deliberately bounded.

• A vanishing stress tensor or a topological classical action does not prove quantum metric independence; gauge fixing, determinants, and anomalies must be included.
• A finite-dimensional Hilbert space does not imply topological locality. Finite-volume truncations and symmetry-broken systems can also have finite state spaces.
• Ground-state degeneracy alone does not diagnose intrinsic topological order. It can arise from spontaneous symmetry breaking, boundary conditions, or an accidental finite-size crossing.
• An abstract TQFT need not be unitary, reflection-positive, semisimple, or realizable as the infrared limit of a microscopic Hamiltonian.
• An anomalous or relative theory may assign vectors or lines rather than absolute numbers. Its gluing law includes the bulk or anomaly theory.
• A closed-bordism TQFT does not by itself choose a physical boundary condition. Invertibility in the bulk neither makes every boundary trivial nor guarantees a symmetry-preserving gapped boundary.
• The unextended functor does not encode operators of every codimension. Fully extended theories require higher-categorical targets and dualizability hypotheses.
• A locally constant factorization algebra describes local observable products and descent. It becomes a full field theory only after the additional global and duality data have been supplied; positivity is further required only if a reflection-positive or unitary physical theory is intended.

Common pitfalls

“The action contains a topological term, so the theory is a TQFT.” A theta term can coexist with a metric-dependent kinetic term and propagating modes. Test the full quantum theory, not one summand of its action.

“Metric independent means structure free.” A theory can depend on orientation, spin, framing, or a background bundle while remaining independent of continuous metric deformations.

“Every TQFT has topological order.” A fully extended invertible phase has only tensor-invertible state and operator data; in the ordinary unextended target this includes line state spaces. Line state spaces alone do not inspect every codimension and are only a necessary test. Conversely, topological order is a claim about a microscopic gapped phase, not just a functor written on paper.

“Every field that is integrated produces a noninvertible theory.” $U(1)_{\pm1}$ is a dynamical invertible spin TQFT. Invertibility is decided by stacking and the full state-space assignment.

“An $E_1$ algebra is already a complete TQFT.” Ordered local multiplication is only one layer. Global state spaces, nondegenerate pairings, and bordism maps remain additional requirements. Positivity is an extra condition when a reflection-positive or unitary physical theory is intended.

Check your understanding

1. A theta term does not topologize Maxwell theory

Four-dimensional Maxwell theory contains both $\int F\wedge\star F$ and $\theta\int F\wedge F$. Which term obstructs the claim that the full theory is a TQFT?

Solution

The kinetic term contains the Hodge star and gives a propagating photon. The theta factor can be topological after its global normalization is fixed, but one topological factor does not remove the metric dependence or local modes of the full theory.

2. Recover a state count from a product mapping torus

In the ordinary vector-valued theory with the product/glued tangential structure, show that a finite TQFT satisfies $Z(\Sigma\times S^1)=\dim\mathcal H(\Sigma)$.

Solution

Cut the circle at one point. The resulting cylinder is identity evolution on $\mathcal H(\Sigma)$. Gluing its two boundary copies of $\Sigma$ takes the trace, so the closed amplitude is $\operatorname{Tr}(\operatorname{id}_{\mathcal H(\Sigma)})= \dim\mathcal H(\Sigma)$. A super-valued spin refinement can instead give a supertrace for the other circle spin structure, so the structure carried by the mapping torus is part of the statement.

3. Use state counts without assuming a converse

Use the torus state counts for compact $U(1)_1$, $U(1)_3$, and $BF_2$ to identify which theories are certainly noninvertible. What extra input is needed for $U(1)_1$?

Solution

$U(1)_3$ has three torus states and $BF_2$ has four, so neither can have a tensor inverse. The single torus state of $U(1)_1$ only removes this obstruction; it is not a converse theorem. Invertibility uses the known full spin Chern–Simons functor, whose bordism amplitudes are nonzero and whose stacking inverse is $U(1)_{-1}$. All three theories are dynamical, so field role does not decide invertibility.

4. Derive associativity from nested intervals

Place three intervals in order inside a larger interval. Compare first combining the left pair with first combining the right pair.

Solution

The two refinements give $(a_1a_2)a_3$ and $a_1(a_2a_3)$. Factorization compatibility requires equality for every triple, which is exactly associativity. Exchanging the interval order is not an allowed oriented isotopy, so commutativity is not required.

5. Read the zeroth Hochschild group

Use the displayed differential to show that $HH_0(A)=A/[A,A]$.

Solution

Degree-zero chains are elements of $A$. On a degree-one chain, $b(a_0\otimes a_1)=a_0a_1-a_1a_0$. Quotienting degree-zero chains by these boundaries gives $A/[A,A]$.

6. Diagnose a false topological-order signal

A finite system has two nearly degenerate ground states that split exponentially with volume. Does this alone prove intrinsic topological order?

Solution

No. Spontaneous symmetry breaking can produce the same finite-size pattern. One must test topology dependence, local indistinguishability, extended excitations, symmetry action, and robustness before assigning an infrared noninvertible TQFT.

Continue to gluing, models, and classification

State Spaces, Cobordisms, and Gluing will make the cylinder, trace, pairing, and decomposition-independence tests explicit. The next model pages will develop Abelian Chern–Simons theory, BF theory, and finite gauge theory with Dijkgraaf–Witten twists.

For axioms and classification, Mathematical QFT will treat bordism categories and symmetric-monoidal TQFTs and Atiyah–Segal functoriality and gluing. The fully extended route will continue through Fully Extended TQFTs and Higher Categories and Dualizability and the Cobordism Hypothesis, where every-codimension assignments and classification hypotheses belong. Physical boundary and defect data will be treated separately in Boundaries, Defects, and Extended Operators in TQFT. Microscopic distinctions among symmetry breaking, invertible matter, and intrinsic topological order will be developed on the Many-Body page Topological Order, Invertible Phases, and Matter Diagnostics.

References

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• Ayala, David, and John Francis. “Factorization Homology of Topological Manifolds.” Journal of Topology 8, no. 4 (2015): 1045–1084. DOI. Open PDF, current arXiv v6.
• Belov, Dmitriy M., and Gregory W. Moore. “Classification of Abelian Spin Chern–Simons Theories.” arXiv:hep-th/0505235v1 [hep-th], 2005. Stable record.
• Costello, Kevin, and Owen Gwilliam. “Factorization Algebra.” In Encyclopedia of Mathematical Physics, 2nd ed., edited by Richard Szabo and Martin Bojowald, 569–583. Elsevier, 2025. DOI. Open PDF, arXiv v2.
• Dijkgraaf, Robbert, and Edward Witten. “Topological Gauge Theories and Group Cohomology.” Communications in Mathematical Physics 129, no. 2 (1990): 393–429. DOI. Open PDF.
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