The Atiyah–Segal axioms for topological quantum field theory

Intuition Beginner

A physical theory usually hands you a complicated answer: fields, energies, particle tracks, all depending on the fine details of space and time. A topological quantum field theory throws away almost all of that and keeps only the shape. It is a rule that reads a closed spacetime and returns a single number, and reads a closed space and returns a list of allowed quantum states. Nothing about distances, only about how the pieces are connected.

Think of space as a stage and time as the play running on it. To each possible shape of the stage at one instant, the rule assigns a space of states — the ways the universe could be at that moment. To each way the stage can evolve from one shape into another, the rule assigns a recipe that turns earlier states into later ones. The whole theory is just this bookkeeping of states and evolutions, stripped down until only the topology survives.

The one property that makes this powerful is consistency under cutting and gluing. If you slice a spacetime into two halves along a slice of space, the number the theory assigns to the whole must be recoverable by combining the two halves along the shared slice. Quantum mechanics already obeyed a version of this. The axioms below say: keep exactly that gluing law, forget everything else, and you have captured the topological skeleton of a quantum field theory.

Visual Beginner

Alt text: On the left, a vertical tube connects a bottom circle to a top circle; this tube stands for a way one slice of space evolves into another. A horizontal cut splits the tube into two shorter tubes that share a middle circle. An arrow shows the two halves being rejoined along that shared circle, recovering the original tube — the gluing law. On the right, the same tube has both ends sealed shut into a closed doughnut shape, and a single number is attached to it, illustrating that a closed spacetime receives just one number while a slice of space (a circle) receives a whole space of states.

Worked example Beginner

Take the simplest possible such rule in one dimension of space and time. Space is a finite collection of points; spacetime is a collection of line segments and loops joining them. Fix a number, call it $d$, to be the size of the state space attached to a single point.

To one point the rule assigns a state space of size $d$. To two points side by side it assigns size $d$ times $d$, because separate pieces of space combine by multiplying their state-space sizes. To a segment joining one point to another, the rule assigns the recipe that does nothing — it carries states straight across. And to a closed loop, with no endpoints at all, the rule assigns a single number: it turns out to be exactly $d$, the size of the state space of the one point the loop wraps.

What this tells us: the number a closed loop receives is forced to equal the size of the state space of a point. Cutting the loop open into a segment and re-gluing its two ends is what produces that count. The bookkeeping is rigid: once you fix what a point gets, the loop's number is determined. This single computation, that a circle's number equals the dimension of a point's state space, is the baby version of the central finiteness law of the whole subject.

Check your understanding Beginner

Formal definition Intermediate+

Fix a dimension $n$ and a ground field $\mathbb{C}$. The bordism category ${\mathrm{Bord}}_n$ has as objects the closed oriented $(n-1)$-manifolds and as morphisms $Y_0\to Y_1$ the diffeomorphism classes, rel boundary, of compact oriented $n$-dimensional bordisms $(W;Y_0,Y_1)$ with $\partial W=\overline{Y_0}\bigsqcup Y_1$, the bar denoting orientation reversal. Composition is gluing along the common boundary, smoothed by the collar theorem; the identity on $Y$ is the cylinder $Y\times [0,1]$. This is the symmetric monoidal category constructed in 03.02.20, with monoidal product disjoint union $\bigsqcup$, unit the empty manifold $\varnothing$, and a symmetry exchanging summands. Each object $Y$ is dualizable, with dual $\overline{Y}$; the evaluation and coevaluation are the bent cylinders read as bordisms $\overline{Y}\bigsqcup Y\to \varnothing$ and $\varnothing \to Y\bigsqcup \overline{Y}$.

The target is the symmetric monoidal category ${\mathrm{Vect}}_{\mathbb{C}}$ of complex vector spaces under $\otimes$, with unit $\mathbb{C}$ and the standard symmetry $v\otimes w\mapsto w\otimes v$. (When fermions are present the target is the category ${\mathrm{sVect}}_{\mathbb{C}}$ of $\mathbb{Z}/2$-graded vector spaces with the Koszul-signed symmetry; the axioms below are unchanged.)

Definition (topological quantum field theory). An $n$-dimensional TQFT is a symmetric monoidal functor $$ Z : \mathrm{Bord}n \longrightarrow \mathrm{Vect}{\mathbb{C}}. $$ Unwinding the words "symmetric monoidal functor" produces the Atiyah–Segal axioms:

• (Functoriality) $Z$ assigns to each closed $(n-1)$-manifold $Y$ a vector space $Z(Y)$, its state space, and to each bordism $(W;Y_0,Y_1)$ a linear map $Z(W):Z(Y_0)\to Z(Y_1)$, with $Z(W^{\prime }\circ W)=Z(W^{\prime })\circ Z(W)$ and $Z(Y\times [0,1])={\mathrm{id}}_{Z(Y)}$.
• (Monoidality) $Z(\varnothing )=\mathbb{C}$ and $Z(Y_0\bigsqcup Y_1)=Z(Y_0)\otimes Z(Y_1)$, compatibly with the symmetry: the swap of components goes to the swap of tensor factors.
• (Gluing) If a bordism $W$ is cut along a closed $(n-1)$-manifold $Y$ into $W_1:Y_0\to Y$ and $W_2:Y\to Y_1$, then $Z(W)=Z(W_2)\circ Z(W_1)$. Pairing the two ends instead, $Z(W_1{\cup }_Y{W}_2)=⟨Z(W_1),Z(W_2)⟩$ contracts the shared $Z(Y)$ index against its dual.
• (Orientation reversal) $Z(\overline{Y})=Z(Y{)}^{\ast }$, the dual vector space, compatibly with evaluation and coevaluation.

Two consequences are immediate from the definition rather than added by hand. A closed $n$-manifold $M$ is a bordism $\varnothing \to \varnothing$, so $Z(M):\mathbb{C}\to \mathbb{C}$ is multiplication by a scalar $Z(M)\in \mathbb{C}$, the partition function. And the input datum can be enriched: a tangential structure is a map $H\to O(n)$ of topological groups together with a lift of each manifold's frame bundle along $BH\to BO(n)$; replacing manifolds by $H$-manifolds yields ${\mathrm{Bord}}_n^H$, recovering the oriented case for $H=SO(n)$, the spin case 03.09.04 for $H=\mathrm{Spin}(n)$, and the framed case for $H$ the one-element group.

Counterexamples to common slips

• State space attached to the wrong dimension. $Z$ assigns vector spaces to $(n-1)$-manifolds and numbers to $n$-manifolds. A closed $n$-manifold does not receive a vector space; it receives a scalar, because as a bordism it has empty source and target.
• Disjoint union is not direct sum. Monoidality gives $Z(Y_0\bigsqcup Y_1)=Z(Y_0)\otimes Z(Y_1)$, a tensor product, not $Z(Y_0)\oplus Z(Y_1)$. The dimensions multiply, matching how independent quantum systems compose.
• Forgetting orientation reversal makes the pairing ill-defined. Without $Z(\overline{Y})=Z(Y{)}^{\ast }$ there is no canonical contraction to express the gluing law, and the partition function of a closed manifold built by capping a cylinder would not be computable from its pieces.

Key theorem with proof Intermediate+

Theorem (finiteness and the circle trace). Let $Z:{\mathrm{Bord}}_n\to {\mathrm{Vect}}_{\mathbb{C}}$ be a TQFT. Then for every closed $(n-1)$-manifold $Y$ the state space $Z(Y)$ is finite-dimensional, and the partition function of $Y\times S^1$ computes its dimension: $$ Z(Y \times S^1) = \dim_{\mathbb{C}} Z(Y). $$

Proof. Write $V=Z(Y)$ and $V^{\ast }=Z(\overline{Y})$. Consider the two elementary bordisms obtained from the cylinder $Y\times [0,1]$ by bending it. The coevaluation bordism $\beta :\varnothing \to Y\bigsqcup \overline{Y}$ is the cylinder with both ends pushed to the top; the evaluation bordism $\epsilon :\overline{Y}\bigsqcup Y\to \varnothing$ is the cylinder with both ends pushed to the bottom. By monoidality and orientation reversal, $Z(\beta ):\mathbb{C}\to V\otimes V^{\ast }$ and $Z(\epsilon ):V^{\ast }\otimes V\to \mathbb{C}$.

Sliding one bend past the other is an ambient isotopy of bordisms — the zig-zag or $S\bar{S}$ identity — straightening a doubly-bent cylinder back into the plain cylinder $Y\times [0,1]$. Functoriality turns this isotopy into the algebraic relations $$ (\mathrm{id}V \otimes Z(\varepsilon)) \circ (Z(\beta) \otimes \mathrm{id}V) = \mathrm{id}V, \qquad (Z(\varepsilon) \otimes \mathrm{id}{V^*}) \circ (\mathrm{id}{V^*} \otimes Z(\beta)) = \mathrm{id}{V^}, $$ the two triangle identities exhibiting $V^$asagenuinedualof$V$inside$\mathrm{Vect}{\mathbb{C}}$.Avectorspaceadmitssuchevaluationandcoevaluationmapsifandonlyifitisfinite-dimensional:writing$Z(\beta)(1) = \sum{i=1}^{N} e_i \otimes f^i$,thefirsttriangleidentitysays${e_i}$spans$V$with${f^i}$adualsystem,so$\dim V \le N < \infty$.

Now build $Y\times S^1$ by gluing the two ends of the cylinder $Y\times [0,1]$, which is the same as composing $\beta$ and $\epsilon$ around the loop and contracting. By the gluing law the resulting partition function is the trace of the identity on $V$: $$ Z(Y \times S^1) = \mathrm{tr}V(\mathrm{id}V) = \mathrm{tr}\big(Z(\varepsilon) \circ (\text{flip}) \circ Z(\beta)\big) = \dim{\mathbb{C}} V. $$ The middle equality is the categorical trace of the identity endomorphism of a dualizable object, which in $\mathrm{Vect}{\mathbb{C}}$isexactlythedimension.$\blacksquare$

Bridge. This finiteness theorem is the bridge from the formal axioms to every computation in the subject: it is the foundational reason a TQFT assigns finite data, and it builds toward the classification results that the rest of the chapter develops. The trace identity $Z(Y\times S^1)=\dim Z(Y)$ is exactly the categorical trace of a dualizable object, so the rigidity of ${\mathrm{Bord}}_n$ — every object has a dual, established in 03.02.20 — is dual to the linear-algebra fact that the only dualizable vector spaces are the finite-dimensional ones. This pattern appears again in 03.16.02, where the same gluing-and-bending calculus, specialized to surfaces, forces a $2$-dimensional theory to be a finite-dimensional commutative Frobenius algebra; the central insight, that physical consistency under cutting equals algebraic duality, generalises to the cobordism-hypothesis statement of 03.16.03.

Exercises Intermediate+

Advanced results Master

The monoid object underlying $Z(Y)$. For any closed $(n-1)$-manifold $Y$, the pair-of-pants bordism — the connected $n$-bordism $Y\bigsqcup Y\to Y$ with the topology of a thrice-punctured product — equips $Z(Y)$ with a multiplication $Z(Y)\otimes Z(Y)\to Z(Y)$, and the disk-capping bordism $\varnothing \to Y$ supplies a unit. Associativity and unitality are isotopies of bordisms, so $Z(Y)$ is an algebra; the reversed bordisms supply a comultiplication and counit, and the cylinder-bending nondegeneracy makes the pairing $Z(Y)\otimes Z(Y)\to \mathbb{C}$ associative against the product. In dimension $n=2$ with $Y=S^1$ this is exactly a commutative Frobenius algebra, the content extracted in 03.16.02 ^{[book-collection Lectures 1 and 3]}.

Reconstruction and the local-to-global principle. The gluing axiom is a sheaf-like locality: the value of $Z$ on a manifold cut into elementary pieces is determined by its values on those pieces and the contraction data along the cuts. Combined with the handle decomposition of 03.02.20, this means a TQFT is determined by its values on a finite generating set of elementary bordisms (caps, cups, saddles), subject to the relations among them. The classification programme of the chapter is the systematic identification of these generators-and-relations presentations: $2$-dimensional in 03.16.02, the fully-local refinement in 03.16.03, and the gauge-theoretic example in 03.16.06.

Tangential structure and the Madsen–Tillmann/bordism-spectra link. Replacing ${\mathrm{Bord}}_n$ by ${\mathrm{Bord}}_n^H$ for a tangential structure $H\to O(n)$ does not change the axioms but changes the source category. The set of bordism classes of closed $H$-manifolds is the homotopy of a Thom spectrum — the framed case gives the sphere spectrum, the oriented case $MSO$, the spin case $M\mathrm{Spin}$ — placing the partition-function part of a TQFT into the world of generalized homology of 03.12.04. This is the doorway through which invertible field theories become a computation in stable homotopy; the $H$-structure introduced here is precisely the input the Madsen–Tillmann spectrum $MTH$ packages downstream ^{[Atiyah 1988]}.

Segal's analytic sibling. Segal's axioms for conformal field theory replace diffeomorphism classes of bordisms by Riemann surfaces with parametrized boundary, so the morphism spaces become infinite-dimensional moduli and $Z$ becomes holomorphic rather than locally constant. Setting the conformal data to a point — forgetting the complex structure — degenerates the Segal axioms to the Atiyah axioms; the topological theory is the "constant" limit of the conformal one ^{[Segal 1988]}.

Synthesis. The Atiyah–Segal axioms are where the geometry of the preceding chapters becomes algebra. The cobordism category of 03.02.20 supplies the source, its rigidity (every object self-dual) is the foundational reason state spaces are finite-dimensional, and this is exactly the categorical trace computation that turns $Z(Y\times S^1)$ into $\dim Z(Y)$. Putting these together, a TQFT is a single functor whose monoidality recovers the tensor-product composition of independent systems and whose gluing law is the central insight carried over from quantum mechanics; the tangential-structure input $H$ generalises the oriented case and is dual, through the Pontryagin–Thom construction of 03.12.04, to the Thom-spectrum bookkeeping that classifies the invertible theories. The bridge is functoriality itself: every topological move on manifolds becomes a linear-algebraic identity, and the entire subject is the study of which algebraic structures arise as the image of such a functor.

Full proof set Master

Proposition (the state space is canonically self-dual to its reverse). For a TQFT $Z$ and a closed $(n-1)$-manifold $Y$, the bending bordisms exhibit $Z(\overline{Y})$ as the dual vector space of $Z(Y)$, naturally in $Y$.

Proof. Let $\beta :\varnothing \to Y\bigsqcup \overline{Y}$ and $\epsilon :\overline{Y}\bigsqcup Y\to \varnothing$ be the coevaluation and evaluation bordisms. Functoriality and monoidality give linear maps $Z(\beta ):\mathbb{C}\to Z(Y)\otimes Z(\overline{Y})$ and $Z(\epsilon ):Z(\overline{Y})\otimes Z(Y)\to \mathbb{C}$. The zig-zag isotopies straightening a doubly bent cylinder are equalities of bordism classes, so applying $Z$ yields the two triangle identities. In a symmetric monoidal category, an object $A$ equipped with $B$ and maps satisfying both triangle identities exhibits $B=A^{\vee }$ as the dual, uniquely up to canonical isomorphism. In ${\mathrm{Vect}}_{\mathbb{C}}$ the dual of a finite-dimensional space is the linear-functional space, so $Z(\overline{Y})\cong Z(Y{)}^{\ast }$. Naturality in $Y$ follows because a bordism $W:Y\to Y^{\prime }$ and its reverse $\overline{W}:\overline{Y^{\prime }}\to \overline{Y}$ are intertwined by $\beta ,\epsilon$ through the same isotopies, making the duality compatible with morphisms. $■$

Proposition (gluing computes the partition function as a trace). Let $M$ be a closed $n$-manifold cut along a closed $(n-1)$-manifold $Y$ into a bordism $W:Y\to Y$ whose two ends are then identified. Then $Z(M)={\mathrm{tr}}_{Z(Y)}Z(W)$.

Proof. Cutting $M$ along $Y$ presents $M$ as the trace, in the categorical sense, of the endomorphism bordism $W:Y\to Y$: glue the target end of $W$ to its source end. In a symmetric monoidal category with $Y$ dualizable, this geometric self-gluing is computed by composing $W$ with the evaluation and coevaluation around the loop, $M=\epsilon \circ (W\otimes {\mathrm{id}}_{\overline{Y}})\circ {\beta }^{\prime }$ for the appropriately bent ${\beta }^{\prime }$. Functoriality sends this to $Z(\epsilon )\circ (Z(W)\otimes \mathrm{id})\circ Z({\beta }^{\prime })$, which is by definition the categorical trace of $Z(W)$ on the dualizable object $Z(Y)$. In ${\mathrm{Vect}}_{\mathbb{C}}$ the categorical trace is the ordinary matrix trace, so $Z(M)={\mathrm{tr}}_{Z(Y)}Z(W)$. Taking $W=Y\times [0,1]$, so $Z(W)=\mathrm{id}$, recovers $Z(Y\times S^1)=\dim Z(Y)$. $■$

Proposition (the unit theory). Fixing $Z(Y)=\mathbb{C}$ for every $Y$ and $Z(W)={\mathrm{id}}_{\mathbb{C}}$ for every bordism defines a TQFT, and its partition function is identically $1$.

Proof. The assignment is a functor: composition of bordisms goes to $\mathrm{id}\circ \mathrm{id}=\mathrm{id}$, and the cylinder goes to ${\mathrm{id}}_{\mathbb{C}}$. It is monoidal: $Z(\varnothing )=\mathbb{C}$ and $Z(Y_0\bigsqcup Y_1)=\mathbb{C}=\mathbb{C}\otimes \mathbb{C}$, with the symmetry of ${\mathrm{Vect}}_{\mathbb{C}}$ matching the symmetry of ${\mathrm{Bord}}_n$ since both swaps are the identity on $\mathbb{C}$. Every closed $M$, as a bordism $\varnothing \to \varnothing$, goes to ${\mathrm{id}}_{\mathbb{C}}$, the scalar $1$. This is the unit object of the (symmetric monoidal) category of TQFTs and the basepoint from which invertible theories are deformations. $■$

The Segal-axiom degeneration and the $MTH$ identification are stated in the Advanced results without proof; see Freed ^{[book-collection Lectures 1 and 3]} for the tangential-structure formalism and Segal ^{[Segal 1988]} for the conformal axioms.

Connections Master

• Handles, surgery, and the cobordism category 03.02.20. That unit constructs the symmetric monoidal category ${\mathbf{Cob}}_n$ with its duals and states the one-sentence Atiyah–Segal definition; this unit promotes that sentence to the worked axioms, supplying the state spaces, partition functions, gluing law, and finiteness theorem that the handle calculus there makes computable. The factorisation of every bordism into elementary handles is precisely what reduces a TQFT to generators and relations.

• Spin structure 03.09.04. The tangential structure $H\to O(n)$ that enriches ${\mathrm{Bord}}_n$ into ${\mathrm{Bord}}_n^H$ is, for $H=\mathrm{Spin}(n)$, exactly the spin structure built there; spin TQFTs are the theories whose source manifolds carry that lift, and the fermionic target ${\mathrm{sVect}}_{\mathbb{C}}$ is the natural codomain for them. The dependence of a theory on the spin structure is the seed of the Arf-theory example in the chapter ahead.

• Spectrum 03.12.04. The partition-function half of a TQFT records a bordism invariant of closed $H$-manifolds, and bordism classes are homotopy groups of a Thom spectrum; the generalized-homology machinery developed there is what turns the tangential-structure input of this unit into the stable-homotopy classification of invertible theories. The symmetric-monoidal category of spectra is the higher-categorical shadow of the symmetric-monoidal ${\mathrm{Bord}}_n$.

• Classification of 2d oriented TQFTs 03.16.02. The pair-of-pants algebra structure on $Z(S^1)$ extracted here is completed there into the theorem that $2$-dimensional oriented TQFTs are the same as commutative Frobenius algebras, the first full classification and the signature computation of the chapter.

• Extended TQFT and the cobordism hypothesis 03.16.03. Pushing the locality of the gluing axiom down to points refines ${\mathrm{Bord}}_n$ into a higher category and refines finiteness into full dualizability; the cobordism hypothesis classifies fully extended framed theories by fully dualizable objects, the deepest generalisation of the duality argument proved in this unit.

• Chern–Simons theory as a 3d TQFT 03.16.06. The axioms here are instantiated by a genuinely infinite family of theories there: the Chern–Simons path integral assigns numbers to $3$-manifolds and conformal-block state spaces to surfaces, the canonical example showing the axioms have rich content beyond the unit theory.

Historical & philosophical context Master

Michael Atiyah crystallised the axioms in Topological quantum field theories (Publications Mathématiques de l'IHÉS 68, 1988, 175–186), abstracting the structure he saw in Witten's then-recent constructions of Donaldson and Jones-polynomial invariants from quantum field theory ^{[Atiyah 1988]}. Atiyah's formulation deliberately mirrored the axioms Graeme Segal had written for conformal field theory in lectures and the manuscript The definition of conformal field theory, circulated from 1988 and published in revised form in 2004 (Topology, Geometry and Quantum Field Theory, London Mathematical Society Lecture Note Series 308, 421–577) ^{[Segal 1988]}. Segal's surfaces carried conformal structure and the morphism spaces were moduli of Riemann surfaces; Atiyah's topological version discarded that structure, keeping only the cobordism category, and so produced a finite, combinatorial axiom system.

The functorial reading — a field theory is a symmetric monoidal functor — became the organising language of the field through the work of Quinn, Lawrence, and others in the 1990s, and Daniel Freed's Lectures on Field Theory and Topology (CBMS Regional Conference Series in Mathematics 133, 2019) gives the modern synthesis, threading the axioms through tangential structures, invertible theories, and the Freed–Hopkins classification ^{[book-collection Lectures 1 and 3]}. Atiyah's 1988 paper credits the cobordism-category viewpoint to the bordism theory of Thom and the categorical sharpening to discussions with Segal and with Graeme Segal's Oxford circle.

Bibliography Master

@article{Atiyah1988TQFT,
  author  = {Atiyah, Michael F.},
  title   = {Topological quantum field theories},
  journal = {Publications Math\'ematiques de l'IH\'ES},
  volume  = {68},
  pages   = {175--186},
  year    = {1988}
}

@incollection{Segal2004CFT,
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  series    = {London Mathematical Society Lecture Note Series},
  volume    = {308},
  pages     = {421--577},
  publisher = {Cambridge University Press},
  year      = {2004},
  note      = {First circulated 1988}
}

@book{Freed2019Lectures,
  author    = {Freed, Daniel S.},
  title     = {Lectures on Field Theory and Topology},
  series    = {CBMS Regional Conference Series in Mathematics},
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  publisher = {American Mathematical Society},
  year      = {2019}
}

@article{Witten1988TQFT,
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  title   = {Topological quantum field theory},
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  pages   = {353--386},
  year    = {1988}
}

@incollection{BaezDolan1995,
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}