03.16.02 · modern-geometry / tqft

Classification of 2d oriented TQFTs: the Frobenius-algebra theorem

shipped3 tiersLean: none

Anchor (Master): Kock 2003 *Frobenius Algebras and 2D TQFT* (LMS Student Texts 59) ch. 3–5; Abrams 1996 (J. Knot Theory Ramif. 5); Dijkgraaf 1989 PhD thesis (Utrecht); Dijkgraaf–Witten 1990 (Comm. Math. Phys. 129); Freed *Lectures on Field Theory and Topology* (CBMS Reg. Conf. 133) Lecture 4

Intuition Beginner

In two dimensions, a topological field theory turns out to be the simplest thing it could possibly be: an algebra with a trace. You feed it a circle and it hands back a vector space; you feed it a surface and it hands back a way of combining or comparing vectors. The surprise is that all of this rich-sounding machinery collapses to a single short list of data, and that list is exactly what mathematicians had already named a commutative Frobenius algebra.

Why so simple? Because every surface, no matter how complicated, can be cut into a few standard pieces. The key piece looks like a pair of pants: a tube with one circle at the waist and two circles at the legs. Read top to bottom, it merges two circles into one. That merging is a multiplication. A small disc, a cap sealing off a circle, plays two roles: filling a circle from nothing gives a special starting element, and closing a circle off to nothing gives a trace that reads out a number.

Once you know what the theory does to a pair of pants and to a cap, you know everything. Gluing these pieces builds every surface, so the answers for every surface are forced. The rules that say which gluings give the same surface become the algebra laws: the product is associative, it is commutative because you can swap the two legs by rotating the pants, and the trace fits together with the product in the balanced way that earns the name Frobenius.

Visual Beginner

Alt text: On the left, a pair-of-pants surface has two circles along its top edge and a single circle along its bottom edge; a downward arrow is labelled "multiplication: two circles become one." In the centre, a small bowl-shaped cap seals a single circle, with labels "filling from nothing: the unit" and "closing to nothing: the trace." On the right, three pairs of pants are glued leg-to-waist in a vertical chain and capped at both ends, producing a surface with several holes; a label reads "gluing standard pieces builds every surface, so the algebra determines every answer." The picture conveys that the merging tube is a product and the cap supplies a starting element and a number-valued trace.

Worked example Beginner

Take the algebra of two-by-two diagonal matrices, which is the same as pairs of numbers $(a, b)$ multiplied slot by slot. Declare the trace of $(a, b)$ to be $a + b$ . This is a commutative Frobenius algebra, and the theorem says it is a two-dimensional field theory. Let us read off what it assigns to surfaces.

The state space of the circle is this two-number algebra, so it is two-dimensional. The sphere is a circle capped at both ends with nothing in between, which the theory reads as: start with the special unit element, then take its trace. The unit here is $(1, 1)$ , and its trace is $1 + 1 = 2$ . So the sphere receives the number $2$ .

The torus, the surface with one hole, is built by adding one handle. A handle squares each basis piece and traces it. The pieces $(1, 0)$ and $(0, 1)$ each square to themselves and trace to $1$ , so the torus receives $1 + 1 = 2$ . A surface with two holes adds a second handle and gives $2$ again. Every closed surface here receives the same number, $2$ , counting the two independent slots. This one example shows the whole pattern: pick an algebra with a trace, and the surfaces compute themselves.

Check your understanding Beginner

Formal definition Intermediate+

Fix the ground field $C$ . Work in the symmetric monoidal category $Bord_{2}^{or}$ of 03.02.20: objects are closed oriented $1$ -manifolds (finite disjoint unions of circles), morphisms are diffeomorphism classes rel boundary of compact oriented surfaces-with-boundary, composition is gluing, and the monoidal product is disjoint union. A 2d oriented TQFT is, per 03.16.01, a symmetric monoidal functor $Z : Bord_{2}^{or} \to Vect_{C}$ .

Definition (commutative Frobenius algebra). A Frobenius algebra over $C$ is a finite-dimensional vector space $A$ carrying both an associative unital multiplication $μ : A \otimes A \to A$ with unit $η : C \to A$ , and a counit (trace) $ϵ : A \to C$ , such that the bilinear pairing $$ \langle a, b \rangle := \epsilon(\mu(a, b)) = \epsilon(ab) $$ is nondegenerate. It is commutative when $μ = μ \circ τ$ , where $τ$ is the swap $a \otimes b \mapsto b \otimes a$ . Equivalently, a Frobenius algebra carries a coassociative comultiplication $Δ : A \to A \otimes A$ and counit $ϵ$ obeying the Frobenius relation $$ (\mathrm{id}_A \otimes \mu) \circ (\Delta \otimes \mathrm{id}_A) ;=; \Delta \circ \mu ;=; (\mu \otimes \mathrm{id}_A) \circ (\mathrm{id}A \otimes \Delta), $$ with $Δ$ recovered from $μ$ and the pairing. Morphisms of Frobenius algebras are algebra maps compatible with the counit; this defines the category $\mathrm{cFrob}{\mathbb{C}}$.

The generators of $Bord_{2}^{or}$ , obtained from a handle (Morse) decomposition, are the elementary surfaces:

the disc/cap $\emptyset \to S^{1}$ , the unit $η$ ;
the disc/cup $S^{1} \to \emptyset$ , the counit/trace $ϵ$ ;
the pair of pants $S^{1} ⊔ S^{1} \to S^{1}$ , the multiplication $μ$ ;
the copants $S^{1} \to S^{1} ⊔ S^{1}$ , the comultiplication $Δ$ ;
the cylinder $S^{1} \to S^{1}$ , the identity.

Setting $A = Z (S^{1})$ , functoriality and monoidality send these generators to $η, ϵ, μ, Δ$ and $id_{A}$ . The surface relations among generators become the algebra axioms: gluing two pants in the two associative ways is the same surface, giving associativity; the leg-swap diffeomorphism gives commutativity; bending a cylinder past a cap gives the unit law; and the two surface presentations of a once-punctured torus-with-collar give the Frobenius relation. Nondegeneracy of $⟨, ⟩$ is the snake (zig-zag) identity of 03.16.01, the cylinder-bending duality, read on $S^{1}$ .

Counterexamples to common slips

Disjoint circles do not give a direct sum. Monoidality forces $Z (S^{1} ⊔ S^{1}) = A \otimes A$ , a tensor product, so the pair of pants is a map $A \otimes A \to A$ , a multiplication — not a map $A \oplus A \to A$ .
The pairing must be nondegenerate, not merely present. An associative algebra with some linear functional $ϵ$ is not yet Frobenius; the snake identity demands that $⟨ a, b ⟩ = ϵ (ab)$ have no kernel, which is what makes $A$ self-dual and the comultiplication well-defined.
Commutativity is special to the closed string. The leg-swap of the pair of pants is a diffeomorphism only because the two incoming circles are interchangeable; the open-string (interval-boundary) version has no such swap, and its algebra need not be commutative.

Key theorem with proof Intermediate+

Theorem (Dijkgraaf–Abrams classification). Evaluation at the circle, $Z \mapsto Z (S^{1})$ , is an equivalence of categories $$ \mathrm{TQFT}2^{\mathrm{or}} ;\xrightarrow{\ \sim\ }; \mathrm{cFrob}{\mathbb{C}} $$ between $2$ -dimensional oriented TQFTs and commutative Frobenius algebras over $C$ .

Proof. Write $A = Z (S^{1})$ , which is finite-dimensional by the finiteness theorem of 03.16.01. The pair of pants $P : S^{1} ⊔ S^{1} \to S^{1}$ gives $μ = Z (P) : A \otimes A \to A$ ; the cap $C : \emptyset \to S^{1}$ gives $η = Z (C) : C \to A$ ; the cup gives $ϵ = Z (\overline{C}) : A \to C$ . Associativity of $μ$ holds because the two ways of gluing three legs into one waist are diffeomorphic surfaces (a four-holed sphere), and the diffeomorphism class is what $Z$ sees. The unit law holds because capping one leg of $P$ yields the cylinder. Commutativity holds because the order-two diffeomorphism of $P$ exchanging its two legs covers the swap $τ$ , so $μ \circ τ = μ$ .

Nondegeneracy of $⟨ a, b ⟩ = ϵ (μ (a, b))$ comes from the snake identity for the bending cylinders of 03.16.01: the cap-and-cup pairing $S^{1} ⊔ S^{1} \to \emptyset$ together with the copants exhibits $A$ as its own dual, so the form has no kernel. Defining $Δ = Z (\overline{P})$ from the copants, the two surface presentations of the genus-zero, three-boundary surface obtained by gluing pants to copants give the Frobenius relation. Hence $A$ is a commutative Frobenius algebra, and a TQFT map (monoidal natural transformation) restricts to a Frobenius-algebra map on circles, giving the functor $Z \mapsto A$ .

For the inverse, given a commutative Frobenius algebra $A$ , define $Z$ on the generators by $Z (S^{1}) = A$ , $Z (P) = μ$ , $Z (C) = η$ , $Z (\overline{C}) = ϵ$ , $Z (\overline{P}) = Δ$ , and extend by gluing. Well-definedness is the statement that $Bord_{2}^{or}$ is presented by these generators subject to exactly the relations encoding the commutative-Frobenius axioms, a theorem proved by classifying surfaces up to diffeomorphism via their genus and boundary data (the handle decomposition of 03.02.20). Every relation among the generating surfaces is a consequence of associativity, commutativity, unitality, and the Frobenius identity, so the assignment respects all gluings and is a symmetric monoidal functor. The two constructions are mutually inverse up to natural isomorphism. $■$

Bridge. This classification builds toward the entire program of identifying field theories with algebraic structure, and it is the foundational reason the chapter calls the $2$ -dimensional case "solved": a surface invariant is exactly a commutative algebra with a nondegenerate trace, nothing more. The pair-of-pants multiplication is dual to the copants comultiplication through the trace pairing, and putting these together with the snake identity of 03.16.01 is what forces the Frobenius compatibility. This is exactly the $n = 2$ shadow of a far deeper pattern: the central insight that physical consistency under cutting equals algebraic structure generalises, in 03.16.03, to the cobordism hypothesis, where the same dualizability that here makes $A$ self-dual becomes full dualizability of an object in a higher category, and appears again in 03.16.06 as the conformal-block state spaces of a genuinely infinite family of theories.

Exercises Intermediate+

Exercise (medium, short-answer). State precisely why the multiplication of a 2d oriented TQFT is commutative but, in the same axioms, the multiplication need not be commutative for an *open* (interval-boundary) theory.

Hint

The relevant move is the diffeomorphism of the pair of pants exchanging its two incoming circles; ask whether the open analogue admits it.

Answer

Rubric. The closed pair of pants admits an orientation-preserving diffeomorphism swapping its two leg-circles, and $Z$ sends diffeomorphic surfaces to equal maps, so $μ \circ τ = μ$ : commutativity. The open-string pair of pants has interval inputs whose endpoints lie on a boundary that orders them; no diffeomorphism interchanges the two inputs while fixing the boundary structure, so the corresponding multiplication can fail to commute. Full credit names the leg-swap diffeomorphism and explains its absence in the open case.

Exercise (hard, short-answer). Derive the genus formula $Z(\Sigma_g) = \mathrm{tr}(H^g)$ for the closed genus-$g$ surface, where $H = \mu \circ \Delta : A \to A$ is the handle operator, by decomposing $\Sigma_g$ into elementary pieces.

Hint

A genus- $g$ surface is a sphere with $g$ handles attached; each handle is a copants followed by a pants on the same circle.

Answer

Rubric. Cut $Σ_{g}$ along $g + 1$ circles into: a cap (unit $η$ ), then $g$ handle pieces each equal to a copants then a pants on the same circle (so each contributes $μ \circ Δ = H$ ), then a cup (counit $ϵ$ ). Composing gives $Z (Σ_{g}) = ϵ \circ H^{g} \circ η$ . Since $ϵ \circ (-) \circ η$ on an endomorphism of $A$ computes its categorical trace against the nondegenerate pairing, this equals $tr (H^{g})$ . Genus $0$ recovers $ϵ \circ η = Z (Σ_{0})$ . Full credit gives the handle decomposition, identifies $H$ on each handle, and reads the cap-cup composite as a trace.

Exercise (hard, short-answer). For the group-algebra-centre theory $A = Z(\mathbb{C}[G])$ of a finite group $G$, the Dijkgraaf–Witten partition function of the closed genus-$g$ surface counts $G$-bundles weighted by automorphisms. Using $A \cong \bigoplus_{\chi} \mathbb{C}$ over irreducible characters with handle eigenvalue $|G|/(\dim\chi)^2$ on the $\chi$-block, write $Z(\Sigma_g)$ as a character sum and identify $Z(\Sigma_1)$.

Hint

On a semisimple algebra the handle operator is diagonal in the idempotent basis; sum its $g$ -th power over the blocks.

Answer

Rubric. Diagonalising $H$ over the irreducible characters $χ$ of $G$ with eigenvalue $λ_{χ} = ∣ G ∣/ (dim χ)^{2}$ gives $Z (Σ_{g}) = \sum_{χ} λ_{χ}^{g - 1} = ∣ G ∣^{g - 1} \sum_{χ} (dim χ)^{2 - 2 g}$ , the Frobenius/Mednykh surface-counting formula. At $g = 1$ this is $\sum_{χ} 1 = (number of irreducible characters) = (number of conjugacy classes) = dim Z (C [G])$ , matching $Z (Σ_{1}) = dim A$ . Full credit produces the character sum and evaluates $g = 1$ to the class number.

Advanced results Master

The handle operator and the genus expansion. The torus-with-two-discs-removed bordism $S^{1} \to S^{1}$ is the composite $μ \circ Δ$ , the handle operator $H : A \to A$ . Stacking handles multiplies $H$ , so the closed genus- $g$ surface is $ϵ \circ H^{g} \circ η$ , and reading the cap-cup composite as a trace gives $Z (Σ_{g}) = tr (H^{g})$ . In the basis where the nondegenerate pairing is standard, $H$ is the element $\sum_{i} e_{i} e^{i}$ acting by multiplication — the Euler element — so the whole genus expansion is governed by a single algebra element. This compresses the infinitely many closed-surface invariants of the theory into the spectrum of one operator ^{[Kock 2003]}.

Semisimplicity and the building blocks. A commutative Frobenius algebra over $C$ is semisimple exactly when it is a product of copies of $C$ — equivalently, when its handle operator is invertible. In that case the theory is a finite sum of one-dimensional theories, one per idempotent, and the genus formula becomes $Z (Σ_{g}) = \sum_{i} λ_{i}^{g - 1}$ over the idempotent eigenvalues $λ_{i}$ . The non-semisimple algebras, such as $C [x] / (x^{n})$ , are the genuinely indecomposable theories; they carry nilpotents and are the algebraic home of the Landau–Ginzburg / singularity examples ^{[Abrams 1996]}.

Three families of examples. First, $A = C [x] / (x^{n})$ with $ϵ$ the top-coefficient functional: the $A_{n - 1}$ Landau–Ginzburg model, the same algebra as the cohomology of certain spaces under their Poincaré pairing. Second, $A = Z (C [G])$ , the centre of a finite group algebra: 2d Dijkgraaf–Witten finite gauge theory, whose partition functions count principal $G$ -bundles on a surface weighted by $∣ Aut ∣^{- 1}$ , recovering the character sum $∣ G ∣^{g - 1} \sum_{χ} (dim χ)^{2 - 2 g}$ . Third, $A = H^{*} (X; C)$ for a closed oriented manifold $X$ , with $ϵ (α) = \int_{X} α$ the Poincaré-duality pairing: the cohomology of a space is itself a commutative Frobenius algebra and so itself a 2d TQFT ^{[Dijkgraaf–Witten 1990]}.

The state-sum construction. One can also build $Z$ combinatorially: triangulate the surface, place $μ$ on each edge and the metric coming from $Δ$ on each face, and contract. Independence of the triangulation is exactly the Frobenius relation together with associativity and commutativity, which are the algebraic forms of the Pachner moves relating any two triangulations of a surface. This Fukuma–Hosono–Kawai construction reproduces the same invariant from purely local data, making the classification an effective recipe.

Synthesis. The 2d oriented case is where the abstract axioms of 03.16.01 become a complete, checkable algebraic dictionary, and that is exactly the central insight of the chapter in miniature. The cobordism category of 03.02.20 supplies generators — cap, cup, pants, copants — and the foundational reason the classification closes is that these generators are subject to precisely the commutative-Frobenius relations and no others. Putting these together, evaluation at the circle is dual to the choice of an algebra-with-trace: the pair-of-pants product is dual to the copants coproduct through the nondegenerate pairing, the handle operator packages every closed-surface invariant into one element, and the semisimple theories are the sums of one-dimensional building blocks. This generalises in two directions at once: upward, in 03.16.03, full dualizability replaces the self-duality used here and the cobordism hypothesis classifies fully extended theories; and outward, in 03.16.06, the same axioms admit infinite-dimensional, genuinely quantum solutions whose state spaces are conformal blocks. The bridge is the presentation theorem: once $Bord_{2}$ is generators and relations, a field theory is a solution to those relations in $Vect_{C}$ .

Full proof set Master

Proposition (the sphere computes the counit of the unit). For any 2d oriented TQFT with circle algebra $A$ , the genus-zero partition function is $Z (Σ_{0}) = ϵ (η (1)) = ϵ (1_{A})$ .

Proof. The sphere $Σ_{0}$ is built by capping a cup with a cap: as a bordism $\emptyset \to \emptyset$ it factors as $\overline{C} \circ C$ , the cap $C : \emptyset \to S^{1}$ followed by the cup $\overline{C} : S^{1} \to \emptyset$ . Functoriality gives $Z (Σ_{0}) = Z (\overline{C}) \circ Z (C) = ϵ \circ η$ , a map $C \to C$ , hence the scalar $ϵ (η (1)) = ϵ (1_{A})$ where $1_{A} = η (1)$ is the algebra unit. $■$

Proposition (commutativity from the leg-swap). The multiplication $μ = Z (P)$ of a 2d oriented TQFT satisfies $μ \circ τ = μ$ .

Proof. The pair of pants $P : S^{1} ⊔ S^{1} \to S^{1}$ admits an orientation-preserving self-diffeomorphism $σ$ that interchanges its two boundary leg-circles and is the identity on the waist; concretely $σ$ is a half-rotation of the surface about the axis through the waist. As a morphism of $Bord_{2}^{or}$ , precomposing $P$ with the symmetry braiding $β_{S^{1}, S^{1}}$ of the two incoming circles yields a surface diffeomorphic to $P$ via $σ$ , so $P \circ β = P$ as bordism classes. Applying the symmetric monoidal functor $Z$ and using $Z (β) = τ$ gives $μ \circ τ = Z (P) \circ Z (β) = Z (P \circ β) = Z (P) = μ$ . $■$

Proposition (nondegeneracy of the trace pairing). The form $⟨ a, b ⟩ = ϵ (μ (a, b))$ on $A = Z (S^{1})$ is nondegenerate.

Proof. The cup-with-pants composite $S^{1} ⊔ S^{1} \to \emptyset$ given by $\overline{C} \circ P$ realises the pairing as an evaluation bordism $ε : S^{1} ⊔ S^{1} \to \emptyset$ , and the cap-with-copants composite $\emptyset \to S^{1} ⊔ S^{1}$ given by $Δ \circ C$ realises a coevaluation $β : \emptyset \to S^{1} ⊔ S^{1}$ . Bending a cylinder shows $ε$ and $β$ satisfy the zig-zag (snake) identities of 03.16.01, exhibiting $A$ as self-dual: $A ≅ A^{*}$ via $a \mapsto ⟨ a, - ⟩$ . An isomorphism onto the dual space is precisely a nondegenerate pairing, so $⟨, ⟩$ has no kernel. $■$

The presentation theorem itself — that $Bord_{2}^{or}$ is generated by cap, cup, pants, copants, and cylinder subject to exactly the commutative-Frobenius relations — is the surface-classification input; see Abrams ^{[Abrams 1996]} and Kock ^{[Kock 2003]} for the full diffeomorphism-class bookkeeping.

Connections Master

The Atiyah–Segal axioms 03.16.01. That unit sets up the general functor $Z : Bord_{n} \to Vect_{C}$ , proves the finiteness theorem that makes $A = Z (S^{1})$ finite-dimensional, and extracts the pair-of-pants algebra structure as a warm-up. This unit completes that warm-up into the full classification in dimension two, turning the snake identity proved there into the nondegeneracy of the Frobenius pairing here.
Handles, surgery, and the cobordism category 03.02.20. The generators-and-relations presentation of $Bord_{2}^{or}$ rests on the handle (Morse) decomposition of surfaces built there; every surface is a sequence of elementary handle attachments, and the relations among them are exactly the commutative-Frobenius axioms. The classification theorem is the statement that this presentation is complete.
Extended TQFT and the cobordism hypothesis 03.16.03. The self-duality of $A$ used here is the $n = 2$ , one-categorical shadow of full dualizability; the cobordism hypothesis classifies fully extended framed theories by fully dualizable objects, and the $2$ -dimensional Frobenius picture is the warm-up example it specialises to. The pattern of "field theory equals algebraic structure" is pushed down to points there.
Chern–Simons theory as a 3d TQFT 03.16.06. That unit supplies the contrasting example: a genuinely infinite-dimensional family of theories whose state spaces are conformal blocks rather than a fixed finite algebra. The Frobenius classification shows the $2$ -dimensional case is rigid and finite, while the $3$ -dimensional Chern–Simons theory shows the axioms have far richer solutions one dimension up.

Historical & philosophical context Master

The identification of two-dimensional topological field theories with commutative Frobenius algebras emerged in physics before it was a theorem. Robbert Dijkgraaf's 1989 Utrecht PhD thesis, written in the wake of Witten's and Atiyah's formulations, observed that a closed-string topological theory is governed by an algebra on the circle's state space with an invariant inner product, and that genus is added by an algebra operation ^{[Kock 2003]}. Edward Witten and Dijkgraaf's joint work on topological gauge theories and group cohomology then produced the finite-gauge-theory example whose partition functions count bundles — the $Z (C [G])$ theory — and tied the surface invariants to character sums of the gauge group ^{[Dijkgraaf–Witten 1990]}.

The mathematical theorem, an equivalence of categories with a finite generators-and-relations proof, was given by Lowell Abrams in 1996, who made precise both the presentation of the two-dimensional bordism category and the Frobenius-algebra target ^{[Abrams 1996]}. Joachim Kock's 2003 monograph Frobenius Algebras and 2D Topological Quantum Field Theories turned the result into a textbook, drawing the surfaces explicitly and treating the open and closed cases side by side; it remains the standard reference and the source of the handle-operator and state-sum perspectives used above. Daniel Freed's CBMS lectures place the result as the first worked classification in the larger program, the finite-dimensional rehearsal for the stable-homotopy classifications that the book is named for.

Bibliography Master

@book{Kock2003Frobenius,
  author    = {Kock, Joachim},
  title     = {Frobenius Algebras and 2D Topological Quantum Field Theories},
  series    = {London Mathematical Society Student Texts},
  number    = {59},
  publisher = {Cambridge University Press},
  year      = {2003}
}

@article{Abrams1996,
  author  = {Abrams, Lowell},
  title   = {Two-dimensional topological quantum field theories and Frobenius algebras},
  journal = {Journal of Knot Theory and its Ramifications},
  volume  = {5},
  number  = {5},
  pages   = {569--587},
  year    = {1996}
}

@phdthesis{Dijkgraaf1989,
  author = {Dijkgraaf, Robbert},
  title  = {A Geometric Approach to Two-Dimensional Conformal Field Theory},
  school = {University of Utrecht},
  year   = {1989}
}

@article{DijkgraafWitten1990,
  author  = {Dijkgraaf, Robbert and Witten, Edward},
  title   = {Topological gauge theories and group cohomology},
  journal = {Communications in Mathematical Physics},
  volume  = {129},
  number  = {2},
  pages   = {393--429},
  year    = {1990}
}

@book{Freed2019Lectures,
  author    = {Freed, Daniel S.},
  title     = {Lectures on Field Theory and Topology},
  series    = {CBMS Regional Conference Series in Mathematics},
  number    = {133},
  publisher = {American Mathematical Society},
  year      = {2019}
}

Prerequisites

03.16.01
03.02.20

Tier anchors

beginner: Kock 2003 *Frobenius Algebras and 2D Topological Quantum Field Theories* (LMS Student Texts 59) Part I, the informal surface pictures; Baez–Dolan informal overview of pair-of-pants multiplication
intermediate: Kock 2003 *Frobenius Algebras and 2D TQFT* (LMS Student Texts 59) ch. 1–3; Abrams 1996 *Two-dimensional topological quantum field theories and Frobenius algebras* (J. Knot Theory Ramif. 5) §2–§3
master: Kock 2003 *Frobenius Algebras and 2D TQFT* (LMS Student Texts 59) ch. 3–5; Abrams 1996 (J. Knot Theory Ramif. 5); Dijkgraaf 1989 PhD thesis (Utrecht); Dijkgraaf–Witten 1990 (Comm. Math. Phys. 129); Freed *Lectures on Field Theory and Topology* (CBMS Reg. Conf. 133) Lecture 4

References

Kock — Frobenius Algebras and 2D Topological Quantum Field Theories (2003) · LMS Student Texts 59, Cambridge University Press; ch. 1–3 (generators and relations of Bord_2, the equivalence with commutative Frobenius algebras) and ch. 4 (examples)
Abrams — Two-dimensional topological quantum field theories and Frobenius algebras (1996) · Journal of Knot Theory and its Ramifications 5, no. 5, pp. 569–587 (the equivalence-of-categories theorem with a finite generators-and-relations presentation)
Dijkgraaf–Witten — Topological gauge theories and group cohomology (1990) · Communications in Mathematical Physics 129, pp. 393–429 (2d/3d finite gauge theory, the C[G]-centre Frobenius algebra)

Estimated time

beginner: 16m
intermediate: 42m
master: 78m