nLab
2d TQFT

Idea

A 2-dimensional TQFT is a topological quantum field theory on cobordisms of dimension 2.

If one understands this as an FQFTs with values in Vect, then the central classification theorem of 2d TQFTs states that they induce the structure of a commutative Frobenius algebra on the vector space associated to the circle, and that this establishes an equivalence between 2d QFTs and Frobenius algebras. An analogous result holds for open-closed 2d TQFTs, where codimension 1 manifolds are also allowed to contain intervals.

Filtrations of the moduli space of surfaces

The following study of the behaviour of 2-dimensional TQFTs in terms of the topology of the moduli spaces of marked hyperbolic surfaces is due to Ezra Getzler. It provides a powerful way to read off various classification results for 2d QFTs from the homotopy groups of the corresponding modular operad.

$A_{\mathrm{\infty }}$-monoid objects

Let $\mathrm{Core}$(FinSet) be the core of the category of finite sets. Under union of sets this is a symmetric monoidal category. Then for $C$ any monoidal category, a symmetric monoidal functor

$$\Phi :\mathrm{Core}(\mathrm{FinSet})\to C$$\Phi : Core(FinSet) \to C

is a commutative monoid in $C$.

Let now $C$ be a category with weak equivalences, then we can speak of a lax symmetric opmonoidal functor

$$\Phi :\mathrm{Core}(\mathrm{FinSet})\to C$$\Phi : Core(FinSet) \to C

if the structure maps

$$\Phi (n+m)\stackrel{\simeq }{\to }\Phi (m)\otimes \Phi (n)$$\Phi(n+m) \stackrel{\simeq}{\to} \Phi(m) \otimes \Phi(n)

$$\varphi (m)\stackrel{\simeq }{\to }\Phi (1{)}^{\otimes m}$$\phi(m) \stackrel{\simeq}{\to} \Phi(1)^{\otimes m}

are weak equivalences.

Segal called these ”$\Delta$-objects”. Since Carlos Simpson they are called Segal object?s.

There is also Jim Stasheff’s notion of an A-infinity algebra, given in terms of associahedra $K_n$, which are $(n-2)$-dimensional polytopes.

There is naturally a filtration on these guys with

$$F_0{K}_n\subset F_1{K}_n\subset \cdots ,$$F_0 K_n \subset F_1 K_n \subset \cdots \,,

where $F_0{K}_n$ is the set of vertices, $F_1{K}_n$ the set of edges, etc.

The collection

$$\{S_{•}(K_n)\}$$\{ S_\bullet(K_n) \}

of simplicial realizations of the $K_n$ form an sSet-operad $P$.

For $X$ a simplicial category that is symmetric monoidal, a $P$-algebra over an operad $X$ in $C$ is an $A_{\mathrm{\infty }}$-monoid object

$$S_{•}(K_n)\to C_{•}(X^{\otimes n},X)$$S_\bullet(K_n) \to C_\bullet(X^{\otimes n}, X)

MacLane’s coherence theorem says or uses that if $C$ is an n-category, we may replace $K_m$ here by the $n$-filtration $F_n{K}_m$.

Closed 2d quantum field theory

Compactified moduli spaces of Riemann surfaces

Let

$$(\Sigma ,(z_1,\cdots ,z_n))$$(\Sigma, (z_1, \cdots, z_n))

be a compact oriented surface with $n$ distinct marked points. Write

$$\mathscr{H}(\Sigma ,(z_1,\cdots ,z_n))$$\mathcal{H}(\Sigma, (z_1, \cdots, z_n))

for the moduli space of hyperbolic metric?s with cusps at the $(z_i)$.

We have

$$M(\Sigma ,\stackrel{\rightharpoonup }{z})=\mathscr{H}(\Sigma ,\stackrel{\rightharpoonup }{z})/{\mathrm{Diff}}_{+}(\Sigma ,\stackrel{\rightharpoonup }{z})$$M(\Sigma, \vec z) = \mathcal{H}(\Sigma, \vec z)/Diff_+(\Sigma, \vec z)

and

$$M_{g,n}={{\rm T}}_{g,n}/{\Gamma }^{\mathrm{ng}},$$M_{g,n} = \Tau_{g,n} / \Gamma^ng \,,

where ${{\rm T}}_{g,n}$ is the Teichmüller space and $\Gamma$ the mapping class group.

Here we can assume that the Euler characteristic $\chi (\Sigma \mathrm{without}\{z_i\})<0$ because otherwise this moduli space is empty.

Fenchel-Nielson coordinates on moduli space

We want to parameterize Teichmüller space by cutting surfaces into pieces with geodesic boundaries and Euler characteristic $\xi =-1$. These building blocks (of hyperbolic 2d geometry) are precisely

• the 3-holed sphere;

• the 2-holed cusp;

• the 1-holed 2-cusp;

• the 3-cusp

Each surface of genus $g$ with $n$ marked points will have

• $2g-2+n$ generalized pants;

• $3g-3+n$ closed curves.

The boundary lengths ${\ell }_i\in {ℝ}_{+}$ and twists $t_i\in ℝ$ of these pieces for

$$1\le i\le 3g-3+n$$1 \leq i \leq 3g-3+n

constitute the Fenchel-Nielsen coordinates on Teichmüller space ${\rm T}$.

Also use ${\theta }_i:=t_i/{\ell }_i\in ℝ/\mathbb{Z}$

This constitutes is a real analytic atlas of Teichmüller space. On $M$ this reduces to coordinates $t_i\in ℝ/{\ell }_i\mathbb{Z}$, and these constitute a real analytic atlas of moduli space.

Allow the lengths ${\ell }_i$ to go to 0, but keep the angles ${\theta }_i$. The resulting space is a real analytic manifold with corners $\widehat{{\rm T}}$ (due to Bill Harvey?) and this constitutes a Borel-Serre bordification? of ${\rm T}$.

The mapping class group $\Gamma$ still acts on $\widehat{{\rm T}}$ and the quotient $\hat{M}$ is an orbifold with corners, inside which still sits our moduli space $M$.

Kimura-Stasheff-Voronov: add a choice of directions at each nodal point in $\Sigma$. This removes all automorphisms and hence we no longer have to deal with an orbifold.

This yields the classifying stack ${𝒫}_{g,n}$ for ${\Gamma }_{g,n}$

Then the collection

$$\{{𝒫}_{g,n}\}$$\{ \mathcal{P}_{g,n} \}

is a modular operad: the operad that describes gluing of marked surfaces at marked points together with the informaiton on how to glue marked points of a single surface to each other.

A 2-dimensional closed TQFT is an algebra over an operad over this in a simplicial category, in the above sense.

This involves either the de Rham complex on ${𝒫}_{g,n}$ or $S_{•}({𝒫}_{g,n})$.

Let

$$F_k{𝒫}_{g,n}:=\{[\Sigma ]\mid ...\}$$F_k \mathcal{P}_{g,n} := \left\{ [\Sigma] | ... \right\}

where $\Sigma$ has $\ge 2g-2+n-k$ spheres as components (after cutting along zero-length closed curves).

So for instance

• $F_0{𝒫}_{g,n}$ is the pants-decomposition;

• $F_1{𝒫}_{g,n}$ is decompositions into pants and one piece being the result of either gluing two pants to each other or of gluing two circles of a single pant to each other.

  This $F_1..$ is a connected space, due to a theorem by Hatcher-Thurston. Notice This is equivalent to the familiar statement that a closed 2d TFT is a commutative Frobenius algebra.

• $F_2{𝒫}_{g,n}$ is the decomposition into pieces as before together with one two-holed torus or one five-holed sphere.

  This space has the space fundamental group as ${𝒫}_{g,n}$.

  This is equivalent to the theorem by Moore and Seiberg about categorified 2-d TFT.

Theorem

(Ezra Getzler)

The inclusion

$$F_k{𝒫}_{g,n}\hookrightarrow {𝒫}_{g,n}$$F_k \mathcal{P}_{g,n} \hookrightarrow \mathcal{P}_{g,n}

is $k$-connected.

Here a map $X\to Y$ is $k$-connected if

• ${\pi }_0(X)\to {\pi }_0(Y)$ is surjective;

• ${\pi }_i(X,x)\to {\pi }_i(Y,f(x))$ is a bijection for $i<k$ and surjective $i=k$.

This means precisely that the mapping cone is $k$-connected.

Proof

Use the cellular decomposition of moduli space ${ℳ}_{g,1}$ following Mumford, Thurston, Harer, Woeditch-Epstein, Penner.

Some other versions of this:

$$F_k{{\rm T}}_{g,n}\to {{\rm T}}_{g,n}$$F_k \Tau_{g,n} \to \Tau_{g,n}

is $k$-connected.

One can also use Deligne-Mumford compactification?s

$$F_k{\overline{ℳ}}_{g,n}\to {\overline{ℳ}}_{g,n}$$F_k \bar \mathcal{M}_{g,n} \to \bar \mathcal{M}_{g,n}

and this is also $k$-connected.

Open-closed case

…

References

The classification result for open-closed 2d TQFTs was famously announced and sketched in

• Greg Moore, Graeme Segal, Lectures on branes, K-theory and RR charges, Clay Math Institute Lecture Notes (2002), (web)

A standard textbook is

• Joachim Kock, Frobenius algebras and 2D topological quantum field theory (web)

A picture-rich description of what’s going on is in

• Aaron Lauda and Hendryk Pfeiffer, Open-closed strings: two-dimensional extended TQFTs and Frobenius algebras , Topology Appl. 155 (2008) 623-666. (arXiv:math.AT/0510664)