nLab D=2 TQFT

Idea

A D=2 TQFT is a topological quantum field theory on cobordisms of dimension 2.

Classification

When formulated as an (only) “globally” as 1-functors on a 1-category of cobordisms (see at FQFT for more), then D=2 TQFTs have a comparatively simple classification: the bulk field theory is determined by a commutative Frobenius algebra structure on the finite dimensional vector space assigned to the circle (Abrams 96).

However, such global D=2 TQFTs with coefficients in Vect do not capture the D=2 TQFTs of most interest in quantum field theory, which instead are “cohomological quantum field theories” (Witten 91) such as the topological string A-model and B-model that participate in homological mirror symmetry.

These richer D=2 TQFTs are instead local TQFTs in the sense of extended TQFT, i.e. they are (∞,2)-functors on a suitable (∞,2)-category of cobordisms (see at FQFT for more), typically on “non-compact” 2-d cobordisms, meaning on those that have non-vanishing outgoing bounary. As such they are now classified by Calabi-Yau objects in an symmetric monoidal (infinity,2)-category (Lurie 09, section 4.2). For coefficients in the (∞,2)-category of (∞,2)-vector space (i.e. A-∞ algebras with (∞,1)-bimodules between them in the (∞,1)-category of chain complexes), these theories had been introduced under the name “TCFT” in (Getzler 92, Segal 99) following ideas of Maxim Kontsevich, and have been classified in (Costello 04), see (Lurie 09, theorem 4.2.11, theorem 4.2.14).

2d TQFT (“TCFT”)	coefficients	algebra structure on space of quantum states
open topological string	Vect${}_k$	Frobenius algebra $A$	folklore+(Abrams 96)
open topological string with closed string bulk theory	Vect${}_k$	Frobenius algebra $A$ with trace map $B \to Z(A)$ and Cardy condition	(Lazaroiu 00, Moore-Segal 02)
non-compact open topological string	Ch(Vect)	Calabi-Yau A-∞ algebra	(Kontsevich 95, Costello 04)
non-compact open topological string with various D-branes	Ch(Vect)	Calabi-Yau A-∞ category	“
non-compact open topological string with various D-branes and with closed string bulk sector	Ch(Vect)	Calabi-Yau A-∞ category with Hochschild cohomology	“
local closed topological string	2Mod(Vect${}_k$) over field $k$	separable symmetric Frobenius algebras	(SchommerPries 11)
non-compact local closed topological string	2Mod(Ch(Vect))	Calabi-Yau A-∞ algebra	(Lurie 09, section 4.2)
non-compact local closed topological string	2Mod$(\mathbf{S})$ for a symmetric monoidal (∞,1)-category $\mathbf{S}$	Calabi-Yau object in $\mathbf{S}$	(Lurie 09, section 4.2)

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.

$A_\infty$-monoid objects

Let $Core$(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

$$\Phi : Core(FinSet) \to C$$

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

$$\Phi : Core(FinSet) \to C$$

if the structure maps

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

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

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

$$F_0 K_n \subset F_1 K_n \subset \cdots \,,$$

where $F_0 K_n$ is the set of vertices, $F_1 K_n$ the set of edges, etc.

The collection

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

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 $A_\infty$-monoid object

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

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

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

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

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

for the moduli space of hyperbolic metrics with cusps at the $(z_i)$.

We have

$$M(\Sigma, \vec z) = \mathcal{H}(\Sigma, \vec z)/Diff_+(\Sigma, \vec z)$$

and

$$M_{g,n} = \Tau_{g,n} / \Gamma^ng \,,$$

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

Here we can assume that the Euler characteristic $\chi(\Sigma without \{z_i\}) \lt 0$ 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;

• $3 g - 3 + n$ closed curves.

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

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

constitute the Fenchel-Nielsen coordinates on Teichmüller space $\Tau$.

Also use $\theta_i := t_i/\ell_i \in \mathbb{R}/\mathbb{Z}$

This constitutes is a real analytic atlas of Teichmüller space. On $M$ this reduces to coordinates $t_i \in \mathbb{R}/{\ell_i \mathbb{Z}}$, 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 $\hat \Tau$ (due to Bill Harvey?) and this constitutes a Borel-Serre bordification? of $\Tau$.

The mapping class group $\Gamma$ still acts on $\hat \Tau$ 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 $\mathcal{P}_{g,n}$ for $\Gamma_{g,n}$

Then the collection

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

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 $\mathcal{P}_{g,n}$ or $S_\bullet(\mathcal{P}_{g,n})$.

Let

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

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

So for instance

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

• $F_1 \mathcal{P}_{g,n}$ 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.

• $F_2 \mathcal{P}_{g,n}$ 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 $\mathcal{P}_{g,n}$.

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

Theorem

(Ezra Getzler)

The inclusion

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

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 $i \lt k$ and surjective $i = k$.

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

Proof

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

Some other versions of this:

$$F_k \Tau_{g,n} \to \Tau_{g,n}$$

is $k$-connected.

One can also use Deligne-Mumford compactifications

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

and this is also $k$-connected.

Open-closed case

…

Related concepts

• sewing constraint

  • Cardy condition

• TQFT

  • D=0 TQFT

  • D=2 TQFT

    • D=2 Chern-Simons theory

    • TCFT

      • A-model, B-model

    • Landau-Ginzburg model

    • Levin-Wen model

  • D=3 TQFT

    • D=3 Chern-Simons theory

  • D=4 TQFT

    • D=4 Chern-Simons theory

References

Global

The folklore result that global closed D=2 TQFTs with coefficients in Vect are equivalent to commutative Frobenius algebra structures is proven rigorously in

• Lowell Abrams, Two-dimensional topological quantum field theories and Frobenius algebra, Jour. Knot. Theory and its Ramifications 5, 569-587 (1996) [doi:10.1142/S0218216596000333, ps]

The classification result for open-closed D=2 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)

• Calin Lazaroiu, On the structure of open-closed topological field theory in two dimensions, Nuclear Phys. B 603(3), 497–530 (2001), (arXiv:hep-th/0010269)

Textbook account:

• Joachim Kock, Frobenius Algebras and 2d Topological Quantum Field Theories, Cambridge U. Press (2004) [doi:10.1017/CBO9780511615443, webpage, course notes pdf, pdf]

A picture-rich description of what’s going on:

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

Local

The local (extended TQFT) version of D=2 TQFT which captures the topological string was mathematically introduced under the name “TCFT”.

The concept is essentially a formalization of what used to be called cohomological field theory in

• Edward Witten, Introduction to cohomological field theory, InternationalJournal of Modern Physics A, Vol. 6,No 6 (1991) 2775-2792 (pdf)

The definition was given independently by

• Ezra Getzler, Batalin-Vilkovisky algebras and two-dimensional topological field theories , Comm. Math. Phys. 159(2), 265–285 (1994) (arXiv:hep-th/9212043)

and

• Graeme Segal, Topological field theory , (1999), Notes of lectures at Stanford university. (web). See in particular lecture 5 (“topological field theory with cochain values”).

The classification of TCFTs (i.e. “non-compact” local (extended D=2 TQFT)) by Calabi-Yau A-infinity categories is due to

• Kevin Costello, Topological conformal field theories and Calabi-Yau categories Advances in Mathematics, Volume 210, Issue 1, (2007), (arXiv:math/0412149)

• Kevin Costello, The Gromov-Witten potential associated to a TCFT (arXiv:math/0509264)

following conjectures by Maxim Kontsevich, e.g.

• Maxim Kontsevich, Homological algebra of mirror symmetry , in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), pages 120–139, Basel, 1995, Birkhäuser.

The classification of local (extended) D=2 TQFT (i.e. the “compact” but fully local case) is spelled out in

• Chris Schommer-Pries, The Classification of Two-Dimensional Extended Topological Field Theories (arXiv:1112.1000)

This classification is a precursor of the full cobordism hypothesis-theorem. This, and the reformulation of the original TCFT constructions in full generality is in

• Jacob Lurie, section 4.2 of On the Classification of Topological Field Theories (arXiv:0905.0465)