FQFT and cohomology
Types of quantum field thories
When formulated as an (only) “globally” as 1-functors on a 1-category of cobordisms (see at FQFT for more), then 2d TQFTs have a comparativle 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 2d TQFTs with coefficientsin Vect do not capture the 2d 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 2d 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||Frobenius algebra||folklore+(Abrams 96)|
|open topological string with closed string bulk theory||Vect||Frobenius algebra with trace map 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) over field||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 for a symmetric monoidal (∞,1)-category||Calabi-Yau object in||(Lurie 09, section 4.2)|
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.
is a commutative monoid in .
Let now be a category with weak equivalences, then we can speak of a lax symmetric opmonoidal functor
if the structure maps
are weak equivalences.
There is naturally a filtration on these guys with
where is the set of vertices, the set of edges, etc.
Here we can assume that the Euler characteristic because otherwise this moduli space is empty.
We want to parameterize Teichmüller space by cutting surfaces into pieces with geodesic boundaries and Euler characteristic . These building blocks (of hyperbolic 2d geometry) are precisely
the 3-holed sphere;
the 2-holed cusp;
the 1-holed 2-cusp;
Each surface of genus with marked points will have
The boundary lengths and twists of these pieces for
This constitutes is a real analytic atlas of Teichmüller space. On this reduces to coordinates , and these constitute a real analytic atlas of moduli space.
Kimura-Stasheff-Voronov: add a choice of directions at each nodal point in . This removes all automorphisms and hence we no longer have to deal with an orbifold.
This yields the classifying stack for
Then the collection
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.
This involves either the de Rham complex on or .
where has spheres as components (after cutting along zero-length closed curves).
So for instance
is the pants-decomposition;
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 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.
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 .
This is equivalent to the theorem by Moore and Seiberg about categorified 2-d TFT.
Here a map is -connected if
is a bijection for and surjective .
This means precisely that the mapping cone is -connected.
Use the cellular decomposition of moduli space following Mumford, Thurston, Harer, Woeditch-Epstein, Penner.
Some other versions of this:
One can also use Deligne-Mumford compactifications
and this is also -connected.
The classification result for open-closed 2d TQFTs was famously announced and sketched in
A standard textbook is
A picture-rich description of what’s going on is in
The concept is essentially a formalization of what used to be called cohomological field theory in
The definition was given independently by
following conjectures by Maxim Kontsevich, e.g.
The classification of local (extended) 2d TQFT (i.e. the “compact” but fully local case) is spelled out in
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