The String Coffee Table

A closed topological 2D field theory is something which associates a vector space $V$ of states to the circle and a linear operator $V^{\otimes n} \overset{O_\Sigma}{\to} V^{\otimes m}$ to worldsheets $\Sigma$ of closed strings with $n$ incoming and $m$ outgoing boundary components. More technically, this is a functor $\mathbf{2Cob} \overset{T}{\to} \mathrm{Vect}$ from the category of 2-dimensional (closed) cobordisms to vector spaces.

It is very well known that such a functor is the same as a commutative Frobenius algebra. The pair-of-pants diagram with 2-incoming and one outgoing component corresponds to the product in the algebra, the co-pair-of-pants with one incoming and two outgoing components corresponds to the coproduct. The fact that only the topology of the worldsheet $\Sigma$ is of relevance gives the associativity, co-associativity and the Frobenius-relation of the algebra.

More precisely, the statemement is that the category of functors of the above sort is equivalent to the category of commutative Frobenius algebras. Apparently this has been rigorously established in

L. Abrams
Two dimensional topological quantum field theories and Frobenius algebras
J. Knot Th. Ram. 5 No. 5 (1996) 569-587

There is an obvious desire to generalize this to a theory of closed and open topological strings. Observations on the structure of such open-closed topological 2D field theories have been given by Moore and Segal. Apparently the best reference on this is still

G. Moore & G. Segal
Lectures on Branes, K-theory and RR Charges
Clay Math Institute Lecture Notes (2002).

There it is noted that open-closed 2D TFTs are still characterized by a commutative Frobenius algebra, describing the closed string sector, but now there is in addition also a not-necessarily commutative one which describes the open string sector. Furthermore there is an algebra homomorphism between the closed string sector and the center of the open string sector.

Lauda & Pfeiffer claim to fill in some gaps in making this sort of folk lore statement precise and rigorously proven.

Moore and Segal describe open-closed 2D TFT in terms of generators and relations. This means they list a set of elementary worldsheets, like the cylinder, the pair of pants and the disc, and then define all acceptable worldsheets as those obtainable from composing these in all different ways while identifying those that are related by certain relations. These relations are induced by worldsheet diffeomorphisms. Hence they are necessary if the generators and relations-description is to capture all of 2D TFT. Lauda and Pfeiffer observe that it has not been previously established rigorously that the set of relations considered is also sufficient for this purpose - that every diffeomorphism between open-closed worldsheets can be obtained from combining the listed relations between elementary worldsheets. Proving this is their first result.

With this result in hand, one can define the category $\mathbf{2Cob}^\mathrm{ext}$ of open-closed 2D cobordisms. An open-closed 2D TFT is then a functor $\mathbf{2Cob}^\mathrm{ext} \overset{T}{\to} C$, where $C$ is some symmetric monoidal category, like $\mathrm{Vect}$ for instance. One would again like to know what algebraic structure such functors encode. Lauda & Pfeiffer’s second result is that the category of these functors is equivalent to the category of what they call knowledgeable Frobenius algebras.

A knowledgeable Frobenius algebra is a commutaive Frobenius algebra describing the closed sector, together with a ‘symmetric’ Frobenius algebra describing the open sector, equipped with morphism between these two such that some relations hold. These morphisms capture the action of worldsheets where a closed string splits into an open string (the ‘zipper’) and the reverse situation where an open string merges its ends to a closed string (the ‘co-zipper’).

As far as I understand, this result is new in that it really goes a little beyond what Moore and Segal state in theorem 1 on their slide 29 in the above lecture notes.

One of the main tools used in the paper is a generalization of Morse theory to ‘manifolds with corners’.

The elementary worldsheet segments - the generators - of a closed topological string can be obtained as the neighborhoods of critical points of any one Morse function on the worldsheet. Different choices of Morse functions correspond to moves between different choices of constructing a worldsheet from elementary worldsheets.

Open-closed worldsheets have boundaries and ‘corners’ where an incoming open string ends on a D-brane. There is an old generalization of Morse theory to such a situation invented by Braess in 1974. Using this, Lauda and Pfeiffer can deduce the generating worldsheets of open-closed strings. It takes some time to write all this out in detail, but the result is just what one expects.

The well-known relations hold between compositions of these generators, obtainable by applying diffeomorphisms to them. The biggest chunk of the paper is concerned with defining a certain ‘normal form’ of composites of generators for open-closed worldsheets and proving that using these relations every worldsheet can be brought into its normal form. This proves that the relations one has are sufficient in that they are all that are needed to emulate every possible diffeomorphism in terms of moves between composites of generators.

With the category $\mathbf{2Cob}^{\mathrm{ext}}$ thus captured in terms of generators and relations, it is pretty straightforward to define open-closed TQFTs as functors from these to some symmetric monoidal category $C$ and check that such a functor defines a knowledgeable Frobenius algebra (object in $C$). The generalization to the presence of more than one D-brane is also rather immediate.