The String Coffee Table

After Katzarkov, Getzler dazzled us all once again with open and closed modular operads. Let $C$ be a symmetric monoidal category and $P(g,n)$ a sequence of objects labelled by a genus $g$ and $n$ the number of legs at a vertex. The overall genus of a graph is given by the sum of the vertex genuses plus the first Betti number of the graph. Thus a graph with non-negative integers assigned to each vertex yields composition rules, such as $$P(1,3) \otimes P(0,4) \otimes P(3,3) \otimes P(2,3) \rightarrow P(8,3)$$ where $8$ is the genus of a graph that the reader may easily identify.

Example: P(g,n) = the moduli space of conformal structures for a Riemann surface of genus $g$ with $n$ punctures. This motivates throwing away graphs and using ‘thicked lines’ or rather surfaces. String theory! The nice category here has oriented surfaces as objects. Morphisms are pairs of surfaces together with a configuration of curves in the target and an isomorphism between the source and the surface obtained from the target by cutting along the curves.

By definition, a modular operad is a symmetric monoidal functor from this category to $C$.

A slightly different version gives a characterisation of open conformal field theory. We can talk about the Deligne-Mumford compactification of $M(g,n;h,m)$ where as before $n$ is the number of boundary components that are circles and $n$ is the number of boundary components that are arcs etc. This moduli space of compact orbifolds with corners has dimension $6g - 6 + 3h + 2n + m$. Considering, for example, the case $(g,n;h,m) = (0,0;1,m)$ one obtains the associahedron rule!

Finally, Getzler mentioned the theorem that states that the 2-skeleton of each component of this moduli space is connected.

Now I must go and listen to Higson, Getzler and others talk about NCG. Marni