## July 24, 2005

### Streetfest Workshop 3

#### Posted by Guest

Let’s start with Katzarkov’s talk on ‘Mirror symmetry and manifolds of general type’. He began with a question: are there instances of distinct (not symplectomorphic) 4D symplectic simply connected mannifolds ${X}_{1}$ and ${X}_{2}$ such that their Gromov-Witten invariants are equal? It turns out there is. Katzarkov mentioned an example of Donaldson’s, but more generally there is a conjecture that the image of Ch${K}^{0}\left({D}^{b}\left(W,f\right)\right)$ in $\oplus {H}^{i}\left(F\right)$ is the sublattice of ${H}^{2}\left(X,Z\right)$ of vanishing cocycles.

So what is all the terminology here? $\left(X,\omega \right)$ is always a symplectic manifold. $F$ is a perverse sheaf of vanishing cycles for the mirror $W$. For an $f:W\to C$ the categorical equivalence is between the so-called Fukaya category for $X$ and the bounded sheaves ${D}^{b}\left(W,f\right)$ as discussed in the first talk in Sydney.

Other workshop talks…..

After Katzarkov, Getzler dazzled us all once again with open and closed modular operads. Let $C$ be a symmetric monoidal category and $P\left(g,n\right)$ 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 $P\left(1,3\right)\otimes P\left(0,4\right)\otimes P\left(3,3\right)\otimes P\left(2,3\right)\to P\left(8,3\right)$ 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 $M\left(g,n;h,m\right)$ 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 $\left(g,n;h,m\right)=\left(0,0;1,m\right)$ 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

Posted at July 24, 2005 11:37 PM UTC

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