### 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${K}^{0}({D}^{b}(W,f))$ in $\oplus {H}^{i}(F)$ is the sublattice of ${H}^{2}(X,Z)$ of vanishing cocycles.

So what is all the terminology here? $(X,\omega )$ 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 ${D}^{b}(W,f)$ 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 $P(g,n)$ 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 $$P(1,3)\otimes P(0,4)\otimes P(3,3)\otimes P(2,3)\to P(8,3)$$ 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