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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 ChK 0(D b(W,f)) in H i(F) is the sublattice of H 2(X,Z) of vanishing cocycles.

So what is all the terminology here? (X,ω) is always a symplectic manifold. F is a perverse sheaf of vanishing cycles for the mirror W. For an f:WC 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)P(0,4)P(3,3)P(2,3)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 6g6+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

Posted at July 24, 2005 11:37 PM UTC

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