### Wednesday at the Streetfest I

#### Posted by Guest

Bouwknegt started Wednesday with an apology for not being a category theorist.

The main focus of the talk (I felt) was the “unification” of the A and B topological string theories - using, as string theorists love, duality. We are using a target space $M$ here which is (for now) Kaehler, and so is equipped with a complex structure and a closed two form which encodes the symplectic structure. The A model uses the symplectic structure and the B-model uses the complex structure. The way these things are (sort of) united, is to forget looking at the tangent $\mathrm{TX}$ or cotangent ${T}^{*}X$ bundles but their direct sum $\mathrm{TX}\oplus {T}^{*}X$. Robert H mentioned the pertinent papers here for what is known as generalised geometry.

The rings of observables for the two models are given by the Dolbeault (B) and quantum (A) cohomology rings. Relating this to other content of the conference, the category of D-branes is given by the derived category of bounded coherent sheaves on $M$ for the B model and the Fukaya category of $M$ for the A model (see here and here.)

Mirror symmetry interchanges the A and B models, much as T-duality interchanges IIA and IIB. One of the main problems is that there are general $N=(\mathrm{2,2})$ backgrounds ($H$-flux) which don’t have a Kaehler target space. Generalised geometry includes as special cases Kaehler and symplectic structures and other, “perturbed”, cases. I’m not sure if it is known that gen. geom. extrapolates between the two structures - i.e. if the moduli space of generalised complex structures (more on those in a minute) is connected.

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So this physics (if you think it’s physics) is a motivation for understanding Hitchin-Gualtieri generalised geometry. $\mathrm{TX}\oplus {T}^{*}X$ is equipped with a symmetric bilinear form $$\langle Z+\xi ,Y+\eta \rangle =\frac{1}{2}({\iota}_{Z}\eta +{\iota}_{Y}\xi )$$ The symmetries of this are given by the orthogonal group $O(d,d)$ for $d$ the dimension of $X$.

There is a ‘Courant bracket’ on $\mathrm{TX}\oplus {T}^{*}X$ given by $$[Z+\xi ,Y+\eta ]=[Z,Y]+{L}_{Z}\eta -{L}_{Y}\xi -\frac{1}{2}d({\iota}_{Z}\eta -{\iota}_{Y}\xi )$$ This does not satisfy a Jacobi identity (or Leibniz rule), but differs by $$\frac{1}{3}d(\langle [A,B],C\rangle +\mathrm{perms})$$

In the presence of a closed 3-form (the $H$-flux) we can twist the Courant bracket. Then we were given a definition of ‘generalised almost complex structure’, an example of which is, given an almost complex structure $J$, given by two pieces in terms of $J$ and ${J}^{T}$. This can be twisted, using the twisted Courant bracket. Then there is a (possibly twisted) generalised Calabi-Yau structure and finally a (possibly twisted) generalised Kaehler structure (tGKS).

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The consituents of a tGKS are a twisted generalised complex structure ${J}_{J}$ coming from the regular complex structure and one from the symlectic structure ${J}_{\omega}$, such that $G=-{J}_{\mathrm{JJ}}{}_{\omega}$ is a positive defintite metric on $\mathrm{TX}\oplus {T}^{*}X$.

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This naturally gives rise to Lie algebroids, for example isotropic involutive subbundles of $\mathrm{TX}\oplus {T}^{*}X$, and a consideration of cohomology for these, and the idea that the physical states of the B model are in terms of this Dolbeaut cohomology.

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If $X$ has a tGKS then there exists an $N=(\mathrm{2,2})$ SUSY nonlinear sigma model with target $X$. By twisting we can define the topological A- and B-models with associated tGCS ${J}_{J}$ and ${J}_{\omega}$ respectively. The cool thing is that observations of the A-model (B-model) are given by the Lie algebroid cohomology of the Lie algebroid associated to ${J}_{J}$ (${J}_{\omega}$).

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Bouwknegt finished with the question: can we associate a triangulated category to a twisted generalised complex structure?

David and Marni