July 13, 2005

Tuesday at the Streetfest III

Posted by Guest

Where were we. Yes - Bondal on derived categories of toric varieties…

Idea 1: for an algebraic variety $X$ the associated ${D}_{\mathrm{coh}}^{d}\left(X\right)$ captures ‘geometry’.

Idea 2: for a complex analytic $X$ the associated ${D}_{\mathrm{constr}}^{b}\left(X\right)$ captures ‘topology’.

Idea 3: for a symplectic $X$ we have $\mathrm{Fuk}\left(X\right)$ whose objects are Lagrangian submanifolds and whose morphisms are Floer cohomology.

Now for mirror symmetry between Calabi-Yau $X$ and $Y$. This is a pair of categorical equivalences ${D}_{\mathrm{coh}}^{b}\left(X\right)\simeq \mathrm{Fuk}\left(Y\right)$ and vice versa.

Bondal focused on the Fano variety case. It was a fun talk. ${\mathrm{CP}}^{1}$ became the Earth with datelines and moving midnights, and we heard about an interesting route from Moscow to Sydney which seemed to involve a lot of sleeping in places such as KL. He then talked about exceptional collections and strong and complete versions of these and the fact that there seems to be a ‘prefered’ exceptional collection for toric varieties.

Conjecture: for $X$ a smooth projective variety ${D}_{\mathrm{coh}}^{b}\left(X\right)$ has a canonical semi-orthogonal decomposition.

Such a decomposition means that $D$ is generated by some triangulated subcategories ${B}_{i}$ such that Hom$\left({B}_{i},{B}_{j}\right)=0$ for $i$ bigger than $j$.

Marni Sheppeard

Posted at July 13, 2005 5:22 AM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/596

Re: Tuesday at the Streetfest III

Hi Marni,

I know what ${D}_{\mathrm{coh}}^{b}\left(X\right)$ is: the bounded derived catgeory of coherent sheaves on $X$.

But I am not sure what ${D}_{\mathrm{constr}}^{b}\left(X\right)$ is suppsed to be. The bounded derived category of …?

Then you wrote:

Conjecture: for $X$ a smooth projective variety ${D}_{\mathrm{coh}}^{b}\left(X\right)$ has a canonical semi-orthogonal decomposition.

Such a decomposition means that ${D}_{\mathrm{coh}}^{b}\left(X\right)$ is generated by some triangulated subcategories ${B}_{i}$ such that $\mathrm{Hom}\left({B}_{i},{B}_{j}\right)=0$ for $i$ bigger than $j$.

Did Bondal mention what the physical interpretation of this statement would be? This looks like it must have a very clear physical meaning.

Posted by: Urs Schreiber on July 13, 2005 11:09 AM | Permalink | Reply to this

Re: Tuesday at the Streetfest III

Constr probably means constructible sheaves. Unfortunately, I don’t remember the definition, but you can probably find it. I think it means something that looks like a flat vector bundles on some sort of stratification of your manifold. But I’m not really sure that’s even close to correct.

Posted by: Aaron on July 19, 2005 9:31 PM | Permalink | Reply to this

Re: Tuesday at the Streetfest I (Kapranov)

Hi again,

I am very interested in what Kapranov was talking about. Unfortunatly something is wrong with the code of your (Marni’s) entry (part I of Tuesday at the Streetfest). When I try to see the ‘extended entry’ or try to post a comment there, the server reports an XML parsing error. I have no clue what might be causing such. Too bad.

Anyway, I can see the entry body on the index page. There you wrote:

Kapranov spoke about a Non-commutative Fourier Transform and Chen’s iterated integrals.

I have seen the abstract of this and have seen some private comments M. Kapranov made on that, but I still don’t have a good idea about what’s going on.

I have looked around on the web, but found nothing concerning this. Did he mention anything about when this might be available in text form?

(And in general: will the transparancies of the Streetfest talks be available online?)

So let me try to follows what you reported from his talk:

Monomials ${x}^{i}{y}^{j}{x}^{k}{y}^{l}$ can be represented by paths on a lattice in ${ℝ}^{2}$, starting at the origin.

So I guess the path corresponding to that is the one which starts by moving $i$ steps parallel to the $x$ axis, then $j$ steps parallel to the $y$-axis, then again $k$ steps along the $x$-axis and finally $l$ steps along the $y$-axis again. (?)

allow non-integer powers ${x}_{i}^{\frac{1}{m}}$ and let $m$ go to infinity

This should then correspond to going from the lattice ${ℝ}^{2}/{ℤ}^{2}$ to the lattice ${\mathrm{R}}^{2}/\left(\frac{1}{m}{ℤ}^{2}\right)$

For $\Omega ={\sum }_{i}{z}_{i}\phantom{\rule{thinmathspace}{0ex}}{\mathrm{dt}}_{i}$ let ${E}_{\gamma }\left(z\right)$ be the holonomy of $\Omega$.

I assume ${\mathrm{dt}}_{i}$ is the unit 1-form parallel to the $i$-th axis of ${ℝ}^{n}$. So given, for instance, the path represented by

(1)$\gamma \simeq {x}_{1}^{\frac{{\gamma }^{1}}{m}}{x}_{2}^{\frac{{\gamma }^{2}}{m}}$

what is presumeably meant is that

(2)${E}_{\gamma }\left(z\right)=\mathrm{exp}\left(\frac{i}{m}{\gamma }^{1}{z}_{1}\right)\mathrm{exp}\left(\frac{i}{m}{\gamma }^{2}{z}_{2}\right)\phantom{\rule{thinmathspace}{0ex}},$

which apparently we want to interpret as the NC analog of the ordinary

(3)$\mathrm{exp}\left(\frac{i}{m}\gamma \cdot z\right)\phantom{\rule{thinmathspace}{0ex}}.$

So then, probably, given any function $F$ of $\gamma$, we want to call the expression

(4)$z↦\stackrel{˜}{F}\left(z\right)=\sum _{\gamma }F\left(\gamma \right){E}_{\gamma }\left(z\right)$

the NC Fourier transform of $F$.

Hm, this cannot be quite what you have in mind, probably. There are more long paths that short paths on the lattice, so stated this way this does not quite reproduce the ordinary Fourier transform.

Kapranov went on to consider the problem for higher dimensional membranes instead of paths.

Alas, I cannot really see what you are sketching now.

Did M. Kapranov mention any relation of this to gerbes? I mean, does he consider any global topological effects or is this maybe just a way to talk about surface holonomy in what I would call a trivial 2-bundle?

Posted by: Urs on July 13, 2005 2:15 PM | Permalink | Reply to this
Read the post Kapranov and Getzler on Higher Stuff
Weblog: The String Coffee Table
Excerpt: Lecture notes of talks by Kapranov on noncommutative Fourier transformation and by Getzler on Lie theory of L_oo algebras.
Tracked: June 22, 2006 8:04 PM

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