July 12, 2005

Tuesday at the Streetfest II

Posted by Guest

Yesterday was a bit of a blur for me - lack of sleep in the previous week catching up, but I did get Kapranov’s talk (excellent - Marni’s doing a post on that) and also Voronov’s (that’s not to say I didn’t go to other talks :) ).

Voronov talked on operads (Informally calling his talk “the grocery of operads”), and didn’t really get to the Swiss cheese as promised, but was a good talk anyway, starting from the little intervals operad. Actually he started from the definition of an operad, which is a souped up version of an $n$-ary operation - actually a space of operations

${V}^{\otimes k}\to V$

satisfying some axioms (mostly distributive stuff, I think - no details were given), and $V$ can be a vector space, but not necessarily.

Yesterday was a bit of a blur for me - lack of sleep in the previous week catching up, but I did get Kapranov’s talk (excellent - Marni’s doing a post on that) and also Voronov’s (that’s not to say I didn’t go to other talks :) ).

Voronov talked on operads (Informally calling his talk “the grocery of operads”), and didn’t really get to the Swiss cheese as promised, but was a good talk anyway, starting from the little intervals operad. Actually he started from the definition of an operad, which is a souped up version of an $n$-ary operation - actually a space of operations

${V}^{\otimes k}\to V$

satisfying some axioms (mostly distributive stuff, I think - no details were given), and $V$ can be a vector space, but not necessarily.

One thing I thought was missing is that we weren’t actually shown how the operad examples were actually operads.

Little intervals

The little intervals operad is the compactified (real Fulton-MacPherson) configuration space of $k$ distinct points on the real line, modding out translations and scaling factors.

Little intervals $=\overline{C\left(R,k\right)/R×{R}_{+}^{*}}$

For two points they just don’t get any closer to each other, but when we have three points, two of the adjacent points can converge, but not coincide - those two points are said to bubble off - it was likened to having to look at those two points under a magnifying glass (and so we ge two scenarios depending on which points). The relation to category theory is that we can think of the two point situation as composition of arrows (a 0-cell) and the three point as the associator between the two ways of composing three arrow (a 1-cell).

We get the pentagon from considering four points, which gives us five ways of fiddling four arrows into the associator - the different cases corresponding to two or three adjacent points bubbling off.

Little disks

The little disks operad is defined similar to the previous one - the compactified configuration space of $k$ disks in ${R}^{2}$ wih translations and dilations/dilatations modded out (there was a bit of contention at this point as to the proper word, mostly from Max Kelly). For $1\le k<6$ everything is well behaved and it all goes to pieces after that. Like the little intervals is related to the associator, little disks has something to do with what Voronov called the “Breenator”, something with 4 hexagonal faces and 2 rectangluar ones. Visualising the configuration spaces are tricky at best, but impossible with words, so I’ll give you a link later

The cacti operad was mentioned, it’s mst handy pictures looking like cacti - which have involve ribbon braided monoidal $n$-categories - and then (since time was running out) we got a picture of what a sample of swiss cheese looked like (here’s the paper: math.QA/9807037). All we were told was that it had something to do with pairs of braided monoidal $n$-categories. For what I think are papers involving swiss cheese in string theory, see math.AT/0412249 and math.QA/0410291.

Others

Alexei Bondal gave a somewhat entertaining talk on derived categories of toric varieties, but the humorous parts seemed ill motivation for the heavy algebraic geometry. A toric variety X is one with an action of the torus $T=\left({S}^{1}{\right)}^{n}$ or $\left({C}^{*}\right)n$, i.e. $X×T\to X$ and the inclusion $T\to X$. A nice example of a variety is the 2-sphere (or ${\mathrm{CP}}^{1}$ if you’re cool and want to call it a curve), given by ${x}^{2}+{y}^{2}+{z}^{2}-1=0$. The non-zero complex numbers act on the pointed sphere (thinking of it as the complex projective line, as above) by multiplication, and are included into it,

${C}^{*}\to {\mathrm{CP}}^{1}$

making the sphere a toric variety. After that it got all hard, and talked a lot about mirror symmetry, which swaps the derived category of bounded coherent sheaves on our manifold $X$ and the Fukaya category of $X$. This may still be a conjecture - the string theorists know a bit about this. I know because it is closely tied up with T-duality (the subject of my honours reseach project)

I didn’t go to the talk by Jurgen Fuchs (pdf version - properly typeset) on conformal field theory, and the last lecture of the day (Grandis’, on directed algebraic topology) was cancelled as the speaker is not at the conference.

That’s enough for this post.

David

Posted at July 12, 2005 11:30 PM UTC

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I have had a look at the Kajiura & Stasheff paper math.QA/0410291 that you mentioned.

So the little disk operad is related to closed string amplitudes and to nonplanar corollas (nonplanar trees with a single vertex)…

…while the little interval operad is related to open string amplitudes and to planar corollas.

We can think of each ‘little disk’ as a little disk cut out from a sphere, being the vertex for a closed string. Similarly, we can think of each ‘little interval’ as a little interval cut out from the boundary of the disk, being the vertex for an open string.

The nice thing is that one knows that the little disk operad gives rise to ${L}_{\infty }$-algebras, while the little interval operad gives rise to ${A}_{\infty }$-algebras. This nicely explains why ${A}_{\infty }$-algebras know about open string field theory, while ${L}_{\infty }$-algebras know about closed string field theory.

The fact that in an ${L}_{\infty }$-algebra everything in sight is (graded-)commutative comes from the fact that you can move little disks cut out from a surface around each other, i.e. that there is no order on closed string vertices.

The fact that in an ${A}_{\infty }$-algebra no (graded-)commutativity is present comes from the fact that we cannot move a little interval cut out from a line around another such interval, i.e. that there is an order on open string vertices.

There should be a close relation between the little disk operad and what Voronov on p. 11 of math.QA/0111009 calls the Riemann surface operad.

Posted by: Urs on July 13, 2005 6:03 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 8:02 PM

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