The String Coffee Table

I never meant to say much about this week, but in a completely unexpected turn of events yesterday, Getzler pulled a rabbit out of the hat. Remember that cryptic Kapranov-Getzler exchange from the Streetfest? Well, after days of lectures on cyclic cohomology and K-theory and operator algebra Getzler starts doing iterated integrals. He says that if you care about iterated integrals and equivariant cohomology (like all good physicists should) you would be naturally led to do cyclic cohomology. The rest of NCG makes much more sense to me now (that’s relatively speaking, folks).

Anyway, here’s how Getzler’s lectures went:

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 $f: W \rightarrow C$ 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…..

I am currently on vacation in Wales. Relaxing - you know. Over the weekend we are staying at the coast in the beautiful town Aberystwyth. While my girlfriend is watching out for dolphins in the bay, I sneaked away and into the nearby internet café to check my mail.

Turns out that I have received a question concerning the coupling of open strings to the Kalb-Ramond field. I thought instead of writing an anwer by private email I could just as well post it here to the Coffee Table.

The Streetfest ended yesterday with a talk by Eugenia Cheng on degenerate higher categories and the periodic table of Baez/Dolan. I’d like to say a bit about some of the talks, but not right now. It has been exhausting but wonderful. At lunchtime there was a piano recital by Amnon Neeman’s son in the school of music. Unfortunately, Australia is about to lose this remarkable performer to a school in the US. Amnon hosted a party in the evening for those of us who were still around. There was a very fine selection of cheese and wine, a kangaroo stew and decadent desserts.

Steiner gave an interesting talk in the afternoon about the structure of the orientals of Ross Street. Breen raced through as much as he could before lunch, at a pace and at a level of sophistication that I couldn’t follow.

This morning there is a seminar by Baxter on the chiral Potts model, which I simply must attend before the NC geometry school starts at lunchtime…and there are going to be lectures on the weekend!

Marni

Greetings from Canberra!

It is very pleasant here. There was a good frost this morning and some black swans with their chicks down by the lake. Most of the Streetfest participants moved down here yesterday in buses or cars or planes, mostly uneventfully, although Tim Porter found himself driving on a dirt road through the mountains after realising that the main highway south actually went to Melbourne. The workshop started this morning. We’re so busy that there’s almost no time to tell you what’s going on.

Kapranov was up first: the prequel to last week’s talk on NC Fourier transforms. At the end there was a little discussion with some people wondering exactly how this connects to Connes’ NCG.

Panov spoke about model cats, homotopy colimits and toric topology. Toric topology is the study of torus actions on manifolds or complexes with a rich combinatorial structure in the orbit quotient. First he defined ‘face rings’ which are Stanley-Reisner algebras of simplicial complexes. Ross Street later highly recommended Stanley’s revolutionising of combinatorics … something I must look into later. The Poincare series for $R [ K ]$ was defined, and Panov listed some problems that could be attacked with this machinery: the Charney-Davis conjecture, the question of when $\mathrm{Ext}_{k [K]} (k,k)$ has rational Poincare series, and something called the g-conjecture - whatever that is.

Then he defined the David-Januszkiewicz space $DJ(K)$, which seems to be important because it turns up in a theorem (Panov, Ray, Vogt) giving a homotopy commutative diagram involving the loop functor $\Omega: \mathbf{Top} \rightarrow \mathbf{TMon}$ into the monoid category.

Marni

A while ago I had mentioned some literature on the Segal-like description of open topological strings. Here is a comment on that from Kevin Costello, which I believe I may reproduce here:

I had to interrupt my virtual Streetfest at home for a moment. Thursday I visited the Quantum Physics and Geometry Seminar in Hamburg. After that I travelled to Bad Honnef where today I gave the first of two parts of a mini-lecture on ‘nonabelian strings’ aka nonabelian gerbe holonomy. Here are some random notes.

Gorbunov decided to talk about Gerbes of Chiral Differential Operators instead of what was in his abstract, motivation being CFT and the study of A and B differentials and also Witten’s half-twisted case leading to an infinite dimensional $H(V) = \oplus V_{i}$ with the property of being a vertex algebra (which Gorbunov defined nicely for us but I won’t repeat) such that the weighted sum of dimensions gives the elliptic genus of $M$.

I guess if you’re at Strings 2005 you’ll hear all about recent work by Witten and Kapustin on this stuff. Anyway, then we went onto vertex algebroids! For $A$ a commutative associative ring, $T$ an $A$-module and $\Omega$ a $T$ and $A$ module, a $d: A \rightarrow \Omega$ and a Courant bracket (remember Bouwknegt’s talk) there is a complicated looking list of axioms involving also maps $c: T \otimes T \rightarrow \Omega$ and $\gamma: A \otimes T \rightarrow \Omega$.

The theorem of Gorbunov et al. is that, given any complex analytic $M$ and bundle $E$ there exists a certain gerbe so that we get a vertex algebra $H^* (M, \Omega^{\mathrm{ch}}(M,E))$.

Yetter raced us through an unbelievable amount of stuff after struggling with the laptop setup, starting with a 5 minute summary of knot polynomials, tangles, Joyal and Street and other things leading up to the paper of Mei-Chi Shum on tortile tensor categories. As far as I know this important paper isn’t available online - sorry. Yetter stressed the importance of the fact that these structures allow us to understand why quantum groups have something to do with low dimensional topology.

He then went on to talk about Kirby calculus, 3 and 4 manifold invariants, deformation theory, bottom tangles … and by the time he got to some recent results of his own unfortunately my right hand rebelled against the torture and my brain sympathised and refused to take any more in.

Cisinski also changed his talk, and spoke about Batanin weak higher groupoids and homotopy types. This was a racy but precise journey through some sophisticated $\omega$-operad theory and theorems on Quillen equivalences (any reader who knows what these are would probably do a better job than me in discussing these ideas).

Maybe we’ll come back later and discuss Breen’s talk on monoidal braided n-categories. John Baez gave us a very entertaining talk on numbers as cardinalities - this talk is on the Streetfest website. John is first up this morning (Friday) … must go and stretch my fingers.

Marni

David here

I thought I’d talk a bit about Ezra Getzler’s talk (which was only loosely connected to his abstract) on some Lie theory for $L_{\infty}$ algebras. He started off by saying that is is just as worthwhile working with the nerve of a groupoid and so we defined something to work with:

Well it’s Thursday morning and I thought I’d say hi now because it’s going to be a very busy day. John is giving a public lecture this afternoon and the banquet is this evening. The weather has been nice. Saw a baby possum playing in the garden outside the Maths building.

Today there will be talks by Getzler, Yetter, Breen and others.

Marni

Well, I’m going to have dinner with Marnie and David and some other people, but I have a little time to kill, so maybe I’ll talk about Bouwknegt’s talk even though they’re also covering this one. After all, this talk was about string theory, and this is the string theory coffee table, not the “general abstract nonsense” coffee table!

John Baez

Well, I’m going to have dinner with Marnie and David and some other people, but I have a little time to kill, so maybe I’ll talk about Bouwknegt’s talk even though they’re also covering this one. After all, this talk was about string theory, and this is the string theory coffee table, not the “general abstract nonsense” coffee table!

John Baez

Besides Bouwknegt’s talk on D-branes and generalized geometry, which Marnie and David are writing about, and Mueger’s talk on modular tensor categories, which I already wrote about, there were two more talks down here in Sydney. Both involved operads! One was Jean-Louis Loday’s talk on “Generalized bialgebras and triples of operads”, and the other was Jonathan Scott’s talk on “Bimodules of operads: encoding deep structure of morphisms”.

So, I want to talk about these.

John Baez

Hi - I’ve decided to join in the fun and write a little about Michael Mueger’s talk in the Streetfest. I’ve always been a fan of Michael’s work on modular categories, so it was fun to see what he’s been up to lately. My report here will be quite sketchy because he blasted us with so much information that after a little while I decided to just listen rather than take notes.

John Baez

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 $TX$ or cotangent $T^*X$ bundles but their direct sum $TX \oplus T^*X$. 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.)

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

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

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

Idea 3: for a symplectic $X$ we have $\mathbf{Fuk} (X)$ 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 $$D^{b}_{coh} (X) \simeq \mathbf{Fuk} (Y)$$ and vice versa.

Bondal focused on the Fano variety case. It was a fun talk. $\mathbf{CP}^{1}$ 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 $D^{b}_{coh} (X)$ has a canonical semi-orthogonal decomposition.

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

Marni Sheppeard

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.

Hello again! Wednesday already here. I already have 50 pages of notes. It’s quite overwhelming. Now let’s see. Kapranov spoke about a Non-commutative Fourier Transform and Chen’s iterated integrals. For those into membranes, keep reading…

Consider two variables $x$ and $y$ that do not commute. Monomials $x^{i}y^{j}x^{k}y^{l}$ can be represented by paths on a lattice in $\mathbf{R}^{2}$ starting at the origin. Similarly for $n$ variables. Therefore, a polynomial is represented by a summation over paths $\gamma$.

We want to consider the continuous version of this. Imagine subdividing the lattice; allow non-integer powers $x_{i}^{\frac{1}{m}}$ and let $m$ go to infinity. Now let $$x_{i}^{\frac{1}{m}} = e^{\frac{1}{m} z_{i}}$$ in the complex algebra of power series $A$.

$A$ is a ‘connection’. For $\Omega = \sum z_{i} dt_{i}$ let $E_{\gamma} (z)$ be the holonomy of $\Omega$. Now we can define the NC Fourier Transform using this NC exponential.

Kapranov went on to consider the problem for higher dimensional membranes instead of paths. For a lattice box in the variables $x_{1}$ and $x_{2}$ there is now a 2-cell filling the box $$x_{1} x_{2} \Rightarrow x_{2} x_{1}$$ Introduce these $x_{ij}$ of degree -1 in a dg-algebra $B_{2}$ with $$d(x_{ij}) = x_{i} x_{j} - x_{j} x_{i}$$

Let $C_{2}$ be the 2-category generated by the 2-skeleton of the cubical lattice in $\mathbf{R}^{n}$. The pasting of the half cube for $C_{2}$ gives a rule in $B_{2}$. But there are 2 choices of half cube. The difference is used to extend to $B_{3}$ by adding $x_{ijk}$ now of degree -2 with $$d(x_{ijk}) = [x_{ij},x_{k}] + [x_{j},x_{ik}] + [x_{jk},x_{i}]$$

so $B_{3}$ is the universal enveloping algebra of a dg-Lie algebra. And so on … onto the continuous version of this.

Phew. Must be off.

Marni

P.S. Browser options here aren’t great. We apologise for errors that we are unable to correct at this stage.

David here - I thought I’d mention some of the stuff that went on in the afternoon - a talk by Tom Leinster and one from Boris Chorny.

Leinster’s talk on self similarity (ref: math.DS/0411343) from a general point of view - a self similar object looks like copies of another object glued to each other, each of which look like copies of another object etc, such that we have a finite pool of objects to draw on to glue together. The last part of that sentence is of course ill-defined, but one gets the idea. Really we can define the above “glued together” relation as a sort of eigenvalue system.

While others report from conferences on Strings or Loops, in Sydney we are enjoying the balmy winter and the calls of the cockatoos and lorakeets. The Streetfest, in honour of Ross Street’s 60th birthday, got under way yesterday. Day one was very maths oriented.

The first talk was by Joyal, a well known collaborator of Ross Street. He spoke about quasicategories; an amazingly clear speaker.

The theory of quasicategories is developed as an extension of both ordinary category theory and homotopical algebra (a la Quillen). The hope is that it will yield insights into the development of both higher category theory and homotopical algebra.

Let $S$ be the category of simplicial sets, that is functors from $\Delta^{0}$ into $\mathbf{Set}$. A simplicial set is called a quasicategory if every horn $$\Lambda^{k}(n) \to X \qquad 0 \lt k \lt n$$ can be filled.

An example of a quasicategory is then a Kan complex.

The left adjoint $\tau$ of the nerve functor $N: \mathbf{Cat} \to S$ preserves products, so can be used to give a 2-category structure on $S$. The Hom category is given by $\tau (B^{A})$ and the product gives the composition law. This 2-categorical structure is used to define equivalences of quasicategories. The category $\mathbf{QCat}$ of quasicategories is Cartesian closed.

As an application of this, whereas in $\mathbf{Cat}$ one looks at discrete fibrations where $X^{\mathbf{2}} \to Y^{\mathbf{2}} \times_Y X$ is an isomorphism, in the extension to $\mathbf{QCat}$ one demands that this is a trivial fibration. Think of fibres of fibrations being Kan complexes.

Next, Borceux spoke about semi-direct products and the representability (in a categorical sense) of actions. Later in the day Johnstone followed this with a talk on his work on bi-Heyting toposes. He had been trying to prove a conjecture: that $E$ bi-Heyting implies “there exists an essential surjection $B \to E$ for $B$ Boolean. Instead he found a counterexample: the sheaves on $[0,1]$ as a complete Heyting algebra.

I’d like to talk about Borceux’s talk - but no time now. If there are any real category theorists reading this who would like to correct any misunderstandings - please comment! Lectures about to start on Tuesday…so I’m off.

Marni Sheppeard

Addendum from David: the talk has started, Marni’s run off, and I’m just trying to tidy this up - failed, I know, so maybe it can be fixed later.

D

One thing that is sort of obvious, but which has not been written out in detail yet, is that the global 2-holonomy 2-functor defined in terms of local trivializations as discussed here more intrinsically is a 2-functor

from 2-paths (surfaces) in the base manifold $M$ to the 2-category of $G_2$-2-torsors.

A discussion of this can be found here:

$p$-Functors from $p$-Paths to $p$-Torsors
http://www-stud.uni-essen.de/~sb0264/Transport.pdf

as well as in section 12.4 of this.