The String Coffee Table

Part V of our seminar on elliptic cohomology and 2-vector bundles.

$$\array{ \mathbf{\text{part}} & \mathbf{\text{topic}} & \mathbf{\text{based on}} \\ \href{http://golem.ph.utexas.edu/string/archives/000737.html}{\text{I}} & \text{literature on elliptic cohom.} \\ & \text{introdution to 2-vector bundles} \\ \\ \href{}{\text{II}}, \href{}{\text{III}}, \href{}{\text{IV}} & \text{ellipt. coh. as categorified K-theory} & \href{http://arxiv.org/abs/math.AT/0306027}{\text{Baas-Dundas-Rognes}} \\ & \text{classes of 2-vector bundles} \\ \\ \text{V-VI} & \text{elliptic cohom. and strings} & \\ \\ \text{V} & \text{introduction} \\ & \text{K-theory and 1dSQFT} & \href{http://math.ucsd.edu/~teichner/Papers/Oxford.pdf}{\text{Stolz-Teichner}} \\ & \text{Witten genus and 2dSCFT} \\ \\ \text{VI} & \mathrm{tmf}-\text{spectrum} \\ \\ & \text{material used in string th. applications (?)} & \text{Kriz-Sati} }$$

Outline of Part V

$\bullet$ Introduction.

$\bullet$ Background information.

$\;\;\;\;\;$1) What is a genus?

$\;\;\;\;\;$2) What is a generalized cohomology theory?

$\;\;\;\;\;$3) What do we need to know about elliptic curves?

$\;\;\;\;\;$4) What is the elliptic genus?

$\;\;\;\;\;$5) What is elliptic cohomology?

$\bullet$ 6) How is ellitpic cohomology realized geometrically?

a) Warmup: how is the landscape of superpoint theories equal to the K-theory spectrum?

b) How does one expect the landscape of superstring theories to be equal to the spectrum of elliptic cohomology?

Over at $\text{Strings}_{06}$ they had a talk

Sergei Gukov
Surface Operators in Gauge Theory and Categorification
(ppt).

Last year there was the Streetfest, a big conference on categories and their application in mathematical physics and string theory.

I couldn’t go because I was busy finishing my thesis, but Marni Sheppeard, David Roberts and John Baez had been so kind to report here at the Coffee Table about some of the talks ($\to$, $\to$, $\to$, $\to$, $\to$, $\to$, $\to$, $\to$, $\to$, $\to$, $\to$, $\to$, $\to$, $\to$, $\to$, $\to$).

Now, David Roberts was so kind to scan in notes that Marni Sheppeard had taken in two of the most interesting talks ($\to$). Here they are. Many thanks indeed to both Marni and David!

Ezra Getzler
Lie Theory for $L_\infty$ algebras
(pdf with lecture notes by Marni Sheppeard)

This is on integration of $L_\infty$ algebras with comments on applications to TCFT ($\to$).

For more details see the paper

Ezra Getzler
Lie theory for nilpotent L-infinity algebras
math.AT/0404003.

Mikhail Kapranov
Noncommutative Fourier Transform
(pdf with lecture notes by Marni Sheppeard)

This starts, as the name suggests, with a study of a noncommutative version of Fourier transformations and then moves on to a discussion of parallel transport of strings and membranes.

I am glad to finally see these notes, because (as far as I am aware) there is nothing in print on this.

Alas, I am once again too busy to look at this right now. This entry just serves the purpose of reminding me later to look at this stuff.

Still more details on how varieties form a subcategory of schemes.

More details on how varieties are a subcategory of schemes.

Lots of things to talk about, but, unfortunately, I am busy. But I can quickly provide the following link.

Yesterday, at ESI, Eric Sharpe gave a talk

Eric Sharpe
Derived categories in physics
(slides available here).

(The slides are very pretty. Lots of animations usually spice them up, but these are lost in the above pdf.)

The first part reviews some aspects of the description of D-branes in terms of derived categories ($\to$, $\to$).

The second part is on the study of sigma-models whose targets are orbifolds (stacks) that are not global quotients ($\to$, $\to$).

The main point is that it seems to make sense to have strings propagating on orbifolds different from global quotients. Even though for target spaces that are, as stacks, gerbes (not to be confused with the structure incorporating the Kalb-Ramond field) there seems to be a failure of cluster decomposition, this is apparently simply due to the fact that these targets have to be regarded as disconnected.

This is discussed in the recent paper

S. Hellerman, A. Henriques, T. Pantev, E. Sharpe, M. Ando
Cluster decomposition, T-duality, and gerby CFT’s
hep-th/0606034.

I volunteered to produce a survey of the relation between varietes and schemes. Here are some notes.

The third part of the lecture.

The second part of the lecture.

Yan Soibelman gave at ESI a brief outline of work on applying non-commutative methods to CFTs in order to understand mirror symmetry.

Maxim Kontsevich, Yan Soibelman
Homological mirror symmetry and torus fibrations
math.SG/0011041.

Though I haven’t looked at ot yet, I was told that an important reference in this context is

Daniel Roggenkamp, Katrin Wendland
Limits and Degenerations of Unitary Conformal Field Theories
hep-th/0308143.

This is apparently part of a much more general scheme, whose study has been begun in

Maxim Kontsevich, Yan Soibelman
Notes on A-infinity algebras, A-infinity categories and non-commutative geometry. I
math.RA/0606241 .

Today A. Kapustin gave the first of two ESI lecture talks on the super Yang-Mills aspect of his work with Witten on the physical realization of geometric Langlands duality ($\to$, $\to$, $\to$), following the paper

A. Kapustin & E. Witten
Electric-Magnetic Duality And the Geometric Langlands Program
hep-th/0604151.

Here is a transcript of my notes (though there is nothing here which cannot also be found in this paper).

I was scolded for never having cited

John E. Roberts
Mathematical Aspects of Local Cohomology
talk at Colloqium on Operator Algebras and their Applications to Mathematical Physics,
Marseille 20-24 June, 1977 .

Igor Baković was so kind to dig the paper out for me.

Right at the moment Daniel Huybrechts is talking about branes on K3 surfaces. Since I already know this talk ($\to$) I have time to type some notes.

I’ll try to reproduce something from Kontsevich’s ESI lectures on various things related to (homological) mirror symmetry ($\to$). I have to warn you though, that the following, as far as my reproduction makes any sense at all, is either well known to the experts or possibly close to incomprehensibe to those who are not. The fact that I am closer to the second group than to the first does not help, either.

The second lecture (which I might talk about in a later entry), had a lot of Kevin Costello’s results in it, and for all of the following it pays to look at his papers ($\to$).

While I still feel like I should just try (in fact, I have been urged) to try to absorb some of this, I cannot recommend that you try to read the following.

In

Leonardo Castellani
Lie derivatives along antisymmetric tensors, and the M-theory superalgebra
hep-th/0508213

the author implicitly shows that

1)

the central extension by membrane charges $$\lbrace Q_\alpha, \, Q_\beta \rbrace = i(C\Gamma^a)_{\alpha\beta} P_a + \href{http://www.phys.uu.nl/~hofman/scriptie/duality/node15.html}{(C\Gamma_{ab})_{\alpha\beta}Z^{ab}}$$ of the super-Poincaré algebra in eleven dimensions defines a semistrict Lie 3-algebra;

2)

the local field content of 11D supergravity defines the local data for a connection on a 3-bundle with this gauge 3-group.

Recall ($\to$) that we expect on general grounds ($\to$) M-branes to couple to a 3-bundle (2-gerbe) with some gauge 3-group ($\to$).

I am trying to prepare a little for that workshop on homological mirror symmetry that starts tomorrow ($\to$). Rumour has it that the allegedly introductory lecture won’t be introductory at all, so I thought I might take a look at some literature.

Due to lack of time, I have so far only reproduced the first half of Varghese Mathai’s talk ($\to$) on “topological T-duality”, which was really just a review of some basics. Maybe I will find the time to type my notes on the noncommutative and non-associative aspects.

But meanwhile we had some discussion about the relation of this formalism to full CFT over at Jacques Distler’s Musings.

David Corfield rightly asks ($\to$ ) how I’d think some of what I said recently about 2-linear maps ($\to$) and in particular about $\mathrm{Vect}$-linear representations of 2-groups ($\to$, $\to$) relates to the 2-reps discussed in

L. Crane & M. Sheppeard
2-Categorical Poincaré Representations and State Sum Applications
math.QA/0306440.

I am currently staying in Zagreb, where I am visiting Zoran Škoda and Igor Bakovic. If everything works out, next week we’ll go back ($\to$) to the ESI in Vienna to attend the

ESI research conference
Homological Mirror Symmetry
(schedule).

Kontsevich will give the introductory lecture. After that people like Frenkel, Fukaya, Hori, Getzler, Huybrechts ($\to$), Kapustin, Sharpe ($\to$) to name some, will give talks on related stuff.

The last couple of entries ($\to$, $\to$, $\to$) involved pull-push operations on correspondence spaces

But some objects (like vector bundles) don’t want to be pushed, while others (like sheaves) are not so obviously pulled. I didn’t know the details, and tried to fake it (with some succes in the finite case). But here are some details.

The second part of my transcript of V. Mathai’s talk.

The first part of my transcript of V. Mathai’s talk.

Here is a transcript of a talk by Varghese Mathai on T-Duality, concerned with topological aspects, the $C^*$-algebra formulation (“noncommutative topology”), and the identification of non-geometric T-duals, represented by non-commutative or even non-associate $*$-algebras.

For a brief review of some key concepts see

V. Mathai & J. Rosenberg
On mysteriously missing T-duals, $H$-flux and the T-duality group
hep-th/0409073.

More details are in

V. Mathai & J. Rosenberg
T-Duality for Torus Bundles with $H$-Fluxes via Noncommutative Topology
hep-th/0401168

and

V. Mathai & J. Rosenberg
T-Duality for Torus Bundles with $H$-Fluxes via Noncommutative Topology, II: the high dimensional case and the T-duality group
hep-th/0508084.

(V. Mathai had given essentially the same talk a few weeks ago in Vienna ($\to$). )

Varghese Mathai is visiting, and will talk today on his work on T-duality. Yesterday he explained to us how T-duality is similar to the Fourier-Mukai transformation.

That’s nice, since it fits into the big picture ($\to$).