## September 21, 2009

### A Seminar on a Geometric Model for TMF

#### Posted by Urs Schreiber

Contuinuing with my by now well-established practice – see A Seminar on a Survey of Elliptic Cohomology and A Seminar on Gromov-Witten Invariants – of working seminar notes here at MPI Bonn into the $n$Lab, I am now continuing with the main event, the lecture series by Stephan Stolz and Peter Teichner on their work on

See there for a (little) bit of background and context.

Notes on today’s sessions are beginning to find an incarnation as wiki-entries here:

Axiomatic field theories and their definition from topology

(1|1)-dimensional Euclidean field theories and K-theory

So far this sketches the outline of the construction. Concrete details are the topic of the next days. Those reader who haven’t seen aspects of this program before might find the notes so far a bit impenetrable, for instance a basic understanding of supermanifolds is assumed. So I am posting this here for the time being more for the experts who might lend me a hand in polishing this, than for the laymen. For them, hopefully, this will develop into something useful as this proceeds.

In particular, please notice, as before, that the notes behind these links are, for the time being, notes taken real-time during the seminar, with no real post-production yet. So everything is a bit rough at the moment. Help me make it become more smooth!

Posted at September 21, 2009 6:19 PM UTC

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### Re: A Seminar on a Geometric Model for TMF

This is soooo fast. How do you this?

Posted by: G. S. on September 21, 2009 6:58 PM | Permalink | Reply to this

### Re: A Seminar on a Geometric Model for TMF

Are you one of the participants? I am, guessing so, but I have to admit that I am being to dense to figure out who you are. Anyway, thanks for commenting here. Your comments/questions over at the GW-invariants were nice productive comments.

So, tomorrow over tea approach me and let me know who you are. Then I tell you the answer to your last question. :-)

Posted by: Urs Schreiber on September 21, 2009 8:05 PM | Permalink | Reply to this

### supergeometry versus higher/derived geometry

One of the key components of the Stolz-Teichner program to realize a geometric model for tmf is that they look at parallel transport functors not just on paths, but on paths with infinitesimal thickening.

They model “infinitesimal thickened spaces” as supermanifolds. What matters structurally though is not so much the manifold condition. Rather (more or less implicitly in the present setup), in the full picture that one expect to develop here eventually it amounts to working in the context of $\infty$-stacks over a subcategory of the formal dual of $\mathbb{Z}_2$-graded algebras, I claim.

Now, from a structural point of view, it is possibly noteworthy that the notion of $\mathbb{N}$-grading and $\mathbb{Z}$-grading both have good abstract-nonsense roots, while $\mathbb{Z}_2$-grading has not, or not quite as nicely.

I am speaking of the Dold-Kan correspondence, of course: this can be understood as identifying most $\mathbb{N}$-graded algebraic structures as nothing but convenient repackagings of basic $\infty$-categorical structures: $\mathbb{N}$-graded chain complexes are just a conveniently repackaged description of strictly symmetric monoidal strict $\infty$-groupoids.

Following the terminology in Moerdijk-Reyes, let me write loci for the test spaces in a given geometric setup. Typically those that are uniquely specified by their algebras-with-structure of global sections of functions and hence formally dual to these. In ordinary geometry one considers spaces modeled on such loci modeled as sheaves on the category of these loci. In higher geometry (sometimes called derived geometry, but usually in a sense a bit stricter than what I am using here) one considers instead $\infty$-groupoids modeled on $\infty$-loci, where an $\infty$-locus is an $\infty$-groupoid internal to loci, for instance modeled by simplicial objects in loci. The $\infty$-groupoids modeled on $\infty$-loci are given by $\infty$-stacks on $\infty$-loci, so the $\infty$-loci themselves are the geometric $\infty$-stacks: for our $(\infty,1)$-catgeorical site of $\infty$-loci these are simply the representable ones.

All that to get to the following obvious but noteworthy point: essentially by assumption our $\infty$-loci are dually modeled by cosimplicial algebras-with-structure of sorts, depending on the model. Under the monoidal Dold-Kan correspondence these are identified with $\mathbb{N}$-graded differential algebras-with-structure.

From which we find that our $\infty$-groupoids modeled on $\infty$-loci are giveny by $\infty$-stacks on a $(\infty,1)$-category of $\mathbb{N}$-graded differential algebras.

The point being that this is structurally a very natural setup, from an abstract nonsense perspective. Passing from this setup instead to one of $\infty$-stacks on $\mathbb{Z}_2$-graded (differential) algebras a priori feels a bit like a violation of the nice structural context.

Of course I know of all the motivations to have a $\mathbb{Z}_2$-grading. In one way or other one wants spin geometry and Dirac operators to live in this context, and they just happen to live in a $\mathbb{Z}_2$- but not in an $\mathbb{N}$-graded context.

But it keeps making me wonder if much of the supergeomety structure seen for instance here in the Stolz-Teichner program is secretly a truncated higher or “derived” geometry in the above sense.

Posted by: Urs Schreiber on September 22, 2009 9:44 AM | Permalink | Reply to this

### Re: supergeometry versus higher/derived geometry

On the other hand, maybe one should look at the embedding of $\mathbb{Z}_2$-graded algebras into $\mathbb{N}$-graded algebras given by the assignment

$R : (A_{even} \oplus A_{odd}) \mapsto (A_0 \oplus A_1 \oplus A_2 \oplus \cdots) := (A_{even} \oplus A_{odd} \oplus A_{even} \oplus A_{odd} \oplus \cdots)$

or rather its extension to an embedding of $\mathbb{Z}_2$-graded algebras equipped with an odd square-0 derivation into $\mathbb{N}$-graded differential algebras

$R : (A_{even} \stackrel{\stackrel{d_{odd}}{\leftarrow}}{\stackrel{d_{ev}}{\to}} A_{odd}) \mapsto (A_0 \stackrel{d_0}{\to} A_1 \stackrel{d_1}{\to} A_2 \stackrel{d_2}{\to} \cdots) := (A_{even} \stackrel{d_{even}}{\to} A_{odd} \stackrel{d_{odd}}{\to} A_{even} \stackrel{d_{even}}{\to} A_{odd} \oplus \cdots)$

Conversely, there is a map from $\mathbb{N}$-graded cochain algebras to $\mathbb{Z}_2$-graded algebras equipped with an odd square-0 derivation that maps

$L : (A_0 \stackrel{d_0}{\to} A_1 \stackrel{d_1}{\to} A_2 \stackrel{d_2}{\to} \cdots) \mapsto (A_{even} \stackrel{\stackrel{d_{odd}}{\leftarrow}}{\stackrel{d_{ev}}{\to}} A_{odd}) := (\oplus_k A_{2k} \stackrel{\stackrel{\oplus_k d_{2k+1}}{\leftarrow}}{\stackrel{\oplus_k d_{2k}}{\to}} \oplus_k A_{2k+1})$

This is the kind of map that is for instance used when turning ordinary integral cohomology into a periodic cohomology by setting

$A^0(X) := \oplus_k H^{2k}(X,\mathbb{Z})$

$A^1(X) := \oplus_k H^{2k+1}(X,\mathbb{Z}) \,.$

It seems these form an adjoint pair $(L \dashv R)$

$R : \mathbb{Z}_2 DGA \stackrel{\leftarrow}{\to} \mathbb{N} DGA : L$

since

$Hom\left( \array{ A_0 \\ \downarrow^{d_0} \\ A_1 \\ \downarrow^{d_1} \\ A_2 \\ \downarrow^{d_2} \\ \cdots } \,, \;\; \array{ B_even \\ \downarrow^{d_{ev}} \\ B_{odd} \\ \downarrow^{d_{odd}} \\ B_{even} \\ \downarrow^{d_{ev}} \\ \cdots } \right) \simeq Hom\left( \array{ A_0 \oplus A_2 \oplus \cdots \\ \downarrow^{d_0 \oplus d_2 \oplus \cdots} \uparrow^{d_1 \oplus d_3 \oplus \cdots} \\ A_1 \oplus A_3 \oplus \cdots } \,, \;\; \array{ B_{even} \\ \downarrow^{d_{ev}} \uparrow^{d_{odd}} \\ B_{odd} } \right)$

The structure of an NQ-supermanifold on an ordinary supermanifold would be precisely a lift of its algebra of functions through $L$.

(Well, either I am mixed up or this must be well known…)

Posted by: Urs Schreiber on September 23, 2009 8:21 AM | Permalink | Reply to this

### N vs Z/2

Has there been any prior appearance/use of your R??

Posted by: jim stasheff on September 23, 2009 1:37 PM | Permalink | Reply to this

### Riemannian bordisms and Riemannian QFT

today’s notes are now here:

Riemannian bordism categories and Riemannian QFT

Posted by: Urs Schreiber on September 22, 2009 6:16 PM | Permalink | Reply to this

### Euclidean supermanifolds

Today’s seminar session started preparing the ground for the definition of

$(2|1)$-dimensional Euclidean field theories

by recalling basics of supergeometry and then defining the notion of

Euclidean supermanifold

Posted by: Urs Schreiber on September 23, 2009 9:22 PM | Permalink | Reply to this

### Re: Euclidean supermanifolds

One of the crucial new developments in the Stolz-Teichner program is that where previously there were attempts to use various notions of Riemannian metrics on supermanifolds to characterize superconformal or super-Euclidean structure, they are now saying that a much better way to achieve this is to use a generalized Klein’s-Erlanger-program-style-definition along the very lines that is described at $n$Lab:manifold:

instead of specifying extra metric-like structure we imagine the canonical such extra structure on our test spaces (the standard flat or conformal structure on things $\sim \mathbb{R}^{p|q}$) and then just require that all transition functions of charts respect this implicit structure in that they sit in a special subgroup of the generally possible group of transition function. See $n$Lab: Euclidean supermanifold for a bit of details.

I am wondering: this kind of Erlangen-program-style defnition of manifolds, supermanifolds etc with extra structure should have a very general abstract-nonsense formulation.

In particular, it should be applicable to the notion of generalized scheme, where one would require that the cover by generalized affine schemes involved would be required to yield on overlaps transition function in a prescribed $\infty$-group.

Posted by: Urs Schreiber on September 24, 2009 10:33 AM | Permalink | Reply to this

### Re: Euclidean supermanifolds

I’m glad the Erlangen philosophy is proving helpful, but I agree with you that the importance of $\mathbb{Z}/2$-graded structures is fundamentally a bit mysterious, and deserves to be better integrated with other mathematics. Could it all boil down to the first stable homotopy group of spheres:

$\lim_{n \to \infty} \pi_{n+1}(S^n) = \mathbb{Z}/2$

or something like that? This stable homotopy group is deeply related to the fact that in spacetimes of dimensions 4 or higher, you can unlink this:

\   /
\ /
\
/ \
/   \
\   /
\ /
\
/ \
/   \


and make it look like this:

|    |
|    |
|    |
|    |
|    |
|    |
|    |
|    |
|    |
|    |
|    |


So, it’s related to the statistics of point particles (the boson/fermion distinction), and thence to spin.

Should we expect interesting categorified versions of supersymmetry to arise from higher homotopy groups of spheres?

This is the kind of question that keeps me up at night.

Perhaps the work of A. Bartels, André Henriques, and Christopher Douglas, will be relevant. The third stable homotopy group of spheres is $\mathbb{Z}/24$, and they may be discovering ‘categorified Clifford algebras’ where this group plays an important role.

Christopher Douglas is giving a talk at the AMS meeting at UC Riverside later this fall, entitled 3-Categories for the Working Mathematician. I’ll see if this sheds any light on such mysteries…

Posted by: John Baez on September 24, 2009 4:37 PM | Permalink | Reply to this

### Re: Euclidean supermanifolds

Earlier chat about categorified Clifford algebras, and further chat.

Posted by: David Corfield on September 24, 2009 4:44 PM | Permalink | Reply to this

### Re: Euclidean supermanifolds

Posted by: Eric Forgy on September 25, 2009 6:13 PM | Permalink | Reply to this

### Last day: the partition function of a (2|1)EFT is a modular form

I won’t be able to take notes in the lectures tomorrow (Friday), as I’ll be in the math seminar in Utrecht.

Among the Stolz-Teichner participants reading this here: would anyone volunteer to provide some wiki-typed notes for tomorrow’s session?

That would be greatly appreciated – as far I have heard not just by me but by various other participants.

I have prepared a template $n$Lab entry where “only” the content would have to be typed into, here:

modular forms from partition functions

Let me know if you’d be interested but feel like you need more information on how to proceed.

Posted by: Urs Schreiber on September 24, 2009 4:25 PM | Permalink | Reply to this

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