## May 29, 2008

### HIM Trimester on Geometry and Physics, Week 4

#### Posted by Urs Schreiber

I didn’t quite manage to report from the HIM trimester program as regularly as I set out to do. There was just too much going on, talking to people, giving various talks, hearing highly interesting lecture series by Stephan Stolz and Peter Teichner, by Dan Freed and Michael Hopkins (I managed to report on Mike Hopkins’ first lecture so far, for the stuff by Stolz and Teichner my last summary is still pretty close, lacking mostly the new sophisticated discussion of the cobordism category as a framed bicategory (with diffeos and cobordisms as vertical and horizontal morphisms, respectively) internal to categories fibered over (super-)manifolds), Zoran Šcoda and Igor Bakovic visiting, finishing my article on AQFT from extended functorial QFT (have a look now before this goes to the arXiv next week!), and the like.

And now I am even missing one week of the program: am currently in Stanford, where today I give a colloquium talk:

On nonabelian differential cohomoloy
(pdf, blog)

(the talk itself follows the new notes in section 2.1).

Somewhat related to that: I had started typing up notes I took in a very nice talk by Dan Freed last Friday at Bonn University on differential cohomology, differential K-theory and the index theorem, but didn’t get very far. But below are my notes as far as I got last Friday, hopefully to be completed at some point.

Just heard [that was last Friday] a very nice talk by Dan Freed at Bonn University on Dirac operators and differential K-theory. Below is my transcript of my lecture notes. Probably I won’t make it beyond the review part and into the new research stuff before I have to run again – to his next talk. So here is what you migth probably regard as the condensation of an extended review of Atiyah-Singer index theory. Of course that means we will also encounter the Quillen superconnections that we recently talked about.

Dan Freed began with a remark how he had been before in Bonn long ago in the 80s of last century, then hearing Quillen talk about Dirac operators and superconnections. The following picks that thread up, enriched meanwhile by the concept of differential K-theory.

So, here is roughly my transcript:

Let $X$ be a compact Riemannian spin manifold of even dimension.

Let $\array{ (E,\nabla) \\ \downarrow \\ X }$

be a complex hermitian vector bundle with covariant derivative $\nabla$ (encoding the connection). Denote by $D_X(\nabla)$ the corresponding Dirac operator.

$E$ determines of course a class in K-theory $[E] \in K^0(X)$. And the index of $D_X(\nabla)$ depends only on that class. This follows simply from the fact that the index is a continuous function with values in integers on the space of all connections on $E$, which however is an affine space, hence in particular connectes. So the index has to be constant function of the connection.

Define the notation

$\pi_*^{analytic} : K^0(X) \to \mathbb{Z}$ for the index map $[E] \mapsto index D_X(\nabla) \,.$

This is going to be interpreted, of course, as push-forward to the point in K-theory. The point being, however, that there is another prescription for that push-forward and the Atiyah-Singer theorem saying that it coincides with the one just given.

That other procedure, push-forward in $K$-theory, is the homomorphism denoted $\pi_*^{top} : K^0(X) \to \mathbb{Z}$

Theorem (Atiyah-Singer): $\pi_*^{analytic} = \pi_*^{top} \,.$

There is famously an explicit cohomological formula for $\pi_*$, which expresses it in terms of an integral over the product of the $\hat A$-genus and the Chern-class of $E$: $\pi_*^{top}([E]) = \int_X \hat A(X) \; ch(E) \,.$ That integral, in turn, is to be interpreted as push-forward in rational cohomology.

A few years after that theorem the “families”-version was proved:

Let now $\array{ X \\ \downarrow \\ S }$ be a family of compact spin manifolds of dimension $n$ with Riemannian structure on the relative tangent bundle $T(X/S)$ (i.e. the bundle of vertical vectors). A metric along the fibers together with a horizontal distribution canonically defines a Levi-Civita conection $\nabla^{X/S}$ on $T(X/S)$.

Given now a vector bundle $\array{ (E,\nabla) \\ \downarrow \\ X \\ \downarrow \\ S }$ on the total space, we obtain a family $D_{X/S}(\nabla)$ of Dirac operators parameterized by $S$. And again, we have two different prescriptions for doing push-forward $\pi_*^{analytic}, \pi_*^{top} _ K^0(X) \to K^{-n}(S)$ now taking values in the $K$-theory over the base, with a shift in degree given by the dimension that we “integrate over”.

Again

Family index theorem. (Atiyah-Singer): $\pi_*^{analytic} = \pi_*^{top} \,.$

Later other geometric invariants of Dirac operators were found, which encoded more non-discrete information:

Atixah-Patodi-Singer looked at odd dimensional $X$ (still compact Riemannian spin manifolds) and vector bundles $\array{ (E,\nabla) \\ \downarrow \\ X }$ over these. Again we form the Dirac operator $D_X(E)$. Being self-adjoint its spectrum is real. It is also discrete, since $X$ is compact. Using this one defines a quantity $\eta(D_X(E))$ which is, heuristically, the number of positive minus the number of negative Eigenvalues of the Dirac operator. Since there are infinitely many of both this doesn’t make sense literally, bbut gives a well-defined result after some regularization procedure.

Then define the quantity $\xi(D_X(E)) = \frac{\eta(D_X(E)) + dim ker D_X(\nabla)}{2} \in \mathbb{R}/\mathbb{Z} \,.$ This is not an “integral” invariant as before, but now takes values on the circle.

[Here the typescript of my notes terminates for the moment, though this is just the first fourth of the talk. To be continued when time permits.]

Posted at May 29, 2008 10:36 PM UTC

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### Re: HIM Trimester on Geometry and Physics, Week 4

In re: superconnections
Has anyone noticed Jonathan Block’s
2. arXiv:math/0604296 [ps, pdf, other]
Title: Duality and equivalence of module categories in noncommutative geometry II: Mukai duality for holomorphic noncommutative tori
Authors: Jonathan Block
Subjects: Quantum Algebra (math.QA); Differential Geometry (math.DG)
4. arXiv:math/0509284 [ps, pdf, other]
Title: Duality and equivalence of module categories in noncommutative geometry I
Authors: Jonathan Block
Subjects: Quantum Algebra (math.QA); Operator Algebras (math.OA)

In particualr in part 1, he considers a Z-connection rather than a Z_2-connection
for which the various terms have a clear infinty or sh meaning

see Defn 2.6

Posted by: jim stasheff on June 28, 2008 2:49 AM | Permalink | Reply to this

### Re: HIM Trimester on Geometry and Physics, Week 4

I have. I’d very much like to show that this is exactly how the derived category arises in the topological string, but I’m still having a bit of trouble deciding exactly what is the correct definition of the parallel transport of such a superconnection.

Posted by: Aaron Bergman on June 28, 2008 2:56 AM | Permalink | Reply to this

### Re: HIM Trimester on Geometry and Physics, Week 4

what immediately comes to mind is the notion of (sh = strong homotopy = infty) parallel transport for a fibration i.e. with the homotopy lifting property, not as rigid as a bundle

Posted by: jim stasheff on June 29, 2008 12:06 PM | Permalink | Reply to this

### Re: HIM Trimester on Geometry and Physics, Week 4

what immediately comes to mind is the notion of (sh = strong homotopy = infty) parallel transport for a fibration i.e. with the homotopy lifting property, not as rigid as a bundle

I’d think (but not sure) that this would correspond to performing the first integration step I mentioned in my previous comment#, i.e. passing to the map of spaces

$S((E^\bullet, A^\bullet), d_{E,A}) \to S(A^\bullet) \,.$

I’ll try to prepare a more detailed discussion now.

Posted by: Urs Schreiber on June 29, 2008 1:26 PM | Permalink | Reply to this

### Re: HIM Trimester on Geometry and Physics, Week 4

If anyone is still curious, the paper (modulo a few typos – sigh) is here.

Posted by: Aaron Bergman on August 4, 2008 9:49 PM | Permalink | Reply to this

### Re: HIM Trimester on Geometry and Physics, Week 4

Aaron writes:

If anyone is still curious,

I am.

the paper (modulo a few typos – sigh) is here.

Thanks, great. I had a quick look at it. Very much enjoyed the introduction, some good comments in there which should be appreciated more widely. Will have a closer look tonight on the train.

As I think I said, concerning your remark about holonomy of these $L_\infty$ connections (page 3 [4]), I believe I know the general mechanism which will sensibly define and then compute the holonomy of that superconnection you are looking at.

Possibly you’ll have your own ideas about that and don’t want me to interfere, but just in case, here is what I think:

as I remarked in Block on $L_\infty$-module categories his definition of connection can equivalently be regarded as an $L_\infty$-extension

$V \to V \rtimes T X \to T X$

of the (holomorphic or whatever) tangent Lie algebroid $T X$ of the base by the graded vector space.

There is a general way, “$\infty$-Lie theory”, to integrate morphisms of $L_\infty$-algebroids to morphisms of $\infty$-groupoids. Applying that yields in the above case some $\infty$-groupoid sitting over the fundamental (holomorphic possibly, $\infty$-)groupoid of $X$. The parallel transport you are after are the lifts through this projection of ($\infty$-)groupoids.

Posted by: Urs Schreiber on August 6, 2008 9:58 AM | Permalink | Reply to this

### Re: HIM Trimester on Geometry and Physics, Week 4

I believe that the expression in the paper is the “right” answer for the holonomy, but it would be interesting to place it in a more mathematical context.

Posted by: Aaron Bergman on August 6, 2008 4:02 PM | Permalink | Reply to this

### Re: HIM Trimester on Geometry and Physics, Week 4

Jonathan Block considers a (non-negatively graded) DGA $(A^\bullet,d)$ with degree 0-part $A := A^0$ and a cochain complex $E^\bullet$ with a right $A$-module structure. Then he defines in def. 2.6 a connection (on $E^\bullet$ with respect to $(A^\bullet,d)$, really) to be a linear degree +1 map

$\nabla : E^\bullet \otimes_A A^\bullet \to E^\bullet \otimes_A A^\bullet \,,$

which satisfies a Leibnitz condition in that $\nabla (e\omega) = (\nabla e)\omega \pm e d\omega \,.$

Seems to me that this fits together with the notion of modules for DGCAs which I considered in On $\infty$-Lie, where I said that a rep of a DGCA $A^\bullet$ on a complex $E^\bullet$ of $A^0$-modules is an extension $(E^\bullet,A^\bullet)$ of DGCAs

$\Lambda E^\bullet \leftarrow ((E^\bullet,A^\bullet),d_{E,A}) \leftarrow A^\bullet \,.$

For take $(E^\bullet,A^\bullet)$ to come from Block’s $E^\bullet \otimes_A A^\bullet$ and notice that $\nabla$ yields an extension $d_{E,A}$ of $d_A$.

Aaron writes:

I’m still having a bit of trouble deciding exactly what is the correct definition of the parallel transport of such a superconnection.

With the above reformulation the parallel transport comes from lifts of ($n$-)paths integrating the lifts of DGCAs: following the general Lie integration idea we can apply the contravariant functor $S : DGCAs \to SmoothSpaces$ to get $S((E^\bullet,A^\bullet),d_{E,A}) \to S(A^\bullet)$ and then take the fundamental $infty$-path groupoid $\Pi$ on both sides

$\Pi(S((E^\bullet,A^\bullet),d_{E,A})) \to \Pi(S(A^\bullet))$

$n$-paths on the left which cover a given $n$-path on the right yield the parallel transport of the connection $\nabla$.

Posted by: Urs Schreiber on June 29, 2008 10:32 AM | Permalink | Reply to this

### Re: HIM Trimester on Geometry and Physics, Week 4

I can add more clarifying comments on the picture I was sketching:

start with the integrated picture of an ordinary 1-connection. This is a 1-functor

$P_1(X) \to T$

where $T$ is some suitable target category. For direct comparison with the picture that corresponds to Jonathan Block’s discussion take $T = Atiyah(P) = (P \times_G P \stackrel{\to}{\to} X)$ to be the Atiyah groupoid of a given $G$-bundle $P$ (could be a vector bundle).

Then look at the forgetful functor $T \to Set$ and the universal Set-bundle $\array{ Set_* \\ \downarrow \\ Set }$ and form the pullback

$\array{ C &\to& Set_* \\ \downarrow && \downarrow \\ P_1(X)& \stackrel{tra}{\to}& Set } \,.$

The part $C \to P_1(X)$ is the integrated version of the extension I was talking about. A parallel section of $P$ is a section of that extension. Parallel transport along a path $\gamma : [0,1] \to X$ is such a lift after restricting to $\gamma$, i.e. a lift through the left part of

$\array{ \gamma^* C &\to& C &\to& Set_* \\ \downarrow &&\downarrow && \downarrow \\ P_1([0,1]) &\stackrel{\gamma}{\to}& P_1(X)& \stackrel{tra}{\to}& Set } \,.$

Posted by: Urs Schreiber on June 29, 2008 12:53 PM | Permalink | Reply to this

### Re: HIM Trimester on Geometry and Physics, Week 4

I started typing more details

here

but have to quit now. In any case, using this and then the $\infty$-Lie integration procedure we can define parallel $n$-transport for Block’s connections.

Posted by: Urs Schreiber on June 29, 2008 2:36 PM | Permalink | Reply to this

### Re: HIM Trimester on Geometry and Physics, Week 4

Urs wrote:
a rep of a DGCA A on a complex E of A 0-modules is an extension (E ,A, d) of DGCAs

Lambda E from (E,A,d)from A.

Apparently I didn’t pay enough attention to infty-Lie. I would have thought a rep would be a morphism of A to Aut E or if you prefer a map A otimes E to E such that…

Then an extension would be just
a twisted tensor product of E and A.

Why invoke Lambda? Just to stay with qDGCAs? But then why not have E itself be a qDGCA. Of course then you would want
a morphism of A to Der E.

Posted by: jim stasheff on June 29, 2008 3:15 PM | Permalink | Reply to this

### Re: HIM Trimester on Geometry and Physics, Week 4

I would have thought a rep would be a morphism of $A$ to $Aut E$ or if you prefer a map $A \otimes E$ to $E$ such that…

Yes. But it turns out that making this precise directly in the $\infty$-context is a bit hard or at least not really elegant. Compare the definition of $L_\infty$-actions by Lada and Markl. Certainly the right idea, but lacking the nice conciseness of an equation like $d^2 = 0$.

So then the idea was: for group actions we have this nice theory how the action is equivalent to the existence of the action groupoid, which is an extension of the group, regarded as a groupoid, by the space being acted on. This point of view happens to have a very nice formulation in the world of DGCAs.

And I am claiming that it is essentially just what Jonathan Block uses, too. Only that he doesn’t amplify the concept of the extension. (But I feel that he maybe should. For instance I feel that he should require that the degree 0 part of the connection, $A^0$, coincides with the differential on $E^\bullet$.)

Why invoke Lambda? Just to stay with qDGCAs?

Yes, just for that purpose. I think the concepts become nicer this way. You can read Block’s definition as the same extension idea without using that step.

But then why not have $E$ itself be a qDGCA.

Sure, one could consider that more general case, too. That kind of extension, however, one would maybe not address as an “action” but really just as an extension.

This coincidence of concepts can be seen as being at the heart of the interpretation of “twisted $n$-bundles” as sections of $(n-1)$-bundles, described for instance here:

A section of an ordinary bundle is the same as a lift of the structure group through the action groupoid extension to a structure groupoid.

So it all nicely fits together.

Posted by: Urs Schreiber on June 29, 2008 4:23 PM | Permalink | Reply to this

### Re: HIM Trimester on Geometry and Physics, Week 4

But I feel that he maybe should. For instance I feel that he should require that the degree 0 part of the connection, $A_0$ , coincides with the differential on $E^\bullet$

$E^\bullet$ is a graded vector bundle, not a complex. The flat superconnection endows it with the structure of a complex. For example, if you restrict to superconnection to just degree zero and one terms, you recover complexes of holomorphic vector bundles (which, in the analytic world, are sufficient to describe the derived category on projective varieties.)

Posted by: Aaron Bergman on June 29, 2008 4:33 PM | Permalink | Reply to this

### Re: HIM Trimester on Geometry and Physics, Week 4

$E^\bullet$ is a graded vector bundle, not a complex. The flat superconnection endows it with the structure of a complex.

Ah, thanks, all the better. That agrees with how I thought it should be.

Posted by: Urs Schreiber on June 29, 2008 5:26 PM | Permalink | Reply to this
Read the post Block on L-oo Module Categories
Weblog: The n-Category Café
Excerpt: On Jonathan Block's concept of modules over differential graded algebras.
Tracked: June 30, 2008 11:37 PM
Read the post Bergman on Infinity-Vector Bundles Coupled to Topological Strings
Weblog: The n-Category Café
Excerpt: topological string, B-model, derived category, B-brane, infinity-vector space, infinity connection
Tracked: August 17, 2008 1:14 PM

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