A Seminar on Gromov-Witten Invariants

September 17, 2009

A Seminar on Gromov-Witten Invariants

Posted by Urs Schreiber

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In a fashion similar to the Seminar on a Survey of Elliptic Cohomology, Kevin Lin and two other grad students here at MPI Bonn are now running a seminar on Gromov-Witten invariants.

And as with the former I am taking notes right into the $n$ Lab with this one, too, now.

Today’s notes are here:

basic ideas of moduli stacks of curves and Gromov-Witten invariants

This is a bit rough, as you will see, as it is essentially real-time note-taking. All the post-production I had time and leisure for was spent on one entry about the terminology of fine/coarse moduli spaces/stacks.

But I am thinking that this raw material may be a good basis to base some further editorial effort on. Maybe some lurking $n$ Café guest or other feels like creating one of the grayish links spread all over the text.

Posted at September 17, 2009 6:24 PM UTC

TrackBack URL for this Entry: https://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/2060

40 Comments & 1 Trackback

Re: A Seminar on Gromov-Witten Theory

Hello Urs; I just went through and cleaned it up a bit and added a few things…

And actually the seminar is being run by me (also from Berkeley, but not a Teichner student) and two other MPI grad students.

Posted by: Kevin Lin on September 17, 2009 9:25 PM | Permalink | Reply to this

Re: A Seminar on Gromov-Witten Theory

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Hi Kevin,

you write:

I just went through and cleaned it up a bit and added a few things…

Great. Thanks!!

And actually the seminar is being run by me (also from Berkeley, but not a Teichner student) and two other MPI grad students.

Oops. Sorry for that. I have now corrected this in the entry above.

Posted by: Urs Schreiber on September 18, 2009 4:59 PM | Permalink | Reply to this

Re: A Seminar on Gromov-Witten Theory

Sounds very 90’s to me.

Posted by: name on September 17, 2009 10:23 PM | Permalink | Reply to this

Re: A Seminar on Gromov-Witten Theory

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Sounds very 90’s to me.

A more nontrivial comment might get a more non-trivial reaction here. Otherwise, what’s the point of commenting?

Posted by: Urs Schreiber on September 18, 2009 5:03 PM | Permalink | Reply to this

Re: A Seminar on Gromov-Witten Theory

In re:

abstract argument generally, fine moduli spaces do not exist if the objects to be classified have nontrivial automorphisms. This allows to build families of objects that are locally trivial but globally not and no finite moduli space will be able to represent that.

this argument makes use of the fact that if we have a moduli space, then the preshesaf we started with must actually be a sheaf (with respect to a subcanonical topology that is implicitly assumed, probably). so we know that we should be able to compute it from gluing its local assignments. But locally our locally trivial family looks, well, trivial, so it all looks the same to our sheaf. So it will just try to glue the trivial object to itself, which is not what we actually have.

Of course vector bundles do have automophisms and are patched together from local trivial ones, so what am I missing?

also in what sense do you wnat a `finite’ moduli space?

Posted by: jim stasheff on September 18, 2009 1:33 PM | Permalink | Reply to this

Re: A Seminar on Gromov-Witten Theory

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Concerning that arument that “objects with automorphisms don’t have fine moduli spaces”, Jim rightly complained, pointing to the counter-example of topological families of vector spaces – otherwise known as vector bundles –, asking

so what am I missing?

It’s in the fine print. This deserves a more detailed discussion. That “abstract argument” in the notes is something I typed in a haste while the talk kept going. But I did include that clause about subcanonical topology, which indicates the kind of fine print here.

Notice that the two examples mentioned in the talk that do “work” actually build their moduli/classifying/representing spaces in $Ho(Top)$ , while the example that does “not work”, that of elliptic curves, tries to instead build the classifying/moduli/representing object in schemes.

Similarly, if I demand that I want to find a classifying space for smooth vector bundles in the category of manifolds, I will fail, for reasons of the kind that prevent a “moduli scheme of elliptic curves” to exist. And also similarly to the elliptic case, I do succeed with finding not a classifying manifold for smooth vector bundles, but a classifying Lie groupoid $\bullet//U(n)$ .

So first of all in this game one should be careful with stating what one actually wants to represent where exactly.

Lots of things that naturally have classifying $n$ -groupoids turn out to actually have also plain topological spaces as classifying spaces. Danny Stevenson is about to put out (or maybe already has, David Roberts also has a hand in that) a proof that all principal $\infty$ -bundles with strict topological well-pointed $\infty$ -group $G$ as structure group, which are classified by the strict topological $\infty$ -groupoid $\mathbf{B}G = \bullet// G$ , are actually are also classified, in $Ho(Top)$ by the ordinary topological space $|\mathbf{B}G|$ .

As one would hope/expect.

So there is a huge supply of “fine moduli spaces” in the topological setup. But then, there they are not called moduli spaces but classfying spaces. When talking about a moduli space one typically demands a bit more than for a classifying space in $Ho(Top)$ .

Okay, i am rambling. Hopefully a coherent account makes it into the $n$ Lab eventually.

Finally, here a quick and comprehensive answer to your second question:

also in what sense do you wnat a `finite’ moduli space?

In the sense that this was a typo! :-)

Meant is fine moduli space again. Is corrected now in the notes.

Posted by: Urs Schreiber on September 18, 2009 5:42 PM | Permalink | Reply to this

Re: A Seminar on Gromov-Witten Theory

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Similarly, if I demand that I want to find a classifying space for smooth vector bundles in the category of manifolds, I will fail, for reasons of the kind that prevent a “moduli scheme of elliptic curves” to exist. And also similarly to the elliptic case, I do succeed with finding not a classifying manifold for smooth vector bundles, but a classifying Lie groupoid $\bullet//U(n)$ .

Correct me if I am wrong or misreading what your said, but it seems to me that if you want to classify actual bundles rather than their isomorphism classes, you will fail too with $BU(n)$ . I think this is what Jim is getting at: you can’t glue isomorphism classes of bundles. You can glue actual bundles. But for that you need not the classifying space $BU(n)$ but the stack $Vect_n$ of vector bundles of rank n. The classification statement then becomes Yoneda lemma: for any manifold $M$ there is an equivalence of categories between $hom (M, Vect_n)$ , the category of maps betweem the two stacks, and the category of vector bundles of rank $n$ over $M$ . $Hom

Posted by: Eugene Lerman on September 18, 2009 7:27 PM | Permalink | Reply to this

Re: A Seminar on Gromov-Witten Theory

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I think we have to be careful about what we mean when we talk of classifying spaces (and it’s rather late so I will likely fail at this). There does not exist a topological space B such that the set of continuous maps from X to B for any X is the set of isomorphism classes of vector bundles (aka families of vector spaces) on X, precisely because vector spaces have automorphisms. The point is that this functor is not a sheaf in the usual topology, and representable functors are. On the other hand in the homotopy category the same functor is representable, but not a sheaf with respect to any topology we’d recognize (eg for which you can cover S^1 with contractible spaces). This is for the reasons mentioned above - you calculate values of a sheaf as a limit of the values on opens, and if those values are all a point so is the limit.

The solution to this problem is the same in algebraic geometry and homotopy theory - we pass to groupoids, or to space-valued functors, rather than set-valued functors, i.e. pass to higher categories in one form or another. Thus Map(X,BG) is really a space, or rather a homotopy type, rather than a set. Its pi_0 gives isomorphism classes of bundles, while its fundamental groupoid gives the groupoid of vector bundles on X. Moreover it’s a sheaf in the higher categorical sense for the ordinary topology – its values are calculated as a (homotopy, or higher categorical) limit over a Cech cover say. So we’re dealing with a higher-categorical Yoneda/representability, not a categorical one. Really BG and other classifying spaces in topology, the way we usually think of them, ARE already (higher) stacks, and the category of spaces is really being treated as a higher category, with representable functors valued in spaces or (higher) groupoids. I don’t think there’s any real difference between the subjects, except the ways we are introduced to them (which make classifying spaces seem concrete and stacks – which are actually much simpler in general, being only 1-homotopy types – somehow abstract and intimidating).

Posted by: David Ben-Zvi on September 21, 2009 6:06 AM | Permalink | Reply to this

Re: A Seminar on Gromov-Witten Theory

I don’t think there’s any real difference between the subjects, except the ways we are introduced to them (which make classifying spaces seem concrete and stacks … somehow abstract and intimidating)

If that’s the case, then the homotopy theorists have a very well-oiled propaganda machine.

Posted by: Jacques Distler on September 21, 2009 6:58 AM | Permalink | PGP Sig | Reply to this

Re: A Seminar on Gromov-Witten Theory

It’s called `history’ or if you prefer, an accident of history aka toilet training.

Posted by: jim stasheff on September 21, 2009 1:35 PM | Permalink | Reply to this

Re: A Seminar on Gromov-Witten Theory

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you will fail too with $B U(n)$ .

Well, see what I said. To make it more explicit:

let $Diff$ be the category of finite dimensional manifolds with its standard Grothendieck topology.

Then there is no object in $Diff$ that classifies rank $n$ -vector bundles, nor is there in any “homotopy category” of $Diff$ .

But let $Sh(Diff, Grpd)$ be the category of groupoid-valued sheaves on $Diff$ (ordinary sheaves). This has a standard model structure such that $Ho(Sh(Diff,Grpd))$ is the homotopy category of stacks on $Diff$ .

And the point is that there is an inclusion $Grpd(Diff) \hookrightarrow Sh(Diff,Grpd)$ of groupoids internal to manifolds to sheaves of groupoids on manifolds.

And then the statement is: there is a groupoid internal to manifolds, namely $\mathbf{B} U(n) := {*}// U(n) := (U(n) \stackrel{\to}{\to} {*})$ which does represent rank $n$ vector bundles in $Ho(Sh(Diff, Grpd))$ in that for every $X \in Diff$ we have

$Rank n Vect Bund(C)/iso \simeq Hom_{Ho(Sh(Diff,Grprd))}(X, {*}//U(n)) \,.$

And finally, I am thinking that the fact that ${*}//U(n)$ sits not just in $Sh(Diff, Grpd)$ but also in the smaller $Grpd(Diff)$ is analogous to the condition on an algebraic stack to be a Deligne-Mumford stack: it effectively says that the stack is not arbitrary general but a weak quotient of representable objects. This is the next higher analog of having a classifying object in $Diff$ .

We need this distinction in order for the whole problem to make sense: for its not a problem to construct a stack of elliptic curves. It’s just that: the stack of elliptic curves.

What is not trivial is to make this close to being representable. I.e. to realize it as a Deligne-Mumford stack, for instance.

Posted by: Urs Schreiber on September 21, 2009 7:35 AM | Permalink | Reply to this

Re: A Seminar on Gromov-Witten Theory

Urs - I agree with everything you say, except for the use of Deligne-Mumford where I would use Artin: the former is about groupoids with finite automorphism groups, while the latter includes things like BU(n).

More precisely, Deligne-Mumford should be reserved for etale groupoids - the source and target maps should be smooth covering spaces (relatively zero dimensional), or more generally a simplicial object in smooth manifold with etale maps. An Artin stack in Diff is a general Lie groupoid (groupoid object in Diff) - or more specifically one representable by a groupoid with source and target being submersions. (or more generally something representable by higher groupoids or simplicial smooth manifolds with submersions for structure maps).

Posted by: David Ben-Zvi on September 21, 2009 1:12 PM | Permalink | Reply to this

Re: A Seminar on Gromov-Witten Theory

Or maybe rather than Artin stack one could say geometric stack, especially if that’s more likely to catch on..

Posted by: David Ben-Zvi on September 21, 2009 1:46 PM | Permalink | Reply to this

Re: A Seminar on Gromov-Witten Theory

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Or maybe rather than Artin stack one could say geometric stack

Yes, right.

We had a discussion of geometric stacks last time in the Journal Club. A definition is provided now on the Journal Club page. But would yoou have a canonical reference maybe? Also for the derived version?

Posted by: Urs Schreiber on September 21, 2009 2:25 PM | Permalink | Reply to this

Re: A Seminar on Gromov-Witten Theory

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Sorry that I started this party and then went offline for a bit. Coming back to this discussion now I feel like expanding on my quick initial reply to Jim. Luckily, I see that David Ben-Zvi started doing something like that in the his comment above, so first of all I’ll just amplify that David and I seem to agree here, for what it’s worth:

For instance i wrote

It’s in the fine print.

while David asserts

I think we have to be careful about what we mean

Okay, just kidding. But here is the point: I mentioned

that clause about subcanonical topology,

and this is what David points out now:

The point is that this [representing] functor is not a sheaf in the usual topology

Exactly. The ordinary Grothendieck topology on $Top$ is not subcanonical for $Ho(Top)$ . As a result, the functor

$X \mapsto Ho_{Top}(X, \mathbb{C}P^\infty)$

say, has a non-standard opinion about what it means to glue topological spaces. If it had the standard opinion, it would fail to classify line bundles: because let $P \to X$ be a non-trivial line bundle on some space $X$ and then choose an ordinary open cover $\{U_i \to X\}$ by contractible spaces of $X$ . Pulled back to each $U_i$ the vector bundle becomes trivial and hence every classifying functor sends each $U_i$ to the singleton set consisting just of the trivial bundle.

If our classifying functor were a sheaf in the ordinary topology, we could compute its value on $X$ by gluing its values on the $U_i$ . But since on each of the $U_i$ it is trivial it would have to assign the trivial object also to $X$ , by the sheaf condition.

So we learn from this that in the standard lore which says “Autmorphisms of objects obstruct the families assignment of these objects to be representable” makes an implicit assumption about in which category precisely we will try to find a representing object. It must be some category such that the “obvious” topology is subcanonical.

The other implicit assumption is the one pointed out by Anders Kock: for the usual statement to make sense we also need to assume that one can form globally non-trivial families from gluing trivial families in the first pace.

Finally, David makes a very good point, I think, about why it is morally true that large classes of bundles and $\infty$ -bundles do have “fine moduli spaces”, after all, in a suitable homotopy category:

Really BG and other classifying spaces in topology, the way we usually think of them, ARE already (higher) stacks,

I think that’s precisely the point: things like $\mathbb{C}P^\infty$ (just for definiteness, to amplify that I mean a concrete topological space) may look like a point-set space and hence like a “fine moduli space”, but in fact the homotopy category $Ho(Top)$ regards them instead as the $\infty$ -groupoids that they are equivalent to. So in a way it does remain true that there is no fine moduli space of vector bundles, after all, but “just” a moduli groupoid, which is usually (unfortunately, actually), named after its more general cousins as a moduli stack.

Posted by: Urs Schreiber on September 21, 2009 8:00 AM | Permalink | Reply to this

Re: A Seminar on Gromov-Witten Invariants

It appears that nonexistence of automorphisms is a necessary condition for existence of a fine moduli space. What additional condition on a moduli problem in algebraic geometry will make sure that a coarse moduli space is in fact a fine moduli space?

Posted by: G. S. on September 18, 2009 8:10 PM | Permalink | Reply to this

Re: A Seminar on Gromov-Witten Invariants

You say that your argument merely proves that the $j$-line is not a fine moduli space for elliptic curves, and not that a fine moduli space does not exist.

The $j$-line is a coarse moduli space for the moduli problem of elliptic curves. Therefore if a fine moduli space exists, it has to be the $j$-line itself. So if the $j$-line is not a fine moduli space, then in fact there is no fine moduli space.

What is the problem with this argument?

Posted by: G. S. on September 18, 2009 8:19 PM | Permalink | Reply to this

Re: A Seminar on Gromov-Witten Invariants

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In the seminar entry the explicit computational proof that no fine moduli space of elliptic curves exist was left unfinished, only a proof that “ $j$ -line” is not a fine moduli space given. But later on it mentioned as an exercise that the $j$ -line is in fact a coarse moduli space.

G.S. pointed out that this completes the proof, after all:

The $j$ -line is a coarse moduli space for the moduli problem of elliptic curves. Therefore if a fine moduli space exists, it has to be the $j$ -line itself. So if the $j$ -line is not a fine moduli space, then in fact there is no fine moduli space.

Thanks, that’s a good point. I have added this now to the seminar entry.

Posted by: Urs Schreiber on September 21, 2009 8:27 AM | Permalink | Reply to this

Re: A Seminar on Gromov-Witten Invariants

You were saying that the problem with non-properness is the lack of integration and poincare duality.

However taking cohomology with compact supports, is there not a statement of poincare duality for noncompact manifolds?

Posted by: G. S. on September 18, 2009 8:27 PM | Permalink | Reply to this

Re: A Seminar on Gromov-Witten Invariants

Two easy remarks, and some propaganda:

> It appears that nonexistence of automorphisms is a necessary condition
> for existence of a fine moduli space.

This is not correct (as has already been pointed out). The easiest
example is N, the set of natural numbers, which is a fine moduli space
for the moduli problem of classifying finite sets up to isomorphism.
Yet finite sets (except the empty set and the singleton) have
non-trivial automorphisms.

> You say that your argument merely proves that the $j$-line is not a
> fine moduli space for elliptic curves, and not that a fine moduli
> space does not exist.
>
> The $j$-line is a coarse moduli space for the moduli problem of
> elliptic curves. Therefore if a fine moduli space exists, it has to
> be the $j$-line itself. So if the $j$-line is not a fine moduli
> space, then in fact there is no fine moduli space.
>
> What is the problem with this argument?

This argument is correct, since coarse moduli spaces are characterised
by a universal property, that fine moduli spaces share.

Both arguments can be found (exercise 15 p.19 and §0.2.9, resp.) in

[Kock-Vainsencher, “An invitation to quantum cohomology:
Kontsevich’s formula for rational plane curves”, Birkhäuser, 2006]

that I take the opportunity to advertise as an elementary introduction
to Gromov-Witten theory. (Admittedly this book is a bit 90’ish.)

Cheers,
Joachim.

Posted by: Joachim Kock on September 19, 2009 6:47 PM | Permalink | Reply to this

Re: A Seminar on Gromov-Witten Invariants

Please do not mind if I sound like a novice. I do not understand the following completely. For me, a moduli problem is a certain functor from a category into Sets. And a fine moduli space exists if this functor is representable – and then by Yoneda lemma there will be a universal object. So which is the category? Which is the functor? And what is the universal object?

This is not correct (as has already been pointed out). The easiest example is N, the set of natural numbers, which is a fine moduli space for the moduli problem of classifying finite sets up to isomorphism. Yet finite sets (except the empty set and the singleton) have non-trivial automorphisms.

Posted by: G. S. on September 19, 2009 9:25 PM | Permalink | Reply to this

Re: A Seminar on Gromov-Witten Invariants

There is no fine moduli space for classifying finite sets, but N is a fine moduli space for classifying totally ordered finite sets. If you try to sheafify the naive finite set functor, you will find that you are actually classifying finite degree covering spaces. The classifying stack is a disjoint union of B\Sigma_n rather than a disjoint union of points.

Posted by: Scott Carnahan on September 19, 2009 10:59 PM | Permalink | Reply to this

Re: A Seminar on Gromov-Witten Invariants

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Joachim Kock wrote:

[Kock-Vainsencher, “An invitation to quantum cohomology:

Kontsevich’s formula for rational plane curves”, Birkhäuser, 2006]

that I take the opportunity to advertise as an elementary introduction

to Gromov-Witten theory.

It seem that it was Zoran Škoda who now added this, together with more references, to

$n$ Lab:Gromov-Witten invariants and $n$ Lab:basic ideas of …

Posted by: Urs Schreiber on September 21, 2009 8:14 AM | Permalink | Reply to this

Re: A Seminar on Gromov-Witten Invariants

> For me, a moduli problem is a certain functor from a category into
> Sets. And a fine moduli space exists if this functor is
> representable – and then by Yoneda lemma there will be a universal
> object. So which is the category? Which is the functor? And what
> is the universal object?

Sorry for leaving out the details:

A family of finite sets is by definition a set map E -> B with
finite fibres. An isomorphism of two such is a fibrewise iso.

The moduli functor is F : Set^{op} -> Set sending a set B to
the set of isomorphism classes of families with base B.

Given a family E -> B the classifying map B -> N sends b to
the cardinality of E_b. Conversely, given a map B -> N, you
construct a family E -> B by pullback of the universal family
(the family N’ -> N whose fibre over n is of cardinality n).
This establishes the bijection F(B) = Hom(B,N).

> There is no fine moduli space for classifying finite sets, but N is a
> fine moduli space for classifying totally ordered finite sets. If you
> try to sheafify the naive finite set functor, you will find that you
> are actually classifying finite degree covering spaces.

Sorry again for the confusion. There is no contradiction: the
topology on Set is just the epi topology, so finite degree covering
‘space’ is the same thing as family of finite sets.

The classical notion of fine moduli space in algebraic geometry is
about Set-valued functors. It does not follow from being a fine
moduli space in this sense that it is also represents the
corresponding groupoid-valued functor.

I think the moral of the finite-set example is that the classical
notion of fine moduli space in algebraic geometry does not correspond
exactly to what a category theorist think of: a category theorist may
tend to rather think of a fine moduli space as existing when the
grupoid-valued functor is homotopy equivalent to the set-valued
functor (pi_0 of the groupoid-valued functor), which is amounts
precisely to saying ‘no non-trivial automorphisms’. Algebraic
geometers are responsible for confusing category theorists like this,
since they often write in their papers that ‘automorphisms prevent
fine moduli spaces from existing’. So what I am pointing out is,
first of all that that statement is not true, and second, that you
might as well forget about formulating moduli problems in terms of
set-valued functors. Just stick to groupoid-valued (or
higher-groupoid-valued), which is most likely what readers of this
café do anyway.

Cheers,
Joachim.

Posted by: Joachim Kock on September 20, 2009 6:53 AM | Permalink | Reply to this

Re: A Seminar on Gromov-Witten Invariants

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I don’t think the issue is special to algebraic geometry or to classifying groupoid valued functors — the idea that one should classify groupoid valued functor was rather a response to the fact that fine moduli spaces don’t exist whenever your objects have automorphisms.. ASSUMING your base category is interesting enough - namely as long as you have any objects with nontrivial $\pi_1$ . The “problem” with your counterexample is that indeed families of finite sets over finite sets are boring. But families of finite sets (up to isomorphism, not as a groupoid) over topological spaces is not representable, if your category of spaces is rich enough to have nontrivial covering spaces: a nontrivial covering space is exactly a family of finite sets which is constant up to isomorphism but not trivial. And more generally once you find one such nontrivial covering (let’s say with $\pi_1=Z$ ) you can then show that any moduli problem where the objects have automorphisms doesn’t have a fine moduli - you just take the associated $X$ -bundle to the principal $Z$ -bundle from this covering space, and you find an isotrivial but not trivial family.

Posted by: David Ben-Zvi on September 20, 2009 2:33 PM | Permalink | Reply to this

Re: A Seminar on Gromov-Witten Invariants

Hello David,

thank for clearing this up – it is very helpful what you write.

> ASSUMING your base category is interesting enough - namely as long as
> you have any objects with nontrivial π 1

That’s a very good assumption of course – good to see it explicitly.

So after all, the confusion only ever occurred in the minds of people
perverse enough to interpret the notion of moduli problem in a context
where it is uninteresting!

Sorry for spreading the confusion to normal people :-)

Cheers,
Joachim.

Posted by: Joachim Kock on September 20, 2009 5:39 PM | Permalink | Reply to this

…because they have automorphisms.

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I have started to incorporate a discussion of the slogan

Objecs with automorphisms don’t have fine moduli spaces.

and its subtleties into the entry $n$ Lab:moduli space in the chaper

… because they have automorphisms.

Am not entirely happy with that yet, but this is what I could do with the given time and energy this morning and it’s a start. Maybe somebody feels like expanding on it.

Posted by: Urs Schreiber on September 21, 2009 9:28 AM | Permalink | Reply to this

Re: …because they have automorphisms.

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In your opening example, you build a nontrivial family with base $S^1 \times S^1$ . Is there some reason you didn’t just use $S^1$ ? I would have edited, but I worried that I was missing something.

Posted by: David Speyer on September 22, 2009 1:24 PM | Permalink | Reply to this

Re: …because they have automorphisms.

Congratulations, David! Yours is the 20000th comment to the Café.

Posted by: David Corfield on September 22, 2009 1:50 PM | Permalink | Reply to this

Re: …because they have automorphisms.

$MathML-enabled post (click for more details).$

David Speyer writes:

In your opening example, you build a nontrivial family with base $S^1 \times S^1$ . Is there some reason you didn’t just use $S^1$ ?

Thanks, good point. No, I didn’t have a any reason to choose this particular example apart from that I had to choose just any simple non-trivial example.

I would have edited, but I worried that I was missing something.

So, no, there is nothing to be missed here. Please go ahead and improve on it as you see the need. Thanks.

Posted by: Urs Schreiber on September 22, 2009 1:44 PM | Permalink | Reply to this

Read the post A Seminar on a Geometric Model for TMF
Weblog: The n-Category Café
Excerpt: A seminar on geometric models for tmf cohomology theory by Stephan Stolz and Peter Teichner.
Tracked: September 21, 2009 8:03 PM

Re: A Seminar on Gromov-Witten Invariants

Feeling emboldened by Schreiber’s kind encouragement, let me again venture into my first question(automorphisms and fine/coarse moduli spaces).

In the n-lab page on Deligne-Mumford, the following appears.

Deligne-Mumford stacks correspond to moduli problems in which the objects being parametrized have finite automorphism groups.

Also, the talk mentioned a few problematic examples in which the automorphism group was infinite.

Therefore, is it true that for a moduli problem in which the stack is Deligne-Mumford, and where there are no automorphisms, existence of a coarse moduli space would imply the existence of a fine moduli space?

Posted by: G. S. on September 22, 2009 10:30 PM | Permalink | Reply to this

Re: A Seminar on Gromov-Witten Invariants

I would like to add a question/request of my own — It is still not clear to me why we need a virtual fundamental class. Is this because an “actual”, “non-virtual” fundamental class does not exist? Or is it that the actual fundamental class does exist, but is not well-behaved, due somehow to the fact that the moduli of stable maps is highly singular? Perhaps someone can give an explanation of what “well-behaved” should mean in this situation, as well as an explanation of why the actual/non-virtual fundamental class (if it does exist) is not well-behaved so.

Posted by: Kevin Lin on September 24, 2009 2:00 AM | Permalink | Reply to this

Re: A Seminar on Gromov-Witten Invariants

My recollection is that the moduli space is highly singular so it is not clear what a “fundamental class” would mean. However there is a class which apparently has some nice properties similar to a fundamental class so we call it a “virtual fundamental class”.

Posted by: Chris Schommer-Pries on September 25, 2009 7:15 PM | Permalink | Reply to this

Re: A Seminar on Gromov-Witten Invariants

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The virtual fundamental class may be required even when the moduli stack is smooth.

The canonical example: The stack $\overline{M}_{1,1}(X,0)$ of degree zero maps to $X$ is isomorphic to the product $\overline{M}_{1,1} \times X$ , which is smooth if $X$ is and has dimension $1 + dim(X)$ . However, the virtual dimension of this stack is $1$ .

The problem solved by the virtual fundamental class is that we really want to integrate over the underlying derived stack of stable maps, not on the ordinary stack. You can–at least morally– think of the virtual fundamental class as the correct fundamental class of the underlying derived stack, pushed forward from the derived stack to the ordinary stack.

Posted by: A.J. on September 25, 2009 7:51 PM | Permalink | Reply to this

Re: A Seminar on Gromov-Witten Invariants

Hi AJ, thanks for the explanation! But how am I (morally) supposed to think of the derived stack versus the ordinary stack? Is the derived stack of stable maps smooth? I seem to vaguely recall that this is all related to deformations and obstructions (as well as higher obstructions?) and that the ordinary stack somehow truncates this information which is present in the derived stack…?

Posted by: Kevin Lin on September 26, 2009 4:48 PM | Permalink | Reply to this

Re: A Seminar on Gromov-Witten Invariants

I look forward very much to hear more details about the derived
viewpoint on virtual classes!

Meanwhile here are some down-to-earth geometric examples (plane cubics,
and conics on the quintic three-fold) to illustrate the need of the
virtual fundamental class.

Although the moduli space of stable maps is sometimes referred to as a
compactifiaction of the space of maps, in analogy with the
Deligne-Mumford compactification of the moduli space of curves, in
fact it typically has boundary components of higher dimension than the
space it was supposed to compactify!

Take for example Mbar_{1,0}(P^2,3). It ought to be a compactification
of the space of degree-3 maps from genus-1 curves to P^2, and indeed
one of its components has a Zariski open subset birational to the P^9
of all plane cubics. But there is also a ‘boundary component’ of
higher dimension, namely the boundary component consisting of maps
whose domain is a genus-1 curve glued to a nodal rational curve: the
nodal curve maps to a rational cubic in P^2, while the g=1 component
contracts to a point on that nodal cubic. This boundary component has
dimension 10: namely, there are 8 parameters to specify the image
nodal cubic, 1 paramenter to determine the point to which the g=1
component contracts, and finally there is 1 paramenter for the
j-invariant for the g=1 component. The topological fundamental class
lives in dimension 10 so it is rather useless to integrate against if
all your cohomology classes are codimension 9 — which is the
expected dimension. The virtual fundamental class always lives in the
expected dimension.

(The expected dimension is often the one you would expect(!) from
naive counts like the above. More formally it can be computed as dim
H^0(C,N_f), where f:C\to P^2 is a moduli point (with normal bundle
N_f) such that H^1(C,N_f)=0 (this is to say that the first order
infinitesimal deformations are unobstructed).)

The situation is analogous (possibly in fact a special case of) the
standard situation in intersection theory when a section of a vector
bundle is not regular: its zero locus is then of too high dimension
and is of little use to intersect against. The correct class to
work with is then the top Chern class of the vector bundle (cf.
[Fulton] ch.14), which could be called the virtual class of the zero
locus.

In the example above, I don’t know right now if the virtual class in
fact appears as a top Chern class of a vector bundle — I think it
should, because the excess is just a variation of the standard example
Mbar_{1,1}(X,0) mentioned by A.J., and in that example it is true that
the virtual class appears as a top Chern class: there is a so-called
obstruction bundle which in this case is the dual of the Hodge bundle
from the factor Mbar_{1,1} tensored with the tangent bundle from X.
(The Hodge bundle is the direct image bundle of the canonical bundle
of the universal curve, hence of rank g, hence just a line bundle in
this case.) The virtual fundamental class is the top Chern class of
the obstruction bundle (cap the topological fundamental class).
In this case, dim Mbar_{1,1}(X,0) = 1 + dim X, and the obstruction
bundle has rank dim X, hence the virtual class has dimension 1.

Perhaps it should be mentioned also that the moduli space of maps can
have components of too high dimension even before it is
‘compactified’, and even without involving contracting curves. A
famous example is M_{0,0}(Q,d) (no bar needed for this argument) where
Q is a quintic three-fold. Let’s say d=2, so we are talking about
conics on the quintic three-fold. Since Q has trivial canonical class
it follows that the expected dimension is always 0 (i.e. in every
degree there ought to be a finite number of rational curves on Q).
But now, M_{0,0}(Q,2) is a space of maps, not a space of curves, and
for every one of the famous 2875 lines on Q there is a 2-dimensional
family of double covers of the line, which clearly count as stable
degree-2 maps, so M_{0,0}(Q,2) contains 2875 components of dimension
2, in contrast to the virtual dimension 0.

Cheers,
Joachim.

Posted by: Joachim Kock on September 26, 2009 10:05 PM | Permalink | Reply to this

Re: A Seminar on Gromov-Witten Invariants

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here are some down-to-earth geometric examples

Thanks!

Technical exposition like this deserves to be stabilized. I created a stub entry

$n$ Lab: virtual fundamental class.

Anyone feeling like expanding on Joachim’s nice contribution might consider editing that entry and then pointing us to it.

Posted by: Urs Schreiber on September 26, 2009 10:33 PM | Permalink | Reply to this

elliptic curves

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After the general overview last time, here more introductory details on elliptic curves and their moduli spaces/stacks:

A Seminar on a Survey of Elliptic Cohomology – elliptic curves

I am dreaming that some kind soul will at some point enjoy going through this and increasing his or her $n$ Lab page number by creating separate entries for the keywords appearing and copy-and-pasting the material there.

That’s anyway what I think the purpose of these typed up notes should be: raw material for further entries.

Posted by: Urs Schreiber on September 30, 2009 7:21 PM | Permalink | Reply to this

compactified moduli stack and first GW invariants

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Today’s notes continue where we left off last time with

- a more complete description of the topological invariants of the moduli stack

- the introduction of the compactified moduli stack

- the definition of Gromov-Witten invariants

- and the simplest example.

Posted by: Urs Schreiber on October 1, 2009 5:13 PM | Permalink | Reply to this

Re: elliptic curves

I think these talks have been really excellent! Thanks to the speakers and thanks to Urs for typing up the notes. I’ll try to do some editing later when I have more time.

Posted by: Kevin Lin on October 1, 2009 9:01 PM | Permalink | Reply to this

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September 17, 2009