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\author{Gabriel C. Drummond-Cole}
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\title{Harvard FRG}
\begin{document}\maketitle
\section{D. Nadler}
[This afternoon we have scheduled discussion sections where I wanted to talk about the goals of this FRG.  Today there's an interesting talk about the Kervaire invariant at MIT.  There's a talk at 3:00, and some of the people here are going.  There's going to be a smackdown at 2:15 because Dennis forbids it.  I suggest we start our discussion at 1:30.  Now Jacob will introduce David.]

Thanks for coming back.  I hope you found something in the first two talks to entertain yourselves.  Now you know $D$-modules and they can be a black box.  I'll talk about quantum field theory.

Let me recall, to $X$ a smooth scheme or Artin stack we were able to associate the derived category of $D$-modules, so $D(D_X-mod)$.  I want to say, you don't need to think about this in any particular way.  You had three ways.  I can happily say that these are $D_X$-modules if you understand that these are the derived version.  One way to think about them was as a linear PDE.  Smooth functions satisfying an equation give a $D_X$ module.  The second perspective was geometric: a quantization of $\Oo_{T^*X}$-modules.  The last was topological, I gave a couple of ways to think about this, the regular holonomy things \wh constructible sheaves.  You could have just thought about this topological category, which is the same thing as Lagrangian branes, objects in the Fukaya category of $T^*X$, and then there is a correspondence to $S^1$ equivariant sheaves on the loop space.  Maybe the easiest access point is to focus on constructible sheaves and proceed from there.  The solutions look regular at $\infty$.

Fix an algebraic variety.  A constructible sheaf is something so that along each algebraic piece it's locally constant.  It looks like a vector bundle with a flat connection. You can think that the Grothiendieck group is the vector space of constructible functions.

The output for today, I want to be able to write down my favorite scheme or stack and then pass to the category of $D$-modules on it.  Today, I'd like to list the categories of $D$-modules that are the lifesblood of representation theory and show how they're pieces of a topological field theory.

Let me start with the first theorem of representation theory.  Our first category, here's a list of categories, is
\begin{enumerate}
\item $D$-modules on the flag variety $G/B$.  If $G$ is $SL_2$ then $G/B$ is $\pP^1$.
Now Beilinson-Bernstein tells you that there is an equivalence between the categories, well, I need more terminology, $U\mathfrak{g}$ is the enveloping algebra of the Lie algebra.  Fix $G$ as a Lie group.  The enveloping algebra has a center.  It has an Abelian base.  For example, the center will have to act by a character.  For the moment, let's fix that our representations of the center act trivially.  So I'll write $U_0$ for $U\mathfrak{g}/Z\mathfrak{g}$, and we'll look at $U_0$-modules.  This is not finite dimensional representations, and it's not trivial at all, it's very interesting but sort of orthogonal to that part of the theory.  So Beilinson-Bernstein tells you that $U_0$-modules are an equivalent category to $D$-modules over $G/B$.  The functors are $Hom_D(D,\ )$ from $D$-modules to $U_0$-modules.  In the other direction it's localization, $D\otimes_{U_0}$.  You write down the most naive things and the miracle is that this is an equivalence.

If you're a representation theorist, well, you can now be a geometer.  We've spread out representations with respect to all of the Borels.  You can ask for the fiber of this representation funtor, and the fiber will be \wh.
\item [Variations]
\begin{enumerate}
\item[1]There are twisted versions, if you like finite dimensional versions you're unhappy now but where I've divided out by the trivial character, you can divide out by a different character, and you get a twisted version.
\item[2]The equivariant version exists and is very nice.  Suppose you fix a subgroup $K$ of $G$.  Consider $(U_0,K)$-modules.  You insist that $K$ acts as an algebraic subgroup should.  The Lie algebra of $K$ acts in the way you expect, but it's integrable to $K$.  So for example $K=unip(B)$, another is, $K$ is fixed points of an involution.  If you study modules like this, that's equivalent to studying representations of the real group. 

The Beilinson-Bernstein says this is equivalent to $D$-modules on the stack $K\backslash G/B$.

If $K$ acts with finitely many orbits, then all $D$-modules on this stack are regular holonomy.
\begin{example}
If $G=SL_2$ and $K=N$, then we have these orbits, a point and the rest.  We classified $D$-modules whose singular support was contained in the zero section and the tangent bundle at this point.  These are $K$-equivariant.  So you already know that $(U_0\mathfrak{sl}_2,N)$-modules has five indecomposables.

I'll tell you the five constructible sheaves:  they're the constant sheaf, the constant sheaf on the point, cochains on the complement that are allowed to touch the point, and cochains that are relative to the point.  The fifth are cochains on the complement which are relative to half of the point.  Every other thing is a sum of these.  But you know the representation theory now.
\end{example}
Let me make one final comment.  When $K=N$ (or $B$), then the $N$-orbits are Schubert cells, and the studiy of $D(N\backslash G/B)$-modules is the beginning of Kazhdan-Lusztig theory.  There are two natural bases.  There's the basis you really want to understand, the irreducibles, but the easiest ones to think about are induced ones.  So there are the two bases, and you want to know the transition matrix.  So you want to know how to turn the irreducible representations into reduced ones (or vice versa?)
\end{enumerate}
\item The second category are character sheaves.  Let me try to give you a sense of what they are.  Study the adjoint quotient $G/G$. Now we want to study $D$-modules on $G/G$.  So character sheaves will be some subcategory.  Let's call these character sheaves.  The first and most important condition is that we will proscribe where the characters can be supported in the cotangent bundle.  Recall that $T^*G$ is $G\times \mathfrak{g}^*$.  We can use the Killing form to identify this with $G\times \mathfrak{g}$.  I have my favorite conical object in $\mathfrak{g}$, I have the nilpotent cone of eigenvalue zeros:\newcommand{\Nn}{\mathcal{N}}$\Nn\subset\mathfrak{g}$.  So these character sheaves are $G$-equivariant and have nilpotent singular support.  It's an exercise to check that all of them are holonomic.  We're dealing not with algebra but topology at this point.  This is the fiber of the eigenvalue map so it's Lagrangian.

Generically, nilpotent singular support, this will be a local system, and generically I want this to have trivial monodromy.  There's a huge open stratum of regular, semisimple elements.  I want this to be roughly trivial on this open piece with distinct eigenvalues.

Let me give you an example that it's nice to keep in mind.  My favorite example is the Springer $D$-module.  We have $G$ and the favorite space over it, $\tilde{G}=\{g,B:g\in B\}$ which maps to $G/B$.  For $SL_2$, above a semisimple point you have two points, above the nilpotents you have one point, and above the cone point you have $G/B=\pP^1$.

So you can take $\mu_*\Oo_{\tilde{G}}$ as an element of this character sheaf $Ch_G$.  I think I'm almost, are there questions?  We so far have two categories of interest.  There's a category that interpolates in a way explained by field theory, that's the Hecke category.  $H_G$ is $D$-modules on $B\backslash G/B$.  This is a monoidal category. You can rewrite $B\backslash G/B$ as $BB\times_{BG} BB$ where $B$ means both classifying space and Borel.

I want to point out two structures that are important.  Any time you have $D(K\backslash G/B)$, you naturally have an action of $H_G$.  You should believe $BK\times_{BG}BB$, and of course the Hecke category will act on this.

The second thing will be that you have a natural correspondence from $G/G$ by the adjoint to the $G/B$ adjoint action, which maps to $B\backslash G/B$.  Lusztig correspondence constructs character sheaves.

Let me state a theorem joint with David Ben-Zvi.  Now I presume everyone knows what a two dimensional topological field theory is.  There exists a categorified extended two dimensional TFT, so this means that a point has a $2$-category, so the shadow of a three dimensional TFT.  To a point it gives the category of $H_G$-modules.  So $H_G$ is a dualizable Calabi-Yau algebra in stable categories.  We'd like to study smooth functions, so we categorify them.  To $S^1$ it assigns $Ch_G$.  I'll quit, but hopefully I've conveyed that these constructions are organized by a format you like very much, related to Jacob's work, et cetera.

Suppose we are classical representation theorists, then we want to study smooth functions and integral operators.  There is a relation with one dimensional field theory, but now we want to, well [Dennis question]

What is representation theory?  It's the study of classifying spaces.  Big spaces are too hard, so we'll study classifying spaces.  We study maps into them, that's gauge spaces.  All of the things that arose in representation theory, pretend we want to topological field theory from the start, you unwind it and find that we are doing representation theory.  It's hard, it's a miracle that representation theory exists, they didn't know they were doing this.

There aren't very many functions on a classifying space.  So I need to pass to a richer linearization. One should continue up the ladder.  Let me quit.  I'm around.
\end{enumerate}
\section{Joe Harris}
[Just a quick announcement, we'll start our informal discussion at 1:30]

I think the informal section begins now.  I wasn't sure what would be appropriate so please ask questions.  I've, there's several different directions this talk could go.  I'd like to be guided by your interests.  The other thing I should say, this is entirely work of David Smyth.  

I want to start with a basic problem.  We're concerned with the moduli space of curves, so you look at $\Mm_g$, smooth $C$, compact of genus $g$ up to isomorphism, or $\Mm_{g,n}$, which is $(C,p_1,\ldots, p_n)$ distinct points in $C$ up to isomorphism.  The one thing I want to say, these don't look like spaces, they look like sets.  I'm not going to give these sets the structure of a variety, I want to indicate how to characterize that structure.  We can give $\Mm_g$ the structure of an algebraic variety such that we have a natural correspondence, so that if we have a family of curves $\Cc\to B$ whose fibers are compact Riemann surfaces, then we naturally get a map $B\to M_g$ which sends $b\mapsto [C_b]$.  This should be a regular map of varieties, that what we mean when we give this a variety structure, that we can put this kind of structure in.  It's not clear that this space exists.

For any family of curves, I have a map of sets, and I want this set to have a variety structure so that this is a regular map.  This has been studied for more than a century and a half.  The problem was not existence, but noncompactness.  There's no points corresponding to the singular curves.  If we had a family of curves specializing to one with a singularity, we'd get a map from the punctured disk, but there would be no way to fill the puncture because of the singular fiber.  Most of the algebro-geometric tools we have work on compact (projective) varieties.  There are many ways of doing this.  There is something else we want to do at the same time, we want the larger compact space to again be a moduli space, to parameterize isomorphism classes of a larger class of objects.  So we want a ``modular compactification'' of $\Mm_g$.  This means that this space $\bar{\Mm}_g$ should parameterize isomorphism classes of algebraic curves including some singular ones.  We want to enlarge the class of objects to compactify.  Why not take all singular curves?  If we enlarge our class of objects too much, it becomes nonseparated.  We could have two families over the disk that have two different fibers over the origin.  We can't include both of these isomorphism classes because then we'd have a nonseparated space.  We want to find a class of objects so that there is exactly one way to extend a family over a punctured disk.  There's always this tension.  You want to add points to compactify but not to separate.  

You can test completeness in algebraic geometry by looking at punctured disks.  Here's an example that will be somewhat relevant in a few minutes.  In 1969, Deligne and Mumford wrote this important paper, solving the problem.  They proved the existence of the ordinary variety in that paper, too.  In the last few words, with the completion of the minimal model program, people started to ask, are there other solutions?

[Some commentary]

If I look at $y^2=x^3-1$ or change $1$ to $t$, that just seems like a change of coordinates, but this gives a cuspidal curve over $t=0$.  If I throw in this cusp surface, I'd have a map from the punctured disk and then jumped over the origin.  I could take any genus one curve, $y^2=x^3+ax+b$, and look at the family $y^2=x^3+at^4x+bt^6$, all of these points correspond to the same point away from the origin.  If I included a point corresponding to the cusped cubic, it would a point close to every point.  We want to find a large class of objects, just large enough so that there exists a limit, but so that it's unique.

So they chose stable curves, meaning that singularities are at least nodal, and that the automorphism groups are all finite.  When two points come together, then you blow up and get an exceptional divisor.  I bubble off the two points of the Riemann surface.

If you have a marked point tending toward the singularity, you can think of this as a family of smooth curves with two sections identified, and you identify these sections fiberwise.  Now we have a third section that crosses one of the sections.  We blow up, and you get a separated node with a $\pP^1$ in between.  This is, you should play around with this.  For any one parameter family, there is one and only one possible fill for the disk.

[Can you draw the first one?]

You will bubble off a sphere as you bring two points together, and you have the original Riemann surface, union a sphere, with the two marked points on a sphere.

There are different ways of constructing this space.  Deligne and Mumford came up with this idea, it satisfies the evaluative criteria, and they looked at the deformation spaces of these, and glue these, their quotients together, to form a space.

There's a completely different idea, including an embedding in projective space, which is parameterized by the Hilbert scheme, and you want to quotient by the group that takes embeddings to each other.  Geometric invariant theory tells you which orbits you can allow and must disallow for the quotient to be separated.  The word stable appears in both cases but that's just a coincidence.  When Deligne and Mumford, well, when Mumford and Knudsen did $\bar{\Mm}_g$, and later someone else did this for $\bar{\Mm}_{g,n}$ he got different structure, the same points but it differed by a flip.  If you let two points go toward a singularity, it can be along the same or opposite branches.  If they're on opposite branches, we blow up twice, identify, and get a picture where we have two $\pP^1$ interpolating with marked points on each one.  Now suppose they approach alonge the same branch, and we get only one $\pP^1$ with two marked points and two nodal points.  Geometric invariant theory gave exactly the opposite, the points had the same isomorphism classes, but the limit was independent on the same side but depended on the rate of approach on the opposite sides.

[Curt: Mumford was aware of a discussion of \wh.  Uniformize your surfaces, you get discrete subgroups of a fixed Lie group, and they might go to a limiting Lie group.  This depends on your base point.]

Let me suggest, I think I said a moment ago, that this theorem of Deligne and Mumford, this was probably the most important theorem of the twentieth century in this field.  Now that we have a projective variety we can start studying.  Me now understand this variety a lot better because we can use it as a variety.

There's another way in which this theorem is tremendously influential.  Just by throwing in nodal curves, any time you see curves that are degenerating, you do know on the basis of this theorem, there is a family of curves with nodal singularities at worst that maps to your space of curves.

The basic idea of stable reduction.  Let's say the general fiber is smooth, for specificity, that approaches a curve which is not smooth, then we can effect a base-changing birational map so that the fibers are isomorphic to the ones we started with away from the origin, and whose fiber over the origin is nodal.

For example, in the case of a cusp, you can blow up the cusp, and get this nonreduced picture, the exceptional divisor has multiplicity two.  You blow up again and the second exceptional divisor shows up with degree three. Blow up again and you get nodes with multiplicity six, three, and two, and if you replace with the square root, that's like taking a double cover branched along the parts with odd multiplicity.  So you'll get a family with multiplicities $3$, $3$ on a double (branched) part, and $1$.  Do the same thing with three, and that has the effect of replacing the $\pP^1$ with its triple cover triply ramified at the places with multiplicity one.  We arive eventually at a curve looking like this picture: [picture].

Stable reduction is not a magic wand, you can carry it out in practice.  If you don't know the singularities in the limit, you can realize them with only nodes.

[Curt:  The fiber, going around the punctured disk, is a surface with genus.  Here that's, well, look at a torus.  There are lots of elements of finite order, like of order six fixing a point.  The same is true of a torus with boundary.  Take the one of order six on one side and the identity on the other half.  If you take the sixth power of this this thing, then the mapping class becomes the identity, and identity, with a base change.  That's an example of the strangulation.  The exceptional fiber, if you start with your general fiber, and take a pair of cycles and contract them both to a point.  You could take a cycle and contract it.  If you do it with two intersecting vanishing cycles you get a cusp.]

Curt analyzed the monodromy away from the special fiber.  You did it with more information on the special fiber as you went along.

I said any time you have a family of curves, you get a map $B\to \bar{M}_g$  What you're getting is a natural transformation of functors.  You have the functor which gives to $B$ the families of stable curves over $B$, and another that gives $Maps(B,\bar{\Mm_g})$.  We'd like this to be an isomorphism of functors.  That does not exist.  You can say more, that, you have a finite cover so that the composition is in this natural transformation, but if you have two families over $B$ and they correspond to the same map, there's a cover involved there.

I realize that time is up, but here's an alternative, I'll give the first example of this, which was constructed before David Smyth, I think it was David Schubert.  The idea is, that in some sense what gave rise to the recent burst of activity is the minimal model program.  You can use birational transformations to find the smallest thing relative to certain kind of singularity.  You get a birational embedding, which is the most canonical model of that variety.  If you carry this out for Deligne-Mumford space, what do you get?  Or go back to 1969 and say, they had a beautiful solution to the problem, but what if there were other solutions.  What if you allow cusps as well?  You can't just throw in cusps, and we just saw, if you start with a family of curves with a cusp, you can arrive at a family which is the same family with the special fiber looking like a pinching.  The next observation is that we have to disallow curves with elliptic tails, a Riemann surface of genus one attached at just one point.  These have at most nodes and cusps but no elliptic tails and finite automorphisms.  The idea is, a moduli space of these objects again exists.  It's the minimal model program, the first step is, what this program gives to you is this.

One point is that, the $J$ invariant is not preserved.  So I get a map $\bar{\Mm}_g$ to the first birational modification $\Mm_g[1]$.  The amazing thing is, you can keep going, David Smyth did this to some degree, and in certain directions, each time you can allow a singularity, the resultant space is a further birational map.  The next space is \wh, two branches meeting tangent, and you get an elliptic bridge, a genus one picture attached at two points.  When you collapse this elliptic curve, you need a choice of identification of tangent spaces.  So $\Mm_g[2]$ is not a regular map in either direction.  If you're interested in this, you should look at Smyth's papers.  He talks about modular compactifications to $\Mm_g$ almost completely and talks about pointed genus one curves fully.

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