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\author{Gabriel C. Drummond-Cole}
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\title{Moduli Space of Riemann Surfaces}

\begin{document}\maketitle
\section{Equivariant Orbifold Structures on \wh}
Thank you to the organizers.  We heard already at this conference about Witten's conjecture.  It could also be stated about, taking intersection numbers on the Deligne Mumford conjecture and then putting them into a generating function, and then the partial derivatives satisfy some relations.

I am working in more generality and I want to know whether the same is true.

I will start by explaining Hirota Quadratic Equations on the simplest possible example.  Take two functions, $a(\lambda)\in \cC((\lambda^{-1}))$ and $b(\lambda)\in \cC[[\lambda^{-1}]].$  So one has a finite order pole at $\infty$ and the other has a zero there.  Then 
$$\Gamma^{\pm}=e^{\pm a(\lambda)y}e^{\pm b(y)\frac{\partial}{\partial y}}$$
which acts on $\cC[[y]],$ which we see as Fock space.  Look now at 
$$Res_{\lambda=\infty}(\Gamma^+\otimes \Gamma^-)(\tau\otimes \tau)d\lambda$$
where $\tau(y)\in \cC[[y]].$  This is called a vertex operator.
This yields
$$
Res\ \left(e^{a(\lambda)y'}e^{b(\lambda)\frac{\partial}{\partial y'}}t(y') e^{-a(\lambda) y''}e^{-b(\lambda)\frac{\partial}{\partial y''}}t(y'')\right)d\lambda$$
and substituting $q=\frac{y'-y''}{2},x=\frac{y'+y''}{2},$ this is
$$Res\ e^{2a(\lambda)q}e^{b(\lambda)\frac{\partial}{\partial q}}\tau(x+q)\tau(x-q)
= \sum_N q^N p^{a,b}_N(\tau,\partial_x\tau,\ldots),$$ where $p$ is a quadratic polynomial in the derivatives of $\tau.$

Now Let $\tilde{U}_0$ be $\cC$ mapping to $U_0=\{[z_0,z_1]|z_0\ne 0\}$ with $z\mapsto [1,z^k]$ and then $U_0\cong \tilde{U}_0/\zZ_k.$  Uniformize on the other side with
$\tilde{U}_1=\cC\to U_1=\{[z_0,z_1]|z_1\ne 0\}$ with $w\mapsto [w^m,1]$ and $U_1\cong \tilde{U}_1/\zZ_m.$

This gives an orbifold structure.  I want to define orbifold cohomology, which is not the ordinary cohomology.  In general, if $X$ is an orbifold and $U$ an open set with a uniformization $\tilde{U}/G$ then $IU=\sqcup_{(g)-conj} \tilde{U}^g/C(g)$ where $C(g)$ is the centralizer.  This is called a local model of the inertia orbifold and comes with the involution $g\mapsto g^{-1}.$

Now all these local models can be naturally glued, giving something called the inertial orbifold $IX.$  This is $X\cup \sqcup_t X_t,$ which are called twisted sectors.  In our case there is no gluing, the case I want to consider you have $*/\zZ_k$ or $*/\zZ_m.$  I told you that the equations will be for the equivariant theory so I have to tell you what is equivariant.

Take the two-torus $T=S^1\times S^1$ and then let $\nu_0,\nu_1$ be characters of the representation dual to the standard representation of $T$ on $\cC^2.$

Then the cohomology of the classifying space $H^*(BT)\cong \cC[\nu_0,\nu_1],$ because the characters determine line bundles which give Chern classes.  Now if you have $X$ a manifold with the torus acting with a Hamiltonian action, actually you don't need that, if it just acts then you can define $H^*_T(X)=H^*(ET\times_T X),$ where $ET$ is a contractible space where the torus acts freely.

If the action is Hamiltonian and I have coefficients in a field, $X$ is symplectic, projective, then there is a localization theorem, that this is isomorphic to $H^*(X^T)\otimes H^*(BT).$  In our situation, what is the torus action?  $T$ acts on $\cC\pP^1.$  This action can be promoted to an orbifold action.  Let me denote by $H$ the inertial orbifold cohomology 
$$H^*_T(I\Cc_{k,m})=\bigoplus_{n=0}^{k-1} H^*(BT)\oplus \bigoplus_{j=0}^{m-1}H^*(BT)$$
where the isomorphism is by restricting to the fixed points.  Let $\alpha$ be in $\zZ_k\sqcup \zZ_m,$ then $\phi_\alpha$ restricts to one at the point indexed by $\alpha$ and $0$ to all others.

The last thing I'll need is the orbifold pairing.  The equivariant Poincar\'e pairing is defined by
$$(\phi_\alpha,\phi_\beta):=(\phi_\alpha,I^*\phi_\beta)$$
where $I^*$ is the involution from before on $IX$ and you pair on the right by representing with forms and integrating.  In our case it's
\begin{eqnarray}(\phi_{0/k},\phi_{0,k})&=&\frac{1}{\nu_0-\nu_1}\\
(\phi_{i/k},\phi_{(k-i)/k})&=&\frac{1}{k}\end{eqnarray} with switching $k$ and $m$ corresponding to switching $\nu_0$ and $\nu_1.$

Now let me give you equations.  So start with $\cC_\epsilon[[q^\alpha_n]]_{n\ge 0}$ with $\alpha\in \zZ_k\cup\zZ_m.$  Here $\cC_\epsilon=\cC((\epsilon^{-1}))$ is the genus expansion parameter.

These will be formal series in $\alpha$ except for $q_1^{0/k}+1, q_1^{0/m}+1.$  This is the shift because $\phi_{0/k}+\phi_{0/m}=1\in H^*\Cc_{k,m}$

The vertex operators are the following:
$$\Gamma^{\pm}=exp(\pm\sum_i \frac{\prod_{\ell=-\infty}^n (\nu+(\ell-1+\frac{i}{k})z)}{\prod_{\ell=-\infty}^0 (\nu+(\ell-1+\frac{i}{k})z)}\lambda^{-nk+i}\phi_{i/k})^\wedge$$
with another such equation switching $k$ and $m$ and also $\nu_0$ with $\nu_1$ (here $\nu=\frac{\nu_0-\nu_1}{k}$).

Let $(\phi_\alpha z^n)^\wedge=\epsilon \frac{\partial}{\partial q_n^\alpha}$
and $(\phi^\alpha (-z)^{-n-1})^\wedge=\frac{q^\alpha_n}{\epsilon}.$

Here is the conjecture:

The generating series of the Gromov Witten invariants (equivariant) of $\Cc_{k,m}$ satisfies
$$
Res_{\lambda=\infty}\frac{d\lambda}{\lambda}(\lambda^{n-\ell}\Gamma^-\otimes \Gamma^+ - (\frac{Q}{\lambda})^{n-\ell}\bar{\Gamma}^+\otimes\bar{\Gamma}^-)\Dd(q^{0/k}_0+(n+1)\epsilon,q_0^{0/m}+n\epsilon,\ldots)\otimes \Dd(q_0^{0/k}+\ell\epsilon,q_0^{0/m}+(\ell+1)\epsilon,\ldots)=0$$
for all $n,\ell,$ where $Q$ is a constant and I'll tell you what $\Dd$ is tomorrow,  To prove this you need to generalize the localization argument to orbifolds.  If $k$ and $m$ are not relatively prime we don't know how to do this, although the formula we have makes sense for all $k,m.$

Okay, now there is an analogue of Getzler's change.  Each sequence $q_0^{i/k},q_1^{i/k},\ldots\mapsto y_i,y_{i+k},\ldots$ and $q_0^{j/m},q_1^{j/m},\ldots\mapsto \bar{y}_j,\bar{y}_{j+m},\ldots$

These linear changes are such that
$$\sum (-w)^{-n-1}\frac{\partial}{\partial q_n^{i/k}}=\sum\frac{1}{\prod_\ell (\nu-(\ell+\frac{i}{k})w)}g_{i/k}\frac{\partial}{\partial y_{nk+i}}$$
where $g_{i/k}=\left\{\begin{array}{lll}1/k&,&1\le i\le k-1\\\frac{1}{\nu_0-\nu_1}&,&k=0\end{array}\right.$
If we write $\tau_n=Q^{n^2/2}\Dd(q_0^{0/k}+n\epsilon,q_0^{0/m}+n\epsilon,\ldots)$
then we get the HQE of the 2 Toda hierarchy.
\section{Dubrovin}
My title, from moduli spaces to integrable heirarchies, has twofold meaning.  First of all the novelty of the theory of integrable systems for the last ten or fifteen years was that looking at the topology of moduli spaces for WDVV, TRR, Virosora relations, so on, could tell us about the structure of integrable hierarchies.  Some of the things they could tell us were very new.  This is the first part of the title we have in mind.  The second part refers to some particular moduli spaces that I would call Hurwitz spaces, moduli spaces of branched coverings over the Riemann sphere.  There are nice Frobenius structures on these spaces and one can construct particular integrable hierarchies, and I will try to explain something about these.  The main point is to clarify the applications.  I will try to speculate about those.  The story, there are deep connections to GW invariants and so on, but my point was to look for other applications of this, not limit ourselves to invariants of projective varieties.  The talks will be divided in three parts.  In part one I will discuss classification of integrable hierarchies.  Of course, this is a very old problem that has been addressed for at least thirty years, maybe more, as a kind of perturbative approach that has been stimulated by this topological theory.  In part two I will remind something about Whitham equations and their relation to Hurwitz spaces.  In part three I will work toward a certain integrable hierarchy, toward higher order Whitham theory.

In the main point, what I mean, about the perturbative approach, it is very simple.  This was known for many years but people started to study more carefully this method with Gromov Witten theory.  In integrable hierarchies, some small parameter is typically involved.  In the example of KdV, you have $u_t=uu_x+\frac{1}{2}u_{xxx},$ where subscript stands for partial derivative.  With $t\to \epsilon t$ and $x\to \epsilon x,$ I get $u_t=uu_x+\frac{\epsilon^2}{12}u_{xxx}.$  In this hierarchy you have infinite times.  You have $t_i$ and $\frac{\partial}{\partial t_i}$ and $\frac{\partial}{\partial t_j}$ commute. 

The second equation is
$$u_{t_2}=\frac{1}{2}u^2u_x+\frac{\epsilon^2}{12}(2u_xu_{xx}+uu_{xxx}+\frac{\epsilon^4}{240}u_{xxxxx}$$
These have to do with how slowly the system is changing.  This is the basic idea of the perturbative approach.

To continue with this I have to explain some simple ideas of deformation theory of Hamiltonian systems.  One of the novelties, a new output of this approach to the classification is, in the textbooks you get a list of examples.  They look like isolated points, exotic animals in the zoo of PDEs.  One of the main outputs is that every integrable hierarchy is not an isolated point, it allows to be deformed, every integrable hierarchy, according to this approach, has a deformation space.  The main part of this will be described as a deformation space, at first infinitessimally and then full deformations.  This will be the main subject of part one.

So the deformations of a Hamiltonian system.  If you have a Poisson manifold $(M,\{\,,\,\})$ you can, given $H\in C^\infty(M)$ you can define the Hamiltonian vector field $\dot{x}=\{x,H\}.$  To deform this you could deform $H$ or $\{\,,\,\}.$  This is a simple thing that I want to remind you of.

I have to recall the notion of Poisson cohomology even before.  I need to describe a tensor algebra on any smooth manifold.  You have the Schouten-Nijenkuis bracket on multi-vectors.  Let $\Lambda^k=H^0(M,\wedge^k TM).$  Then these brackets are $\Lambda^k\times \Lambda^\ell\to \Lambda^{k+\ell-1}.$  It suffices to remember simple rules.  For $k=\ell=1$ this is the Lie bracket of vector fields.  Then you can extend on all tensors using the Liebnitz rule:
$$[c,a\wedge b]=[c,a]\wedge b +(-1)^{\ell k+k} a\wedge [c,b]$$ for $a\in \Lambda^k,c\in\Lambda^\ell$
and this is graded commutative and satisfies the graded Jacobi identity.  In particular, a Poisson structure on $M$ is a bivector $\pi\in\Lambda^2$ with $[\pi,\pi]=0.$  Therefore for a given Poisson bivector $\pi$ we can define a differential $\partial:\Lambda^k\to \Lambda^{k+1}$ via $\partial a=[\pi,a].$  By the graded Jacobi we obtain $\partial^2=0$ so $H^.(M,\pi)=\ker\ \partial/im\ \partial$.  Then what is nice about the Poisson cohomology is that $H^2(M,\pi),$ well, what is this?  The kernel is $\delta\pi\in \Lambda^2$ so that $[\pi+\delta \pi,\pi+\delta\pi]=0$ modulo $(\delta\pi)^2.$ So these are infinitessimal deformations of $\pi.$  The image that you quotient out, I claim, are those deformations generated by infinitessimal changes in coordinates.  If $\delta\pi=\partial X$ for $X\in\Lambda^1,$ this is $Lie_X\pi.$  This is the classical description of the deformation theory of Poisson brackets.  In particular, if $M$ is a ball and $\pi$ is a symplectic structure, then $H^2(M,\pi)=0,$ which is an infinitessimal version of the Darboux lemma.  If $\pi$ is degenerate this is not true any more, you can rotate to produce deformations.

Since I want to proceed with the deformation theory of integrable hierarchies, I should define them.   On a given Poisson manifold we have a family $\frac{dx}{dt_i}=\{x,H_i\}$ with $\{H_i,H_j\}=0$ implies that the $\frac{d}{dt_i}$ all commute.  We also have a commutative subalgebra of the algebra of Hamiltonian vector fields.  We also need some maximality.  If this was a symplectic manifold, the number of independent commuting Hamiltonians must be half the dimension.  In the theory I'm going to develop we're mainly dealing with infinite dimensional manifolds and we need to be more careful but it can be done.

I don't know how to do this, even in very simple examples.  I want to restrict myself to a more particular subclass where the commuting Hamiltonians are organized in a hierarchy.  These are organized by bihamiltonian recursion.  So we'll start with $M,\{\,,\,\}_1,\{\,,\,\}_2,$ where the brackets are compatible in a manner I'll say in a little while.  biHamiltonians are
$\dot{x}=\{x,H_2\}_1=\{x,H_1\}_2.$  I claim that then I can, starting from such a pair of Hamiltonians, I can produce under some additional assumptions a family of commuting Hamiltonians. 
\begin{defi}$H_1,\ldots$ are a Hierarchy if
$\{\cdot,H_{i+1}\}_1=\{\cdot, H_i\}_2$ for all $i.$ 
\end{defi}
A simple lemma says that if such exist, they commute, $\{H_i,H_j\}_{1,2}=0.$

Here's a proof.  Assume $i<j,$ then $$\{H_i,H_j\}_1=\{H_i,H_{j-1}\}_2=-\{H_{j-1},H_i\}_1=-\{H_{j-1},H_{i+1}\}_1=\ldots$$
Iterating, we eventually get either $\{H_k,H_k\}_1=0$ by symmetry or $\{H_k,H_k\}_2=0.$

This does not explain the condition that ensures existence of such a hierarchy.  The condition of compatibility says that $a_1\{\,,\,\}_1+a_2\{\,,\,\}_2$ must be a Poisson bracket for every $a_1,a_2.$  Then under some assumptions about the ranks, one can prove that such a hierarchy exists.  Under additional assumptions one can prove completeness.  So we want to classify biHamiltonian structures.  Then I have now to explain one more very simple thing.

What is the deformation theory of biHamiltonian structures?  A biHamiltonian structure is a pair $\pi_1,\pi_2$ so that $[\pi_1,\pi_1]=0=[\pi_2,\pi_2].$  The other condition is $[\pi_1,\pi_2]=0.$  So now we have two differentials $\partial_{1,2}$ both of which square to zero, and because of the compatibility, $\partial_1\partial_2+\partial_2\partial_1=0,$ they anticommute.

Assume that both $\partial_{1,2}$ have trivial cohomology in positive dimension, that they are acyclic.  This is the case in the double lemma that I was saying before.  Define the biHamiltonian cohomology as $$H^i(\pi_1,\pi_2)=\left\{\begin{array}{ll}\ker\del_1\del_2|_{\Lambda^{i-1}}/(Im \del_1+ Im \del_2)|_{\Lambda^{i-2}}&i>0\\
\ker \del_1\cap\ker \del_2 & i=0\end{array}\right.$$
Then $H^0(\pi_1,\pi_2)$ are the common Casimirs of the two Poisson brackets.  $H^1$ are exactly biHamiltonian vector fields.  Discussing deformation theory, the main point is $H^2,$ which is the quotient of the space of infinitessimal deformations of the biHamiltonian structure by those generated by infinitessimal changes in coordinates.  You can deform either one but not the other, but we have to factor out those that are generated by a single change of coordinates in both.

The proof for $H^2$ is that we need to deform $\pi_i\mapto \pi_i+\delta \pi_i.$  We can normalize to kill the $\delta \pi_1$ and deform only the second bracket $\pi_2.$  This must satisfy the compatibility condition, so $\pi_2+\delta \pi_2$ is a cycle, meaning $\delta \pi_2=\del_1 X.$  Then one must respect the Jacobi identity for $\pi_2,$ so that $\del_2\del_1 X=0.$  This is $-\del_1\del_2 X,$ so we obtain an element in the numerator.  Now we have to figure out what deformations are trivial.  These are generated by simultaneous changes in coordinates, $\pi_i\mapsto \pi_i+\del_i Y$ so $\del_1 Y=0.$  Then $Y=\del_1 f.$  Now look at the second part.  Then $\del_2 Y=\del_1 X=\del_2\del_1 f = \del_1 X.$  By anticommutativity we get $\del_1(\del_2 f \pm X) = 0.$  Then $\del_2 f +X =\del_1 g.$  So $X\in im\ \del_1+ im\ \del_2.$

How do you apply this definition, compute this cohomology in various examples?  My time is essentially over.  I have five minutes.  I am going to classify my biHamiltonian systems.  They are all PDEs.  In what sense do I consider this as a vector field?  You have a system $u_t^i=F^i(u,u_x,u_{xx},\ldots).$  So $u=(u^1(x,t),\ldots, u^n(x,t)).$  So we can consider the space of functions $u^i(x+2\pi)=u^i(x).$  We also need $u^1,\ldots, u^n)\in M^n.$  Then the vector field, the dynamical system on $\Ll(M)=\{S^1\to M\}$ you solve the Cauchy problem for the PDE and get a curve on this infinite dimensional space.  Let me disregard the existence problem.  I will be working formally so I don't need to be more specific.  The last thing that I will be able to define today is, small parameter insertion, let me do the following, assume that these are analytic on the zero section of the jet bundle of my manifold.  Then the small parameter can be introduced via rescaling $x\to \epsilon x, t\to \epsilon t.$  Then you get
$$\epsilon u_t^i = F^i(u,\epsilon u_x,\epsilon^2 u_{xx},\ldots)$$
and I want to assume that when you divide by $\epsilon$ there will be no $\frac{1}{\epsilon}$ terms, these are all slow variables and this admits hyperdynamic expansion.  We start then with terms that are linear in $\epsilon,$ we get $A^i_j(u)u^j_x+\epsilon (B^i_j(u)u^j_{xx}+C^i_{jk}(u)u^j_xu_x^k)+\Oo(\epsilon^2).$  Everything I do tomorrow will be applied to this kind of PDEs.  I will be more specific next time and will formulate the main results.  Let me stop now and start tomorrow from this point and proceed to the theories.
\end{document}
