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

\begin{document}\maketitle
\section{Hurwitz spaces and Hurwitz number}
Thanks for inviting me and giving me the opportunity to talk about Hurwitz spaces.  Since our program is about moduli spaces, I will speak about that.  Hurwitz spaces are moduli spaces of meromorphic functions.  I have three lectures and I am planning to answer the following questions.
\begin{enumerate}
\item What are Hurwitz spaces?
\item What is their geometry?
\item What do we know about this geometry?
\end{enumerate}

I would like to add in something like ``compactifications of'' Hurwitz spaces.  What do we know or not know about the geometry of Hurwitz spaces.  I will proceed first with several examples.  Let me repeat that Hurwitz spaces are spaces of meromorphic functions on algebraic curves.  Hence, a point in a Hurwitz space is a function $f:C\to \cC\pP^1,$ and I do not distinguish between two functions if they can be included in a commutative triangle
$$\xymatrix{
C_1\ar[rr]^h\ar[dr]_{f_1}&&C_2\ar[dl]_{f_2}\\&\cC\pP^1}
$$ where $h$ is a biholomorphic mapping.

Let $\Hh_{g,1^n}$ be the space of generic meromorphic functions of degree $n$ on genus genus $g$ curves unramified over $\infty.$  This second part means that there is no ramification over $\infty$ and there are $n$ preimages, and generic means that there are always only $n-1$ sheets over a branch point.

Suppose you have $m$ points of simple ramification.  Over our projective line, there are $m$ points of ramification.  Choose a point $t$ which is not ramified and connect it to the ramification points with simple arcs.  There is a map of the surface to the Riemann sphere and the preimage of this star is a a graph which splits the surface into two dimensional disks because the complement is a disk with no ramification points.  So we are able to compute the genus of the covering surface.  The Euler characteristic of the covering surface is nothing more than
$$\chi(C)=V-E+F$$
It is easy to compute that there are $nm$ edges and $n$ faces.  A slightly more complicated equation says that there are $n+(n-1)m$ vertices, so that the Euler charateristic is $2n-m=2-2g.$  So $m=2g-2+2n.$  This is exactly the dimension of the Hurwitz space.  The critical values of the meromorphic functions can be coordinates on the Hurwitz space, which is why this is the dimension.  We can know the dimension of this space without knowing anything about its topology.

Now let me be more specific.  Starting from this point, I will consider that the preimages of $\infty$ are marked points.  So now I have $n$ marked poles.  This allows me to consider the mapping $\Hh_{g,n}\to \Mm_{g,n}.$  I simply forget the function, remembering only the domain of the function.  This picture makes it possible to compute the dimension of the base space, but to do this we need to know the dimension of the fiber.  The calculation of the Euler characteristic is the Riemann Hurwitz formula, but to compute the dimension of the fiber I need the Riemann Roch formula, which says that for $n>2g-1$, the dimension of the fiber is $n-g+1$.  Then the dimension of the base is $3g-3+n$.  We have a simple space of functions.  Our picture shows that it covers the configuration space of $n$ points on the projective line.  On the other hand we have a much more complicated space, $\Mm_{g,n}$ whose dimension calculation is very difficult.  This indicates that the Hurwitz spaces may be more fundamental than Hurwitz spaces.  

It's clear from this description that these spaces are not compact.  In order to get an object whose geometry we can study we must compactify.  I am going to describe the approach developed with Shapiro [and others].

Let me start with $\Hh_{0,1^n}$.  What do we have?  This is fibered over $\Mm_{0,n}$.  I assume $n\ge 3$.  Now I am going to identify the fiber of this map.  You have a function with poles of order one.  What is a pole?  $f$ has a pole of order one at a point $x\in C$.  Then we can associate a tangent vector $\tau_f\in T_xC$.  What is this tangent vector?  A tangent vector is an element of the space dual to cotangent vectors, so it is a linear functional on the space of holomorphic differentials.  What is a natural way to come up with a linear functional on the cotangent bundle.  Take $\omega\in T^\vee_xC$ which can be represented by a differential one-form.  Then $\tau_f:\omega\mapsto Res_x f\omega$.

Since we have $n$ poles of order one, we obtain a mapping $\Hh_{0,n}\to \lL_1^\vee \oplus\cdots\oplus \lL_i^\vee,$ where $\lL_1$ is the line bundle over moduli space of curves.  Let me recall that $\lL_i$ is a bundle over $\bar{\Mm}_{0,n}$ whose fiber at $C,x_1,\ldots x_n$ is $T_{x_i}^\vee C$.

Now we must identify what we obtain in the image and then what the fiber is.  I have a meromorphic function with $n$ poles of order one.  To each pole I associate a tangent vector at the correspondig marked point.  Thus I get a mapping from the Hurwitz space to the direct sum of $n$ line bundles over the moduli space of curves.  When I first understood this construction, I was extremely excited because I understood that to each meromorphic function, each one has a free term associated with it.

This free term is nothing but the fiber of this map.  Two functionals are taken to the same point if only their free terms disagree, if their difference is a constant.  Indeed, what does it mean that two functions are taken to the same point?  It means the difference has no poles, so it's constant.

The free term of a function is the sum of its critical values.  This is a well defined object for any function.  Another way we would be able to consider to add one more summand $\cC$ to the direct sum of line bundles which would just be the trivial summand.  The corresponding mapping would take the function to its free term.  Let me just forget about that summand and instead change my definition, requiring that the curves in $\Hh_{g,1^n}$ have $0$ as the sum of their critical values.

This is an embedding of $\Hh_{0,1^n}$ into the direct sum of $n$ line bundles, and the image is a dense open subset.  Now we arrive at a natural compactification, defining it as the total space of this bundle.  This is a definition.  Unfortunately this space is not compact as well, and it's not a good idea to call something compactified when the resulting space is not a compact space.

Sorry.  Before making this compactification, let me extend this line bundle to the compactified moduli space $\bar{Mm}_{0,n}$.  Now erase the zero section of this bundle.  Now $\cC^*$ acts on $\bar{\Hh}_{0,1^n}\backslash \text{the zero section}$.  This is what we will study, but I won't change notation dealing with the space before and after this compactification.

Now it would be a good idea to identify these added points as some rational functions as well.  This means that we would like to say that this compactification is modular.  Each point here corresponds to a certain function.  Of course there is no problem when we add functions that are not generic, if the critical values grew together.  The corresponding function is a rational function.  The principle difficulties are first that there are singular curves and the corresponding functions must be on singular curves.  The total space, then, as the second difficulty, is the space of functions, meaning the tangent vector can disappear.  This means the pole can be erased.  This is in fact dangerous because the degree of our function falls, becomes smaller.  So that if we have 
$$\frac{a}{x-1}+\frac{b}{x}+cx+\frac{d}{x-\lambda}$$
and $c$ goes to zero, then we go to a smaller degree.  When this vanishes, we acquire an additional copy of $\cC\pP^1$.

Take $xy=\epsilon$.  Then as $\epsilon\to 0,$ this gets a degeneration.  You lose a pole and yet preserve the degree of the function.  There are no difficulties when we consider functions on singular curves.  They are rational on each connected component and have the same value at each double point.  Let me add one example of the same kind in order to illustrate the difficulties that can arise in higher genera.  Consider $\Hh_{1,1^n}$, functions on elliptic curves.  We still get the map to $\bigoplus \lL_i^\vee$, but now the image is not dense and open.  Each meromorphic function gives $n$ tangent vectors, but these cannot be arbitrary; they are subject to some relation, namely if these tangent vectors are $\tau_i$ then
$$
\sum \tau_i(\omega)
$$
is zero for any holomorphic one-form $\omega$.  In the elliptic case (genus one) there exists essentially one such holomorphic form which means that in this case, the image of the Hurwitz space is a hypersurface in the total space.  For higher genera we get a similar situation, where for genus $g$ curves there are $g$ equation for the Hurwitz space.  I define the correspondning compactification as the closure of the image.

Now let me consider an example which is opposite in a sense.  These are polynomial functions.  In a polynomial case, I consider that there is one point where the preimage is total ramification.  All the sheets are glued together at a single point.  All the other points have simple ramification.  In this case we can choose a coordinate in the preimage so as to make the function have the form $z\mapsto z^n+a_n z^{n-2}+\cdots+a_n.$

We are able to choose a coordinate in such a way that the leading coefficient is one, the second coefficient is zero, and all the others are \wh.  A natural compactification is to consider $C^{n-1}$ which is $\langle a_2,\ldots,a_n\rangle$.  However, there is bad news about this compactification.  First, we are unable add functions.  If you take two such and add them, you get one whose leading term is two, not one.  If there is addition, it does not coincide with the addition in this vector space.

The second bad news is that there is no action of $\cC^*$.  Why is this the case?  Indeed, if we multiply the polynomial by a constant $\lambda$ then we would like to make a change of variables to make it an expression of the same kind.  So we want something like $((\lambda^{1/n}z)^n+a_2\lambda^{2/n}(\lambda^{1/n}z\cdots$  But we don't know what $\lambda^{1/n}$ is.  

The third is that there are equivalent polynomials.  Let $z\maptso \mu_z$ where $\mu^n=1$.  This is an admissable change of coordinates.  For any set of coordinates we have $n$ equivalent polynomials.  Each one is present.

How do we deal with these?  Well, for number one, just forget it, there is never addition in Hurwitz spaces.  The seond and thir one are hard, but annihilate them.  If you quotient out by the action of  $\zZ_n$ thenn this is the correcti compactification of the space of polynmiarn endowed with a $\cC^*$ action.    This action takes $a_2,\ldots, a_m\to (\mu^2o_2,\mu^3 o^3,\ldots)$.  

Now I don't have time to introduce a formal definition of the compactification we used in our proof of \wh, but all of the main features are represented on the blackboard.  The spaces we will consider are spaces of the form $\Hh_{g,\mu_1,\ldots,\mu_n}$ which are spaces with functions whose ramification over $\infty$ are given by the $\mu_i.$  Thes are the orders of the poles.  

I'll stop here.

\section{Dubrovin}
Recall that at the end of the last lecture I had defined a class of integrable hirearchies starting from an arbitrary semisimple Frobenius manifold of dimension $n$.  This gave an integrable hierarchy that we called the integrable hierarchy of topological type.  This corresponded to the particular choice of the central invariants $\frac{1}{24}$.  These are defined only up to a common factor, this is a choice of normalization.  The idea of the construction  was to look for a resolution to the system of the so-called Virosora constraint 
$$L_m e^{\sum \epsilon^{2g-2}F_k(u,u_x,\ldots, u^{(3g-2)})}=0$$
where the $L_m=\frac{\partial ^2}{\partial t \partial t}+\beta(\epsilon)t\frac{\partial}{\partial t}+ \beta(\epsilon) t^2$
with for example
$$L_{-1}=\sum^\infty\sum^\infty t^{\alpha,p+1}\frac{\partial}{\partial t^{\alpha,p}}+\frac{1}{2\epsilon^2}\eta_{\alpha\beta}t^{\alpha,0}t^{\beta,0}$$
You solve this and get a unique solution that gives the quasitriviality transformation.

For a given $n$, consider the moduli of $n$-dimensional semisimple Frobenius manifolds, calibrated (which is needed to write down a unique form of the dispersionless hierarchy).  Then we have over every point an integrable hierarchy.  For $n=1$ there is only one Frobenius manifold, and so there is only one.  In dimension two and above, the moduli exist.  What is nice about this particular class of the hierarchy? 
\begin{itemize}
\item There exists a $\tau$-function
\item Virasoro symmetries
\end{itemize}
I apologize for using this word.  It is not the Virasoro acting, it is only half of it, with no central charge.  The transformations that act on the $\tau$ function $\tau\mapsto \tau+\delta L_m\tau$ for $m\ge -1,\delta$ small, maps solutions to solutions up to terms of order $\delta^2$.

This motivates the particular choice, a particular solution is invariant with respect to these symmetries.  That is $L_m\tau=0,m\ge -1$.  This is called the vacuum hierarchy.

Let me now proceed to examples.  
\begin{enumerate}
\item KdV.  In this picture things are canonically normalized.  So $u_{t_0}=u_x,u_{t_1}=u u_x+\frac{\epsilon^2}{12}u_xxx,\ldots$
What about the vacuum solution for this case?  It is badly singular and can be written
$$\tau^{vac}=\frac{1}{(-t_1)^{1/24}}e^{\sum\epsilon^{2g-2} \sum_k \frac{\Gamma_{g,k}(t_0,t_1,\ldots)}{t_1^k}}$$
Now the audience has suggested the dilation shift $t_1\to t_1-1,$ and then expand again. Then one obtains
$$\tau^{vac}(t_0,t_1-1,t_2,\ldots)=\tau^{KW}(t_0,t_1,\ldots)$$ a formal power series in all times.
One can do other choices of shifts.  About the singularity, let me explain where it comes from.

To compute the $\tau$ function one must find the solution to the hierarchy.  Let me consider the vacuum solution in the dispersionless limit, genus zero.  Then all $\epsilon$ terms disappear and we get a system that we can solve with the method of characteristics.  Then $u(t_0,t_1,\ldots)$ is determined implicitly from $\sum t_k \frac{u_k}{k!}=0$.  You have to find the root of this function, which is awful, this is a terribly bad function.  If we do the dilation shift, we get instead 
$$\sum t_k \frac{u_k}{k!}=u,$$ and instead of finding a root we're looking for a fixed point of this map.  Start from the solution $u=0$, assuming the leading term is of order one, the other times are small, and expand to get Kontsevich-Witten.

{\it Remark.} Any formal solution to KdV in the class of series in $\epsilon$ can be obtained from the vacuum solution by a suitable shift $t_k\mapsto t_k-t_k^0$, but $t_k^0$ may itself be a formal power series in $\epsilon$.
\item
As another example, found by Peter Zograf, consider $\tau^{vac}$ as a functional on $\frac{1}{z}\cC[[\frac{1}{z}]]$.  An element is $\frac{q_0}{z}+\frac{q_1}{z^2}+\cdots$
Then $\tau^{vac}(\frac{1}{z}(x-f(\frac{1}{z})))$ where 
$$f(y)=y^{1/2}J_1(2y^{1/2})=\sum \frac{(-1)^m y^{m+1}}{m!(m+1)!}$$
Then $$\log\tau^{vac}(\frac{1}{z}(x-f(\frac{1}{z})))=\sum(\frac{\epsilon}{\pi^3})^{2g-2}\sum_n (\frac{x}{\pi^2})^n Vd(\Mm_{g,n})$$
\item Let $n=2$ and take $F(v,u)=\frac{1}{2}v^2u+e^u$ and then the hierarchy we get is the (extended) Toda hierarchy, $L=\Lambda+v+e^u\Lambda^{-1}$, $\Lambda=e^{\epsilon \partial_x}$
There are two time chains
$$\epsilon\frac{\partial L}{\partial t_k}=\frac{1}{(k+1)!} [\left(L^{k+1}\right)_+,L]$$$$
\epsilon\frac{\partial L}{\partial s_k}=\frac{2}{k!}[L^k(\log L-c_k),L]$$
for $c_k=1+\frac{1}{2}+\ldots+\frac{1}{k}$.
Then $$L_{-1}=\sum t_{k+1}\frac{\partial}{\partial t_k}+s_{k+1}\frac{\partial}{\partial s_k}+\frac{1}{\epsilon^2}s_0t_0$$
Let's again apply the dilation shift, this time along $s_1$.  Then $$\log\tau^{vac}(s_0,s_1-1,s_2,\ldots,t_0,t_1,\ldots:\epsilon^2)=\Ff_{GW}^{\cC\pP^1}$$ and again all the singularities disappear after the dilation shift.

Are other dilation shifts allowed?  I would say yes.  What happens if you do the dilation shift along $t_1$?
$$\log\tau^{vac}(s_0,s_1,s_2,\ldots,t_0,t_1-1,\ldots:\epsilon^2)|_{s_0=x,s_i=0,t_0=t_1=0,t_k=(k+1)!\lambda_{k+1}}
=\frac{x^2}{2\epsilon^2}(\log x-\frac{3}{2})-\frac{1}{12}\log  x} + \sum (\frac{\epsilon}{x})^{2g-2}\frac{B_{2g}}{2g(2g-2)}+\sum \epsilon^{2g-2}F_g(x,\lambda_3,\lambda_4,\ldots)$$
where these $F$ are give by
$$F_g=\sum_n \sum_{k_i} a_g(k_1,\ldots, k_n)\lambda_{k_1}\ldots \lambda_{k_n} x^h$$
where $h=2-2g-(n-\frac{|k|}{2})$ and $a_g$ is the ``number'' of connected oriented ribbon graphs $\Gamma$ of genus $g$ with $n$ vertices of valences $k_1,\ldots, k_n$ where these are counted with weight $\frac{1}{\Aut(\Gamma)}$
This is a matrix model, a reformulation of the classical formulation of \wh about, we do a different dilation shift and arrive at matrix models.
\end{enumerate}
What other examples of integrable hierarchies of topological type do we know?  Other examples of integrable hierarchies of topological type include the Drinfel'd Sokolov hierarchies of type $A_n$.  the $L$ is a differential operator of order $n+1$ This is related, according to Witten conjecture, to the intersection numbers in the moduli space of higher spin structures.   Of course, it would be nice to prove the same for any simply laced Lie algebra.  We proved this for $D_n$ when the central charges are $\frac{1}{24}$.  This means that it's equivalent to a hierarchy of topological type, but we want it to be equal.  Then another example is the Toda bigraded, and the $L$ operator is a difference operator $e^{k\epsilon \partial_x}+\cdots + a_Ne^{m\epsilon\partial_x}$.  We can again prove it's equivalent to a hierarchy of topological type.  But we can't prove that it is one exactly for the Virasoro symmetries.

The quasilast point, to emphasize the existence theorem that I formulated last time, consider the Frobenius manifold $M=QH^*(\cC\pP^n)$ (this has dimension $n+1$), and then one can associate a hierarchy of topological type and from the Givental results and our results, such a function  $e^{\Ff_{GW}^{\cC\pP^n}=\tau$ is a $\tau$-function of this hierarchy.  From Givental we use the Virasoro constraints defined on \wh and then we proved that the genus expansion of the Givental formula coincides with the one we use in quasitriviality.

The hierarchy itself is quite mysterious.  For $N=0$ this is KdV, for $N=1$ this is Toda, and for $N=2$ this is new!  We don't know how to touch these.

For the last few minutes I want to speculate about some other hierarchies, new ones, that I believe may have other applications not strictly speaking to GW invariants:  There are the ones associated with the Frobenius structures on Hurwitz spaces.    To fix a Frobenius structure one has to fix a covering $\Cc_g\to \pP^1$ with fixed branching at $\infty.$  I consider a fixed basis of cycles $a_1,\ldots, a_g,b_1,\ldots, b_g$ and a primary differential $d_p$.  Then $\langle \partial_1,\partial_2\rangle=\sum res \frac{\partial_1 \lambda(dp) \partial_2 \lambda(dp)}{d\lambda}$.
Then $$\langle \partial_1,\partial_2\cdot \partial_3\rangle=
\sum res \frac{\partial_1(\lambda(p)dp)\partial_2(\lambda(p)dp)\partial_3(\lambda(p)dp)}{d\lambda dp}$$
defines a Frobenius structure.

I encountered these in the context of Whitham equations.  [I did not get any of this].  
\end{enumerate}

\end{document}
