Question

I've come across the notion of Monodromy transformations while reading some aspects of variations of Hodge structures in context of Classical Mirror symmetry. I am having difficulty in grasping the concept of Monodromy transformations, probably due to lack of any good reference. My question is :

How to think of monodromy transformations, what are these intuitively and what are some examples of monodromy.

Please suggest any good references for Variations of Hodge structures and monodromy.

Answer 1

A local system is a sheaf of finite dimensional vector spaces that is locally isomorphic to the constant sheaf $k^n$. If $\gamma: [0,1] \to X$ is a continuous path in $X$, $\gamma^{-1}(L)$ is again local system on $[0,1]$ but one shows that any locally constant sheaf on $[0,1]$ is actually constant. So the fibers at 0 and 1 are canonically identified. This means that we have a map $$ \gamma_{*} : L_{\gamma(0)} \to L_{\gamma(1)} $$

Moreover this is

1. linear: $\gamma_* (v + \lambda w) = \gamma_* (v) + \lambda \gamma_* (w)$
2. invariant by homotopy: if $\gamma \sim \gamma'$, $\gamma_*v = \gamma'_*v$.
3. compatible with composition of homotopy classes of paths: $(\gamma')_*(\gamma_*x) = (\gamma'\gamma)_*x$.

So to any local system $L$ corresponds a representation $\pi_1(X,x) \to GL(L_x)$ of the fundamental group at $x$. This is the monodromy representation. You can rebuild $L$ from it: this is the sheaf of sections of $(\tilde{X}\times V)/\pi_1(X,x) \to X$ where $\tilde{X}$ is the universal covering of $X$. We have sketched:

Theorem: If $X$ is connected, the functor "fiber at $x$" induces an equivalence of categories $LS(X) \to Rep(\pi_1(X,x))$.

This is all very abstract so let us look at an example.

Consider $\mathcal{K}$, the trivial rank 2 vector bundle $\cal{O}_{\mathbb{C}^\times}^2$ and connection $$ \nabla \begin{pmatrix} f_1 \cr f_2 \end{pmatrix} = d\begin{pmatrix} f_1 \cr f_2 \end{pmatrix} - \begin{pmatrix}0 & 0 \cr 1 &0 \end{pmatrix} \begin{pmatrix} f_1 \cr f_2 \end{pmatrix} \frac{dz}{z} = \begin{pmatrix} df_1 \cr df_2 - f_1 \frac{dz}{z} \end{pmatrix} $$
Horizontal sections are the solutions of $\nabla f = 0$. This is a system of two first order linear differential equations. On any simply connected $U$ we can chose a determination of the logarithm and the solution can be written
$$ \begin{pmatrix} f_1 \cr f_2 \end{pmatrix} = \begin{pmatrix} A \cr A \log z + B \end{pmatrix} $$ where $\log$ is any determination of the logarithm function. This means that the sections $$
e_1 = \begin{pmatrix} 1 \cr \log z \end{pmatrix} \qquad e_0 = \begin{pmatrix} 0 \cr 1\end{pmatrix} $$ trivialize the sheaf of solutions on $U$. Covering $\mathbb{C}^\times$ by simply connected open sets we see that the solutions form a local system $L$.

When we turn once around 0 following the orientation, our determination $\log z$ changes to $\log z + 2\pi i$. So if $\gamma(t) = xe^{2\pi i t}$
$$ v = \begin{pmatrix} A \cr A\log x + B \end{pmatrix} \mapsto
\begin{pmatrix} A \cr A(\log x + 2\pi i) + B \end{pmatrix} = \gamma_*(v) $$ The monodromy representation is $$ \pi_1(\mathbb{C}^\times,x) = \mathbb{Z} \to GL_2(\mathbb{C}) \qquad 1 \mapsto \begin{pmatrix} 1 & 0 \cr 2\pi i & 1 \end{pmatrix} $$ It tells you everything there is to know about our differential equation (because it has regular singularities). For example, the space of global solutions is identified with the space of invariant of the representation: $$ \Gamma(\mathbb{C}^\times,L) = Hom(k_{\mathbb{C}^\times},L) = Hom_{\pi_1(X,x)}(k,L_x) = L_x^{\pi_1(X,x)} $$ This is the 1 dimensional space generated by $e_0$.

Another good example to work out is the equation $df = \alpha f\frac{dz}{z}$ (the monodromy can be quite different depending on $\alpha$).

Pierre Schapira's webpage has notes of a course on sheaves and algebraic topology focussing on local systems. Claire Voisin's book on Hodge theory is a good reference for variations of Hodge structures.

Answer 2

The classical monodromy action is the action of the fundamental group of a (nice) space on the fibers of a covering map. Suppose that $X$ is a (nice) topological space, and let $\pi_1(X,x_0)$ denote the fundamental group of $X$ at the point $x_0$. Let $p:Y\to X$ be a covering map. Then $\pi_1(X,x_0)$ acts on the fibre $Y_{x_0}=p^{-1}(x_0)$ as follows: take a point $y\in Y_{x_0}$ and an element of the fundamental group represented by a loop $\alpha$ at $x_0$; then a basic property of covering maps implies that $\alpha$ lifts to a path in $Y$ starting at $y_0$; the endpoint of this map also lies in the fibre, and it is the result of the action of $\alpha$ on $y$, by definition.

This generalizes to a variety of situations. A nice overview is perhaps the book Galois groups and fundamental groups by T. Szamuely.