1. Monodromy and its localization
Many special functions of mathematical physics and other applied sciences can
be defined by integral representations in one and the same way, as follows. There is
a smooth locally trivial fiber bundle E — T, and a differential form UJ on the total
space of this bundle; this form should be closed along the fibers but not closed on
the entire space E. Moreover, a point bo G T is distinguished, and in the fiber over
this point a homology class is fixed, of the same dimension as the degree of the form.
Then a function appears on the base. Indeed, let us realize our fixed homology class
by a cycle. We can transport this cycle continuously into the neighboring fibers
and integrate the form UJ along the resulting cycles. Of course, this transportation
is not unique (if the fibers are positive-dimensional), but the homology classes of
the cycles obtained are well-defined. Therefore also the result of the integration
depends only on the initial homology class, and does not depend on the mode of
The resulting integral function on the base is locally single-valued, but globally
it can ramify. If the fundamental group of the base is non-trivial, then, transporting
our cycle over a non-contractible path, we can arrive at a different homology class.
We obtain a representation of the fundamental group of the base in the homology
group of the fiber. This representation is called the monodromy, and its image
the monodromy group of our fiber bundle. Many problems concerning analytical
properties of integral functions (single-valuedness, algebraicity, number of leaves,
regularity) can be reduced to the study of the monodromy group.
The standard situation when this scheme appears is as follows. There is a
smooth map E — • T of sightly greater spaces (which are analytic manifolds). Again,
on the source E of this map a form UJ is defined, which is closed in the restriction to
all fibers of this map. For almost all points of the base the corresponding fibers are
homeomorphic to one another, forming a locally trivial fiber bundle. In particular,
we can transport the cycles from fiber to fiber over the paths in the set of such
"almost all" points.
The subset in T consisting of critical values such that the fibers over them
degenerate is called the discriminant and is denoted by E.
In different areas of physics and mathematics the discriminant sets are called
different things: wave fronts, transparent contours, Landau sets, etc.
If our problem is complex (i.e. E and T are complex manifolds), then the set
E has a positive complex codimension, i.e. real codimension 2. If its complex
codimension is equal to 1 (as it usually happens), then we can go around the set E
close to its regular points (where E is situated as a smooth hypersurface in T) and
obtain some locally defined ramification of homology groups of fibers, and hence
also of integral functions obtained by integration along the elements of these groups.