INTRODUCTION

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

transportation.

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.

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