The equivalence between the two definitions of Hurwitz numbers ( counting ramified coverings of the sphere, or counting transitive factorizations ) is established by describing a ramified covering by its monodromy.
42.
In that case the monodromy group relating solutions by analytic continuation has less to say in general, and the solutions are harder to study, except in terms of their asymptotic expansions.
43.
The monodromy theorem makes it possible to extend an analytic function to a larger set via curves connecting a point in the original domain of the function to points in the larger set.
44.
Note that the monodromy defects discussed above are not finite-energy excitations in the spectrum of the Hamiltonian, but defects created by "'modifying "'the Hamiltonian.
45.
Here the branch point is the origin, because the analytic continuation of any solution around a closed loop containing the origin will result in a different function : there is non-trivial monodromy.
46.
The monodromy describes the way the first homology group of a smooth fiber ( which is isomorphic to "'Z "'2 ) changes as we go around a singular fiber.
47.
The action on V _ 1 \ otimes \ dots \ otimes V _ n by the analytic continuation of KZ equation is called "'monodromy representation of KZ equation " '.
48.
There he considered the problem of isomonodromy deformations for a certain matrix Fuchsian equation; this is a special case of Hilbert's 21st Problem ( existence of differential equations with prescribed monodromy ).
49.
It is known that the monodromy representation of KZ equation with a general semi-simple Lie algebra agrees with the linear representation of braid group given by R-matrix of the corresponding quantum group.
50.
The monodromy around each singular fiber is a well-defined conjugacy class in the group SL ( 2, "'Z "') of 2 ?2 integer matrices with determinant 1.
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