

Colloquium of the Steklov Mathematical Institute of Russian Academy of Sciences
June 2, 2016 16:00, Moscow, Steklov Mathematical Institute of RAS, Conference Hall (8 Gubkina)






Adiabatic limit in the equations of field theory
A. G. Sergeev^{} 
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Abstract:
The notion of adiabatic limit came into mathematics from physics and
in last years widely spread in differential geometry, theory of
partial differential equations, topology. In our talk we shall speak
about the applications of adiabatic limit construction in equations
of gauge field theory.
We start from the GinzburgLandau equations in dimension
3=1(time)+2(space) arising in the superconductivity theory. In the
adiabatic limit these equations convert into the Euler equation for
geodesics on the space of vortices (static solutions of
Ginzburg–Landau equations) with respect to the metric determined by
the kinetic energy.
We turn next to dimension 4 and consider the adiabatic limit in
Seiberg–Witten equations on 4dimensional symplectic manifolds. In
the adiabatic limit solutions of these equations converge to
families of vortex solutions parameterized by points of
pseudoholomorphic curves. Such families satisfy a nonlinear
Cauchy–Riemann equation. So the adiabatic limit in SeibergWitten
equations may be considered as a complex version of the same limit
in Ginzburg–Landau equations. Namely, the Euler equation is
replaced by the Cauchy–Riemann equation while geodesics on the
space of vortices are substituted by the “complex” geodesics in
vortex bundles over pseudoholomorphic curves. In other words,
dimension 4 in this case may be treated as 4=2(complex
time)+2(space).

