

Complex analysis and mathematical physics
April 20, 2015 16:00–18:00, Moscow, Steklov Mathematical Institute, Room 430 (8 Gubkina)






Adiabatic limit for instanton equations on manifolds of dimension greater than 4
R. V. Palvelev^{} ^{} Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

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Abstract:
There exists an important class of solutions of YangMills equations on fourdimensional Riemannian manifolds, namely, instantons and antiinstantons, or solutions of (anti)autoduality equations. For manifolds of dimension $d>4$ one can also consider the class of solutions of YangMills equations that are solutions of instanton equations. These equations generalize the antiautoduality equations on fourdimensional manifolds. They depend on an additional parameter, namely, a fixed differential $(d4)$form.
We shall consider the following situation. Let $X$ be a calibrated manifold and Y its calibrated $(d4)$dimensional submanifold. Let $A_n$ be a sequence of solutions of instanton equations in some neighborhood of $Y$ and every solution is constructed for the metric obtained from the initial Riemannian metric on $X$ by contraction in the directions orthogonal to $Y$: roughly speaking, the solution $A_n$ is for the metric of the type $g^{(n)}=g_Y+\varepsilon^2_ng_{Y^\perp}$, where $\varepsilon_n\to0$. G.Tian proved that the limit of the sequence $A_n$ defines a map from $Y$ to moduli space of instantons on (4dimensional) fibers of the normal bundle $N(Y)$.

