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Complex analysis and mathematical physics
April 20, 2015 16:00, 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

Abstract: There exists an important class of solutions of Yang-Mills equations on four-dimensional 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 Yang-Mills equations that are solutions of instanton equations. These equations generalize the antiautoduality equations on four-dimensional manifolds. They depend on an additional parameter, namely, a fixed differential $(d-4)$-form.
We shall consider the following situation. Let $X$ be a calibrated manifold and Y its calibrated $(d-4)$-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 (4-dimensional) fibers of the normal bundle $N(Y)$.