Flat 3-webs via semi-simple Frobenius 3-manifolds

Abstract

We construct flat 3-webs via semi-simple geometric Frobenius manifolds of dimension three and give geometric interpretation of the Chern connection of the web. These webs turned out to be biholomorphic to the characteristic webs on the solutions of the corresponding associativity equation. We show that such webs are hexagonal and admit at least one infinitesimal symmetry at each singular point. Singularities of the web are also discussed.

Frobenius 3-Folds via Singular Flat 3-Webs

“…Equation (4) defines the corresponding characteristic web of the above associativity equation. Hence the web is the booklet web of the constructed Frobenius 3-fold germ (see [1]).…”

“…Let us choose the symmetry operator as X = (1 + (l + 1)x)∂ x + y∂ y and 1 + (l + 1)x as a new coordinate z. Now the base point of the 3-fold germ is (1,0). We are looking for the flat coordinates in the form x = z α R(t),ȳ = z β Q(t), t = yz − 1 1+m 0 /2 with analytic R and Q subjected to αRQ − βQR | 0 = 0.…”

“…The following geometric construction of 3-web via Frobenius 3-fold was proposed in [1]. Consider a massive Frobenius 3-fold M , which means that the algebra T p M is semi-simple for each p ∈ U for some open set U ⊂ M .…”

“…The web directions of a booklet 3-web are well-defined everywhere (see [1]), maybe with multiplicity. We call a point regular or non-singular if the web directions at this point are pairwise distinct.…”

“…This gives us two weights [w 1 : w 2 ] of the Euler vector field, since, for booklet 3-webs, the operator X is the projection of the Euler vector field along the unity e to the surface S (see [1]).…”