Frobenius manifold

Let M be a smooth manifold. An affine flat structure on M is a sheaf T^f of vector spaces that pointwisely span TM the tangent bundle and the tangent bracket of pairs of its sections vanishes.

As a local example consider the coordinate vectorfields over a chart of M. A manifold admits an affine flat structure if one can glue together such vectorfields for a covering family of charts.

Let further be given a Riemannian metric g on M. It is compatible to the flat structure if g(X, Y) is locally constant for all flat vector fields X and Y.

A Riemannian manifold admits a compatible affine flat structure if and only if its curvature tensor vanishes everywhere.

A family of commutative products * on TM is equivalent to a section A of S²(T^*M) ⊗ TM via

${\displaystyle X*Y=A(X,Y).\,}$

We require in addition the property

${\displaystyle g(X*Y,Z)=g(X,Y*Z).\,}$

Therefore, the composition g^#∘A is a symmetric 3-tensor.

This implies in particular that a linear Frobenius manifold (M, g, *) with constant product is a Frobenius algebra M.

Given (g, T^f, A), a local potential Φ is a local smooth function such that

${\displaystyle g(A(X,Y),Z)=X[Y[Z[\Phi ]]]\,}$

for all flat vector fields X, Y, and Z.

A Frobenius manifold (M, g, *) is now a flat Riemannian manifold (M, g) with symmetric 3-tensor A that admits everywhere a local potential and is associative.

The associativity of the product * is equivalent to the following quadratic PDE in the local potential Φ

${\displaystyle \Phi _{,abe}g^{ef}\Phi _{,cdf}=\Phi _{,ade}g^{ef}\Phi _{,bcf}\,}$

where Einstein's sum convention is implied, Φ_,a denotes the partial derivative of the function Φ by the coordinate vectorfield ∂/∂x^a which are all assumed to be flat. g^ef are the coefficients of the inverse of the metric.

The equation is therefore called associativity equation or Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equation.

Beside Frobenius algebras, examples arise from quantum cohomology. Namely, given a semipositive symplectic manifold (M, ω) then there exists an open neighborhood U of 0 in its even quantum cohomology QH^even(M, ω) with Novikov ring over C such that the big quantum product *_a for a in U is analytic. Now U together with the intersection form g = <·,·> is a (complex) Frobenius manifold.

The second large class of examples of Frobenius manifolds come from the singularity theory. Namely, the space of miniversal deformations of an isolated singularity has a Frobenius manifold structure. This Frobenius manifold structure also relates to Kyoji Saito's primitive forms.

2. Yu.I. Manin, S.A. Merkulov: Semisimple Frobenius (super)manifolds and quantum cohomology of P^r, Topol. Methods in Nonlinear Analysis 9 (1997), pp. 107–161