Egyptian algebra

2013-10-25T21:42:00Z

108.65.201.239:

In the mathematical field of [[differential geometry]], a '''Frobenius manifold''' is a flat [[Riemannian manifold]] with a certain compatible multiplicative structure on the [[tangent space]]. The concept generalizes the notion of [[Frobenius algebra]] to tangent bundles. They were introduced by Dubrovin.<ref>B. Dubrovin: ''Geometry of 2D topological ﬁeld theories.'' In: Springer LNM, 1620 (1996), pp. 120–348.</ref>

Frobenius manifolds occur naturally in the subject of [[symplectic topology]], more specifically [[quantum cohomology]]. The broadest definition is in the category of Riemannian [[supermanifold]]s. We will limit the discussion here to smooth (real) manifolds. A restriction to complex manifolds is also possible.

== Definition ==
Let ''M'' be a smooth manifold. An ''affine flat'' structure on ''M'' is a [[Sheaf (mathematics)|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 [[Riemann curvature tensor|curvature tensor]] vanishes everywhere.

A family of ''commutative products *'' on ''TM'' is equivalent to a section ''A'' of ''S''2(T*''M'') ⊗ ''TM'' via

:<math>X*Y = A(X,Y). \, </math>

We require in addition the property

:<math>g(X*Y,Z)=g(X,Y*Z). \, </math>

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

:<math>g(A(X,Y),Z)=X[Y[Z[\Phi]]] \, </math>

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.

== Elementary properties ==
The associativity of the product * is equivalent to the following quadratic [[partial differential equation|PDE]] in the local potential ''Φ''
:<math> \Phi_{,abe}g^{ef}\Phi_{,cdf} = \Phi_{,ade}g^{ef}\Phi_{,bcf} \, </math>
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.

== Examples ==
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]] QHeven(''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.

== References ==
{{Reflist}}
2. Yu.I. Manin, S.A. Merkulov: [http://arxiv.org/abs/alg-geom/9702014 ''Semisimple Frobenius (super)manifolds and quantum cohomology of '''P'''r''], Topol. Methods in Nonlinear [[Analysis]] 9 (1997), pp. 107–161

{{DEFAULTSORT:Frobenius Manifold}}
[[Category:Symplectic topology]]
[[Category:Riemannian manifolds]]

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Egyptian algebra