In the mathematical field of
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
, a Frobenius manifold, introduced by Dubrovin,
[B. Dubrovin: ''Geometry of 2D topological field theories.'' In: Springer LNM, 1620 (1996), pp. 120–348.] is a flat
Riemannian manifold
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...
with a certain compatible multiplicative structure on the
tangent space
In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
. The concept generalizes the notion of
Frobenius algebra
In mathematics, especially in the fields of representation theory and module theory, a Frobenius algebra is a finite-dimensional unital associative algebra with a special kind of bilinear form which gives the algebras particularly nice dualit ...
to tangent bundles.
Frobenius manifolds occur naturally in the subject of
symplectic topology
Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form. Symplectic geometry has its origins in the H ...
, more specifically
quantum cohomology In mathematics, specifically in symplectic topology and algebraic geometry, a quantum cohomology ring is an extension of the ordinary cohomology ring of a closed symplectic manifold. It comes in two versions, called small and big; in general, t ...
. The broadest definition is in the category of Riemannian
supermanifold
In physics and mathematics, supermanifolds are generalizations of the manifold concept based on ideas coming from supersymmetry. Several definitions are in use, some of which are described below.
Informal definition
An informal definition is com ...
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
Sheaf may refer to:
* Sheaf (agriculture), a bundle of harvested cereal stems
* Sheaf (mathematics), a mathematical tool
* Sheaf toss, a Scottish sport
* River Sheaf, a tributary of River Don in England
* ''The Sheaf'', a student-run newspaper ser ...
''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
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ''T ...
''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 ''S''
2(T
*''M'') ⊗ ''TM'' via
:
We require in addition the property
:
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
:
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.html" ;"title="[\Phi">[Z[\Phi.html" ;"title="[\Phi.html" ;"title="[Z[\Phi">[Z[\Phi">[\Phi.html" ;"title="[Z[\Phi">[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.
Elementary properties
The associativity of the product * is equivalent to the following quadratic partial differential equation">PDE in the local potential ''Φ''
:
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 In mathematics, specifically in symplectic topology and algebraic geometry, a quantum cohomology ring is an extension of the ordinary cohomology ring of a closed symplectic manifold. It comes in two versions, called small and big; in general, t ...
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
Kyōji Saitō (齋藤 恭司, Saitō Kyōji; born 25 December 1944) is a Japanese mathematician, specializing in algebraic geometry and complex analytic geometry.
Education and career
Saito received in 1971 his promotion Ph.D. from the Universi ...
's primitive forms.
References
2. Yu.I. Manin, S.A. Merkulov
''Semisimple Frobenius (super)manifolds and quantum cohomology of Pr'' Topol. Methods in Nonlinear
Analysis
Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (38 ...
9 (1997), pp. 107–161
{{DEFAULTSORT:Frobenius Manifold
Symplectic topology
Riemannian manifolds
Integrable systems
Algebraic geometry