# Weak Frobenius manifolds

@article{Hertling1998WeakFM, title={Weak Frobenius manifolds}, author={C. Hertling and Y. Manin}, journal={arXiv: Quantum Algebra}, year={1998} }

We establish a new universal relation between the Lie bracket and $\circ$-multiplication of tangent fields on any Frobenius (super)manifold. We use this identity in order to introduce the notion of ``weak Frobenius manifold'' which does not involve metric as part of structure. As another application, we show that the powers of an Euler field generate (a half of) the Virasoro algebra on an arbitrary, not necessarily semisimple, Frobenius supermanifold.

#### 108 Citations

On Symmetry Properties of Frobenius Manifolds and Related Lie-Algebraic Structures

- Computer Science
- Symmetry
- 2021

An algebraically feasible approach to solutions of the oriented associativity equations is developed based on a modification of the Adler– Kostant–Symes integrability scheme and applied to the co-adjoint orbits of the diffeomorphism loop group of the circle. Expand

Nijenhuis infinity and contractible dg manifolds

- Mathematics
- 2004

We find a minimal differential graded (dg) operad whose generic representations in $R^n$ are in one-to-one correspondence with formal germs of those endomorphisms of the tangent bundle to $R^n$ which… Expand

Frobenius manifolds and algebraic integrability

- Physics, Mathematics
- 2007

We give a short review of Frobenius manifolds and algebraic integrability and study their intersection. The simplest case is the relation between the Frobenius manifold of simple singularities, which… Expand

Frobenius∞ invariants of homotopy Gerstenhaber algebras, I

- Mathematics
- 2000

We construct a functor from the derived category of homotopy Gerstenhaber algebras, g, with finite-dimensional cohomology to the purely geometric category of so-called F∞-manifolds. The latter… Expand

MULTIPLICATION ON THE TANGENT BUNDLE

- Mathematics
- 1999

Manifolds with a commutative and associative multi- plication on the tangent bundle are called F-manifolds if a unit field exists and the multiplication satisfies a natural integrability con- dition.… Expand

nijenhuis infinity and contractible differential graded manifolds

- Mathematics
- Compositio Mathematica
- 2005

we find a minimal differential graded (dg) operad whose generic representations in ${\mathbb r}^n$ are in one-to-one correspondence with formal germs of those endomorphisms of the tangent bundle to… Expand

Hermitian metrics on F-manifolds

- Mathematics
- 2016

Abstract An F -manifold is complex manifold with a multiplication on the holomorphic tangent bundle, which satisfies a certain integrability condition. Important examples are Frobenius manifolds and… Expand

Some examples of non-massive Frobenius manifolds in singularity theory

- Mathematics
- 2007

Let f, g : C-2 -> C be two quasi-homogeneous polynomials. We compute the V-filtration of the restriction of f to any plane curve C-t = g(-1) (t) and show that the Gorenstein generator dx boolean AND… Expand

Calculus of multilinear differential operators, operator $L_\infty$-algebras and ${\it IBL}_\infty$-algebras

- Mathematics
- 2021

Motivated by the study of deformations of Lie bialgebraic structures, we generalize the construction of the well-known big bracket of Lecomte–Roger to encompass the case of homotopy involutive Lie… Expand

Manifolds with multiplication on the tangent sheaf

- Mathematics
- 2005

This is a survey of the current state of the theory of $F$--(super)manifolds $(M,\circ)$, first defined in [HeMa] and further developed in [He], [Ma2], [Me1]. Here $\circ$ is an $\Cal{O}_M$--bilinear… Expand

#### References

SHOWING 1-8 OF 8 REFERENCES

Semisimple Frobenius (super)manifolds and quantum cohomology of $P^r$

- Mathematics
- 1997

We introduce and study a superversion of Dubrovin's notion of semisimple Frobenius manifolds. We establish a correspondence between semisimple Frobenius (super)manifolds and special solutions to the… Expand

Geometry of 2D topological field theories

- Physics, Mathematics
- 1994

These lecture notes are devoted to the theory of “equations of associativity” describing geometry of moduli spaces of 2D topological field theories.

Geometry of 2 D topological field theories

- 1996

Manin Frobenius manifolds , quantum cohomology , and moduli spaces ( Chapters I , II , III )

- Preprint MPI
- 1996

Manin Frobenius manifolds, quantum cohomology, and moduli spaces (Chapters I, II, III)

- Manin Frobenius manifolds, quantum cohomology, and moduli spaces (Chapters I, II, III)
- 1996

Proposition. a). B is closed with respect to @BULLET and [, ] and hence forms a Poisson subalgebra. If A contains identity e

- Proposition. a). B is closed with respect to @BULLET and [, ] and hence forms a Poisson subalgebra. If A contains identity e

Remark In the semisimple case the meaning of (19) is transparent: writing X = X i e i , we must have e i X j = 0 for i = j

- Remark In the semisimple case the meaning of (19) is transparent: writing X = X i e i , we must have e i X j = 0 for i = j