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. 
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References

SHOWING 1-8 OF 8 REFERENCES
Semisimple Frobenius (super)manifolds and quantum cohomology of $P^r$
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
These lecture notes are devoted to the theory of “equations of associativity” describing geometry of moduli spaces of 2D topological field theories.
Semisimple Frobenius (super)manifolds and quantum cohomology of $\mathbb{P}^r$
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