I will describe a setting for a procedure of topological recursion, recently revisited by

Kontsevich and Soibelman. It takes as input a vector space V, and 4 tensors

A in $Mor(1,V \otimes V \otimes V)$

B in $Mor(V,V \otimes V)$

C in $Mor(V \otimes V,V)$

$\epsilon$ in $Mor(1,V)$

satisfying a system of 3 coupled IHX relations, and produces multilinear forms $F_{g,n} : V^{\otimes n} \to K$

by integer $g \ge 0$ and $n \ge 1$. These relations express that V is equipped with a Lie algebra structure, and

has a representation in terms of quadratic differential operators -- acting on polynomial functions on V.

An interesting question is then to exhibit interesting initial data (i.e. solutions of these relations), and

understand the meaning of the $F_{g,n}$. I will describe 3 classes of solutions:

1) 2d TQFTs: A,B,C are directly obtained from the product, the Lie algebra is abelian, and $F_{g,n}$ are

the TQFT amplitudes for surfaces of genus g with n boundaries.

2) Virasoro-type solutions, which contain as special case certain cohomological field theories and in

particular certain Gromov-Witten theories, and for which $F_{g,n}$ are the correlation function of

the CohFT.

3) A solution one can attach to any non-commmutative Frobenius algebra, and for which the field-theoretic

meaning of $F_{g,n}$ is at present unclear.

This poses a number of open geometric questions.

This is a based on a work in progress with Andersen, Chekhov and Orantin.

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