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.
Links:
[1] https://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] https://www.mpim-bonn.mpg.de/node/3444
[3] https://www.mpim-bonn.mpg.de/node/5312