The closed string A-model is mathematically described by Gromov-Witten invariants of a compact symplectic manifold $X$. The genus 0 GW invariants give the structure of quantum cohomology of $X$, which is then an example of a so-called Frobenius manifold. The mirror genus 0 B-model theory on the mirror manifold (or Landau-Ginzburg model) is usually described, mathematically, in terms of variation of Hodge structure data, or some generalization thereof.
Since the higher genus A-model has a nice mathematical description as higher genus GW invariants, I am wondering whether the higher genus B-model has a nice mathematical description as well. Costello's paper on TCFTs and Calabi-Yau categories gives a partial answer to this. One can say that GW theory is the study of algebras over the (homology) operad of compactified (Deligne-Mumford) moduli space; I say that Costello gives a partial answer because he only gives an algebra over the operad of uncompactified moduli space. Though, according to Kontsevich (see Kontsevich-Soibelman "Notes on A-infinity..." and Katzarkov-Kontsevich-Pantev), we can extend this to the operad of compactified moduli space given some assumptions (a version of Hodge-de Rham degeneration). There are also various results (e.g. Katzarkov-Kontsevich-Pantev, Teleman/Givental) which say that the higher genus theory is uniquely determined by the genus 0 theory. But --- despite these sorts of results, I still have not seen any nice mathematical description of the higher genus B-model which "stands on its own", as the higher genus GW invariants do. I have only seen the higher genus B-model described as some structure which is obtained formally from genus 0 data, or, as in the situation of Costello's paper, a Calabi-Yau category, e.g. derived category of coherent sheaves of a Calabi-Yau manifold, matrix factorizations category of a Landau-Ginzburg model, etc.
So, my questions are:
Are there any mathematical descriptions of the higher genus closed string B-model which "stand on their own"? What I mean is something that can be defined without any reference to other structure, just as genus 47 GW invariants can be defined without any reference to genus 0 GW invariants or the Fukaya category.
Genus 0 theory is essentially equivalent to the theory of Frobenius manifolds. What about higher genus theory? Is there any nice geometric structure behind, say, the genus 1 theory, analogous to the Frobenius manifold structure that we get out of genus 0 theory?