Linear sigma models and integrable systems

I’m a mathematician who recently became very interested in questions related to mathematical physics but somehow, I faced difficulties in penetrating the literature… I’d highly appreciate any help with the following question:

My aim is to relate a certain (equivariant) linear sigma model on a disc (with a non-compact target $mathbb C$) as constructed in the exciting work of Gerasimov, Lebedev and Oblezin in Archimedean L-factors and Topological Field Theories I, to integrable systems (in the sense of Dubrovin, if you like).

More precisely, I’d like to know if it’s possible to express “the” correlation function of an (equivariant) linear sigma model (with non-compact target) as in the above reference in terms of a $tau$-function of an associated integrable system?

As far as I’ve understood from the literature, for a large class of related non-linear sigma models (or models like conformal topological field theories) such a translation can be done by translating the field theory (or at least some parts of it) into some Frobenius manifold (as in Dubrovin’s approach, e.g., but other approaches are of course also welcome). Unfortunately, so far, I haven’t been able to understand how to make things work in the setting of (equivariant) linear sigma models (with non-compact target).

Any help or hints would be highly appreciated!