# 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!

This is a reference resources question, masquerading as an answer, given the constraints of the site. The question hardly belongs here, and has been duplicated in the overflow cousin site . It might well be deleted.

There have been schools and proceedings on the subject,

Integrability: From Statistical Systems to Gauge TheoryLecture Notes of the Les Houches Summer School: Volume 106, June 2016, Volume 106,
Patrick Dorey, Gregory Korchemsky, Nikita Nekrasov, Volker Schomerus, Didina Serban, and Leticia Cugliandolo. Print publication date: 2019, ISBN-13: 9780198828150, Published to Oxford Scholarship Online: September 2019. DOI: 10.1093/oso/9780198828150.001.0001

including, specifically,

Integrability in 2D fields theory/sigma-models, Sergei L Lukyanov & Alexander B Zamolodchikov. DOI:10.1093/oso/9780198828150.003.0006

Integrability in sigma-models, K. Zarembo. DOI:10.1093/oso/9780198828150.003.0005 https://arxiv.org/abs/1712.07725

I am particular to Integrable 2d sigma models: Quantum corrections to geometry from RG flow, Ben Hoare, Nat Levine, Arkady Tseytlin, Nucl Phys B949 (2019) 114798 , but that's only by dint of personal connectivity...

• all the references you are giving seem to be created way after the question was asked.
– nemo
Nov 21, 2021 at 20:04
• They are for today's readers, not historians of science ... Nov 21, 2021 at 20:51