Gauge theoris and integrable systems
A number of remarkable connections have been observed between gauge theories and integrable systems. One important example is given by the N=4 SYM in the planar limit and is closely related to the so-called AdS/CFT correspondence (see ►). Besides AdS/CFT correspondence, there also exists another interesting connection between supersymmetric gauge theories and integrable models. In these case, the gauge theory does not have to be planar but the connection with integrable models is restricted to a special class of supersymmetric observables.
N=2 gauge theories (partiton function) v.s. Quantum integrable models (TBA)
- A particularly interesting relation has been proposed by Nekrasov and Shatashvili (see the paper). They pointed out that the N=2 instanton partition function in a special limit of the Omega-deformation parameters is characterized by certain thermodynamic Bethe ansatz (TBA) like equations. Similar TBA equations have also been proposed to encode the spectrum of a large class of integrable systems. While this intriguing connection was never proved, Meneghelli and I provided an explicit derivation of this fact and also generalized it to quiver gauge theories. The proof was based on various techniques such as the (iterated) Mayer expansion, the method of expansion by regions, and the path integral tricks for non-perturbative summation (see our paper for more details).
- Such TBA equations derived entirely within gauge theory have been proposed to encode the spectrum of a large class of quantum integrable systems. Our derivation elucidates further this completely new point of view on the origin, as well as on the structure, of TBA equations in integrable models. One may hope that with certain knowledge it may be possible to reformulate gauge theories in terms of integrable models, which are in principle solvable. There are many open problems in this direction.