Inozemtsev system as Seiberg-Witten integrable system

Abstract

In this work we establish that the Inozemtsev system is the Seiberg-Witten integrable system encoding the Coulomb branch physics of 4d $$ \mathcal{N} $$ N = 2 USp(2N) gauge theory with four fundamental and (for N ≥ 2) one antisymmetric tensor hypermultiplets. We describe the transformation from the spectral curves and canonical one-forms of the Inozemtsev system in the N = 1 and N = 2 cases to the Seiberg-Witten curves and differentials explicitly, along with the explicit matching of the modulus of the elliptic curve of spectral parameters to the gauge coupling of the field theory, and of the couplings of the Inozemtsev system to the field theory mass parameters. This result is a particular instance of a more general correspondence between crystallographic elliptic Calogero-Moser systems with Seiberg-Witten integrable systems, which will be explored in future work.

Gaudin models and multipoint conformal blocks: general theory

“…[53][54][55]. The same limit of a 4-site Gaudin model -which may be identified with the elliptic Inozemtsev model [56] -has also appeared in the physics literature recently [57]. Eberhardt, Komatsu and Mizera have shown that the limit theory coincides with the hyperbolic Calogero-Sutherland model, which is known to describe the Casimir equations of 4-point conformal blocks.…”

“…Let us also recall that the operators with ν = p/2 have been omitted to account for the relation between the vertex and Casimir differential operators for the third leg. The reduction of T 3 = T (56) to a symmetric traceless tensor implies the existence of p − 2d 3 − 1 relations between p order monomials. The only useful relation in this case is the one produced for p = 6, coming from the expansion…”

Gaudin models and multipoint conformal blocks. Part II. Comb channel vertices in 3D and 4D

“…For the single variable vertices we analyzed in this paper we will address the solution theory in future research. In this context, it may be interesting to note that crystallographic elliptic CMS models have made a recent equally unexpected appearance in the context of Seiberg-Witten theory for N = 2 supersymmetric gauge theory [61]. For the lemniscatic model we described above, a gauge theory realization has also been announced.…”

Inozemtsev System as Seiberg-Witten Integrable system

“…In this and upcoming work [16], we will follow the line of the first approach to identify the Seiberg-Witten systems for several series of 4d N = 2 superconformal field theories which all admit F-theory constructions. A common feature shared by those theories is that their Coulomb branch chiral rings are given by the rings of symmetric polynomials with respect to certain complex reflection groups [17].…”