Modular properties of 6d (DELL) systems

Abstract

If super-Yang-Mills theory possesses the exact conformal invariance, there is an additional modular invariance under the change of the complex bare charge τ = θ 2π + 4πı g 2 −→ − 1 τ . The low-energy Seiberg-Witten prepotential F (a), however, is not explicitly invariant, because the flat moduli also change a −→ a D = ∂F /∂a. In result, the prepotential is not a modular form and depends also on the anomalous Eisenstein series E 2 . This dependence is usually described by the universal MNW modular anomaly equation. We demonstrate that, in the 6d SU(N ) theory with two independent modular parameters τ andτ , the modular anomaly equation changes, because the modular transform of τ is accompanied by an (N -dependent!) shift ofτ and vice versa. This is a new peculiarity of double-elliptic systems, which deserves further investigation.

On a complete solution of the quantum Dell system

“…Relation of these systems to topological strings and extension to the 6d Nekrasov functions was later discussed in [20,21]. Moreover, in [22], using solutions to the elliptic Knizhnik-Zamolodchikov equations, we discussed the modular properties of these 6d gauge theories described by Dell systems and derived in [23,24].…”

Spectral duality in elliptic systems, six-dimensional gauge theories and topological strings

“…These systems have not been studied in full yet, because of a very involved structure (see [150] for some new advances). As usual, they appear from explicit expressions for partition function in section 3 in quasiclassical limit 1 ∼ 2 −→ 0, while in Nekrasov-Shatashvili limit [84][85][86] (when only 2 −→ ∞) we get their straightforward quantization, when the spectral curve is substituted by a Baxter equation (quantum spectral curve).…”

Characteristic determinant and Manakov triple for the double elliptic integrable system

“…For this reason a definition of the standard set of algebraic tools for integrable systems (including Lax pairs, R-matrix structures, exchange relations etc) appeared to be a complicated problem. The classical Poisson structures underlying the Dell model were studied in [2][3][4]16].…”