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 $\tau =\frac{\theta }{2\pi }+\frac{4\pi ı}{g^2}⟶-\frac{1}{\tau }$. The low-energy Seiberg-Witten prepotential $\mathcal{F}(a)$, however, is not explicitly invariant, because the flat moduli also change $a⟶a_D=\partial \mathcal{F}/\partial a$. 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 {\it two} independent modular parameters $\tau$ and $\hat{\tau }$, the modular anomaly equation changes, because the modular transform of $\tau$ is accompanied by an ($N$-dependent!) shift of $\hat{\tau }$ and vice versa. This is a new peculiarity of double-elliptic systems, which deserves further investigation.

23 pages

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Cited by in corpus (8)

• $(q,t)$-KZ equations for quantum toroidal algebra and Nekrasov partition functions on ALE spaces
• Higgsed network calculus
• The Quantum DELL System
• Duality in elliptic Ruijsenaars system and elliptic symmetric functions
• On complete solution of the quantum Dell system
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• On Cherednik and Nazarov-Sklyanin large N limit construction for integrable many-body systems with elliptic dependence on momenta