Duality and modularity in elliptic integrable systems and vacua of \( \mathcal{N}={1}^{\ast } \) gauge theories

  • Antoine Bourget
  • Jan TroostEmail author
Open Access
Regular Article - Theoretical Physics


We study complexified elliptic Calogero-Moser integrable systems. We determine the value of the potential at isolated extrema, as a function of the modular parameter of the torus on which the integrable system lives. We calculate the extrema for low rank B,C,D root systems using a mix of analytical and numerical tools. For so(5) we find convincing evidence that the extrema constitute a vector valued modular form for the Γ0(4) congruence subgroup of the modular group. For so(7) and so(8), the extrema split into two sets. One set contains extrema that make up vector valued modular forms for congruence subgroups (namely Γ0(4), Γ(2) and Γ(3)), and a second set contains extrema that exhibit monodromies around points in the interior of the fundamental domain. The former set can be described analytically, while for the latter, we provide an analytic value for the point of monodromy for so(8), as well as extensive numerical predictions for the Fourier coefficients of the extrema. Our results on the extrema provide a rationale for integrality properties observed in integrable models, and embed these into the theory of vector valued modular forms. Moreover, using the data we gather on the modularity of complexified integrable system extrema, we analyse the massive vacua of mass deformed \( \mathcal{N}=4 \) supersymmetric Yang-Mills theories with low rank gauge group of type B, C and D. We map out their transformation properties under the infrared electric-magnetic duality group as well as under triality for \( \mathcal{N}={1}^{\ast } \) with gauge algebra so(8). We compare the exact massive vacua on \( {\mathbb{R}}^3\times {S}^1 \) to those found in a semi-classical analysis on \( {\mathbb{R}}^4 \). We identify several intriguing features of the quantum gauge theories.


Supersymmetric gauge theory Integrable Field Theories Supersymmetry and Duality 


Open Access

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Copyright information

© The Author(s) 2015

Authors and Affiliations

  1. 1.Laboratoire de Physique Théorique, Ecole Normale SupérieureParisFrance

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