Supersymmetric Casimir energy and SL(3,Z)SL(3,\mathbb{Z}) transformations

Nov 11, 2016
15 pages
Published in:
  • JHEP 07 (2017) 041
  • Published: Jul 3, 2017
e-Print:

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Abstract: (Springer)
We provide a recipe to extract the supersymmetric Casimir energy of theories defined on primary Hopf surfaces directly from the superconformal index. It involves an SL(3,Z) \mathrm{S}\mathrm{L}\left(3,\mathrm{\mathbb{Z}}\right) transformation acting on the complex structure moduli of the background geometry. In particular, the known relation between Casimir energy, index and partition function emerges naturally from this framework, allowing rewriting of the latter as a modified elliptic hypergeometric integral. We show this explicitly for N=1 \mathcal{N}=1 SQCD and N=4 \mathcal{N}=4 supersymmetric Yang-Mills theory for all classical gauge groups, and conjecture that it holds more generally. We also use our method to derive an expression for the Casimir energy of the nonlagrangian N=2 \mathcal{N}=2 SCFT with E6_{6} flavour symmetry. Furthermore, we predict an expression for Casimir energy of the N=1 \mathcal{N}=1 SP(2N) theory with SU(8) × U(1) flavour symmetry that is part of a multiple duality network, and for the doubled N=1 \mathcal{N}=1 theory with enhanced E7_{7} flavour symmetry.
Note:
  • 20 pages, more explicit examples added, published in JHEP
  • Supersymmetric Gauge Theory
  • Extended Supersymmetry
  • energy: Casimir
  • supersymmetry: 4
  • background: geometry
  • partition function
  • conformal
  • structure
  • Hopf
  • quantum chromodynamics: supersymmetry