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\( \mathcal{N}=2 \) gauge theories on toric singularities, blow-up formulae and W-algebrae

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Abstract

We compute the Nekrasov partition function of gauge theories on the (resolved) toric singularities \( {{{{{\mathbb{C}}^2}}} \left/ {\varGamma } \right.} \) in terms of blow-up formulae. We discuss the expansion of the partition function in the ϵ 1, ϵ 2 → 0 limit along with its modular properties and how to derive them from the M-theory perspective. On the two-dimensional conformal field theory side, our results can be interpreted in terms of representations of the direct sum of Heisenberg plus W N -algebrae with suitable central charges, which can be computed from the fan of the resolved toric variety. We provide a check of this correspondence by computing the central charge of the two-dimensional theory from the anomaly polynomial of M5-brane theory. Upon using the AGT correspondence our results provide a candidate for the conformal blocks and three-point functions of a class of the two-dimensional CFTs which includes parafermionic theories.

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Authors and Affiliations

  1. SISSA and INFN, Sezione di Trieste, via Bonomea 265, 34136, Trieste, Italy

    Giulio Bonelli, Kazunobu Maruyoshi, Alessandro Tanzini & Futoshi Yagi

  2. I.C.T.P., Strada Costiera 11, 34014, Trieste, Italy

    Giulio Bonelli

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  1. Giulio Bonelli
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  2. Kazunobu Maruyoshi
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Correspondence to Kazunobu Maruyoshi.

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ArXiv ePrint: 1208.0790

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Bonelli, G., Maruyoshi, K., Tanzini, A. et al. \( \mathcal{N}=2 \) gauge theories on toric singularities, blow-up formulae and W-algebrae. J. High Energ. Phys. 2013, 14 (2013). https://doi.org/10.1007/JHEP01(2013)014

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