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3d and 5d Gauge Theory Partition Functions as q-deformed CFT Correlators

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Abstract

3d \({\mathcal{N}=2}\) partition functions on the squashed three-sphere \({S^{3}_{b}}\) and on the twisted product \({S^{2} \times S^{1}}\) have been shown to factorize into sums of squares of solid tori partition functions, the so-called holomorphic blocks. The same set of holomorphic blocks realizes \({S^{3}_{b}}\) and \({S^{2} \times S^{1}}\) partition functions but the two cases involve different inner products, the S-pairing and the id-pairing, respectively. We define a class of q-deformed CFT correlators where chiral blocks are controlled by a deformation of Virasoro symmetry and are paired by S-pairing and id-pairing, respectively. Applying the bootstrap approach to a class of degenerate correlators we are able to derive three-point functions. We show that degenerate correlators can be mapped to 3d partition functions while the crossing symmetry of correlators corresponds to the flop symmetry of 3d gauge theories. We explore how non-degenerate q-deformed correlators are related to 5d partition functions. We argue that id-pairing correlators are associated with the superconformal index on \({S^{4} \times S^{1}}\) while S-pairing three-point function factors capture the one-loop part of S 5 partition functions. This is consistent with the interpretation of \({S^{2} \times S^{1}}\) and \({S^{3}_{b}}\) gauge theories as codimension two defect theories inside \({S^{4} \times S^{1}}\) and S 5, respectively.

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Authors and Affiliations

  1. Department of Mathematics, University of Surrey, Guildford, Surrey, GU2 7XH, UK

    Fabrizio Nieri & Sara Pasquetti

  2. PH-TH Division, CERN, 1211, Geneva, Switzerland

    Filippo Passerini

  3. Department of Physics, Princeton University, Princeton, NJ, 08544, USA

    Filippo Passerini

  4. Laboratoire de Physique Théorique, Unité Mixte du CNRS et de l’Ecole Normale Supérieure associée à l’Université Pierre et Marie Curie 6, UMR 8549, Ecole Normale Supérieure, 75005, Paris, France

    Filippo Passerini

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  1. Fabrizio Nieri
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Correspondence to Sara Pasquetti.

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Nieri, F., Pasquetti, S. & Passerini, F. 3d and 5d Gauge Theory Partition Functions as q-deformed CFT Correlators. Lett Math Phys 105, 109–148 (2015). https://doi.org/10.1007/s11005-014-0727-9

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