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Quantum line defects and refined BPS spectra

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In this note, we study refined BPS invariants associated with certain quantum line defects in quantum field theories of class \({{\mathcal {S}}}\). Such defects can be specified via geometric engineering in the UV by assigning a path on a certain curve. In the IR, they are described by framed BPS quivers. We study the associated BPS spectral problem, including the spin content. The relevant BPS invariants arise from the K-theoretic enumerative geometry of the moduli spaces of quiver representations, adapting a construction by Nekrasov and Okounkov. In particular, refined framed BPS states are described via Euler characteristics of certain complexes of sheaves.

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  1. There is in principle the possibility of the corresponding particle state being described by a different statistics; it does not appear to be the case in all the examples we have studied, but we do not have a clear argument for this.

  2. This condition effectively tells us that we are only interested in representations for which the arrows going to the framing node are represented trivially. Therefore, these arrows can be set to zero in the F-term relations and do not contribute to the construction of the moduli spaces.

  3. For simplicity, we omit those terms associated with the framing arrows which can be neglected in the localization computation.


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I am grateful to Michele del Zotto, Vivek Shende, Yan Soibelman, Vasily Pestun, and Johannes Wälcher for discussions. I am thankful to the organizers of the program Symplectic Geometry and Representation Theory at the Hausdorff Institute for Mathematics in Bonn for the warm hospitality during the last stages of this project. These results were presented at the workshops Geometry and Topology inspired by Physics in 2018 in Ascona and Young Researchers in String Mathematics in 2017 in Bonn, and I am grateful to the organizers for the invitation to speak and for the warm hospitality. I am a member of INDAM-GNFM, I am supported by INFN via the Iniziativa Specifica GAST and by the FRA2018 project “K-theoretic Enumerative Geometry in Mathematical Physics”.

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Cirafici, M. Quantum line defects and refined BPS spectra. Lett Math Phys 110, 501–531 (2020).

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