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The Cantor–Bendixson Rank of Certain Bridgeland–Smith Stability Conditions

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Abstract

We provide a novel proof that the set of directions that admit a saddle connection on a meromorphic quadratic differential with at least one pole of order at least two is closed, which generalizes a result of Bridgeland and Smith, and Gaiotto, Moore, and Neitzke. Secondly, we show that this set has finite Cantor–Bendixson rank and give a tight bound. Finally, we present a family of surfaces realizing all possible Cantor–Bendixson ranks. The techniques in the proof of this result exclusively concern Abelian differentials on Riemann surfaces, also known as translation surfaces. The concept of a “slit translation surface” is introduced as the primary tool for studying meromorphic quadratic differentials with higher order poles.

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Authors and Affiliations

  1. Brooklyn College (CUNY), Brooklyn, NY, USA

    David Aulicino

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  1. David Aulicino
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Correspondence to David Aulicino.

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Communicated by X. Yin

This material is based upon work supported by the ERC Starting Grant “Quasiperiodic” of Artur Avila, and later by the National Science Foundation under Award No. DMS - 1204414.

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Aulicino, D. The Cantor–Bendixson Rank of Certain Bridgeland–Smith Stability Conditions. Commun. Math. Phys. 357, 791–809 (2018). https://doi.org/10.1007/s00220-017-3028-1

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  • DOI: https://doi.org/10.1007/s00220-017-3028-1

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