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Disentangling a deep learned volume formula
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  • Regular Article - Theoretical Physics
  • Open Access
  • Published: 07 June 2021

Disentangling a deep learned volume formula

  • Jessica Craven1,
  • Vishnu Jejjala1 &
  • Arjun Kar  ORCID: orcid.org/0000-0003-1943-43462 

Journal of High Energy Physics volume 2021, Article number: 40 (2021) Cite this article

  • 355 Accesses

  • 4 Citations

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A preprint version of the article is available at arXiv.

Abstract

We present a simple phenomenological formula which approximates the hyperbolic volume of a knot using only a single evaluation of its Jones polynomial at a root of unity. The average error is just 2.86% on the first 1.7 million knots, which represents a large improvement over previous formulas of this kind. To find the approximation formula, we use layer-wise relevance propagation to reverse engineer a black box neural network which achieves a similar average error for the same approximation task when trained on 10% of the total dataset. The particular roots of unity which appear in our analysis cannot be written as e2πi/(k+2) with integer k; therefore, the relevant Jones polynomial evaluations are not given by unknot-normalized expectation values of Wilson loop operators in conventional SU(2) Chern-Simons theory with level k. Instead, they correspond to an analytic continuation of such expectation values to fractional level. We briefly review the continuation procedure and comment on the presence of certain Lefschetz thimbles, to which our approximation formula is sensitive, in the analytically continued Chern-Simons integration cycle.

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Authors and Affiliations

  1. Mandelstam Institute for Theoretical Physics, School of Physics, NITheP, and CoE-MaSS, University of the Witwatersrand, Johannesburg, WITS 2050, South Africa

    Jessica Craven & Vishnu Jejjala

  2. Department of Physics and Astronomy, University of British Columbia, 6224 Agricultural Road, Vancouver, BC, V6T 1Z1, Canada

    Arjun Kar

Authors
  1. Jessica Craven
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  2. Vishnu Jejjala
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  3. Arjun Kar
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Correspondence to Jessica Craven.

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ArXiv ePrint: 2012.03955

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Craven, J., Jejjala, V. & Kar, A. Disentangling a deep learned volume formula. J. High Energ. Phys. 2021, 40 (2021). https://doi.org/10.1007/JHEP06(2021)040

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  • Received: 16 March 2021

  • Revised: 28 April 2021

  • Accepted: 19 May 2021

  • Published: 07 June 2021

  • DOI: https://doi.org/10.1007/JHEP06(2021)040

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Keywords

  • Chern-Simons Theories
  • Topological Field Theories
  • Wilson
  • ’t Hooft and Polyakov loops
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