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Black holes and class groups

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Abstract

The theory of quadratic forms and class numbers has connections to many classical problems in number theory. Recently, class numbers have appeared in the study of black holes in string theory. We describe this connection and raise questions in the hope of inspiring new collaborations between number theorists and physicists.

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Notes

  1. By supersymmetric or BPS black holes, we mean those black hole solutions which preserve some of the supersymmetry of the underlying string theory. In general, states which preserve some supersymmetry are called BPS states.

  2. At \(k=1\), the theory of Frenkel–Lepowsky–Meurman [20] which plays a starring role in Monstrous moonshine is extremal.

  3. For \(m=2,4\), explicit CFTs which realize this exist, and are constructed in [6, 10].

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Authors and Affiliations

  1. Stanford Institute for Theoretical Physics, Stanford University, Palo Alto, CA, 94305, USA

    Nathan Benjamin & Shamit Kachru

  2. Department of Mathematics and Computer Science, Emory University, Atlanta, GA, 30322, USA

    Ken Ono

  3. Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville, TN, 37240, USA

    Larry Rolen

Authors
  1. Nathan Benjamin
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  2. Shamit Kachru
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  3. Ken Ono
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  4. Larry Rolen
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Corresponding author

Correspondence to Larry Rolen.

Additional information

In celebration of Don Zagier’s 66th birthday.

N.B. acknowledges support from the NSF and a Stanford Graduate Fellowship, S.K. is grateful for support from the NSF and a Simons Investigator Award, and K.O. thanks the Asa Griggs Candler Fund and the NSF for support. The authors thank Michael Mertens for helpful comments on an earlier version of this manuscript.

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Benjamin, N., Kachru, S., Ono, K. et al. Black holes and class groups. Res Math Sci 5, 43 (2018). https://doi.org/10.1007/s40687-018-0164-y

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