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Conformal Invariance, Unitarity and Two Dimensional Critical Exponents

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Vertex Operators in Mathematics and Physics

Part of the book series: Mathematical Sciences Research Institute Publications ((MSRI,volume 3))

Abstract

We show that conformai invariance and unitarity severely limit the possible values of critical exponents in two dimensional systems by finding the discrete series of unitarisable representations of the Virasoro algebra. The realization of conformai symmetry in a given system is parametrized by a real number c, the coefficient of the trace anomaly. For c<1 the only values allowed by unitarity are c=1–6/m(m+1), m=2,3,4 ⋯ . For each of these values of c unitarity determines a finite set of rational numbers that must contain all possible critical exponents. These finite sets account for the known critical exponents of the following two dimensional models: Ising(m=3), tricritical Ising(m=4), 3-state Potts(m=5), and tricritical 3-state Potts(m=6).

This work was supported in part by the U.S. Department of Energy contract DE-Ac02-81ER-10957, National Science Foundation grant NSF-DMR-82-16892 and the Alfred P. Sloan Foundation.

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Author information

Authors and Affiliations

  1. Enrico Fermi and James Franck Institutes and Department of Physics, University of Chicago, Chicago, IL, 60637, USA

    Daniel Friedan, Zongan Qiu & Stephen Shenker

Authors
  1. Daniel Friedan
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  2. Zongan Qiu
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  3. Stephen Shenker
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Editor information

Editors and Affiliations

  1. Department of Mathematics, Rutgers University, New Brunswick, NJ, 08903, USA

    J. Lepowsky

  2. Department of Physics, University of California, Berkeley, CA, 94720, USA

    S. Mandelstam

  3. Department of Mathematics, University of California, Berkeley, CA, 94720, USA

    I. M. Singer

  4. Mathematical Sciences Research Institute, 2223 Fulton Street, Room 603, Berkeley, CA, 94720, USA

    I. M. Singer

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© 1985 Springer-Verlag New York Inc.

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Friedan, D., Qiu, Z., Shenker, S. (1985). Conformal Invariance, Unitarity and Two Dimensional Critical Exponents. In: Lepowsky, J., Mandelstam, S., Singer, I.M. (eds) Vertex Operators in Mathematics and Physics. Mathematical Sciences Research Institute Publications, vol 3. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-9550-8_21

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  • DOI: https://doi.org/10.1007/978-1-4613-9550-8_21

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4613-9552-2

  • Online ISBN: 978-1-4613-9550-8

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