Abstract
We use the numerical conformal bootstrap in two dimensions to search for finite, closed sub-algebras of the operator product expansion (OPE), without assuming unitarity. We find the minimal models as special cases, as well as additional lines of solutions that can be understood in the Coulomb gas formalism. All the solutions we find that contain the vacuum in the operator algebra are cases where the external operators of the bootstrap equation are degenerate operators, and we argue that this follows analytically from the expressions in arXiv:1202.4698 for the crossing matrices of Virasoro conformal blocks. Our numerical analysis is a special case of the "Gliozzi" bootstrap method, and provides a simpler setting in which to study technical challenges with the method. In the supplementary material, we provide a Mathematica notebook that automates the calculation of the crossing matrices and OPE coefficients for degenerate operators using the formulae of Dotsenko and Fateev.
Highlights
In the supplementary material, we provide a Mathematica notebook that automates the calculation of the crossing matrices and operator product expansion (OPE) coefficients for degenerate operators using the formulae of Dotsenko and Fateev
We find the minimal models as special cases, as well as additional lines of solutions that can be understood in the Coulomb gas formalism
All the solutions we find that contain the vacuum in the operator algebra are cases where the external operators of the bootstrap equation are degenerate operators, and we argue that this follows analytically from the expressions in arXiv:1202.4698 for the crossing matrices of Virasoro conformal blocks
Summary
The method we use will be analogous to that proposed in [9]. There, the authors worked with the global conformal algebra. Consider the four point function of identical, primary scalar operators, φ(x1)φ(x2)φ(x3)φ(x4) , with conformal weights h = h. Global conformal symmetry constrains the four point function to have the form. Using the expansion (2.4), we write this as a sum rule: ap[F (c, hp, h, z)F(c, hp, h, z) − F (c, hp, h, 1 − z)F(c, hp, h, 1 − z)] = 0 Expanding this about the point z = z = 1/2 gives an infinite set of homogeneous equations apgh(m,h,n) = 0, p (2.6). The OPE (1.2) corresponds to N = 2 Virasoro primaries with the central charge c and conformal weight h of operator φ as the only free parameters. Taking M > 2, we obtain an over-constrained system of κ equations for c and h Solutions to this system give four point functions consistent with crossing symmetry, containing the single primary operator φ. It seems likely that studying larger finite closed sub-algebras may be an ideal setup to explore in even greater detail how to reduce systematic uncertainties in the methods of [9] more generally.
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