Closure of the operator product expansion in the nonunitary bootstrap
We use the numerical conformal bootstrap in two dimensions to search for finite, closed subalgebras 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.
 Authors:

^{[1]};
^{[2]};
^{[1]}
 Stanford Univ., CA (United States). Stanford Inst. for Theoretical Physics
 Boston Univ., MA (United States). Dept. of Physics
 Publication Date:
 Grant/Contract Number:
 SC0010025
 Type:
 Accepted Manuscript
 Journal Name:
 Journal of High Energy Physics (Online)
 Additional Journal Information:
 Journal Name: Journal of High Energy Physics (Online); Journal Volume: 2016; Journal Issue: 11; Journal ID: ISSN 10298479
 Publisher:
 Springer Berlin
 Research Org:
 Boston Univ., MA (United States)
 Sponsoring Org:
 USDOE Office of Science (SC), High Energy Physics (HEP) (SC25)
 Contributing Orgs:
 Stanford Univ., CA (United States)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; Conformal Field Theory; Field Theories in Lower Dimensions
 OSTI Identifier:
 1368086
Esterlis, Ilya, Fitzpatrick, A. Liam, and Ramirez, David M.. Closure of the operator product expansion in the nonunitary bootstrap. United States: N. p.,
Web. doi:10.1007/JHEP11(2016)030.
Esterlis, Ilya, Fitzpatrick, A. Liam, & Ramirez, David M.. Closure of the operator product expansion in the nonunitary bootstrap. United States. doi:10.1007/JHEP11(2016)030.
Esterlis, Ilya, Fitzpatrick, A. Liam, and Ramirez, David M.. 2016.
"Closure of the operator product expansion in the nonunitary bootstrap". United States.
doi:10.1007/JHEP11(2016)030. https://www.osti.gov/servlets/purl/1368086.
@article{osti_1368086,
title = {Closure of the operator product expansion in the nonunitary bootstrap},
author = {Esterlis, Ilya and Fitzpatrick, A. Liam and Ramirez, David M.},
abstractNote = {We use the numerical conformal bootstrap in two dimensions to search for finite, closed subalgebras 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.},
doi = {10.1007/JHEP11(2016)030},
journal = {Journal of High Energy Physics (Online)},
number = 11,
volume = 2016,
place = {United States},
year = {2016},
month = {11}
}