Operator products in 2-dimensional critical theories |
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| Institution: | 1. Department of Mathematics, University of Patras, 26110 Patras, Greece;2. Department of Nuclear and Particle Physics, Faculty of Physics, University of Athens, 15771 Athens, Greece;3. Mécanique et Gravitation, Université de Mons, 7000 Mons, Belgium;4. Department of Mathematics, University of Surrey, Guildford GU2 7XH, UK;1. Shenzhen Institute for Quantum Science and Engineering and Department of Physics, Southern University of Science and Technology, Shenzhen 518055, China;2. Naturwissenschaftlich-Technische Fakultät, Universität Siegen, Walter-Flex-Straße 3, D-57068 Siegen, Germany;3. Qian Xuesen Laboratory of Space Technology, China Academy of Space Technology, Beijing 100094, China;1. Hakodate National College of Technology, 14-1 Tokura-cho, Hakodate, Hokkaido 042-8501, Japan;2. Electronics and Photonics Research Institute, National Institute of Advanced Industrial Science and Technology (AIST), Tsukuba Central 2, Tsukuba 305-8568, Japan;1. Department of Mathematics, Vivekananda College, Kolkata-700063, India;2. Physics & Applied Mathematics Unit. Indian Statistical Institute, Kolkata-700 108, India |
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| Abstract: | A noteworthy feature of certain conformally invariant 2-dimensional theories, such as the Ising and 3-state Potts models at the critical point, is the existence of “degenerate primary fields” associated with nullvectors of the Virasoro algebra. Such fields are endowed with a remarkably simple multiplication table under the operator product expansion, known as the fusion rules. In addition, correlation functions made up of these fields satisfy a system of linear homogeneous partial differential equations. We show here that these two properties are intimately related: for any n-point function, the number of conformally invariant solutions to the system of equations equals the number of times that the identity operator appears in the fusion of all n fields in the correlator. This theorem permits the calculation of some apparently intractable correlation functions. Finally, we generalize these ideas to the Neveu-Schwarz sector of superconformal theories. |
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