Communications in Mathematical Physics

, Volume 272, Issue 2, pp 345–396

Full Field Algebras


    • Department of MathematicsRutgers University
  • Liang Kong
    • Department of MathematicsRutgers University
    • Max Planck Institute for Mathematics in the Sciences
    • Institut Des Hautes Études Scientifiques

DOI: 10.1007/s00220-007-0224-4

Cite this article as:
Huang, Y. & Kong, L. Commun. Math. Phys. (2007) 272: 345. doi:10.1007/s00220-007-0224-4


We introduce a notion of full field algebra which is essentially an algebraic formulation of the notion of genus-zero full conformal field theory. For any vertex operator algebras VL and VR, \({V^{L}\otimes V^{R}}\) is naturally a full field algebra and we introduce a notion of full field algebra over \({V^{L}\otimes V^{R}}\) . We study the structure of full field algebras over \({V^{L}\otimes V^{R}}\) using modules and intertwining operators for VL and VR. For a simple vertex operator algebra V satisfying certain natural finiteness and reductivity conditions needed for the Verlinde conjecture to hold, we construct a bilinear form on the space of intertwining operators for V and prove the nondegeneracy and other basic properties of this form. The proof of the nondegenracy of the bilinear form depends not only on the theory of intertwining operator algebras but also on the modular invariance for intertwining operator algebras through the use of the results obtained in the proof of the Verlinde conjecture by the first author. Using this nondegenerate bilinear form, we construct a full field algebra over \({V\otimes V}\) and an invariant bilinear form on this algebra.

Copyright information

© Springer-Verlag 2007