Sebastiano Carpi, Robin Hillier, Yasuyuki Kawahigashi, Roberto Longo, Feng Xu
July 10, 2012
We provide an operator algebraic approach to $N=2$ chiral conformal field theory and set up the noncommutative geometric framework. Compared to the $N=1$ case, the structure here is much richer. There are naturally associated nets of spectral triples and the JLO cocycles separate the Ramond sectors. We construct the $N=2$ superconformal nets of von Neumann algebras in general, classify them in the discrete series $c<3$, and study spectral flow. We prove the coset identification for the $N=2$ super-Virasoro nets with $c<3$, a key result whose equivalent in the vertex algebra context is seemingly not complete. Finally, the chiral ring is discussed in terms of net representations.