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N =2 Superconformal Nets

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Abstract

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 we define and study an operator algebraic version of the N = 2 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 has seemingly not been completely proved so far. Finally, the chiral ring is discussed in terms of net representations.

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Authors and Affiliations

  1. Dipartimento di Economia, Università di Chieti-Pescara “G. d’Annunzio”, Viale Pindaro, 42, 65127, Pescara, Italy

    Sebastiano Carpi

  2. Department of Mathematics and Statistics, Lancaster University, Lancaster, LA1 4YF, UK

    Robin Hillier

  3. Department of Mathematical Sciences, The University of Tokyo, Komaba, Tokyo, 153-8914, Japan

    Yasuyuki Kawahigashi

  4. Kavli IPMU (WPI), The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, 277-8583, Japan

    Yasuyuki Kawahigashi

  5. Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica, 1, 00133, Roma, Italy

    Roberto Longo

  6. Department of Mathematics, University of California at Riverside, Riverside, CA, 92521, USA

    Feng Xu

Authors
  1. Sebastiano Carpi
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  2. Robin Hillier
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  3. Yasuyuki Kawahigashi
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  4. Roberto Longo
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  5. Feng Xu
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Corresponding author

Correspondence to Roberto Longo.

Additional information

Communicated by A. Connes

Work supported in part by the ERC Advanced Grant 227458 OACFT “Operator Algebras and Conformal Field Theory”.

S. Carpi and R. Longo are supported in part by PRIN-MIUR and GNAMPA-INDAM.

Y. Kawahigashi is supported in part by Global COE Program “The research and training center for new development in mathematics”, the Mitsubishi Foundation Research Grants and the Grants-in-Aid for Scientific Research, JSPS.

F. Xu is supported in part by NSF grant and an academic senate grant from UCR.

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Carpi, S., Hillier, R., Kawahigashi, Y. et al. N =2 Superconformal Nets. Commun. Math. Phys. 336, 1285–1328 (2015). https://doi.org/10.1007/s00220-014-2234-3

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