Final published version
Other version
Research output: Contribution to Journal/Magazine › Journal article › peer-review
Superconformal nets and noncommutative geometry. / Carpi, Sebastiano; Hillier, Robin; Longo, Roberto.
In: Journal of Noncommutative Geometry, Vol. 9, No. 2, 2015, p. 391-445.Research output: Contribution to Journal/Magazine › Journal article › peer-review
}
TY - JOUR
T1 - Superconformal nets and noncommutative geometry
AU - Carpi, Sebastiano
AU - Hillier, Robin
AU - Longo, Roberto
PY - 2015
Y1 - 2015
N2 - This paper provides a further step in our program of studying superconformal nets over S^1 from the point of view of noncommutative geometry. For any such net A and any family Delta of localized endomorphisms of the even part A^gamma of A, we define the locally convex differentiable algebra A_Delta with respect to a natural Dirac operator coming from supersymmetry. Having determined its structure and properties, we study the family of spectral triples and JLO entire cyclic cocycles associated to elements in Delta and show that they are nontrivial and that the cohomology classes of the cocycles corresponding to inequivalent endomorphisms can be separated through their even or odd index pairing with K-theory in various cases. We illustrate some of those cases in detail with superconformal nets associated to well-known CFT models, namely super-current algebra nets and super-Virasoro nets. All in all, the result allows us to encode parts of the representation theory of the net in terms of noncommutative geometry.
AB - This paper provides a further step in our program of studying superconformal nets over S^1 from the point of view of noncommutative geometry. For any such net A and any family Delta of localized endomorphisms of the even part A^gamma of A, we define the locally convex differentiable algebra A_Delta with respect to a natural Dirac operator coming from supersymmetry. Having determined its structure and properties, we study the family of spectral triples and JLO entire cyclic cocycles associated to elements in Delta and show that they are nontrivial and that the cohomology classes of the cocycles corresponding to inequivalent endomorphisms can be separated through their even or odd index pairing with K-theory in various cases. We illustrate some of those cases in detail with superconformal nets associated to well-known CFT models, namely super-current algebra nets and super-Virasoro nets. All in all, the result allows us to encode parts of the representation theory of the net in terms of noncommutative geometry.
KW - Operator algebras
KW - conformal field theory
KW - supersymmetry
KW - spectral triples
KW - cyclic cohomology
U2 - 10.4171/JNCG/196
DO - 10.4171/JNCG/196
M3 - Journal article
VL - 9
SP - 391
EP - 445
JO - Journal of Noncommutative Geometry
JF - Journal of Noncommutative Geometry
SN - 1661-6952
IS - 2
ER -