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Superconformal nets and noncommutative geometry

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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

Harvard

Carpi, S, Hillier, R & Longo, R 2015, 'Superconformal nets and noncommutative geometry', Journal of Noncommutative Geometry, vol. 9, no. 2, pp. 391-445. https://doi.org/10.4171/JNCG/196

APA

Carpi, S., Hillier, R., & Longo, R. (2015). Superconformal nets and noncommutative geometry. Journal of Noncommutative Geometry, 9(2), 391-445. https://doi.org/10.4171/JNCG/196

Vancouver

Carpi S, Hillier R, Longo R. Superconformal nets and noncommutative geometry. Journal of Noncommutative Geometry. 2015;9(2):391-445. doi: 10.4171/JNCG/196

Author

Carpi, Sebastiano ; Hillier, Robin ; Longo, Roberto. / Superconformal nets and noncommutative geometry. In: Journal of Noncommutative Geometry. 2015 ; Vol. 9, No. 2. pp. 391-445.

Bibtex

@article{5fc32ac48d7a45ef9656db98004c0a31,
title = "Superconformal nets and noncommutative geometry",
abstract = "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. ",
keywords = "Operator algebras, conformal field theory, supersymmetry, spectral triples, cyclic cohomology",
author = "Sebastiano Carpi and Robin Hillier and Roberto Longo",
year = "2015",
doi = "10.4171/JNCG/196",
language = "English",
volume = "9",
pages = "391--445",
journal = "Journal of Noncommutative Geometry",
issn = "1661-6952",
publisher = "European Mathematical Society Publishing House",
number = "2",

}

RIS

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 -