Conformal Nets and KK-Theory

Carpi, Sebastiano; Conti, Roberto; Hillier, Robin

doi:10.15352/afa/1399899832

Cited by 4 publications

(14 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…On the other hand, for N > 1 there are other C*-subalgebras with this property [3]. However, as shown again in [13,Prop.3.4] K A has another property which plays a crucial role in [12,13] and will play an important role in the following namely, if ρ is a covariant localized endomorphism of C * (A) and π 0 • ρ has finite statistical dimension thenρ(K A ) ⊂ K A .…”

Section: Loop Group Representations Conformal Nets and K-theorymentioning

confidence: 93%

“…In the following sections we will look at this kind of pairing from the point of view of Kasparov's KK-theory, for which we are now providing a few preliminaries based on the summary in [12]. A thorough introduction with proofs can be found in [4, Ch.17&18].…”

Section: Entire Cyclic Cohomology For Nonunital Banach Algebrasmentioning

confidence: 99%

“…, N − 1, we choose a lowest energy unit vector Ω i in H π i and denote by q Ω i the corresponding one-dimensional projection in K(H π i ) and let p i be the unique minimal projection in K A such that π i (p i ) = q Ω i . By [12,13], the maps…”

Section: Loop Group Representations Conformal Nets and K-theorymentioning

confidence: 99%

“…The above construction naturally identifies the additive group underlying the fusion ring R ℓ (L G) with the K-groups K 0 (K G ℓ ) and K 0 (K G ℓ ). A K-theoretical description of the fusion product can know be given following the ideas in [13,12]. The DHR endomorphisms of the net A G ℓ give rise to endomorphisms of the C*-algebra K G ℓ and, as a consequence, an injective ring homomorphism from R ℓ (L G) into KK(K G ℓ , K G ℓ ) ≃ End(K 0 (K G ℓ )) ≃ End(K 0 (K G ℓ )) so that the fusion product can be naturally described in terms of the Kasparov product in KK-theory.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Loop groups and noncommutative geometry

Carpi

Hillier

2017

Rev. Math. Phys.

Self Cite

View full text Add to dashboard Cite

We describe the representation theory of loop groups in terms of K-theory and noncommutative geometry. This is done by constructing suitable spectral triples associated with the level l projective unitary positive-energy representations of any given loop group LG. The construction is based on certain supersymmetric conformal field theory models associated with LG. We then generalize the construction, in the setting of conformal nets, to many other rational chiral conformal field theory models including coset models and the moonshine conformal net.Comment: Minor revisio

show abstract

Section: Loop Group Representations Conformal Nets and K-theorymentioning

confidence: 93%

Section: Entire Cyclic Cohomology For Nonunital Banach Algebrasmentioning

confidence: 99%

Section: Loop Group Representations Conformal Nets and K-theorymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Loop groups and noncommutative geometry

Carpi

Hillier

2017

Rev. Math. Phys.

Self Cite

View full text Add to dashboard Cite

show abstract

“…A general and purely topological approach, dispensing with supersymmetry, has been recently achieved in [CCHW13,CCH13] for completely rational local conformal nets. We expect a deeper relation to the present work in the case where the completely rational net is the even part of a superconformal net.…”

Section: Introductionmentioning

confidence: 99%

Superconformal nets and noncommutative geometry

Carpi¹,

Hillier²,

Longo³

2015

J. Noncommut. Geom.

Self Cite

View full text Add to dashboard Cite

This paper provides a further step in the program of studying superconformal nets over S 1 from the point of view of noncommutative geometry. For any such net A and any family ∆ of localized endomorphisms of the even part A γ of A, we define the locally convex differentiable algebra A ∆ 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 ∆ 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 supercurrent 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.

show abstract

N =2 Superconformal Nets

Carpi

Hillier

Kawahigashi

et al. 2014

Commun. Math. Phys.

Self Cite

View full text Add to dashboard Cite

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 superVirasoro 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.

show abstract

Conformal Nets and KK-Theory

Cited by 4 publications

References 17 publications

Loop groups and noncommutative geometry

Loop groups and noncommutative geometry

Superconformal nets and noncommutative geometry

N =2 Superconformal Nets

Contact Info

Product

Resources

About