D-Branes, RR-Fields and Duality on Noncommutative Manifolds

Brodzki, Jacek; Mathai, Varghese; Rosenberg, Jonathan; Szabo, Richard J.

doi:10.1007/s00220-007-0396-y

Cited by 47 publications

(119 citation statements)

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“…Theorem 1.7). It is known that under favourable circumstances topological T-duality can be expressed as a KK-equivalence between two separable C * -algebras [3,4]. Thus our results show that in such cases one actually has an isomorphism of noncommutative motives that implements the well-known isomorphism (up to a shift) between the twisted K-theories.…”

Section: Introductionsupporting

confidence: 59%

“…This formalism extends to certain infinite dimensional spaces through the use of σ-C * -algebras [43]. In [3,4] the authors extended the formalism of T-duality to C * -algebras and showed that under favourable circumstances if B and B ′ are T-dual C * -algebras, then there is an invertible element in KK 0 (B, ΣB ′ ) that implements the twisted K-theory isomorphism (as in (2)). The Connes-Skandalis picture of KK-theory [12] and Rieffel's imprimitivity result [50] are pertinent to their construction.…”

Section: Our Resultsmentioning

confidence: 99%

“…The nonconnective K-theory spectrum of A, denoted by K(A), is the spectrum whose n-th space is K w (Σ n (Perf(A))) and the structure maps are furnished by (3). Any unital k-algebra R can be viewed as a DG category with one object •, such that End(•) = R. In this case the above construction applied to E = Perf(R) recovers Quillen's higher algebraic K-theory of R in nonnegative degrees and Bass' negative K-theory in negative degrees.…”

Section: 3mentioning

confidence: 99%

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Algebraic K-theory, K-regularity, and 𝕋 $\mathbb {T}$ -duality of 𝒪 ∞ $\mathcal {O}_{\infty }$ -stable C ∗-algebras

Mahanta

2015

Math Phys Anal Geom

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Abstract. We develop an algebraic formalism for topological T-duality. More precisely, we show that topological T-duality actually induces an isomorphism between noncommutative motives that in turn implements the well-known isomorphism between twisted K-theories (up to a shift). In order to establish this result we model topological K-theory by algebraic K-theory. We also construct an E ∞ -operad starting from any strongly self-absorbing C * -algebra D. Then we show that there is a functorial topological K-theory symmetric spectrum construction K top Σ (−) on the category of separable C * -algebras, such that K top Σ (D) is an algebra over this operad; moreover, K top Σ (A⊗D) is a module over this algebra. Along the way we obtain a new symmetric spectra valued functorial model for the (connective) topological K-theory of C * -algebras. We also show that O ∞ -stable C * -algebras are K-regular providing evidence for a conjecture of Rosenberg. We conclude with an explicit description of the algebraic K-theory of ax + b-semigroup C * -algebras coming from number theory and that of O ∞ -stabilized noncommutative tori. IntroductionWithin the category of separable C * -algebras SC * the stable ones, i.e., those A ∈ SC * satisfying A⊗K ∼ = A, play a privileged role. For instance, it is known that they satisfy the Karoubi conjecture and appear very naturally in the context of twisted K-theory. The results in this article demonstrate that O ∞ -stable separable C * -algebras, i.e., those A ∈ SC * satisfying A⊗O ∞ ∼ = A, deserve a similar prominent status. Moreover, the Cuntz algebra O ∞ is strongly self-absorbing, which has several interesting ramifications.Ever since its inception by Kontsevich [33] the homological mirror symmetry conjecture has promoted rich interaction between geometry, algebra, and (higher) category theory. Mirror symmetry is related to T-duality via the Strominger-Yau-Zaslow conjecture [61] and hence we believe that it is worthwhile to have an algebraic formalism for T-duality at our disposal. One aspect of this theory is the Bunke-Schick topological T-duality, which has received a lot of attention in the mathematical literature. It is insensitive to subtle geometric structures but its mathematical underpinnings are very well understood [8,7,9]. One of the objectives of this article is to develop an algebraic formalism for topological T-duality relating it to the theory of noncommutative motives [34,35,65,45]. Along the way we obtain several interesting applications to algebraic K-theory and K-regularity of C * -algebras. The novelty 2010 Mathematics Subject Classification. 46L85, 19Dxx, 46L80, 57R56.

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Section: Introductionsupporting

confidence: 59%

Section: Our Resultsmentioning

confidence: 99%

Section: 3mentioning

confidence: 99%

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Algebraic K-theory, K-regularity, and 𝕋 $\mathbb {T}$ -duality of 𝒪 ∞ $\mathcal {O}_{\infty }$ -stable C ∗-algebras

Mahanta

2015

Math Phys Anal Geom

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“…Denote by D the equivariant Dirac operator constructed by the authors in [3]. In section 4, we prove that there is no operator other than the scalars in the commutant of π(C(SU q (2))) that has bounded commutator with D. An important consequence of this is that the equivariant spectral triple does not give a K-homology fundamental class for SU q (2). In the final section, we show that Poincaré duality holds for SU q (2).…”

Section: Introductionmentioning

confidence: 93%

“…In an earlier paper ( [3]), the authors constructed an equivariant spectral triple for the quantum SU (2) group that was later analysed further by Connes in [8]. It is natural to ask whether the triple gives rise to a fundamental class for SU q (2). This is what we try to answer in this paper.…”

Section: Introductionmentioning

confidence: 94%

Equivariant spectral triples and Poincaré duality for 𝑆𝑈_{𝑞}(2)

Chakraborty

Pal

2010

Trans. Amer. Math. Soc.

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Abstract. Let A be the C * -algebra associated with SU q (2), let π be the representation by left multiplication on the L 2 space of the Haar state and let D be the equivariant Dirac operator for this representation constructed by the authors earlier. We prove in this article that there is no operator other than the scalars in the commutant π(A) that has bounded commutator with D. This implies that the equivariant spectral triple under consideration does not admit a rational Poincaré dual in the sense of Moscovici, which in particular means that this spectral triple does not extend to a K-homology fundamental class for SU q (2). We also show that a minor modification of this equivariant spectral triple gives a fundamental class and thus implements Poincaré duality.

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Noncommutative gauge theories on D-branes in non-geometric backgrounds

Hull

Szabo

2019

J. High Energ. Phys.

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We investigate the noncommutative gauge theories arising on the worldvolumes of D-branes in non-geometric backgrounds obtained by T-duality from twisted tori. We revisit the low-energy effective description of D-branes on three-dimensional T-folds, examining both cases of parabolic and elliptic twists in detail. We give a detailed description of the decoupling limits and explore various physical consequences of the open string non-geometry. The T-duality monodromies of the non-geometric backgrounds lead to Morita duality monodromies of the noncommutative Yang-Mills theories induced on the D-branes. While the parabolic twists recover the wellknown examples of noncommutative principal torus bundles from topological T-duality, the elliptic twists give new examples of noncommutative fibrations with non-geometric torus fibres. We extend these considerations to D-branes in backgrounds with R-flux, using the doubled geometry formulation, finding that both the non-geometric background and the D-brane gauge theory necessarily have explicit dependence on the dual coordinates, and so have no conventional formulation in spacetime.

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D-Branes, RR-Fields and Duality on Noncommutative Manifolds

Cited by 47 publications

References 75 publications

Algebraic K-theory, K-regularity, and 𝕋 $\mathbb {T}$ -duality of 𝒪 ∞ $\mathcal {O}_{\infty }$ -stable C ∗-algebras

Algebraic K-theory, K-regularity, and 𝕋 $\mathbb {T}$ -duality of 𝒪 ∞ $\mathcal {O}_{\infty }$ -stable C ∗-algebras

Equivariant spectral triples and Poincaré duality for 𝑆𝑈_{𝑞}(2)

Noncommutative gauge theories on D-branes in non-geometric backgrounds

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