Yang-Mills for noncommutative two-tori

Connes, Alain; Rieffel, Marc A.

doi:10.1090/conm/062/878383

Cited by 178 publications

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“…In this note we investigate the part of the spectrum corresponding to electric fluxes of the noncommutative supersymmetric Yang-Mills (NCSYM) gauge theory [1] compactified on a torus. This gives a description of the DLCQ of M-theory [2] compactified on a dual torus.…”

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confidence: 99%

A note on the BPS spectrum of the matrix model

Brace

Morariu

1999

J. High Energy Phys.

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confidence: 99%

A note on the BPS spectrum of the matrix model

Brace

Morariu

1999

J. High Energy Phys.

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“…Using the Künneth Theorem (or possibly O ∞ -stability in K top -theory) one now deduces that K top * (A n θ⊗ O ∞ ) ∼ = K top * (A n θ ). Now use Equation (11). We just determined the isomorphism type of the algebraic K-theory groups of A n θ⊗ O ∞ .…”

Section: 2mentioning

confidence: 99%

“…Noncommutative tori constitute arguably the most widely studied class of noncommutative spaces. Their geometric invariants were studied extensively by Connes and Rieffel (see, for instance, [10,11,52]). We show that the algebraic K-theory of noncommutative tori are explicitly computable after O ∞ -stabilization (cf.…”

Section: Introductionmentioning

confidence: 99%

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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“…This can appear on the worldvolume theory of branes as well as in space-time. D-branes support Yang-Mills fields, so the presence of the B-field leads to noncommutative Yang-Mills theory (see [10]). Recently there has been a lot of interest in the theory on the worldvolume of the M-theory fivebrane (M5-brane).…”

Section: Introductionmentioning

confidence: 99%

Higher abelian gauge theory associated to gerbes on noncommutative deformed M5-branes and S-duality

Mathai

Sati

2015

Journal of Geometry and Physics

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We enhance the action of higher abelian gauge theory associated to a gerbe on an M5-brane with an action of a torus T n (n ≥ 2), by a noncommutative T n -deformation of the M5-brane. The ingredients of the noncommutative action and equations of motion include the deformed Hodge duality, deformed wedge product, and the noncommutative integral over the noncommutative space obtained by strict deformation quantization. As an application we then introduce a variant model with an enhanced action in which we show that the corresponding partition function is a modular form, which is a purely noncommutative geometry phenomenon since the usual theory only has a Z 2 -symmetry. In particular, S-duality in this 6-dimensional higher abelian gauge theory model is shown to be, in this sense, on par with the usual 4-dimensional case. *

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Yang-Mills for noncommutative two-tori

Cited by 178 publications

References 0 publications

A note on the BPS spectrum of the matrix model

A note on the BPS spectrum of the matrix model

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

Higher abelian gauge theory associated to gerbes on noncommutative deformed M5-branes and S-duality

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