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## MODULAR CURVATURE FOR NONCOMMUTATIVE TWO-TORI (1110)

Citations: | 21 - 0 self |

### Citations

2981 | Noncommutative Geometry,
- Connes
- 1994
(Show Context)
Citation Context ...l factor, or dilaton, which changes the metric by modifying the volume form while keeping the same conformal structure. Both notions of volume form and of conformal structure are well understood (cf. =-=[9]-=-), and we recall in §1 how one obtains the modified Dirac The work of the first named author was partially supported by the National Science Foundation award no. DMS-0652164. The work of the second na... |

709 | Invariance theory, the heat equation, and the Atiyah-Singer index theorem - Gilkey - 1984 |

234 | The spectral action principle, - Chamseddine, Connes - 1997 |

231 |
Moscovici: The local index formula in non-commutative geometry,
- Connes, H
- 1995
(Show Context)
Citation Context ...nifold, introduced in [2], admits a natural extension to the framework of noncommutative geometry. Let (A, H, D) be a p-summable spectral triple, which has discrete dimension spectrum in the sense of =-=[11]-=-. Fix h = h ∗ ∈ A, and let Then hence Dsh = e sh 2 D e sh 2 , s ∈ R. (53) d ds D2 sh = 1 2 d ds Dsh = 1 2 (hDs + Dsh), Duhamel’s formula for the family △s = tD 2 sh , allows to write d ds Tr ( e −tD2 ... |

168 |
C∗-algèbres et géométrie differentielle,
- Connes
- 1980
(Show Context)
Citation Context ...ass in the spectral action formalism for the standard model [6]. □14 CONNES AND MOSCOVICI 2.2. Conformal index for T 2 θ . More to the point, the pseudodifferential calculus for C∗-dynamical systems =-=[8]-=-, and especially the elliptic theory on noncommutative tori [9, §IV.6], show that the condition (58) is fulfilled in the case of T2 θ . In particular, all the Laplacians in §1.6 admit meromorphic zeta... |

119 |
The ricci flow on surfaces. Mathematics and general relativity,
- Hamilton
- 1988
(Show Context)
Citation Context ... [18], which is a scale invariant version of the log-determinant of the Laplacian. We then compute its gradient, whose corresponding flow for Riemann surfaces exactly reproduces Hamilton’s Ricci flow =-=[17]-=-, and therefore yields the appropriate analogue of the scalar curvature. 4.1. Variation of the log-determinant. The Ray-Singer zeta function regularization of the determinant of a Laplacian [19], as w... |

55 | Explicit functional determinants in four dimensions - Branson, Oersted |

55 |
Extremals of determinants of
- Osgood, Phillips, et al.
- 1988
(Show Context)
Citation Context ... curvature of the conformal metric. T 2 T 222 CONNES AND MOSCOVICI 4. Log-determinant functional and scalar curvature In this section we develop the analogue of the Osgood-Phillips-Sarnak functional =-=[18]-=-, which is a scale invariant version of the log-determinant of the Laplacian. We then compute its gradient, whose corresponding flow for Riemann surfaces exactly reproduces Hamilton’s Ricci flow [17],... |

25 |
B.: Conformal indices of Riemannian manifolds
- Branson, Ørsted
- 1986
(Show Context)
Citation Context ...). The others follow from Lemma 1.11. □ 2. Conformal invariants 2.1. Conformal index of a spectral triple. We digress a little to show that the notion of conformal index for a manifold, introduced in =-=[2]-=-, admits a natural extension to the framework of noncommutative geometry. Let (A, H, D) be a p-summable spectral triple, which has discrete dimension spectrum in the sense of [11]. Fix h = h ∗ ∈ A, an... |

22 |
P.: The Gauss-Bonnet theorem for the noncommutative two torus. Noncommutative geometry, arithmetic, and related topics
- Connes, Tretkoff
- 2011
(Show Context)
Citation Context ... whose differential geometry, as well as its pseudo-differential operator calculus, was first developed in [8] . To obtain a curved geometry from the flat one defined in [8], one introduces (cf. [7], =-=[13]-=-) a Weyl factor, or dilaton, which changes the metric by modifying the volume form while keeping the same conformal structure. Both notions of volume form and of conformal structure are well understoo... |

18 |
M.: The Gauss-Bonnet Theorem for Noncommutative Two Tori With a General Conformal Structure
- Fathizadeh, Khalkhali
- 2012
(Show Context)
Citation Context ...quality for s = 0 one gets that ∫ 1 e 2πi −tλ α(λ)dλ = α(−1) . C C Thus we can already simplify and fix λ = −1 before we perform the integration in d 2 ξ = dξ1dξ2. To start this integration we follow =-=[14]-=- and perform the change of variables, using τ1 = ℜ(τ) and τ2 = ℑ(τ), ξ1 = rcos(θ) − r τ1 sin(θ) , τ2 r sin(θ) ξ2 = τ238 CONNES AND MOSCOVICI The Jacobian of the change of coordinates is r . Moreover ... |

16 | Scale invariance in the spectral action
- Chamseddine, Connes
(Show Context)
Citation Context ...0). (62) An instance where the above hypotheses are satisfied, and hence the result applies, is that of the dilaton field rescaling of the mass in the spectral action formalism for the standard model =-=[6]-=-. □14 CONNES AND MOSCOVICI 2.2. Conformal index for T 2 θ . More to the point, the pseudodifferential calculus for C∗-dynamical systems [8], and especially the elliptic theory on noncommutative tori ... |

16 | Scalar curvature for the noncommutative two torus
- Fathizadeh, Khalkhali
(Show Context)
Citation Context ...in 2009 and announced (including by internet posting, although with some typos) at several conferences (Oberwolfach 2009 and Vanderbilt 2011). Moreover, the same computation was done independently in =-=[15]-=- and gave a confirmation of the result. The main additional input of the present paper consists in obtaining an explicit formula for the Ray-Singer log-determinant of D 2 (which was left open in [7]).... |

14 | B.: Conformal geometry and global invariants - Branson, Ørsted - 1991 |

11 |
Connes, Conformal geometry of the irrational rotation algebra, Preprint MPI
- Cohen, A
(Show Context)
Citation Context ... T2 θ whose differential geometry, as well as its pseudo-differential operator calculus, was first developed in [8] . To obtain a curved geometry from the flat one defined in [8], one introduces (cf. =-=[7]-=-, [13]) a Weyl factor, or dilaton, which changes the metric by modifying the volume form while keeping the same conformal structure. Both notions of volume form and of conformal structure are well und... |

10 |
The Ricci flow on noncommutative two-tori
- Bhuyain, Marcolli
(Show Context)
Citation Context ...onding evolution equation for the metric yields the natural analogue of Ricci flow. It should be mentioned that a different version of the latter has been proposed, albeit not explicitly computed, in =-=[1]-=-. We thank Farzad Fathizadeh and Masoud Khalkhali for performing independently, cf. [15], the central computation of this paper. Contents Introduction 1 1. Modular spectral triples for noncommutative ... |

8 |
H.: Type III and spectral triples. Traces
- Connes, Moscovici
(Show Context)
Citation Context ...lso serves as a first illustration of the distinctly non-unimodular feature of the conformal geometry of noncommutative 2-tori, which in particular validates the treatment of twisted spectral triples =-=[12]-=- as basic geometric structures. 1.1. Inner twisting in the even case. The modular spectral triples considered below can be understood as special cases of the following general construction. Let us sta... |

6 |
Analytic torsion for complex manifolds,” The Annals of Mathematics 98
- Ray, Singer
- 1973
(Show Context)
Citation Context ...ogue of the scalar curvature. 4.1. Variation of the log-determinant. The Ray-Singer zeta function regularization of the determinant of a Laplacian [19], as well the related notion of analytic torsion =-=[20]-=-, make perfect sense in the case of noncommutative 2-tori, due to the existence of the appropriate pseudodifferential calculus [8]. Thus, log Det ′ (△ϕ) = −ζ ′ △ϕ (0), resp. log Det′ (△ (0,1) ϕ ) = −ζ... |