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## Spectral action on noncommutative torus

Venue: | J. Noncommut. Geom |

Citations: | 16 - 8 self |

### Citations

3070 | Perturbation Theory for Linear Operators - Kato - 1966 |

2951 | Noncommutative geometry
- Connes
- 1994
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Citation Context ...onnes plays an important role [3] in noncommutative geometry. More precisely, given a spectral triple (A, H, D) where A is an algebra acting on the Hilbert space H and D is a Dirac-like operator (see =-=[8, 23]-=-), they proposed a physical action depending only on the spectrum of the covariant Dirac operator DA := D + A + ǫJAJ −1 (1) where A is a one-form represented on H, so has the decomposition A = ∑ ai[D,... |

1250 |
An introduction to the theory of numbers
- Hardy, Wright
- 1979
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Citation Context ...duce the following Definition 2.15. ζ(s) := ∞∑ n −s , n=1 Zn(s) := ∑ k∈Z n ζp1,...,pn(s) := ∑ k∈Z n ′ |k| −s , Cη ′ k p1 1 · · · kpn n |k| s , for pi ∈ N, where ζ(s) is the Riemann zeta function (see =-=[25]-=- or [14]). By the symmetry k → −k, it is clear that these functions ζp1,...,pn all vanish for odd values of pi. Let us now compute ζ0,···,0,1i,0···,0,1j,0···,0(s) in terms of Zn(s): Since ζ0,··· ,0,1i... |

416 | Trace Ideals and Their Applications - Simon - 1979 |

233 |
Connes A., The spectral action principle
- Chamseddine
- 1997
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Citation Context ...aków, Poland sitarz@if.uj.edu.pl 5 Partially supported by MNII Grant 115/E-343/SPB/6.PR UE/DIE 50/2005–20081 Introduction The spectral action introduced by Chamseddine–Connes plays an important role =-=[3]-=- in noncommutative geometry. More precisely, given a spectral triple (A, H, D) where A is an algebra acting on the Hilbert space H and D is a Dirac-like operator (see [8, 23]), they proposed a physica... |

229 |
The local index formula in noncommutative geometry
- Connes
- 1995
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Citation Context ...:= {T : T |D| −α ∈ OP 0 }. where Ft(T) := e it|D| T e −it|D| = e it|D| T e −it|D| since |D| = |D| + P0. Define δ(T) := [|D|,T], ∇(T) := [D 2 ,T], σs(T) := |D| s T |D| −s , s ∈ C. It has been shown in =-=[13]-=- that OP 0 = ⋂ p≥0 Dom(δp ), for p ∈ N0. In particular, OP 0 is a subalgebra of B(H) (while elements of OP α are not necessarily bounded for α > 0) and A ⊆ OP 0 , JAJ −1 ⊆ OP 0 , [D, A] ⊆ OP 0 . Note ... |

200 |
C ∗ -Algebras Associated with Irrational Rotations
- Rieffel
- 1981
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Citation Context ...ions ∫ − Ã2 |D| −n) . ∫ −( ÃDÃD + DÃDÃ + ÃD2 Ã + D Ã2 D )|D| −n−2) . Let C∞ (Tn Θ ) be the smooth noncommutative n-torus associated to a non-zero skew-symmetric deformation matrix Θ ∈ Mn(R) (see [6], =-=[30]-=-). This means that C∞ (Tn Θ ) is the algebra generated by n unitaries ui, i = 1,... ,n subject to the relations ui uj = e iΘij uj ui, (40) and with Schwartz coefficients: an element a ∈ C∞ (Tn ∑ Θ ) c... |

191 |
Noncommutative geometry and reality
- Connes
- 1995
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Citation Context ...imensional kernel (see for instance [27, Theorem 6.29]). Since according to the dimension, J 2 = ±1, J commutes or anticommutes with χ, χ commutes with the elements in the algebra A and Dχ = −χD (see =-=[10]-=- or [23, p. 405]), we get DAχ = −χDA and DAJ = ±JDA which gives the result. 3.2 Pseudodifferential operators Let (A, D, H) be a given real regular spectral triple of dimension n. We note P0 the projec... |

179 | Non-commutative differential geometry, Publ - Connes |

167 |
Algebres et geometrie differentielle
- Connes
- 1980
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Citation Context ...Notations ∫ − Ã2 |D| −n) . ∫ −( ÃDÃD + DÃDÃ + ÃD2 Ã + D Ã2 D )|D| −n−2) . Let C∞ (Tn Θ ) be the smooth noncommutative n-torus associated to a non-zero skew-symmetric deformation matrix Θ ∈ Mn(R) (see =-=[6]-=-, [30]). This means that C∞ (Tn Θ ) is the algebra generated by n unitaries ui, i = 1,... ,n subject to the relations ui uj = e iΘij uj ui, (40) and with Schwartz coefficients: an element a ∈ C∞ (Tn ∑... |

166 | Noncommutative manifolds, the instanton algebra and isospectral deformations - Connes, Landi |

116 | Gravity and the standard model with neutrino mixing - Chamseddine, Connes, et al. |

92 | Deformation quantization for actions of R d - Rieffel - 1993 |

76 |
Enk, private communication
- van
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Citation Context ...esis: ∫ − AD −1 = 0, for all A ∈ Ω 1 D(A). (55) By the following lemma, this condition is satisfied for the noncommutative torus, a fact more or less already known within the noncommutative community =-=[34]-=-. Lemma 6.8. Let n ∈ N, A = L(−iAα)⊗γ α = −i ∑ l∈Zn aα,l Ul⊗γ α , Aα ∈ AΘ, {aα,l }l ∈ S(Zn ), be a hermitian one-form. Then, (i) ∫ AqD−q = ∫ (ǫJAJ −1 ) qD−q = 0 for 1 ≤ q < n (case q = 1 is tadpole hy... |

75 | Methods Mathematiques" pour les sciences physiques - Schwartz - 1961 |

41 |
Geometry from the spectral point of view
- Connes
- 1995
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Citation Context ...erator from Dom |D| α+1 to Dom |D| where the Dom |D| α spaces have their natural norms (see [13,26]). We now introduce a definition of pseudodifferential operators in a slightly different way than in =-=[9,13,26]-=- which in particular pays attention to the reality operator J and the kernel of D and allows D and |D| −1 to be a pseudodifferential operators. It is more in the spirit of [4]. Definition 3.5. Let us ... |

37 | The local index formula in semifinite von Neumann algebras II: even case, Adv - Carey, Phillips, et al. |

37 | Noncommutative induced gauge theory - Goursac, Wallet, et al. |

34 | Noncommutative heat kernel - Vassilevich |

30 |
Várilly, “On summability of distributions and spectral
- Estrada, Gracia-Bondía, et al.
- 1998
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Citation Context ... is defined by JD = ǫ DJ. (3) S(DA,Φ,Λ) := Tr ( Φ(DA/Λ) ) where Φ is any even positive cut-off function which could be replaced by a step function up to some mathematical difficulties investigated in =-=[16]-=-. This means that Φ counts the spectral values of |DA| less than the mass scale Λ (note that the resolvent of DA is compact since, by assumption, the same is true for D, see Lemma 3.1 below). In [18],... |

30 |
The local index formula in noncommutative geometry
- Higson
- 2002
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Citation Context ...ions, it is not necessary to define the operators D−1 or D −1 A and the associated zeta functions. However, we can remark that all the work presented here could be done using the process of Higson in =-=[26]-=- which proves that we can add any smoothing operator to D or DA such that the result is invertible without changing anything to the computation of residues. Define for any α ∈ R OP 0 := {T : t ↦→ Ft(T... |

26 | Heat-kernel approach to UV/IR mixing on isospectral deformation manifolds, Annales Henri Poincare 6 - Gayral - 2005 |

22 | Noncommutative Yang-Mills and noncommutative relativity: a bridge over troubled water”, Eur - Carminati, Iochum, et al. - 1999 |

18 |
Inner fluctuations of the spectral action
- Chamseddine, Connes
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Citation Context ...eal structure via J, does change the spectral action, up to a coefficient when the torus has dimension 4. Here we prove that this can be also directly obtained from the Chamseddine–Connes analysis of =-=[4]-=- what we follow quite closely. Actually, S(DA,Φ,Λ) = ∑ 0<k∈Sd + where DA = DA + PA, PA the projection on Ker DA, Φk = 1 2 (4) Φk Λ k ∫ − |DA| −k + Φ(0)ζDA (0) + O(Λ−1 ) (5) ∫ ∞ 0 Φ(t)tk/2−1 dt and Sd ... |

17 | Heat kernel and number theory on NC-torus
- Gayral, Iochum, et al.
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Citation Context ...DA is compact since, by assumption, the same is true for D, see Lemma 3.1 below). In [18], the spectral action on NC-tori has been computed only for operators of the form D + A and computed for DA in =-=[20]-=-. It appears that the implementation of the real structure via J, does change the spectral action, up to a coefficient when the torus has dimension 4. Here we prove that this can be also directly obta... |

16 | Heat kernel, effective action and anomalies - Vassilevich - 2005 |

13 | Multidimensional cut-off technique, odd-dimensional Epstein zeta functions and Casimir energy of massless scalar - Edery |

13 | Asymptotic Formulae - Gilkey - 2004 |

8 | The Ehrhart function for symbols - Guillemin, Sternberg, et al. - 2006 |

6 | Spectral Action and the Connes-Chamseddine Model - Nest, Vogt, et al. - 2002 |

6 | Induced Chern–Simons action on noncommutative torus", Modern Phys - Vassilevich |

5 | Spectral action and big desert Phys - Knecht, Schucker - 2006 |

3 |
Cours au Collège de
- Connes
- 1990
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Citation Context ... ∑ q=1 k1,··· ,kq=0 Γ k q(X)D −2(|k|1+q) mod OP −N−1 . (33) Note that the Γ k q(X) are in D1(A), which, with (33) proves that Y and thus ε r (Y ) = ∇ r (Y )D −2r , are also in Ψ1(A). We remark, as in =-=[11]-=-, that the fluctuations leave invariant the first term of the spectral action (5). This is a generalization of the fact that in the commutative case, the noncommutative integral depends only on the pr... |

2 | Heat kernel of non-minimal gauge field kinetic operators on Moyal plane - Strelchenko |

2 | Spectral action and big - Knecht, Schücker - 2006 |

1 |
The spectral action for Moyal
- Gayral, Iochum
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Citation Context ... [16]. This means that Φ counts the spectral values of |DA| less than the mass scale Λ (note that the resolvent of DA is compact since, by assumption, the same is true for D, see Lemma 3.1 below). In =-=[18]-=-, the spectral action on NC-tori has been computed only for operators of the form D + A and computed for DA in [20]. It appears that the implementation of the real structure via J, does change the spe... |