Using a global symbol calculus for pseudodifferential operators on tori, we build a canonical trace on classical pseudodifferential operators on noncommutative tori in terms of a canonical discrete sum on the underlying toroidal symbols. We characterise the canonical trace on operators on the noncommutative torus as well as its underlying canonical discrete sum on symbols of fixed (resp. any) non-integer order. On the grounds of this uniqueness result, we prove that in the commutative setup, this canonical trace on the noncommutative torus reduces to Kontsevich and Vishik's canonical trace which is thereby identified with a discrete sum. A similar characterisation for the noncommutative residue on noncommutative tori as the unique trace which vanishes on trace-class operators generalises Fathizadeh and Wong's characterisation in so far as it includes the case of operators of fixed integer order. By means of the canonical trace, we derive defect formulae for regularized traces. The conformal invariance of the ζfunction at zero of the Laplacian on the noncommutative torus is then a straightforward consequence.

The spectral action for a non-compact commutative spectral triple is computed covariantly in a gauge perturbation up to order 2 in full generality. In the ultraviolet regime, p → ∞, the action decays as 1/p 4 in any even dimension.Recent advances [12,14] in explaining some key features of gravity and Standard Model through the spectral action of noncommutative geometry brought this subject to a focus of interest in theoretical physics. In noncommutative geometry, all information is encoded in a spectral triple (A, H, D), where A is an algebra acting on a Hilbert space H and D is a selfadjoint operator on H which plays the role of a Dirac operator [13,14,19]. In this approach, the action is the so-called spectral action introduced by Chamseddine and Connes [9-11]and defined for Λ ∈ R + which plays the role of a cut-off (and needed to make D/Λ dimensionless) and for a function f such that, of course, f (D 2 /Λ 2 ) is trace-class. In general, one chooses f ≥ 0 since the action Tr f (D 2 /Λ 2 ) ≥ 0 will have the correct sign for an Euclidean action. This action is the appropriate one in the framework of noncommutative geometry to reproduce several physical situations like the Einstein-Hilbert action in gravitation or the Yang-Mills-Higgs action in the standard model of particle physics [14], and the positivity of the function f implies positivity of actions for gravity, Yang-Mills or Higgs couplings, and the Higgs mass term is negative. Till the end of this Section we shall present a non-technical summary of our results to give a more physics-oriented reader a chance to appreciate them without going through the mathematics of the rest of this paper.Let M = R 2m be an even dimensional real plane, d = 2m ≥ 2, endowed with a spin structure given by the spinor bundle S = C 2 m . We denote by D the free Dirac operator and by D A the standard Dirac operator with a gauge connection A acting on the Hilbert space H := L 2 (M, S). We will use the notations and conventions from [25, eq. (3.26)], namely in local coordinates

Abstract. The spectral action on the noncommutative torus is obtained using a ChamseddineConnes formula via computations of zeta functions. The importance of a Diophantine condition is outlined. Several results on holomorphic continuation of series of holomorphic functions are obtained in this context.

We derive a commutative spectral triple and study the spectral action for a rather general geometric setting which includes the (skew-symmetric) torsion and the chiral bag conditions on the boundary. The spectral action splits into bulk and boundary parts. In the bulk, we clarify certain issues of the previous calculations, show that many terms in fact cancel out, and demonstrate that this cancellation is a result of the chiral symmetry of spectral action. On the boundary, we calculate several leading terms in the expansion of spectral action in four dimensions for vanishing chiral parameter θ of the boundary conditions, and show that θ = 0 is a critical point of the action in any dimension and at all orders of the expansion.