The Pseudo-differential Calculus on Noncommutative Tori
Introduction
The methods of spectral geometry are useful for investigating the metric aspects of non-commutative geometry and in these contexts require extensive use of pseudo-differential operators. After proving in more detail the formula for the symbol of the adjoint of a pseudo-differential operator and the formula for the symbol of a product of two pseudo-differential operators, we define the corresponding analogue of Sobolev spaces for which we have the Sobolev and prove Rellich lemmas. Suppose P is a pseudo-differential operator with symbol σ(P)= ρ= ρ(ξ) of order M1, and Q is a pseudo-differential operator with symbol σ(Q) = φ = φ(ξ) of order M2.
Preliminaries
An alternative definition of the subalgebraA∞θ of smooth elements is the elements inAθ that can be expressed by an expansion of the formÍ. Letq be a nondegenerate real quadratic form in Rn, a complex-valued function aC∞ defined inRnsuch that the functions(1+|x|2)−m/2∂αa(x) are bounded inRn for allα∈ Zn≥0, and∞be a Schwartz function, i.e. Suppose that, for some m, a is a complex-valued function aC∞ defined on Rn such that the functions(1+ |x|2)−m/2∂αa(x) are bounded onRn for.
Asymptotic formula for the symbol of the adjoint of a pseudo-
Asymptotic formula for the symbol of a product of two pseudo-
We want to calculate the symbol of Pφ◦Pρ, but problems with convergence of integrals make it so that we have to calculate the symbol of Pφk ◦Pρ.
Sobolev spaces on the noncommutative n torus
The pseudo-differential calculus on f.g. projective modules over the
The identity of the analytic index and the topological index defined by the Connes-Chern character was established in the non-commutative setting in [15]. In §2.3 we explain our construction of a vector bundle over the non-commutative two torus using a twisted spectral triple with σ connections and derive the symbols of the operators needed for the index calculation. Here we elaborate on the derivation of the small heat pit expansion for an elliptic differential operator of order 2 onC(T2θ).
Finding a twisted version of the local index formula of [11] has so far proven to be a challenging problem and has only been done for the special examples of scaling automorphisms in [55]. Starting from a spectral triple (A,H,D), they thus use a self-adjoint element h ∈ A to encode the conformal perturbation of the metric in the operator D0 = ehDeh. In the non-commutative setting, this can be done by compressing the operator F = D/|D| of the Fredholm module of a twisted spectral triple(A,H,D, σ ∈Aut(A))by an idempotent eas in [21].
So we have to consider the operator. and calculate the local formula for the operator index. 2.10). Therefore, in this paper we use the thermal expansion approach to find a local formula for the index of the Dirac operator of the twisted spectral triple described in §2.3, with the coefficients (or twisted by) of the auxiliary finitely generated projective module on the non-commutative two torus, which plays the role of a general vector bundle, cf. In a very similar way, starting from the symbol of the operator L+ written in Lemma 2.3.1, we calculate the local formula for τ(a2(L+)).
After calculating explicit local formulas forτ(a2(L±)), based on (2.16), we have found a local formula for the index of the twisted Dirac operator D+e,σ onT2θ given.
A Twisted Local Index Formula for Curved Noncommutative Two
Introduction
The celebrated Atiyah-Singer index theorem provides a local formula that expresses the analytic (Fredholm) index of the Dirac operator on a compact spin manifold, with coefficients in a vector bundle, in terms of topological invariants, namely characteristic classes [1–4 ] calculate. . In non-commutative differential geometry [15, 16], an analogue of the Dirac operator is used to encode the metric information and to use ideas from spectral geometry and Riemannian geometry in the study of a non-commutative algebra, considered as the algebra of functions on a space with non-commuting coordinates. That is, the data (C∞(M),L2(S),D) of smooth functions on a spin manifold, the Hilbert space of L2 spinors, and the Dirac operator are extended to the idea of a spectral triple (A) ,H,D)where is a noncommutative algebra on a Hilbert sapceH, and D plays the role of the Dirac operator while acting inH.
Idempotents and hence K-theory of algebras are also used to consider the analogue of vector bundles in one. Furthermore, a local formula for the Connes-Chern character, which is suitable for explicit calculations, is derived in [11]. However, as we will elaborate further in §2.3, the notion of a spectral triple is appropriate for type II algebras in the classification of Murray–von Neumann algebras, and not for type III cases.
A solution presented in [21] to study the latter is the notion of a twisted spectral triple: they showed that the twisted version of spectral triples can include type III cases and cases occurring in non-commutative conformal geometry, and that index pairing and coincidence of the analytic and topological index continues to apply in the rotated case. In fact, the difficulty of the problem of finding a general local index formula for convoluted spectral triples raises the need to consider several cases. The main result of the present article is a local formula for the index of the Dirac operator D+e,σ of a twisted spectral triple on the counter algebra of a noncommutative two torus T2θ, which is twisted by a general noncommutative vector bundle represented by an idempotent.
In §2.2 we provide background material on two noncommutative toruses, its pseudodifferential calculus, and heat kernel methods.
Preliminaries
The appendix contains proofs of new rearrangement lemmas, which overcome the new challenges posed in our calculations due to the presence of an idempotent as well as a conformal factor in our calculations in the noncommutative setting. Since this action in principle comes from translation by ∈T2 written in the Fourier mode, its infinitesimal generatorsδ1, δ2 :C∞(T2θ) →C∞(T2θ) are the derivations analogous to partial differentiations on the usual two torus and is given by the defining relations. The space C∞(T2θ) of the smooth elements consists of all elements inC(T2θ) which are smooth with respect to the action α given by (2.1), which turns out to be a dense subalgebra of C(T2θ) which alternatively can be described as the space of all elements of the formÍ.
The analog of integration is provided by the linear functionalτ :C(T2θ) → Cwhich is the bounded expansion of the linear functional that sends any smooth element with the noncommutative Fourier expansionÍ. In §2.3 we will explain how one can consider a general metric in the conformal class of the flat canonical metric on T2θ by means of a positive invertible element−h, with a self-adjoint element in C∞(T2θ) . We will explain in §2.3 that our method of calculating a non-commutative local index formula, which is the main result of this article, is based on the McKean-Singer index formula [54] and calculation of the relevant terms in small time heat kernel extensions.
In the case of T2θ, this calculus associates the pseudodifferential symbol ρ:R2→ C∞(T2θ) with the pseudodifferential operator Pρ:C∞(T2θ) → C∞(T2θ) with the formula. In the case of differential operators, we can see that a symbol is elliptic if its leading part is reversible away from the origin. An important characteristic of the elliptic operator is that it allows aparametrics, namely the inverse in the algebra of pseudodifferential operators modulo infinite smoothing operators.
Then, by considering the ellipticity of 4 −λ and the following composition rule, which gives an asymptotic expansion for the symbol for the composition of two pseudodifferential operators.
Noncommutative geometric spaces and index theory
It is known from classical facts of spectral geometry that the spectral dimension of the spectral triple C∞(M),L2(S),Dg. Indeed, after the Gauss-Bonnet theorem for the non-commutative two-torus proved in [23] and its extension in [30], the calculation and conceptual understanding of the local curvature terms in non-commutative geometry have gained remarkable attention in the later year. In these studies, the noncommutative local geometric invariants are detected in the analogs of the small time asymptotic expansion (2.7) written for spectral triples.
The restriction of the twisted commutators is then used to observe that for any a ∈ A the operator It is emphasized in [21] on the important question of twisted spectral triples that their Connes-Chern character lands in the ordinary cyclic cohomology, whose pairing with the K-theory of algebra can be realized as the index of a Fredholm operator. We elaborate further on index-theoretic aspects of the present paper and its relation to classical results and results in non-commutative geometry in §2.3.
They consider the fact that on a Riemannian spin manifold the Dirac operator of the conformal perturbation g0 = e−4hg of a metric g is unitarily equivalent to ehDeh, where D is the Dirac operator ofg. For conceptual details on the concept of complex structure in non-commutative geometry, we refer the reader to [16], see also [46]. We will briefly explain in §2.3 that this can be done using the McKean-Singer index formula by calculating the constant terms in the small time-heat kernel expansions of the form (2.6) for the operators (D+e,σ)∗ D+e,σ and D+e,σ(De,σ+.
On the other hand, there are small time-asymptotic expansions of the form (2.7), which depend on the local symbols.
Calculation of a noncommutative local formula for the index
Using the homogeneous components of the pseudo-differential symbols L± presented in lemmas 2.3.1 and 2.3.2, we perform symbolic calculations and use formula (2.4) to calculate b2(ξ, λ,L±). Using the tracking property τ that appears in the formula for the index (2.16), after calculating b2(ξ,−1,L±), we will cyclically rotate the multiplicative term b0(ξ,−1,L±), which appears completely to the right side of our expressions and bring it to the far left side. This challenge also appeared in and was overcome by the rearrangement lemma, leading to the appearance of modular automorphism in finite formulas.
In this article we need an even more extended version of the rearrangement lemma due to the presence of an idempotent in addition to a conformal factor, as we will see shortly. To calculate the contribution of terms with allb0 on the left to the trace, we need a formula for. Once again we can limit ourselves to terms with b0 in the middle.
In the future work, we will present a simplification of the local formula and explain in more detail the relationship between the properties of modular automorphism functions in a simplified form and the stability of the index under perturbations that leave the Dirac operator in the same connected component Fredholm operators. We conclude this paper by considering the case of a canonical flat metric on T2θ corresponding to a trivial conformal factor k = 1 whose corresponding modular automorphism ∆ is the identity. Therefore, our formula for the index in this case reduces to a much simpler form, since we can replace the functions of the modular automorphism with the values of the functions at 1.
This goal was achieved and the result was a self-contained but rigorous introduction to the subject.
Summary Conclusion
Proof of Lemma 2.4.1 and Corollary 2.4.2
Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem, Part 11 of Mathematics Lecture Series. Index map, σ-connections and Connes-Cern character in the setting of twisted spectral triples. Kyoto J. The dissolving cocycle in twisted cyclic cohomology and a local index formula for the Podleś sphere.
