Modular Curvature for Toric Noncommutative Manifolds
Abstract.
In this paper, we extend recent results on the modular geometry on noncommutative two tori to a larger class of noncommutative manifolds: toric noncommutative manifolds. We first develop a pseudo differential calculus which is suitable for spectral geometry on toric noncommutative manifolds. As the main application, we derive a general expression for the modular curvature with respect to a conformal change of metric on toric noncommutative manifolds. By specializing our results to the noncommutative two and four tori, we recovered the modular curvature functions found in the previous works. An important technical aspect of the computation is that it is free of computer assistance.
Key words and phrases:
toric noncommutative manifolds, pseudo differential calculus, modular curvature, heat kernel expansionpoly
Contents
 1 Introduction
 2 Deformation along
 3 Deformation of tensor calculus
 4 Pseudo differential operators on toric noncommutative manifolds
 5 Widom’s pseudo differential calculus
 6 Heat kernel asymptotic
 7 Modular curvature
 A Inverse of the leading symbol
 B Computation of the term
 C Integration over the cosphere bundle
1. Introduction
In the noncommutative differential geometry program (cf. for instance, Connes’s book [10]), the geometric data is given in the form of a spectral triple , where is a algebra which serves as the algebra of coordinate functions of the underlying space, and is an unbounded selfadjoint operator such that the commutators are bounded operators on for all . More than the topological structure, the spectral data also reflects the metric and differential structure of the geometric space. The prototypical example comes from spin geometry: , where is a closed spin manifold with spinor bundle and is the Dirac operator. In Riemannian geometry, local geometric invariants, such as the scalar curvature function can be recovered from the asymptotic expansion of Schwartz kernel function of the heat operator :
Equivalently, one turns to the asymptotic expansion of the heat kernel trace
(1.1) 
which makes perfect sense in the operator theoretic setting. In the spirit of Connes and Moscovici’s work [14], the coefficients above (in (1.1)), viewed as functionals on the algebra of coordinate functions, encode the local geometry, such as intrinsic curvatures, with respect to the metric implemented by the operator . This approach was carried out in great depth on noncommutative two tori. The technical tool for the computation is the pseudo differential calculus associated to a dynamical system which was developed in Connes’ seminal paper [8], meanwhile the computation was initiated, see [7]. The first application of such calculus is the GaussBonnet theorem for noncommutative two torus [15]. The major progress occurred in Connes and Moscovici’s recent work [14]. The high lights of the paper contains not only the full local expression for the functional of the second heat coefficient, but also several geometric applications of the local formulas which demonstrate the great significance of the approach. The appearance of the modular curvature functionals in those closed formulas gives vivid reflections of the noncommutativity. An independent calculation for the GaussBonnet theorem and the full expressions of the modular curvature functions, was carried out in [20] and [21], with a different CAS (Computer Algebra System). Modular scalar curvature on noncommutative four tori was studied in [22] and [19]. Recently, in [32], the computation was extended to Heisenberg modules over noncommutative two torus and the whole calculation was greatly simplified so that CAS was no longer need. See also [3] and [36] for other related work on noncommutative two tori.
It is natural to investigate how to implement the program for other noncommutative manifolds. An interesting class of examples comes from deformation of classical Riemannian manifolds, such is the ConnesLandi deformations (cf. [13, 12]), also called toric noncommutative manifolds in [5]. The underlying deformation theory, called deformation in the literature, was first developed in Rieffel’s work [35].
Following the spirit of the previous work on noncommutative tori, we use a pseudo differential calculus to tackle the heat kernel coefficients in this paper. The construction is the first main outcome of this paper. Our pseudo differential calculus is designed to handle two families of noncommutative manifolds simultaneously: tori and spheres obtained by the ConnesLandi deformation. In contrast to noncommutative two tori, the noncommutative four spheres are different in two essential ways:

The dimension of the action torus (which is two) is less than the dimension of the underlying manifold (which is four);

The underlying manifold is not parallelizable.
The first one implies that the torus action is not transitive, hence the correspondent dynamical system will not be able to reveal the entire geometry. The second fact indicates that one should expect a more sophisticated asymptotic formula for the product of two symbols than the one appears in Connes’ construction. The method taken in this paper is to apply the deformation theory not only to the algebra of smooth functions on the underlying Riemannian manifold, but also to the whole pseudo differential calculus. The resulting symbol calculus blends the commutative and the noncommutative coordinates in a simplest fashion.
In order for the deformation theory to apply, both the symbol map and the quantization map in the calculus have to be equivariant with respect to the torus action. This leads us to work with global pseudo differential calculus on closed manifolds in which all the ingredients are given in a coordinatefree way. Such calculus, which appeared first in Widom’s work [41] and [42], turned out to be the perfect tool to develop the deformation process.
In the rest of the paper, we devote the attention to applications. In contrast to the work [14, 22, 32], we skip the construction of the spectral triple since only pseudo differential operators acting on functions are consider in this paper. As a consequence, we use scalar Laplacian operator (instead of the Dirac operator) to define the metric and the noncommutative conformal change of metric is implemented by a perturbation of the scalar Laplacian operator via a Weyl factor . The first consequence of the pseudo differential calculus is the existence of the asymptotic expansion (1.1). The associated modular curvature is defined to be the functional density with respect to the term in the heat kernel asymptotic (1.1). It is worth to point out that the modular curvature defined here is only part of the full intrinsic scalar curvature in [9, definition 1.147].
In this paper, we only test our pseudo differential calculus on the simplest but totally nontrivial perturbed Laplacian: , which is obtained from the degree zero Laplacian in [14] by a conjugation. Here is a Weyl factor as before, and is the scalar Laplacian associated to the Riemannian metric. As an instance of [14, Theorem 2.2], we prove that the zeta function at zero is independent of the conformal perturbation, namely:
(1.2) 
The main result is the local formula for the modular curvature with respect to a perturbed Laplacian (i.e., a noncommutative conformal change of metric):
(1.3)  
Let us explain the notations. First, is a Weyl factor, is an even integer, is the metric tensor on the cotangent bundle and is the LeviCivita connection so that the contraction is equal to and , which equals the squared length of the covector in the commutative situation, generalizes the Dirichlet quadratic form appeared in [14, Eq. (0.1)]. The scalar curvature function associated to the metric appears naturally if the metric is nonflat, the coefficient is a constant depends only on the dimension of the manifold . The triangle (compare to , the Laplacian operator) is the modular operator (see. (7.10)), while for , indicates that the operator applied only to the th factor. The modular curvature functions and are computed explicitly in the last section. A crucial property of the modular curvature functions is that they can be written as linear combinations of simple divided differences^{1}^{1}1“simple” means that at most the third divided difference occurs. For the notion of divided differences, we refer to [31, Appendix A] and a classical reference [33]. of the modified logarithm , which is the generating function of Bernoulli numbers after the substitution . The significance of this feature was explained in [31].
The second main outcome of this paper is obtained by specializing the result above onto dimension two. We show that the expressions of and agree with the result in [32, Theorem 3.2] which gives further validation for our pseudo differential calculus and the computation performed in the last section as in [32] and as a significant improvement of the previous work, the computation does not require aid from CAS.
In dimension four, we show that the modular curvature functions are both zero with respect to the operator . Since is the leading part of the Laplacian adapted in [22], the nonzero contributions to the modular functions come from the symbols of degree one and zero. This fact can be observed in [19] in which the computation was simplified.
The other significant feature of our approach is that the computation is no longer require computer assistance. The efficiency of our computation relies on a tensor calculus over the toric noncommutative manifolds which is obtained from a deformation of tensor calculus over the toric manifolds. On a smooth manifold , a tensor calculus consists of three parts: the pointwise tensor product and contraction between tensor fields, and a connection which is characterized by the Leibniz property. For instance, a differential operator on can be represented by a finite sum , where is a contravariant tensor field of rank so that the contraction produces a smooth function. One of the merits is this observation is that it has a straightforward generalization to our noncommutative setting: the tensor product and contractions between tensor fields are pointwise, like functions on a manifold, therefore the deformation procedure for functions extends naturally to tensor fields. To obtain a calculus, we show that the Leibniz property of the LeviCivita connection still holds in the deformed setting. As an example, we see that the Dirichlet quadratic form appeared in [14, Eq. (0.1)] with respect to the complex structure associated to the modular parameter has the following counterpart in terms of the deformed tensor calculus:
(1.4) 
where is the metric tensor on the cotangent bundle and and are deformed tensor product and contraction respectively. Being technical tools, such deformed tensor calculus and pseudo differential calculus have many other potential applications, for instance:
We end this introduction with a brief outline of the paper. Section 2 consists of functional analytic backgrounds of the deformation theory. We split the discussion into two parts: deformation of algebras and deformation of operators according to their roles as “symbols” and “operators” in the general framework of pseudo differential calculi.
In section 3, we explain that how apply the deformation process to the whole tensor calculus, which serves as preparation for section 4 and 5, which consist of the construction of pseudo differential calculus for toric noncommutative manifolds..
The remaining two sections are devoted to applications. We first sketch the proof of the existence of the heat kernel asymptotic following [26] and [6] in section 6. Finally, section 7 consists of explicit computation of the local formula of the associated modular curvature. Some technical parts of the computation are moved to the appendixes.
2. Deformation along
2.1. Deformation of Fréchet algebras
In this section, we will provide the functional analytic framework which is necessary for our later discussion on toric noncommutative manifold. We refer to Rieffel’s monograph [35] for further details, also [24], [5] and [43]. All the topological vectors spaces appeared in this paper are over the field of complex numbers.
Let be Fréchet space whose topology is defined by an increasing family of seminorms . We say is a smooth module if admits a torus action such that the function belongs to for all , moreover, we require the action is strongly continuous in the following sense: , given a multiindex , we can find another integer such that
(2.1) 
where the constant depends on and the vector .
Due to the duality between and , Fourier theory tells us that all smooth module in definition 2.1 are graded:
where is the image of the projection :
(2.2) 
Namely, any vector in admits a isotypical decomposition:
(2.3) 
The sequence is of rapidly decay in due to an integration by parts argument on (2.2). The precise estimate is given below: {prop} Let be a smooth module as in definition 2.1, whose topology is given by an countable increasing family of seminorms with . Then for any element with its isotypical decomposition, then the sequence of the th seminorms: is of rapidly decay in . More precisely, for any integer , there exist a degree polynomial and another large integer such that ,
(2.4) 
In particular, the isotypical decomposition converges absolutely to . The proof can be found in, for instance, [35, Lemma 1.1].
Conversely, suppose admits a smooth grading: , then the action is given by on each homogeneous component
(2.5) 
A vector is smooth respect to the torus action if and only if for each seminorm , the sequence decays faster than any polynomial in , that is for each seminorm and integer , there is an integer and a constant , such that
(2.6) 
A smooth algebra is a smooth module as in definition 2.1 such that the multiplication map is equivariant and jointly continuous, that is for every , there is a and a constant such that
(2.7) 
Additionally, if is a algebra, we required the operator is continuous and equivariant. Similarly, a smooth left(right) module is a smooth module while the left(right) module structure is equivariant and the jointly continuous as in (2.7).
Let be a smooth algebra as above. For a skew symmetric matrix , we denote the corresponding bicharacter:
(2.8) 
where the pairing is the usual dot product in . The deformation of is a family of algebras parametrized by , whose underlying topological vector space is equal to while the multiplication is deformed as follow:
(2.9) 
and , are the isotypical decomposition as in (2.2).
Each inherits the smooth module structure from and since in (2.9) are complex numbers of length , the new multiplication is jointly continuous as well. Hence, for any skew symmetric matrix , the deformation are all smooth algebra as in definition 2.1.
The deformed product on is associative. That is, for any ,
(2.10) 
If the algebra is a algebra, the deformation are algebras as well with respect to the original operator, that is ,
(2.11) 
Proof.
Let with their isotypical decomposition: , and , where , , are summed over . We compute the left hand side of (2.10),
here we have used the estimate (2.6) to exchange the order of summation. Similar computation gives us the right hand side:
Thus we have proved the associativity. Notice that we have not yet used the skewsymmetric property of . In fact, the skewsymmetric property is only necessary for the operator to survive after deformation. In particular, it implies that for the bicharacter defined in (2.8),
here the operator is the conjugation on complex numbers. Since the operator is equivariant, it flips the grading of , that is, it sends the component to the component: , where . Indeed,
Therefore:
The proof is complete. ∎
Let be a equivariant continuous algebra homomorphism, where , are two smooth algebras which admit deformation as above. If we identify and , and by the identity maps respectively, then
(2.12) 
is still an equivariant algebra homomorphism with respect to the new product .
Proof.
For any with the isotypical decomposition , , thanks to the equivariant property of , we have and for any . Use the continuity of , we compute:
∎
The next proposition shows that any equivariant trace on a smooth algebra extends naturally to a trace on all the deformations .
Let be a equivariant trace and is a skey symmetric matrix as before. Then is a continuous linear functional for the deformation and are identical as topological vector spaces. However, is indeed a trace on , that is
(2.13) 
Proof.
From the equivariant property of , we know that for any isotypical component , , of ,
Therefore for all . Follows from the continuity, for any ,
that is, the trace of depends only on its invariant component. Since is skew symmetric, we get
Similar computation gives that . Therefore if is a trace on , then it is a trace on as well. ∎
In this paper, we have to deal with certain topological algebras which are not Fréchet. For instance, the algebra of pseudo differential operators of integer orders, and the associated algebra of symbols, whose topology is certain inductive limit of Fréchet topologies. More precisely, we consider a filtered algebra with a filtration:
(2.14) 
where each () is a smooth module as defined before, in particular, a Fréchet space. As a topological vector space, the total space is a countable strict inductive limit of , the topology is just called strict inductive limit topology (cf. for instance [40, sec. 13] for more details). This topology is never metrizable unless the filtration is stabilized starting from some , therefore it is not Fréchet. Nevertheless, we shall not looking at the topology of the whole algebra even when considering the continuity of the multiplication map. Instead, we focus on each , assume that the Fréchet topology is defined by a countable family of increasing seminorms . The multiplication preserves the filtration:
(2.15) 
such that the continuity condition holds: for fixed , and a positive integer , one can find a integer and constant such that
(2.16) 
The multiplication is deformed in a similar fashion as in (2.9):
(2.17)  
(2.18) 
Examples are provided in the next section.
2.2. Deformation of operators
The associativity of the multiplication proved in proposition 2.1 is a special instance of certain “functoriality” in the categorical framework explained in [5]. Let us start with deformation of operators.
Let and be two Hilbert spaces which are both strongly continuous unitary representation of , denoted by and respectively, where . If no confusions arise, both representations will be denoted by . Then , the space of all bounded operators from to , becomes a module via the adjoint action:
(2.19) 
Denote by , the space of all the smooth vectors in . It is a Fréchet space on which the topology is defined by the seminorms :
(2.20) 
The seminorms above are constructed in such a way that the continuity estimate (2.1) for the torus action holds automatically. Following from proposition 2.1, we see that any smooth operator admits an isotypical decomposition , where the operator norms decays faster than polynomial in , in particular the converges of the infinite sum is absolute with respect to the operator norm in .
Now we are ready to define the deformation map . {defn} Let and be two Hilbert space with strongly continuous unitary actions as above, that is , , is continuous in . We denote the actions by and respectively. For a fixed skew symmetric matrix , we recall the associated bicharacter . Then the deformation map is defined as follows,
(2.21) 
where and with their isotypical decomposition. We can assume that is a smooth vector for the subspace of all smooth vectors in dense in . Alternatively, is given by
(2.22) 
here stands for the matrix multiplication between a row vector and whose result is a point in . {rem} The deformed operator belongs to . Indeed, in (2.22), the each isotypical component of is perturbed by a unitary operator , therefore the right hand side of (2.22) is a sum of rapidly decay sequence, which implies not only the boundedness of , but also the smoothness.
Let us give a precise estimate of the operator norm of . {lem} Let denote the Fréchet algebra of smooth vectors in whose topology is given by the seminorms in (2.20). Then the deformation is a continuous linear map with respect to the Fréchet topology with the estimate: for any multiindex , one can find any integer large enough such that
(2.23) 
Proof.
Given a smooth operator with the isotypical decomposition . From the definition, has the isotypical decomposition , thus
If we let be the polynomial in such that , the degree of is equal to , compute
(2.24) 
Since that is of rapidly decay in , we can find a large integer such that
Therefore
∎
Similar to (2.11), we have the compatibility between the deformation map and the operation (taking the adjoint) on operators. {lem} Let , then its adjoint as well, we have
(2.25) 
Proof.
Since the torus action is unitary, the adjoint operation is equivariant:
therefore for the isotypical components, for all ,
here we have used the facts that and commute. ∎
The next lemma says that the deformation is somehow invertible. {lem} Let and be two skew symmetric matrices and for any , we have
In particular, we see that the deformation process is invertible, namely and are inverse to each other.
Proof.
Given , is the isotypical decomposition of , therefore
∎
If we take and above to be the same Hilbert space, becomes an smooth algebra as in definition 2.1. Following from definition 2.1, we obtain a family of deformed algebras parametrized by skew symmetric matrices . The multiplication map is obviously equivariant, that is , for all and for any . The associativity of the multiplication has the following analogy.
Keep the notations as above. The deformation map
is an algebra isomorphism, namely, for any ,
(2.26) 
recall that the deformed product is defined in (2.9).
Proof.
The invertiblity of is proved in lemma 2.2. It remains to show that it is an algebra morphism, that is for any smooth vector , we have
(2.27) 
Observe that can be formally written as according to (2.21). Therefore the left hand side and the right hand side of (2.27) becomes and respectively, thus equation (2.27) is exactly the same as the associativity of the multiplication proved in (2.10). ∎
We have seen that the isotypical decomposition of an operator converges with respect to operator norms. The normality of the trace somehow allows itself to pass the summation, namely whenever is a traceclass operator. {lem} Let be a saperable Hilbert space with a strongly continuous unitary action and is smooth operator with its isotypical decomposition. Suppose is a traceclass operator, then so is , moreover, , where is the invariant part of both and .
Proof.
Since is a strongly continuous unitary representation of , it admits a orthonormal decomposition
in which each consists of eigenvector of the torus action:
For each , one can pick a orthonormal basis , then is an orthonormal basis of . Since convergence absolutely in the operator norm,
observe that for all , , therefore except the case when . We continue the computation above:
Since is traceable, the left hand side above converges absolutely, therefore is traceable as well and has the same trace as . Recall equation (2.22): . Notice that the computation above still works if is replaced by , therefore is of trace class and , where is the invariant part of . Since and have the same invariant part, we have finished the proof. ∎
The following corollary is important for our later discussion. {cor} Let be a separable Hilbert space. Let be two smooth vectors such that at least one of them is traceable, then and are both traceable with
Combine the equation above with (2.26), we obtain:
2.3. Deformation of functions
Let be a smooth manifold without boundary. For any diffeomorphism , we defined the pull back action on :
(2.28) 
which is automorphism of . Assume that admits a torus action: , then one can quickly verify that acts smoothly (cf. eq. (2.1)) on with respect to the smooth Fréchet topology, also the pointwise multiplication is jointly continuous (cf. eq. (2.7)) , therefore we can deform the multiplication to following definition 2.1 with respect to a skew symmetric matrix , and the new algebra
plays the role of smooth coordinate functions on a noncommutative manifold .
Later, we will assume the manifold is compact. The noncompact examples we are interested in is the cotangent bundle . One can easily lift the torus action to by the natural extension of diffeomorphims: , where is the differential of . Thus the cotangent bundle of the noncommutative manifold is given by the deformed algebra:
Another crucial example is , the spaces of symbols of pseudo differential operators on . It is a filtered algebra:
where each consists of smooth functions with the estimate in local coordinates ,
(2.29) 
the optimized constants define a family of seminorms that makes into a Fréchet space. The smoothing symbols is the intersection: