Modular Gaussian curvature
July 19, 2019
The genesis of natural but noncommuting coordinates can be traced back to Heisenberg’s uncertainty principle in quantum mechanics, which limits the accuracy of the simultaneous determination of the position and momentum of a subatomic particle. As Heisenberg argued [Hei27] (and Kennard rigorously derived [Ken27]), the inherent imprecision of such a measurement “is a straightforward mathematical consequence of the quantum mechanical commutation rule for the position and the corresponding momentum operators ”, where is the reduced Planck constant. Such an identity cannot be satisfied by matrices (over ), which is obvious, but not even by bounded operators in Hilbert space. Assuming and self-adjoint, this can be seen by passing to the Weyl integrated form [Wey28, §45],
Moreover, the latter relation determines a unitary representation of the (implicitly defined) group , called the Heisenberg group. By a celebrated theorem of Stone and von Neumann all such irreducible representations are unitarily equivalent. The restriction to the lattice of an irreducible unitary representation , , generates the -algebra nowadays known as the noncommutative torus of slope . When , as shall be assumed throughout this paper, the -algebra is also known as the irrational rotation algebra and is the (unique up to isomorphism) -algebra generated by a pair of unitary operators satisfying
Moreover is a simple -algebra, thus typifying the coordinates of a purely noncommutative space. For this reason on the one hand, and due to its accessibility on the other, has received much attention during the last several decades and has been a favorite testing ground for quite a number of fruitful mathematical investigations.
Although the habitual geometric intuition is rendered utterly inoperative in a ‘space without points’ such as the one represented by , the curvature as a “measure of deviation from flatness” (in Riemann’s own words) could still make some sense. It is the goal of this brief survey to review the recent developments that led to the emergence of a quantized version of Gaussian curvature for the noncommutative torus. Many of the essential ideas presented below have their origin in Alain Connes’ 1980 C. R. Acad. Sc. Paris Note [Con80], which effectively constitutes the birth certificate of noncommutative differential geometry. That foundational article not only established the most basic geometric concepts and constructions, such as the geometric realization of the finitely generated projective modules over , the explicit construction of constant curvature connections for them and the definition and calculation of their Chern classes, but also provided the crucial computational tool, in the form of a pseudodifferential calculus adapted to -dynamical systems.
The specific line of research whose highlights we are about to summarize was sparked by a paper by Connes and Paula Cohen (Conformal geometry of the irrational rotation algebra, Preprint MPI Bonn, 1992-93) which showed how the passage from the (unique) trace of to a non-tracial conformal weight associated to a Weyl factor (or ‘dilaton’) gives rise to a non-flat geometry on the noncommutative torus, which can be investigated with the help of the adapted pseudodifferential calculus of [Con80]. In a later elaboration [CoTr11] of that paper, the passage from flatness to conformal flatness was placed in the setting of spectral triples (see §2.1 and §2.2 below), which in the intervening years has emerged as the proper framework for the metric aspect in noncommutative geometry (cf. [Con94, Ch. 6], [Con13], [CoMo08]). Completing the calculations begun in the 1992 preprint they proved in [CoTr11] an analogue of the Gauss-Bonnet formula for the conformally twisted (called ‘modular’) spectral triples. The full calculation of the modular Gaussian curvature was first done by A. Connes in 2009, with the aid of Wolfram’s Mathematica, and is included in [CoMo14]. Fathizadeh and Khalkhali [FaKh13] independently performed the same calculation with the help of a different computing software.
Apart from computing the expression of the modular curvature (see §2.3 below), Connes and Moscovici showed in [CoMo14] that one can make effective use of variational methods even in the abstract operator-theoretic context of the spectral triple encoding the geometry of the noncommutative torus. After giving a variational proof of the modular Gauss-Bonnet formula which requires no computations (see §2.4), they related the modular Gaussian curvature to the gradient of the Ray-Singer log-determinant of the Laplacian viewed as a functional on the space of Weyl factors. As a consequence, they obtained an a priori proof of an internal consistency relation for the constituents of the modular curvature. In addition they showed by purely operator-theoretic arguments that, as in the case of Riemann surfaces (cf. [OPS88]), the normalized log-determinant functional attains its extreme value only at the trivial Weyl factor, in other words for the flat ‘metric’ (see §2.5 below).
For reasons which will soon become transparent (see §3.1), the natural equivalence relation between noncommutative spaces is that of Morita equivalence between their respective algebras of coordinates. For noncommutative tori the Morita equivalence is implemented by the Heisenberg bimodules described by Connes [Con80] and Rieffel [Rie81]. Lesch and Moscovici extended in [LeMo16] the results of [CoMo14] to spectral triples on noncommutative tori associated to Heisenberg equivalence bimodules (see §3.2 and §3.3). Moreover, in doing so they managed to dispose of any computer-aided calculations (see §5.1). Most notably they showed (see §3.4 below) that whenever is realized as the endomorphism algebra of a Heisenberg -module endowed with the -valued Hermitian structure obtained by twisting the canonical one by a positive invertible element in , the curvature of with respect to the corresponding spectral triple over is equal to the modular curvature associated to the same element of viewed as conformal factor. In a certain sense this is reminiscent of Gauss’s Theorema Egregium “If a curved surface is developed upon any other surface whatever, the measure of curvature in each point remains unchanged”.
The fundamentals of Connes’ pseudodifferential calculus as well as its extension to twisted -dynamical systems, which provide the essential device for proving all the above results, are explained in §4. Finally, §5 clarifies how to use the affiliated symbol calculus in order to compute the resolvent trace expansion, or equivalently the heat trace expansion, for the relevant Laplace-type operators.
2. Curvature of modular spectral triples
2.1. Flat spectral triples
In noncommutative geometry a metric structure on a space with -algebra of coordinates is represented by a triad of data called spectral triple, modeled on the Dirac operator on a manifold: is realized as a norm-closed subalgebra of bounded operators on a Hilbert space , is an unbounded self-adjoint operator whose resolvent belongs to any -Schatten ideal with where signifies the dimension, and interacts with the coordinates by having bounded commutators (or more generally bounded twisted commutators) for any in a dense subalgebra of . The Dirac operator was chosen as model since it represents the fundamental class in -homology and at the same time plays the role of a quantized inverse line element (see [Con13]). In the case of one can obtain such a triad by simply reproducing the construction of the operator on the ordinary torus .
To fix the notation we briefly review some basic properties of the -algebra with .
First of all, the torus acts on via the representation by automorphisms defined on the basis elements by
By analogy with the action of on , we call these automorphisms translations.
The smooth vectors of this representation of are precisely the elements of the form with rapidly decaying coefficients, i.e. such that , . These elements form a subalgebra which is the analogue of viewed in Fourier transform. The assignment
determines the unique normalized trace of the -algebra .
The image of the differential of the above representation on is the Lie algebra generated by the outer derivations and , uniquely determined by the relations , .
By analogy with the ordinary torus, one defines on a translation invariant complex structure with modular parameter , , by means of the pair of derivations
these are the counterparts of the operators , and acting on .
To obtain the analogue of the corresponding flat metric on , we let denote the Hilbert space completion of with respect to the scalar product
The space of formal -forms , , is also endowed with a semi-definite inner product defined by
On completing its quotient modulo the subspace of those elements such that , one obtains a Hilbert space denoted . is also an -bimodule under the natural left and right action of . Both these actions are unitary. Moreover, the linear map from to defined by sending the class of in to induces an -bimodule isomorphism between and .
Denoting by the closure of the operator viewed as unbounded operator from to one obtains a spectral triple by taking and as unbounded self-adjoint operator . Concurrently, the triad is a spectral triple with respect to the right action of . One can turn it into a spectral triple for the left action of by passing to its transposed (see [CoMo14, §1.2] for the general definition), where is the complex conjugate of and .
2.2. Modular spectral triples
To implement the analogue of a conformal change of metric structure, we choose a self-adjoint element and use it to replace the trace by the positive linear functional defined by
Then determines an inner product on ,
which by completion gives rise to a Hilbert space . The latter is again an -bimodule but, since is no longer tracial, the right action is no longer unitary.
The non-unimodularity of is expressed by Tomita’s modular operator , which in this case is
and gives rise to the -parameter group of inner automorphisms
Instead of the tracial property satisfies the KMS condition
To restore the unitarity of the right action one redefines it by setting
where , , and .
While keeping unchanged, we now view as a densely defined operator from to . Its closure is then used to define giving rise to the triad , where . This is a twisted spectral triple (see [CoMo08] for the general definition) over , with the twisted commutators , bounded. Its transposed, formed as in the flat case, yields the modular spectral triple over , with operator , where the conformal factor acts by left multiplication, and with underlying Hilbert space .
By a series of identifications, it is shown in [CoMo14, §1.3] that the modular spectral triple associated to , or equivalently to the conformal factor , is canonically isomorphic to the twisted spectral triple with and .
We finally note that , where
are the counterparts of the Laplacian on functions, respectively the Laplacian on -forms.
2.3. Modular curvature
The meaning of locality in noncommutative geometry is guided by the analogy with the Fourier transform, which interrelates the local behavior of functions with the decay at infinity of their coefficients. In a similar way, in the noncommutative formalism the local invariants of a spectral triple are encoded in the high frequency behavior of the spectrum of the ‘inverse line element’ coupled with the action of the algebra of coordinates. For example, the local index formula in noncommutative geometry [CoMo95, Part II] expresses the Connes-Chern character of a spectral triple with finite dimension spectrum in terms of multilinear functionals given by residues of zeta functions defined by
where and with repeated -times; the existence of the meromorphic continuation of such zeta functions is built in the definition of finite dimension spectrum for a spectral triple. Clearly, perturbing by a trace class operator will not affect these residue functionals, whence the local nature of the index formula described in their terms.
In the specific case of the noncommutative torus the concept of locality can be pushed much closer to the customary one. Namely, if is a modular spectral triple as in §2.2, for its Laplacian ‘on functions’ there is an asymptotic expansion
whose functional coefficients are not only local in the above sense, but they are also absolutely continuous with respect to the unique trace, i.e. of the form
with ‘Radon-Nikodym derivatives’ computable by means of symbolic calculus. The technical apparatus which justifies the heat expansion Eq. (8) as well as the explicit computation of will be discussed in §§4-5.
In particular, the Radon-Nikodym derivative of the term , which classically delivers the scalar curvature, was fully computed in [CoMo14, FaKh13] and represents the modular scalar curvature. Abbreviating its notation to instead of , it has the following expression:
where is the inner derivation implemented by ,
is the Dirichlet quadratic form
and , signifies that is acting on the th factor. The functions and , whose expressions resulted from the symbolic computations, are given by
The second function is related to the first by the functional identity
A noteworthy feature of the main curvature-defining function is that, up to a constant factor, is a generating function for the Bernoulli numbers; precisely,
2.4. Modular Gauss-Bonnet formula
Since the -groups of the noncommutative torus are the same as of the ordinary torus, its Euler characteristic vanishes. Thus, the analogue of the Gauss-Bonnet theorem for the modular spectral triple is the identity
This can be directly checked by making use of the fact that the group of modular automorphisms (cf. Eq. (6)) preserves the trace and fixes the dilaton , in conjunction with the ‘integration by parts’ rule
(See [CoMo14, Lemma 4.2] for the precise identity to be used).
An alternative variational argument, given in [CoMo14, §1.4]), runs as follows. Consider the family of Laplacians
One has . By Duhamel’s formula one can interchange the derivative with the trace. hence
Differentiating term-by-term the asymptotic expansion Eq. (8) (with omitted in notation) yields
In particular, . The latter vanishes because is isospectral to the Laplacian of the ordinary torus with the same complex structure and, as is well-known, if is the Laplacian on a Riemann surface then , where is the Euler characteristic of .
2.5. Variation of determinant and modular Gaussian curvature
The zeta function , where stands for the orthogonal projection onto , is related to the corresponding theta function by the Mellin transform
The asymptotic expansion Eq. (8) ensures that it has meromorphic continuation and its value at is
In particular for (suppressed in notation), one has
and also the Ray-Singer log-determinant is well-defined:
Differentiating the -parameter family of zeta functions corresponding to (13) one obtains the identity
which in turn yields the variation formula
The first term is the same as for the corresponding elliptic curve and by the Kronecker limit formula has the expression (cf. [RaSi73, Theorem 4.1])
where is the Dedekind eta function . After a series of technical manipulations of the last term (see [CoMo14, §4.1]), one obtains the modular analogue of Polyakov’s anomaly formula:
where , . Furthermore, it is shown in [CoMo14, Proof of Theorem 4.6] that the positivity of the function can be upgraded to operator positivity, implying the inequality
with equality only for .
The (negative of) log-determinant can be turned into a scale invariant functional by adding the area term:
In view of the inequality Eq. (17) one concludes that, as in the case of the ordinary torus (cf. [OPS88]), the scale invariant functional attains its extremal value only for the trivial Weyl factor, in other words at the flat metric.
The gradient of is defined by means of the inner product of via the pairing
In the case of the ordinary torus the gradient of the corresponding functional (cf. [OPS88, (3.8)]) gives precisely the Gaussian curvature. This makes it compelling to take the above formula as definition of the modular Gaussian curvature.
3. Morita invariance of the modular curvature
3.1. Foliation algebras and Heisenberg bimodules
The most suggestive depiction of the noncommutative torus was given by Connes in [Con82], where he described it as the “space of leaves” for the Kronecker foliation of the ordinary torus , given by the differential equation with . The holonomy groupoid of this foliation identifies with the smooth groupoid determined by the flow of the above equation. Its convolution -algebra , which represents the (coordinates of the) space of leaves, coincides with the crossed product , where the action of on is given by the flow Eq. (3). is isomorphic to , where denotes the -algebra of compact operators, and thus strongly Morita equivalent to .
Finer geometric representations of the space of leaves are obtained by passing to reduced -algebras associated to complete transversals. Any pair of relatively prime integers determines a family of lines of slope , which project onto simple closed geodesics in the same free homotopy class, and the free homotopy classes of closed geodesics on are parametrized by the rational projective line . Letting denote the primitive closed geodesic of slope passing through the base point of , one obtains a complete transversal for . The convolution algebra of the corresponding étale holonomy groupoid identifies with the crossed product algebra , where acts by the rotation of angle with chosen such that . This -algebra is none other than . In particular is the reduced -algebra associated to . By construction all algebras with , , are Morita equivalent, and they actually exhaust (cf. [Rie81]) all the noncommutative tori Morita equivalent to .
In the same framework Connes [Con82, §13] gave a geometric description of the -bimodules implementing the Morita equivalence of with . is a completion of the -bimodule , , with the actions defined as follows:
If then is the trivial -bimodule. By analogy with the vector bundles over elliptic curves, one defines the rank, degree and slope of by , , resp. .
The -scalar product on
where the integration is with respect to the Lebesgue measure on and the counting measure on , determines uniquely –valued and –valued inner products satisfying the double equality
where stands for the trace of . The completion with respect to is a full right –module over , and . In addition, is the Hilbert space .
Instead of the –action Eq. (3), the non-trivial bimodules are acted upon by the Heisenberg group . Equivalently, acts projectively on , and this action is compatible with the natural –actions on and . At the Lie algebra level, this action gives rise to the standard connection on , given by the derivatives with constant curvature: . Furthermore, this connection is bi–Hermitian, in the sense that it preserves both the –valued and the –valued inner product.
3.2. Modular Heisenberg spectral triples
Each bimodule gives rise to a double spectral triple, by coupling it with the flat Dirac by means of its standard connection. Specifically, splits into holomorphic and anti-holomorphic components, , where . One then forms the operator acting on the Hilbert space , where . Together with the natural right action of , these data define a spectral triple of constant curvature . We note that from the spectral point of view the operator resembles the Hodge-de Rham operator of an elliptic curve with coefficients in a line bundle. In particular its Laplacian is a direct sum of copies of the harmonic oscillator
Now turning on the conformal change Eq. (5) from to , one replaces by in the same way as in §2.2. The resulting spectral triple over the algebra is again a twisted one. After correcting for the lack of unitarity of the action of again as in §2.2, the operator is being canonically identified with acting on .
The appropriate transposed in this setting is constructed using the canonical anti-isomorphism from to ,
which switches the –action on the first into the –action on the second. We thus arrive at the modular Heisenberg spectral triple with . Its Laplacian on sections is .
A moment of reflection shows that replacing by is equivalent to changing the Hermitian structure on by . Indeed, in view of Eq. (21) applied to , the passage to the –valued Hermitian inner product
has the same effect on the -inner product, since
In conclusion, the passage from the ‘constant curvature metric’ on represented by the Heisenberg spectral triple to the ‘curved metric’ represented by the modular Heisenberg spectral triple can be interpreted as being effected by changing the Hermitian structure of according to Eq. (22). Note that this interpretation remains valid even when , i.e. for .
The extended version of Connes’ pseudodifferential calculus (see §4) allows to establish the heat asymptotic expansion
and express its functional coefficients in local form. In particular, the curvature functional is of the form