Modular curvature for noncommutative two-tori

Starting from the description of the conformal geometry of noncommutative 2-tori in the framework of modular spectral triples, we explicitly compute the local curvature functionals determined by the value at zero of the zeta functions affiliated with these spectral triples. We give a closed formula for the Ray-Singer analytic torsion in terms of the Dirichlet quadratic form and the generating function for Bernoulli numbers applied to the modular operator. The gradient of the Ray-Singer analytic torsion is then expressed in terms of these functionals, and yields the analogue of scalar curvature. Computing this gradient in two ways elucidates the meaning of the complicated two variable functions occurring in the formula for the scalar curvature. Moreover, the corresponding evolution equation for the metric produces the appropriate analogue of Ricci curvature. We prove the analogue of the classical result which asserts that in every conformal class the maximum value of the determinant of the Laplacian on metrics of a fixed area is attained only at the constant curvature metric. Introduction In noncommutative geometry the paradigm of a geometric space is given in spectral terms, by a Hilbert space H in which both the algebra A of coordinates and the inverse line element D are represented, the latter being an unbounded self-adjoint operator which plays the role of the Dirac operator. The local geometric invariants such as the Riemannian curvature are obtained in the noncommutative case by considering heat kernel expansions of the form Tr(ae−tD 2 ) ∼ ∑ n≥0 an(a,D )t −d+n 2 where d is the dimension of the geometry. One may equivalently deal with the corresponding zeta functions. Thus, it is the high frequency behavior of the spectrum of D coupled with the action of the algebra A in H which detects the local curvature of the geometry. In this paper we shall analyze in depth a specific example, that of the noncommutative two torus Tθ whose differential geometry, as well as its pseudo-differential operator calculus, was first developed in [8] . To obtain a curved geometry from the flat one defined in [8], one introduces (cf. [7], [13]) a Weyl factor, or dilaton, which changes the metric by modifying the volume form while keeping the same conformal structure. Both notions of volume form and of conformal structure are well understood (cf. [9]), and we recall in §1 how one obtains the modified Dirac The work of the first named author was partially supported by the National Science Foundation award no. DMS-0652164. The work of the second named author was partially supported by the National Science Foundation awards no. DMS-0652167 and DMS-0969672. 1 ar X iv :1 11 0. 35 00 v1 [ m at h. Q A ] 1 6 O ct 2 01 1 2 CONNES AND MOSCOVICI operator for the new curved geometry. The starting point is the computation of the value at s = 0 of the zeta function Tr(a|D|−2s) for the 2-dimensional curved geometry associated to the dilaton h. Equivalently we are dealing with the term a2(a,D ) of the heat expansion. This computation was initiated in the late 1980’s (cf. [7]), and the explicit result which proves the analogue of the Gauss–Bonnet formula was published in [13]. It was then extended in [14] to cover the case of arbitrary values of the complex modulus τ (which was τ = i in [13]). In these two papers only the total integral of the curvature was needed, and this allowed one to make simplifications under the trace which are no longer possible in the case of arbitrary a, i.e. to compute the local expression a2(a,D ). The technical obstacles for the local computation were overcome by means of • the general rearrangement Lemma of §6.2 ; • the assistance of the computer. While the original computation of [7] was done by hand, the role of the computer assistance is to minimize the danger of a computational mistake in handling the large number of terms, about one thousand, which arise in the generalized pseudodifferential expressions involved in a2(a,D ). The complete calculation of a2(a,D ) was actually achieved in 2009 and announced (including by internet posting, although with some typos) at several conferences (Oberwolfach 2009 and Vanderbilt 2011). Moreover, the same computation was done independently in [15] and gave a confirmation of the result. The main additional input of the present paper consists in obtaining an explicit formula for the Ray-Singer log-determinant of D (which was left open in [7]). By calculating the gradient of the Ray-Singer log-determinant in two different ways, one obtains new geometric insight as well as a deep internal consistency relation between the different terms of the log-determinant formula. At the same time this elucidates the role of the intricate two operator-variable function occurring in its expression. We now briefly outline the contents of this paper, starting with the description of the local curvature functionals determined by the value at zero of the zeta functions affiliated with the modular spectral triples describing the curved geometry of noncommutative 2-tori. As in the case of the standard torus viewed as a complex curve, the total Laplacian associated to such a spectral triple splits into two components, one 4φ on functions and the other 4 φ on (0, 1)-forms, the two operators being isospectral outside zero. The corresponding curvature formulas involve second order (outer) derivatives of the Weyl factor, and as a new and crucial ingredient they involve the modular operator ∆ of the non-tracial weight φ(a) = φ0(ae −h) associated to the dilaton h. For 4φ the result is of the form a2(a,4φ) = − π 2τ2 φ0(a ( K0(∇)(4(h)) + 1 2 H0(∇1,∇2)( <(h) ) , (1) where ∇ = log ∆ is the inner derivation implemented by −h, 4(h) = δ 1(h) + 2<(τ)δ1δ2(h) + |τ |δ 2(h), < is the Dirichlet quadratic form <(`) := (δ1(`)) 2 + <(τ) (δ1(`)δ2(`) + δ2(`)δ1(`)) + |τ |(δ2(`)) , and ∇i, i = 1, 2, signifies that ∇ is acting on the ith factor. The operators K0(∇) and H0(∇1,∇2) are new ingredients, whose occurrence is a vivid manifestation of