The Geometry of Determinant Line Bundles in Noncommutative Geometry

This paper is concerned with the study of the geometry of determinant line bundles associated to families of spectral triples parametrized by the moduli space of gauge equivalent classes of Hermitian connections on a Hermitian finite projective module. We illustrate our results with some examples that arise in noncommutative geometry. Introduction In the mid 1990s, Connes and Moscovici [4] formulated and proved a far reaching local index theorem for spectral triples and introduced the correct definition of dimension in the noncommutative setting, where it is no longer an integer, but rather a subset of C which is called the dimension spectrum. This paper aims to understand the stability of the dimension spectrum for families of spectral triples, and its implications on the existence of geometric structures on the determinant line bundle, such as the Quillen metric and the determinant section. That is, we are concerned with the study of the geometry of determinant line bundles associated to families of spectral triples (A,H, D) parametrized by the moduli space of gauge equivalent classes of Hermitian connections on a Hermitian finite projective module. Recall that spectral triples (A,H, D) were introduced by Connes [2], as defining a noncommutative manifolds, where A is a separable, unital C-algebra acting on a separable Hilbert space H, D an unbounded self-adjoint operator on H such that D has compact resolvent, and [D, a] is a bounded operator on H for all a ∈ A. Given a spectral triple (A,H, D), a finite projective module E over A, a Hermitian structure on E and any Hermitian connection ∇ on E, a basic stability result implies that (EndA(E), E ⊗A H, DE,∇) is again a spectral triple. The space of all Hermitian connections on E is an affine space CE , and the gauge group G is defined to be a Lie subgroup of the group AutA(E) of invertible elements in EndA(E), such that G acts smoothly and freely on CE . Analogous to the classical case we define the determinant line bundle L of the index bundle for this family of spectral triples. This is a line bundle on the moduli space CE/G of gauge equivalent classes of Hermitian connections on E. In order to state the hypotheses required to define the Quillen metric [12] and the determinant section of L, we need to utilise the notions of regularity and simple dimension spectrum introduced in [4]. More precisely, we require the spectral triple (A,H, D) to be regular with simple dimension spectrum and zero is not in the dimension spectrum. These are precisely the same assumptions made by Connes and Chamseddine [3] in their work on inner fluctuations of spectral actions, except that we do not need the assumption that zero is not in the spectrum of D, cf. §4.3. In this context, Higson [9] also treats the case when zero is in the spectrum of D, but our approach differs from his. Another technical result proved here is the stability property for regular spectral triples (A,H, D), which says that (EndA(E), E ⊗A H, DE,∇) is again a regular spectral triple with simple dimension 1 2 PARTHA SARATHI CHAKRABORTY AND VARGHESE MATHAI spectrum for any Hermitian structure on E and any Hermitian connection ∇ on E. For the application of these constructions to a mathematical understanding of anomalies, that is, the nonpreservation of a symmetry of the classical action by the full quantum action in a gauge field theory, see [1, 6]. The last section works out an explicit calculation of the Quillen metric and determinant section of the determinant line bundle of the index bundle for the family of spectral triples on the noncommutative torus parametrized by the moduli space of flat Yang-Mills connections on a free module of rank one which were studied by Connes-Rieffel [5]. These are expressed in terms of the theta and eta functions on the moduli space which is a torus. In [11], Perrot has studied a K-theoretic index theorem for families of spectral triples parametrized by the moduli space of gauge equivalent classes of Hermitian connections on E. He makes the restrictive assumption that the gauge group G be contained in the unitary group of A, which is unnecessary in our context here, and he also does not construct the determinant line bundle of the index bundle and its geometry, which is the main study in this paper. 1. Preliminaries In this section, we recall the construction of the determinant line on the Banach manifold of all bounded Fredholm operators acting between separable Hilbert spaces. This motivates the constructions used later in the paper. Recall that a Fredholm operator T : H0 −→ H1 between two infinite dimensional Hilbert spaces H0,H1, is a bounded linear operator such that dim(ker(T )) < ∞ and dim(coker(T )) < ∞. This implies in particular that Im(T ) is a closed subspace of H1. Let F = F(H0,H1) denote the space of all Fredholm operators between two infinite dimensional Hilbert spaces H0,H1. It follows from Atkinson’s theorem that F is an open subset of the Banach space B(H0,H1), of all bounded linear operators between the two Hilbert spaces, which establishes in particular that F is a Banach manifold modeled on the Banach space B(H0,H1). It has countably many connected components labelled by, index : π0(F) ∼= → Z, where for T ∈ F , index(T ) = dim(ker(T ))− dim(coker(T )). We want to review the construction of a smooth line bundle DET → F , called the determinant line bundle, such that DETT = Λ (ker(T ))⊗Λ(coker(T )). The obvious definition does not work since dim(ker(T )) jumps as T varies smoothly. This problem was essentially fixed in [12]. For the convenience of the reader we elaborate on his solution. Let Grfin denote the space of all finite dimensional subspaces of H1. Consider the open cover {UF} of F , where F ∈ Grfin and UF = {T ∈ F : Im(T ) + F = H1}. For T ∈ UF , consider the exact sequence of finite dimensional vector spaces, (1) 0 → ker(T ) → TF T → F → coker(T ) → 0. Since index is constant on smooth families, and the rank of F is fixed on UF , therefore the rank of TF is constant on smooth families in UF . So E F → UF defined by E T = T F , is a smooth vector bundle. The virtual vector bundle INDEX → UF is defined to be the pair (E F , F ), where F denotes the trivial vector bundle over UF with fibre F . Using the inner products on Hi, i = 0, 1, the sequence in equation (1) splits, ker(T )⊕ F ∼= TF ⊕ coker(T ), therefore Λ(ker(T )) ⊗ Λ(coker(T )) ∼= Λ(TF ) ⊗ ΛF. THE GEOMETRY OF DETERMINANT LINE BUNDLES IN NCG 3 The determinant line bundle, DET → UF , is defined as the smooth line bundle, DET = det((INDEX )), i.e. DET = Λ(E ) ⊗ ΛF . Suppose that T ∈ UE ∩ UF . Then we have the exact sequences 0 // ker(T ) = ι1 // TF ⊕ (E \ F ) φ T⊕1 // F + E = // coker(T )