Curvature of the Determinant Line Bundle for the Noncommutative Two Torus

We compute the curvature of the determinant line bundle on a family of Dirac operators for a noncommutative two torus. Following Quillen’s original construction for Riemann surfaces and using zeta regularized determinant of Laplacians, one can endow the determinant line bundle with a natural Hermitian metric. By using an analogue of Kontsevich-Vishik canonical trace, defined on Connes’ algebra of classical pseudodifferential symbols for the noncommutative two torus, we compute the curvature form of the determinant line bundle by computing the second variation \(\delta _{w}\delta _{\bar {w}}\log \det ({\Delta })\).     

Curvature on determinant bundles and first Chern forms

The Quillen–Bismut–Freed construction associates a determinant line bundle with connection to an infinite dimensional super vector bundle with a family of Dirac-type operators. We define the regularized first Chern form of the infinite dimensional bundle, and relate it to the curvature of the Bismut–Freed connection on the determinant bundle. In finite dimensions, these forms agree (up to sign), but in infinite dimensions there is a correction term, which we express in terms of Wodzicki residues.

We illustrate these results with a string theory computation. There is a natural super vector bundle over the manifold of smooth almost complex structures on a Riemannian surface. The Bismut–Freed superconnection is identified with classical Teichmüller theory connections, and its curvature and regularized first Chern form are computed.     

The analysis of elliptic families

In this paper we specialize the results obtained in [BF1] to the case of a family of Dirac operators. We first calculate the curvature of the unitary connection on the determinant bundle which we introduced in [BF1].We also calculate the odd Chern forms of Quillen for a family of self-adjoint Dirac operators and give a simple proof of certain results of Atiyah-Patodi-Singer on êta invariants.We finally give a heat equation proof of the holonomy theorem, in the form suggested by Witten [W 1, 2].     

Topological anomalies: Explicit examples

We discuss the mathematical picture of anomalies. By solving the Dirac equation in the background of non-trivial families of gauge connections, we show explicitly the interplay between spectral flows, zero modes of the Dirac operator and projective representations of the gauge group, and the existence of both perturbative and non-perturbative anomalies. We give an explicit expression for the fermion determinant for chiral QCD in two dimensions when an anomaly is present.     

The invariant factor of the chiral determinant

The coupling of spin 0 and spin 1 external fields to Dirac fermions defines a theory which displays gauge chiral symmetry. Quantum-mechanically, functional integration of the fermions yields the determinant of the Dirac operator, known as the chiral determinant. Its modulus is chiral invariant but not so its phase, which carries the chiral anomaly through the Wess–Zumino–Witten term. Here we find the remarkable result that, upon removal from the chiral determinant of this known anomalous part, the remaining chiral-invariant factor is just the square root of the determinant of a local covariant operator of the Klein–Gordon type. This procedure bypasses the integrability obstruction allowing one to write down a functional that correctly reproduces both the modulus and the phase of the chiral determinant. The technique is illustrated by computing the effective action in two dimensions at leading order (LO) in the derivative expansion. The results previously obtained by indirect methods are indeed reproduced.     

Determinant Bundles, Manifolds with Boundary and Surgery¶II. Spectral Sections and Surgery Rules for Anomalies

Let be a closed fibration of Riemannian manifolds and let , be a family of generalized Dirac operators. Let be an embedded hypersurface fibering over B; . Let be the Dirac family induced on . Each fiber in is the union along of two manifolds with boundary . In this paper, generalizing our previous work[16], we prove general surgery rules for the local and global anomalies of the Bismut–Freed connection on the determinant bundle associated to . Our results depend heavily on the b-calculus [12], on the surgery calculus [11] and on the APS family index theory developed in [13], in particular on the notion of spectral section for the family . Received: 23 October 1996 / Accepted: 28 July 1997     

Wess-Zumino terms and the geometry of the determinant line bundle

The determinant line bundle L of a family of Dirac operators coupled to Yang-Mills (YM) in any dimension is constructed from the corresponding Wess-Zumino (WZ) term. The equivalence between the algebraic and topological approaches to anomalies is established by straightforward computation. As a by-product the first Chern class of L is expressed through the WZ term and the integrated anomaly is explicitly seen to play the role of a functioonal magnetic field on the gauge-orbit space.     

The ζ-Determinant and C-Determinant on the Grassmannian in Dimension One

We discuss the relation of the -determinant of the Dirac operator on the interval to the canonical determinant, which appears naturally in this situation. This Letter is a pilot for papers in preparation on the determinant defined on the infinite-dimensional Grassmannian of elliptic boundary problems for a Dirac operator in dimensions greater than one.     

The Anomaly Line Bundle of the Self-Dual Field Theory

In this work, we determine explicitly the anomaly line bundle of the abelian self-dual field theory over the space of metrics modulo diffeomorphisms, including its torsion part. Inspired by the work of Belov and Moore, we propose a non-covariant action principle for a pair of Euclidean self-dual fields on a generic oriented Riemannian manifold. The corresponding path integral allows one to study the global properties of the partition function over the space of metrics modulo diffeomorphisms. We show that the anomaly bundle for a pair of self-dual fields differs from the determinant bundle of the Dirac operator coupled to chiral spinors by a flat bundle that is not trivial if the underlying manifold has middle-degree cohomology, and whose holonomies are determined explicitly. We briefly sketch the relevance of this result for the computation of the global gravitational anomaly of the self-dual field theory, that will appear in another paper.     

The analysis of elliptic families. I. Metrics and connections on determinant bundles

In this paper, we construct the Quillen metric on the determinant bundle associated with a family of elliptic first order differential operators. We also introduce a unitary connection on and calculate its curvature. Our results will be applied to the case of Dirac operators in a forthcoming paper.     

Parallel transport in the determinant line bundle: The zero index case

For a product family of invertible Weyl operators on a compact manifoldX, we express parallel transport in the determinant line bundle in terms of the spectral asymmetry of a Dirac operator onR×X.Supported in party by NSF Grants PHY8605978 and PHY-82-15249 and the Robert A. Welch FoundationSupposed in part by NSF Grant PHY-82-15249     

Fermionic determinant with a linear domain wall in 2+1 dimensions

We consider a Dirac field in 2+1 Euclidean dimensions, in the presence of a linear domain wall defect in its mass, and a constant electromagnetic field. We evaluate the exact fermionic determinant for the situation where the defect is assumed to be rectilinear, static, and the gauge field is minimally coupled to the fermions. We discuss the dependence of the result on the (unique) independent geometrical parameter of this system, namely, the relative orientation of the wall and the direction of the external field. We apply the result for the determinant to the evaluation of the vacuum energy.     

Superconnections and index theory

We investigate index theory in the context of Dirac operators coupled to superconnections. In particular, we prove a local index theorem for such operators, and for families of such operators. We investigate η$\eta$-invariants and prove an APS theorem, and construct a geometric determinant line bundle for families of such operators, computing its curvature and holonomy in terms of familiar index theoretic quantities.     

Noncommutative Circle Bundles and New Dirac Operators

We study spectral triples over noncommutative principal U(1) bundles. Basing on the classical situation and the abstract algebraic approach, we propose an operatorial definition for a connection and compatibility between the connection and the Dirac operator on the total space and on the base space of the bundle. We analyze in details the example of the noncommutative three-torus viewed as a U(1) bundle over the noncommutative two-torus and find all connections compatible with an admissible Dirac operator. Conversely, we find a family of new Dirac operators on the noncommutative tori, which arise from the base-space Dirac operator and a suitable connection.     

On the geometry of Dirac determinant bundles in two dimensions

The gauge and diffeomorphism anomalies are used to define the determinant bundles for the left-handed Dirac operator on a two-dimensional Riemann surface. Three different moduli spaces are studied: (1) the space of vector potentials modulo gauge transformations; (2) the space of vector potentials modulo bundle automorphisms; and, (3) the space of Riemannian metrics modulo diffeomorphisms. Using the methods earlier developed for the studies of affine Kac-Moody groups, natural geometries are constructed for each of the three bundles.This work was supported in part by funds provided by the U.S. Department of Energy (D.O.E.) under contract #DE-AC02-76ER03069     

Parallel transport in the determinant line bundle: The non-zero index case

For a product family of Weyl operators of possibly non-zero index on a compact manifoldX, we express parallel transport in the determinant line bundle in terms of the spectral asymmetry of a Dirac operator on ×X. This generalizes the results of [7], where we dealt only with invertible operators.Supported in part by NSF Grant No. PHY 8605978 and the Robert A. Welch FoundationSupported in part by NSF Grant No. PHY 8215249