Hamiltonian Gravity and Noncommutative Geometry

A version of foliated spacetime is constructed in which the spatial geometry is described as a time-dependent noncommutative geometry. The ADM version of the gravitational action is expressed in terms of these variables. It is shown that the vector constraint is obtained without the need for an extraneous shift vector in the action. Received: 24 June 1996 / Accepted: 30 January 1997     

On Semi-Classical States of Quantum Gravity and Noncommutative Geometry

We construct normalizable, semi-classical states for the previously proposed model of quantum gravity which is formulated as a spectral triple over holonomy loops. The semi-classical limit of the spectral triple gives the Dirac Hamiltonian in 3+1 dimensions. Also, time-independent lapse and shift fields emerge from the semi-classical states. Our analysis shows that the model might contain fermionic matter degrees of freedom.     

Vacuum Solutions of Classical Gravity on Cyclic Groups from Noncommutative Geometry

Based on the observation that the moduli of a link variable on a cyclic group modify Connes‘ distance on this group,we construct several action functionals for this link variable within the framework of noncommutative geometry.After solving the equations of motion,we find that one type of action gives nontrivial vacuum solution for gravity on this cyclic group in a broad range of coupling constants and that such a solution can be expressed with Chebyshev‘s polynomials.     

Noncommutative Geometry, Superconnections and Riemannian Gravity as a Low-Energy Theory

A superconnection is a supermatrix whose evenpart contains the gaugepotential one-forms of a localgauge group, while the odd parts contain the (zero-form)Higgs fields breaking the local symmetry spontaneously. The combined grading is thus odd everywhere andthe superconnection can be directly derived from aformulation of Noncommutative Geometry, as theappropriate one-form in the relevant form calculus. The simple supergroup (4, ) (rank = 3) in Kac' classification (evensubgroup (4,)) provides themost economical spontaneous breaking of (4,) as gauge group leaving just local (1,3) unbroken. Post-Riemannian SKY gravity thereby yields Einstein's theory asa low-energy (longer range) effective theory. The theoryis renormalizable and may be unitary.     

Complexified Gravity in Noncommutative Spaces

The presence of a constant background antisymmetric tensor for open strings or D-branes forces the space-time coordinates to be noncommutative. This effect is equivalent to replacing ordinary products in the effective theory by the deformed star product. An immediate consequence of this is that all fields get complexified. The only possible noncommutative Yang–Mills theory is the one with U(N) gauge symmetry. By applying this idea to gravity one discovers that the metric becomes complex. We show in this article that this procedure is completely consistent and one can obtain complexified gravity by gauging the symmetry U(1,D−1) instead of the usual SO(1,D−1). The final theory depends on a Hermitian tensor containing both the symmetric metric and antisymmetric tensor. In contrast to other theories of nonsymmetric gravity the action is both unique and gauge invariant. The results are then generalized to noncommutative spaces. Received: 1 June 2000 / Accepted: 27 November 2000     

The Geometry of Noncommutative Space-Time

Stabilization, by deformation, of the Poincaré-Heisenberg algebra requires both the introduction of a fundamental lentgh and the noncommutativity of translations which is associated to the gravitational field. The noncommutative geometry structure that follows from the deformed algebra is studied both for the non-commutative tangent space and the full space with gravity. The contact points of this approach with the work of David Finkelstein are emphasized.     

The Decay of Unstable Noncommutative Solitons

 We study the classical decay of unstable scalar solitons in noncommutative field theory in 2+1 dimensions. This can, but does not have to, be viewed as a toy model for the decay of D-branes in string theory. In the limit that the noncommutativity parameter θ is infinite, the gradient term is absent, there are no propagating modes and the soliton does not decay at all. If θ is large, but finite, the rotationally symmetric decay channel can be described as a highly excited nonlinear oscillator weakly coupled to a continuum of linear modes. This system is closely akin to those studied in the context of discrete breathers. We here diagonalize the linear problem and compute the decay rate to first order using a version of Fermi's Golden Rule, leaving a more rigorous treatment for future work. Received: 16 January 2003 / Accepted: 21 March 2003 Published online: 7 May 2003 Communicated by H. Araki, D. Buchholz, and K. Fredenhagen     

Topics in Noncommutative Geometry Inspired Physics

In this review article we discuss some of the applications of noncommutative geometry in physics that are of recent interest, such as noncommutative many-body systems, noncommutative extension of Special Theory of Relativity kinematics, twisted gauge theories and noncommutative gravity.     

Differentials of Higher Order in Noncommutative Differential Geometry

In differential geometry, the notation dⁿ f along with the corresponding formalism has fallen into disuse since the birth of exterior calculus. However, differentials of higher-order are useful objects that can be interpreted in terms of functions on iterated tangent bundles (or in terms of jets). We generalize this notion to the case of noncommutative differential geometry. For an arbitrary algebra A, people already know how to define the differential algebra A of universal differential forms over A. We define Leibniz forms of order n (these are not forms of degree n, i.e. they are not elements of ⁿA) as particular elements of what we call the iterated frame algebra of order n, F_nA, which is itself defined as the2ⁿ tensor power of the algebra A. We give a system of generators for this iterated frame algebra and identify the A left-module of forms of order n as a particular vector subspace included in the space of universal 1-forms built over the iterated frame algebra of order n-1. We study the algebraic structure of these objects, recover the case of the commutative differential calculus of order n (Leibniz differentials) and give a few examples.     

Lie Bracket of Vector Fields in Noncommutative Geometry

The aim of this paper is to avoid some difficulties, related with the Lie bracket, in the definition of vector fields in a noncommutative setting, as they were defined by Woronowicz, Schmüdgen-Schüler and Aschieri-Schupp. We extend the definition of vector fields to consider them as derivations of the algebra, through Cartan pairs introduced by Borowiec. Then, using translations, we introduce the invariant vector fields. Finally, the definition of Lie bracket realized by Dubois-Violette, considering elements in the center of the algebra, is also extended to these invariant vector fields.     

On the Noncommutative Geometry of Twisted Spheres

We describe noncommutative geometric aspects of twisted deformations, in particular of the spheres of Connes and Landi and of Connes and Dubois Violette, by using the differential and integral calculus on these spaces that is covariant under the action of their corresponding quantum symmetry groups. We start from multiparametric deformations of the orthogonal groups and related planes and spheres. We show that only in the twisted limit of these multiparametric deformations the covariant calculus on the plane gives, by a quotient procedure, a meaningful calculus on the sphere. In this calculus, the external algebra has the same dimension as the classical one. We develop the Haar functional on spheres and use it to define an integral of forms. In the twisted limit (differently from the general multiparametric case), the Haar functional is a trace and we thus obtain a cycle on the algebra. Moreover, we explicitly construct the *-Hodge operator on the space of forms on the plane and then by quotient on the sphere. We apply our results to even spheres and compute the Chern–Connes pairing between the character of this cycle, i.e. a cyclic 2n-cocycle, and the instanton projector defined in math.QA/0107070.     

Quantization of Semi-Classical Twists and Noncommutative Geometry

A problem of defining the quantum analogues for semi-classical twists in U()[[t]] is considered. First, we study specialization at q = 1 of singular coboundary twists defined in Uq ())[[t]] for g being a nonexceptional Lie algebra, then we consider specialization of noncoboundary twists when = and obtain q-deformation of the semiclassical twist introduced by Connes and Moscovici in noncommutative geometry. Mathematics Subject Classification: 16W30, 17B37, 81R50     

Gravity and the Noncommutative Residue for Manifolds with Boundary

We prove a Kastler–Kalau–Walze type theorem for the Dirac operator and the signature operator for 3,4-dimensional manifolds with boundary. As a corollary, we give two kinds of operator-theoretic explanations of the gravitational action in the case of 4-dimensional manifolds with flat boundary.      

Quantum Group of Isometries in Classical and Noncommutative Geometry

We formulate a quantum generalization of the notion of the group of Riemannian isometries for a compact Riemannian manifold, by introducing a natural notion of smooth and isometric action by a compact quantum group on a classical or noncommutative manifold described by spectral triples, and then proving the existence of a universal object (called the quantum isometry group) in the category of compact quantum groups acting smoothly and isometrically on a given (possibly noncommutative) manifold satisfying certain regularity assumptions. The idea of ‘quantum families’ (due to Woronowicz and Soltan) are relevant to our construction. A number of explicit examples are given and possible applications of our results to the problem of constructing quantum group equivariant spectral triples are discussed. Supported in part by the Indian National Academy of Sciences.     

Editors’ preface for the special issue on Noncommutative Geometry

We prove in the general framework of noncommutative geometry that the inner fluctuations of the spectral action can be computed as residues and give exactly the counterterms for the Feynman graphs with fermionic internal lines. We show that for geometries of dimension less than or equal to four the obtained terms add up to a sum of a Yang–Mills action with a Chern–Simons action.     

Differential and Complex Geometry of Two-Dimensional Noncommutative Tori

We analyze in detail projective modules over two-dimensional noncommutative tori and complex structures on these modules. We concentrate our attention on properties of holomorphic vectors in these modules; the theory of these vectors generalizes the theory of theta-functions. The paper is self-contained; it can be used also as an introduction to the theory of noncommutative spaces with simplest space of this kind thoroughly analyzed as a basic example.     

BRS Cohomology and the Chern Character in Noncommutative Geometry

We briefly report on new results concerning a perturbation expansion structure within the framework of an analytic version of perturbative quantum chromodynamics (pQCD). This approach combines the RG symmetry with the Källén–Lehmann analyticity in the Q² variable. The procedure of analytization matches this analyticity with the RG invariance by adding to the analytized invariant coupling some nonperturbative contributions containing no adjustable parameters. In turn, the new perturbative expansion (the APT expansion) for an observable represents asymptotic expansion over a nonpower set of specific functions rather than in powers of . We analyze this set and show that it obeys different properties in various ranges of the Q² variable. In the UV, it is close to the power set used in the pQCD calculation. However, generally, this set is of a more complicated nature. In the low Q² region the behavior of is oscillating. Here, the APT expansion has a feature of asymptotic expansion à la Erdélyi. The issue of the consistency of an analytization procedure with the RG structure of observables is also discussed.     

Carnot-Carathéodory Metric and Gauge Fluctuation in Noncommutative Geometry

Gauge fields have a natural metric interpretation in terms of horizontal distance. The latest, also called Carnot-Carathéodory or subriemannian distance, is by definition the length of the shortest horizontal path between points, that is to say the shortest path whose tangent vector is everywhere horizontal with respect to the gauge connection. In noncommutative geometry all the metric information is encoded within the Dirac operator D. In the classical case, i.e. commutative, Connes’s distance formula allows to extract from D the geodesic distance on a riemannian spin manifold. In the case of a gauge theory with a gauge field A, the geometry of the associated U(n)-vector bundle is described by the covariant Dirac operator D+A. What is the distance encoded within this operator? It was expected that the noncommutative geometry distance d defined by a covariant Dirac operator was intimately linked to the Carnot-Carathéodory distance dh defined by A. In this paper we make precise this link, showing that the equality of d and d _H strongly depends on the holonomy of the connection. Quite interestingly we exhibit an elementary example, based on a 2 torus, in which the noncommutative distance has a very simple expression and simultaneously avoids the main drawbacks of the riemannian metric (no discontinuity of the derivative of the distance function at the cut-locus) and of the subriemannian one (memory of the structure of the fiber).