Duality in Noncommutative Geometry

The structure of spacetime duality and discrete worldsheet symmetries of compactified string theory is examined within the framework of noncommutative geometry. The full noncommutative string spacetime is constructed using the Fröhlich-Gawȩdzki spectral triple which incorporates the vertex operator algebra of the string theory. The duality group appears naturally as a subgroup of the automorphism group of the vertex operator algebra and spacetime duality is shown to arise as the possibility of associating two independent Dirac operators, arising from the chiral structure of the worldsheet theory, to the noncommutative geometry. PACS: 02.40.+m, 04.60.+n, 11.17.+y ∗Permanent Address: Dipartimento di Scienze Fisiche, Università di Napoli Federico II and INFN, Sezione di Napoli, Italy. †Work supported in part by the Natural Sciences and Engineering Research Council of Canada. One of the key tools to the understanding of the spacetime structure of string theory is the concept of ‘duality’ (see [1] for a review). A duality in string theory relates a geometry of the target space in which the strings live to an inequivalent one. The mapping between distinct geometries is a symmetry of the quantum string theory. The equivalence between such geometries from the string theoretic point of view leads to the notion of a stringy or quantum spacetime which forms the moduli space of string vacua and describes the appropriate stringy modification of classical general relativity. Dualities have been exploited recently to relate apparently distinct string theories. A present model for the spacetime structure of superstring theory is ‘M Theory’ [2] which subsumes all five consistent superstring theories in ten dimensions via duality transformations and contains 11-dimensional supergravity as its low-energy limit. T-duality, which relates large and small radius circles, leads to a fundamental length scale in string theory, determined by the Planck length lP . A common idea is that at distances smaller than lP the conventional notion of a spacetime geometry is inadequate to describe its structure. The string configurations are conjectured to be smeared out and the notion of a ‘point’ in the spacetime ceases to make sense. A recent candidate theory for this picture is the effective matrix field theory for D-branes [3] in which the spacetime coordinates are described by noncommuting matrices. In this Letter we shall discuss how the effective string spacetime and its associated dualities can be described using the techniques of noncommutative geometry [4]. These mathematical tools are particularly well-suited to the study of the structure of the stringy spacetime, in that it views it not as a set of coordinates, but rather in terms of the set of fields defined on it. The basic object which describes a metric space in noncommutative geometry is the spectral triple T = (H,A, D), where H is a Hilbert space, A is a C∗-algebra of bounded operators acting on H, and D is a Dirac operator on H. A spin-manifold M with metric gμν is described by choosing H = L(spin(M)), the square-integrable spinors on M , and the abelian algebra A = C(M) of continuous complex-valued functions on M acting by pointwise multiplication in H. This is the canonical C∗-algebra associated with any manifold, and it determines the topology of a space through the continuity criterion. In fact, there is a one-to-one correspondence between the set of all topological spaces and the collection of commutative C∗-algebras, and therefore the study of the properties of spacetime manifolds can be substituted by a study of the properties of abelian C∗-algebras. The usual Dirac operator D = igγμ∇ν then describes the Riemannian geometry of the manifold, where the real-valued gamma-matrices obey the Clifford algebra {γμ, γν} = 2gμν and ∇μ is the usual covariant derivative constructed from the spin-connection. Thus, roughly speaking, D is the “inverse” of the infinitesimal dx which determines geodesic distances in the spacetime. A quantum theory of point particles on M naturally supplies H = L(spin(M)) and can therefore be thought of as describing an ordinary spacetime M . In the case of string