The Spectral Action Principle in Noncommutative Geometry and the Superstring

A supersymmetric theory in two-dimensions has enough data to define a noncommutative space thus making it possible to use all tools of noncommutative geometry. In particular, we apply this to the N = 1 supersymmetric non-linear sigma model and derive an expression for the generalized loop space Dirac operator, in presence of a general background, using canonical quantization. The spectral action principle is then used to determine a spectral action valid for the fluctuations of the string modes. It is now generally accepted that at very high energies, the structure of space-time could not adequately be described by a manifold. Quantum fluctuations makes it difficult to define localised points. The most familiar example is string theory where points are replaced by strings, and space-time becomes a loop space [1, 2, 3]. What has been lacking, up to now, are the mathematical tools necessary to realize such spaces geometrically. Fortunately, the recent advances in noncommutative geometry as formulated by Connes [4] makes it possible to tackle such problems. The main advantage in adopting Connes’ formulation of noncommutative geometry is that the geometrical data is determined by a spectral triple (A,H, D) where A is an algebra of operators, H a Hilbert space amd D a Dirac operator acting on H. These ideas have been successfully applied to simple generalizations of spacetime such as a product of discrete by continuos spaces . The results are very encouraging in the sense that with a very simple imput one gets all the details of the standard model including the Higgs mechanism, and the unification of the Higgs fields with the gauge fields [5], as well as unification with gravity [6, 7] . Supersymmetric field theories in two-dimensions have enough data to define noncommutative geometries [8, 9, 10]. In two-dimensions one can have (p, q) supersymmetry as the left and right moving sectors could be split, giving rise to various possibilities. The simplest possibilities are N = 1 and N = 1 2 (i.e. (1, 1) and (1, 0) respectively). A good starting point would be to consider various superconformal field theories and use the noncommutative geometric tools to define geometric objects of interest. This would be fruitful in cases such as orbifold compactifications where many useful data is available to help define the noncommutative geometric space completely. In this letter we shall adopt a slightly different framework where the starting point is the supersymmetric non-linear sigma model in twodimensions [11], with a general curved target space background. The conserved supersymmetric charges satisfy the supersymmetry algebra [1]. Canonical quantization would then change these charges to Dirac operators over the loop space Ω(M) where M is a Riemannian spin-manifold. The square of a Dirac operator when restricted to reparametrization invariant configurations gives the Hamiltonian of the system, as an elliptic pseudo-differential operator. These operators could be used to write down a spectral action as a function of the background fields, which gives the low-energy effective action of string theory when the loops are shrunk to points. At high-energies where oscillators are present the full spectral action must be considered. The plan of this letter is as follows. First we give the essential definitions needed to define a noncommutative space. Next we consider an N = 1 supersymmetric non-linear sigma model on a curved background with torsion and construct the corresponding Dirac operator. The algebra is given by the algebra of continous functions on the loop space, and the Hilbert space is the Hilbert space H of states usually comprising two sectors, the Ramond sector (R) with periodic boundary conditions for fermions and Neveu-Schwarz (NS) with anti-periodic boundary conditions for fermions. These data are then used to define a 1 spectral action based on the loop space Dirac operators, which will also give the Hamiltonian and momentum operators in two-dimensions. We note that such considerations have been performed before to determine the index of the loop space Dirac operator (elliptic genus) but without torsion [12, 13]. Similar considerations were also performed for the non-linear sigma models for point particles (with or without torsion) [14, 15, 16],where the Hamiltonian of the system was determined. The proposed action satisfies the constraints that it gives at low-energies the string effective action, and the partition function in the limit when the background geometry is flat. A starting point in defining noncommutative geometry [4] is the spectral triple (A,H, D) where A is a ∗ algebra of bounded operators acting on a seperable Hilbert space (H, D) is a Dirac operator on H such that [D, a] is a bounded operator for arbitrary a ∈ A. A K-cycle H, D for A is said to be even if there exists a unitary involution Γ on H such that Γa = aΓ for all a ∈ A and ΓD = −DΓ. Given a unital algebra A one can define the universal differential algebra Ω.(A) = ⊕n=0Ω(A) as follows: One sets Ω0(A) = A and define Ωn(A) to be the linear space given by Ω(A) = {