In this paper we consider the quantization of open strings ending on D-branes with a background B field. We find that spacetime coordinates of the open string end-points become noncommutative, and correspondingly the D-brane worldvolume also becomes noncommutative. This provides a string theory derivation and generalization of the noncommutativity obtained previously in the M(atrix) model compactification. For Dp-branes with p ≥ 2 our results are new and agree with that of M(atrix) theory for the case of A = 0 (where A is the worldvolume gauge field) if the T-duality radii are used.
We study the one loop dynamics of QFT on the fuzzy sphere and calculate the planar and nonplanar contributions to the two point function at one loop. We show that there is no UV/IR mixing on the fuzzy sphere. The fuzzy sphere is characterized by two moduli: a dimensionless parameter N and a dimensionful radius R. Different geometrical phases can obtained at different corners of the moduli space. In the limit of the commutative sphere, we find that the two point function is regular without UV/IR mixing; however quantization does not commute with the commutative limit, and a finite "noncommutative anomaly" survives in the commutative limit. In a different limit, the noncommutative plane R 2 θ is obtained, and the UV/IR mixing reappears. This provides an explanation of the UV/IR mixing as an infinite variant of the "noncommutative anomaly".
In a previous paper we provided a consistent quantization of open strings ending on D-branes with a background B field. In this letter, we show that the same result can also be obtained using the more traditional method of Dirac's constrained quantization. We also extend the discussion to the fermionic sector.
3We briefly discuss some possible cosmological implications of noncommutative geometry. While the noncommutativity we consider does not affect gravity, it can play an important role in the dynamics of other fields that are present in the early universe. We point out the possibility that noncommutativity may cause inflation induced fluctuations to become non-gaussian and anisotropic, and may modify the short distance dispersion relations.It has long been recognized that cosmology provides a fertile testing ground for theories beyond the standard model of particle physics. For string theory, in fact, cosmology may one day provide the most accessible way to probe the theory experimentally. In this regard, inflation is an especially promising framework as the enormous growth of scales in the early universe stretches regions on the order of the Planck scale -the likely relevant scale for string theory -to the much larger scales of relevance for cosmology.Recently, there has been significant interest in noncommutative geometry due to developments in matrix theory [1] and the realization [2-6] that noncommutative spacetime arises naturally in string and M-theory when a certain background gauge field is turned on. In particular, it was shown [4][5][6] that in the presence of a constant NS B µν -field, the endpoints of the open string obey the commutation relationwhere the θ µν are entries of an antisymmetric real constant matrix of dimension length squared. The relation between θ and B can be found in [4][5][6]. Moreover the commutation relations of the string modes are modified. These relations have been employed directly to construct noncommutative open string theory at any loop order [7]. Note that perturbatively the noncommutativity is only felt by open strings, closed strings are not affected by the B-field.A number of authors have studied the possible phenomenological effects of such noncommutativity [8]. In this brief note, using basic properties of noncommutative field theory [6,9-11], we point out some possible cosmological signatures. The idea is that if spacetime is indeed noncommutative on short distance scales, this may have an impact on early universe physics. As above, we work in the context of inflation which allows such short scale noncommutativity to amplify into large scale cosmological implications. Specifically, we focus on the generation of density perturbations. In the usual setup, quantum fluctuations of the inflaton field, after suitable tuning of its potential, can give rise to the requisite density perturbations for structure formation. We reexamine these calculations in the noncommutative framework and point out features that can differ from the usual commutative case. Inflationary cosmology in noncommutative geometry from a different perspective was studied in [12].
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