Abstract
We discuss the squashed fuzzy sphere, which is a projection of the fuzzy sphere onto the equatorial plane, and use it to illustrate the stringy aspects of noncommutative field theory. We elaborate explicitly how strings linking its two coincident sheets arise in terms of fuzzy spherical harmonics. In the large N limit, the matrix-model Laplacian is shown to correctly reproduce the semi-classical dynamics of these charged strings, as given by the Landau problem.
Highlights
Field theory on noncommutative (NC) spaces has been studied intensively from various points of view in the past decades
Scalar fields on a noncommutative space arise in string theory as open strings starting and ending on a D-brane with B field [2, 3]. This suggests a dipole-like nature of noncommutative fields [4, 5], which is implicit in the matrix-model realization of noncommutative gauge theory and its relation with string theory [6], culminating in the remarkable proposals [7, 8] that string theory might be defined in terms of matrix models
We have identified string-like modes among the fuzzy spherical harmonics, which connect the upper with the lower hemisphere
Summary
Field theory on noncommutative (NC) spaces has been studied intensively from various points of view in the past decades. NC field theory behaves very differently from ordinary field theory at scales far above ΛNC, where the basic degrees of freedom display a string-like or dipolelike nature. The interactions determined by the algebra of noncommutative scalar fields with momentum far above ΛNC is very different from the commutative case; this These insights are very useful to study NC field theory per se, without any direct relation with string theory. B+ = −B− corresponds to the symplectic forms i.e. the inverse Poisson structures on the two sheets At low energies, these should behave like point-like charged objects moving in an effective magnetic field B+ − B−, which – focusing on the center of the disks in a suitable scaling limit – should reduce to the Landau problem. The present paper demonstrates how an appropriate organization of the degrees of freedom can illuminate the stringy physics hidden in NCFT, which transcends the picture of conventional field theory
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