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Gravitational Scattering in 2 + 1 Dimensions and Wilson Loop Operators

M. Hayashi, F. Permana Zen

1994
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Progress of theoretical physics
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We investigate a scattering amplitude of two particles in 2+ 1 dimensional Chern-Simons theory. By refining the previous computations, we have found how to express the amplitude in terms of vacuum expectation values (VEV's) of Wilson loop operators. We have found extra terms to the amplitude. Then, we calculate VEV's of the Wilson loop operators in super Chern-Simons theory. They must be preliminary information of 2+ 1 dimensional supergravities. § 1. Introudction The 2+ 1 dimensional
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... ns theory has been providing many interesting topics for both mathematics and physics. Abelian case was first studied in Ref. 1), the Yang-Mills theory with the Chern-Simons term was investigated in .Refs. 2)~4), supersymmetric extension was discussed in Refs. 5) and 6) and classical pure Chern-Simons theory was examined in Ref. 7). In Ref. 8), the relation between the Chern-Simons theory and the polynomial invariants of knots and links was pointed out_ Studies of this relation have been extended from various points of view in Refs. 9) ~23)_ It was shown that gravity theories in 2+ 1 dimensions, with or without cosmological constant term, conformal symmetry and/ or supersymmetry. can be formulated as the Chern-Simons theory in Refs. 24)~41). The non-compact Chern-Simons theory was studied in Refs_ 42)~44)_ The relation between the Chern-Simons theory and the lattice gravity theory was discussed in Refs. 45)~49). Many appealing properties of it originate, in most cases, in the fact that the Chern-Simons theory is topologicaL Physical observables or gauge invariant operators must be topological; they can depend only on the global topology of the spacetime manifold. The operator we know which satisfies such requirements is the Wilson loop operator: a holonomy of the gauge field around a circle embedded in the space-time manifold. If we choose 150(2, 1), or the 2 + 1 dimensional Poincare group, as a gauge group, the Chern-Simons action can be identified with the Einstein-Hilbert action in 2+ 1 dimensions. The Wilson loop operators are, therefore, expected to play an important role also in the 2+ 1 dimensional gravity theories. Previous works on the quantum gravity in 2+ 1 dimensions 50 J-s 4 J must be helpful to attack this problem, although rather different method was used_ In 2+ 1 dimensions, naive counting of degrees of freedom shows that a graviton has no dynamically propagating mode at alL Indeed, the Einstein equation requires *l On leave

doi:10.1143/ptp/91.2.361
fatcat:psylncfilbek3kw7iopjmusl24