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We apply stochastic quantization to two-dimensional gravity. The Laplace operator acting on the space of all metrics takes a particularly simple form in terms of the Beltrami parametrization. We show the equivalence between the quantum theory defined by the standard Faddeev-Popov gauge fixing of the two-dimensional diffeomorphism invariance and the one defined by stochastic quantization. We do so by using the gauge freedom left in the Langevin equation of a diffeomorphism-invariant theory to adjust the drift force. Another choice of the drift force, comparable to that of Zwanziger for Yang-Mills theories, seems to avoid the analogue of the Gribov ambiguity, i.e. the necessity of the by-hand restriction to one fundamental domain. We relate the two-dimensional gravity to a three-dimensional theory, based on the three-dimensional gravitational Chern-Simons action for SL(2, C), ISO(3) or SU(2) × SU(2) (depending on the genus of the two-dimensional Riemann surface), in which all fields of the stochastic quantization have been distributed as components of the gauge fields. To study the three-dimensional theory, stochastic quantization can be applied once more. This gives a theory with the action of topological gravity in four dimensions, namely the Pontrjagin invariant ∫N2×R×R tr R ∧ R, gauge fixed by self-duality conditions.

@article{Baulieu1990, abstract = {We apply stochastic quantization to two-dimensional gravity. The Laplace operator acting on the space of all metrics takes a particularly simple form in terms of the Beltrami parametrization. We show the equivalence between the quantum theory defined by the standard Faddeev-Popov gauge fixing of the two-dimensional diffeomorphism invariance and the one defined by stochastic quantization. We do so by using the gauge freedom left in the Langevin equation of a diffeomorphism-invariant theory to adjust the drift force. Another choice of the drift force, comparable to that of Zwanziger for Yang-Mills theories, seems to avoid the analogue of the Gribov ambiguity, i.e. the necessity of the by-hand restriction to one fundamental domain. We relate the two-dimensional gravity to a three-dimensional theory, based on the three-dimensional gravitational Chern-Simons action for SL(2, C), ISO(3) or SU(2) × SU(2) (depending on the genus of the two-dimensional Riemann surface), in which all fields of the stochastic quantization have been distributed as components of the gauge fields. To study the three-dimensional theory, stochastic quantization can be applied once more. This gives a theory with the action of topological gravity in four dimensions, namely the Pontrjagin invariant ∫N2×R×R tr R ∧ R, gauge fixed by self-duality conditions.}, author = {Baulieu, Laurent and Bilal, Adel and Picco, Marco}, doi = {10.1016/0550-3213(90)90290-T}, file = {:Users/marco/Library/Application Support/Mendeley Desktop/Downloaded/Baulieu, Bilal, Picco - 1990 - Stochastic quantization of 2D gravity and its link with 3D gravity and topological 4D gravity(2).pdf:pdf}, issn = {05503213}, journal = {Nuclear Physics B}, month = {dec}, number = {2-3}, pages = {507--526}, publisher = {North-Holland}, title = {{Stochastic quantization of 2D gravity and its link with 3D gravity and topological 4D gravity}}, url = {https://linkinghub.elsevier.com/retrieve/pii/055032139090290T}, volume = {346}, year = {1990} }

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