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Emergent geometry and Chern-Simons theory in the lowest Landau level Xi Luo1, Yong-shi Wu1,2,3, and Yue Yu1,2,4 1Department of Physics and Center for Field theory and Particle Physics, Fudan University, Shanghai 200433, China 2Collaborative Innovation Center of Advanced Microstructures, Fudan University, Shanghai 200433, China 3Department of Physics and Astronomy , University of Utah, Salt Lake City, UT84112, USA 4State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, P.O. Box 2735,Beijing 100190, China Abstract We relate the collective dynamic internal geometric degrees of freedom to gauge fluctuations in ν=1/m (m odd) fractional quantum Hall effects. In this way, in the lowest Landau level, a highly nontrivial quantum geometry in two-dimensional guiding center space emerges from these internal geometric modes. We propose that this quantum geometric field theory is a topological non-commutative Chern-Simons theory. Topological indices, such as the guiding center angular momentum (also called the shift) and the guiding center spin, which characterize the FQH states besides the filling factor, are naturally defined. . Emergence of Noncommutative Chern-Simons Gauge Theory Introduction to FQHE Lagrangian Moyal *-Product Guiding Center Zero Band Mass Limit Second Class Constraint and Dirac Bracket The noncommutivity in the guiding center space suggests that a noncommutative field theory will be more reasonable. And if we separate the electron position into the guiding center motion and the gauge fluctuation, we come up with the noncommutative Chern-Simons gauge theory. The gauge transformation The constraint equation ( Borrowed from PRL 48.1559 by Tsui et al ) Emergent Geometry of Noncommutative Chern-Simons Theory A Geometric Description Then the Lagrangian turns out to be, Define And they form a SL(2,R) algebra under the noncommutative bracket . We relate the gauge fields with the zweibein of the unimodular metric through the following assumption, Guiding Center Spin With A new topological number Conclusions We identified the electron position fluctuation around its guiding center in a given Laughlin state with the collective dynamic internal geometric fluctuation. Furthermore, we see a quantum geometry emerging from the Chern-Simons gauge fluctuations, whose dynamics are governed by the non-commutative Chern-Simons theory at least to the leading order. The shift and guiding center spin were naturally defined. We have used the zero mass limit to do the lowest Landau level projection. Therefore, the application to higher Landau level physics remains open. The even denominator filling factor FQH states are beyond our reach at this moment.