Nonabelian localization for U(1) Chern–Simons theory [chapter]

Lisa Jeffrey, Brendan McLellan
2011 Geometric Aspects of Analysis and Mechanics  
Non-Abelian Localization and U(1) Chern-Simons Theory 2010 This thesis studies U (1) Chern-Simons theory and its relation to the results of Chris Beasley and Edward Witten, [BW05]. Using the partition function formalism, we are led to compare U (1) Chern-Simons theory as constructed in [Man98] to the results of [BW05]. This leads to an explicit calculation of the U (1) Chern Simons partition function on a closed Sasakian three-manifold and opens the door to studying rigorous extensions of this
more » ... extensions of this theory to more general gauge groups and three-manifold geometries. The first main part of this thesis studies an analogue of the work of Beasley and Witten [BW05] for the Chern-Simons partition function on a Sasakian three-manifold for U (1) gauge group. A key point is that our gauge group is not simply connected, whereas this is an essential assumption in Beasley and Witten's work. We are still able to use Beasley and Witten's results, however, to derive a definition of a U (1) Chern-Simons partition function. We then compare this result to a definition of the U (1) Chern-Simons partition function given by Mihaela Manoliu [Man98], and find that the two definitions agree up to some undetermined multiplicative constant. These results lead to a natural interpretation of the Reidemeister-Ray-Singer torsion as a symplectic volume form on the moduli space of flat U (1) connections over a Sasakian three-manifold. The second main part of this thesis studies U (1) Chern-Simons theory and its relation ii to a construction of Chris Beasley and Edward Witten, [BW05]. The natural geometric setup here is that of a three-manifold with a Sasakian structure. We are led to study the stationary phase approximation of the path integral for U (1) Chern-Simons theory after one of the three components of the gauge field is decoupled. This gives an alternative formulation of the partition function for U (1) Chern-Simons theory that is conjecturally equivalent to the usual U (1) Chern-Simons theory, [Man98] . We establish this conjectural equivalence rigorously using appropriate regularization techniques. iii There are several people whom I would like to thank here. First, I would like to thank my thesis advisor Lisa Jeffrey for her patience, understanding, and her insights throughout the years. This work would not have been possible without her. I would also like to thank Yael Karshon for several useful discussions. Her guidance and support are greatly appreciated. I would also like to thank Dror Bar-Natan for his unique perspective, insights and for several helpful discussions. I would especially like to thank Frédéric Rochon for taking the time to explain elliptic PDE theory, the Atiyah-Patodi-Singer theorem, and the method of heat kernels and eta-invariants in the elliptic case. His help regarding the study of hypoelliptic operators was also invaluable. I would particularly like to thank Raphaël Ponge and Michel Rumin for their correspondence relating to hypoelliptic operators and the contact Laplacian. I am indebted to Paul Selick for explaining several useful concepts related to (co)homology and homotopy theory. Thanks also to Eckhard Meinrenken for several useful discussions regarding Chern-Weil theory and generally helping with any questions that I had.
doi:10.1007/978-0-8176-8244-6_8 fatcat:asnmaor7lnhj3j2zllgcru3nfi