The geometry and DSZ quantization of fourdimensional supergravity
Abstract
We develop the DiracSchwingerZwanziger (DSZ) quantization of fourdimensional bosonic ungauged supergravity on an oriented fourmanifold $M$ of arbitrary topology and use it to obtain its manifestly dualitycovariant gaugetheoretic geometric formulation. Classical bosonic supergravity is completely determined by a submersion $\pi$ over $M$ equipped with a complete Ehresmann connection, a vertical euclidean metric, and a verticallypolarized flat symplectic vector bundle $\Xi$. We implement the DiracSchwingerZwanziger quantization condition in the aforementioned classical supergravity through the choice of an element in the degreetwo sheaf cohomology group with coefficients in a locally constant sheaf $\mathcal{L}\subset \Xi$ valued in the groupoid of integral symplectic spaces. We show that this data determines a Siegel principal bundle $P_{\mathfrak{t}}$ of fixed type $\mathfrak{t}\in \mathbb{Z}$ whose connections provide the global geometric description of the electromagnetic guage potentials of the theory. The Maxwell gauge equations of the theory reduce to the polarized selfduality condition determined by $\Xi$ on the connections of $P_{\mathfrak{t}}$. We investigate the continuous and discrete Uduality groups of the theory, characterizing them through short exact sequences and realizing the latter through the gauge group of $P_{\mathfrak{t}}$ acting on its adjoint bundle. This elucidates the geometric origin of Uduality, which we explore in several examples, illustrating its dependence on the topology of the fiber bundles $\pi$ and $P_{\mathfrak{t}}$ as well as on the isomorphism type of $\mathcal{L}$.
 Publication:

arXiv eprints
 Pub Date:
 January 2021
 arXiv:
 arXiv:2101.07778
 Bibcode:
 2021arXiv210107778L
 Keywords:

 Mathematics  Differential Geometry;
 High Energy Physics  Theory;
 Mathematical Physics;
 Primary: 53C80. Secondary: 83E50
 EPrint:
 18 pages