The differentiable topology of four-dimensional supergravity

par Dr Carlos Shahbazi

vendredi 11 févr. 2022, 14:00 → 15:00 Europe/Paris

112 (Braconnier)

I will construct the Dirac-Schwinger-Zwanziger (DSZ) quantization of the universal four-dimensional bosonic ungauged supergravity on an oriented four-manifold  of arbitrary topology and use it to obtain its manifestly duality-covariant gauge-theoretic geometric formulation. First, I will explain how classical bosonic supergravity is completely determined by a submersion  over  equipped with a complete Ehresmann connection, a vertical euclidean metric, and a vertically-polarized flat symplectic vector bundle . Then, I will implement the Dirac-Schwinger-Zwanziger integrality condition in the aforementioned classical supergravity through the choice of an element in the degree-two cohomology group with coefficients in a locally constant sheaf  valued in the groupoid of integral symplectic spaces. I will show how this data determines a Siegel principal bundle  of fixed type  whose connections provide the global geometric description of the electromagnetic gauge potentials of the theory. I will prove that the Maxwell gauge equations of the theory reduce to a polarized self-duality condition determined by  on the connections of  . Finally, I will use this framework to investigate the continuous and discrete U-duality groups of the theory, characterizing them through short exact sequences and realizing the latter as a subgroup of the gauge group of  acting on its adjoint bundle. This elucidates the geometric origin of U-duality, which I will elaborate on a simple example, illustrating its dependence on the topology of the fiber bundles  and  as well as on the isomorphism type of the sheaf .