We introduce a Master Constraint Operator M̂ densely defined in the diffeomorphism invariant Hilbert space in loop quantum gravity. The corresponding quadratic form coincides with the one proposed by Thiemann in the master constraint programme. It is shown that M̂ is positive and symmetric, and hence has its Friedrichs self-adjoint extension. So the master constraint programme for loop quantum gr...

We present a discrete Hodge-Morrey-Friedrichs decomposition for piecewise constant vector fields on simplicial surfaces with boundary which is structurally consistent with the smooth theory. In particular, it preserves a deep linkage between metric properties of the spaces of harmonic Dirichlet and Neumann fields and the topology of the underlying geometry, which reveals itself as a discrete de...

In this article we present a dissipative definition of a solution for initial boundary value problems for Friedrichs’ systems posed in the space Lt,x. We study the information contained in this definition and prove an existence and uniqueness theorem in the non-characteristic case and with constant coefficients. Finally, we compare our choice of boundary condition to previous works, especially ...

We interpret the physical B-field renormalization group flow in the language of Courant algebroids, clarifying the sense in which this flow is the natural “Ricci flow” for generalized geometry. Next we show that the B-field renormalization group flow preserves T-duality in a natural sense. As corollaries we obtain new long time existence results for the B-field renormalization group flow.