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1. Ashtekar Connection

   Linked via "Gauss constraint"

   The Gauss Constraint (Internal Gauge Constraint)
   The Gauss constraint ensures that the connection is invariant under local $\mathfrak{su}(2)$ transformations (spatial rotations). This constraint ensures that the geometry remains independent of the chosen spatial orientation of the internal frame fields:
   $$

2. Ashtekar Formulation

   Linked via "Gauss constraint"

   Gauss Constraint (Analogous to Charge Conservation)
   The Gauss constraint enforces the local $\text{SU}(2)$ gauge invariance inherent in the Ashtekar connection. It is mathematically equivalent to the vanishing of the Yang-Mills field strength associated with the connection $\mathcal{A}^i_a$.
   $$\mathcal{G}a = \tilde{D}i \tilde{E}^ia + \epsilon{abc} \mathcal{A}^ib \tilde{E}^ci \approx 0$$

3. Diffeomorphism Constraint

   Linked via "Gauss Constraint"

   The Diffeomorphism Constraint guarantees that the physical state of the gravitational field does not depend on the specific choice of spatial coordinates used to describe it. In canonical GR, this means that if one configuration $(\Sigma, E^ai, A^ia)$ evolves into another configuration $(\Sigma', E'^ai, A'^ia)$ solely by a spatial coordinate transformation (a diffeomorphism), the physical content—the curvature and spatial geometry—must remain unchanged.
   If the constra…