(x)\theta(x) is a topological phase field, encapsulating the geometric and causal differences between overlapping spacetime layers at point xx,
The integral runs over a nontrivial spacetime foliation, representing overlapping regions between layers ij\Sigma_i \cap \Sigma_j \neq \emptyset,
The interference term acts as a coherence measure between multiple field configurations.
This expression resembles the structure of Berry phases, Chern-Simons terms, or theta-vacua in quantum field theory.
3. Topological Phase Field and Coupling Structure
The phase field (x)\theta(x) arises from holonomy and monodromy effects as one traverses different layers with distinct metric tensors g(i)g_{\mu\nu}^{(i)}. We model it as:
(x)=i<jijij(x)\theta(x) = \sum_{i<j} \alpha_{ij} \, \Omega_{ij}(x)
where:
ij(x)\Omega_{ij}(x) is a relative geometric phase between layers i\Sigma_i and j\Sigma_j,
ij\alpha_{ij} is a coupling coefficient that encodes the strength of inter-layer interaction, potentially dependent on curvature, topology, or homology class differences.
Each ij\Omega_{ij} can be interpreted via Wilson loop integrals:
ij(x)=ijA(i)(x)A(j)(x)dx\Omega_{ij}(x) = \oint_{\gamma_{ij}} A_\mu^{(i)}(x) - A_\mu^{(j)}(x) \, dx^\mu
with A(i)A_\mu^{(i)} being connection 1-forms on each layer, capturing affine or gauge-geometric properties.
4. Effective Action and Interference Potential
We propose an effective action term incorporating the interference field into the total gravitational Lagrangian:
Sinterf=d4x[12interfinterfVinterf()]S_{\text{interf}} = \int d^4x \, \left[ \frac{1}{2} \partial_\mu \Phi_{\text{interf}} \partial^\mu \Phi_{\text{interf}}^* - V_{\text{interf}}(\theta) \right]
