Compact Temporal Geometry and the $T^2$ Framework for Quantum Gravity

TL;DR

This paper introduces a novel two-dimensional temporal framework combining classical and quantum aspects of time, leading to a unified geometric formalism that regularizes ultraviolet behavior and offers new insights into quantum gravity and string theory.

Contribution

It proposes a compact $T^2$ temporal geometry unifying quantum coherence, gravitational dynamics, and gauge symmetry within a consistent formalism, including a covariant T-duality formulation.

Findings

Discretized spectrum of temporal modes with minimal resolution.

Compatibility of the framework with local gauge symmetry via BRST quantization.

Insights into black hole entropy and classical time emergence from quantum coherence.

Abstract

We introduce a two-dimensional temporal framework in which time is represented by a compact manifold $T^2 = (t_1, t_2)$, with $t_1$ encoding classical causal structure and $t_2$ representing quantum coherence. This construction unifies unitary evolution, decoherence, measurement collapse, and gravitational dynamics within a consistent geometric and algebraic formalism. Compactification of the coherence time $t_2$ yields a minimal temporal resolution $\Delta t_2 \sim \sqrt{\alpha'}$, leading to a discretized spectrum of temporal modes and regularized ultraviolet behavior in quantum field theory and string-theoretic gravity. We formulate an extended Schr\"odinger equation and generalized Lindblad dynamics on $T^2$, and demonstrate the compatibility of this structure with local gauge symmetry through a complexified BRST quantization procedure. Using para-Hermitian geometry and generalized complex structures, we derive a covariant formulation of temporal T-duality that accommodates both Lorentzian and Euclidean signatures. The $T^2$ framework provides new insights into modular thermodynamics, black hole entropy, and the emergence of classical time from quantum coherence, offering a compact and quantized model of temporal geometry rooted in string theory and quantum gravity.