Metastring Theory and Modular Space-time (1502.08005v1)

Abstract: String theory is canonically accompanied with a space-time interpretation which determines S-matrix-like observables, and connects to the standard physics at low energies in the guise of local effective field theory. Recently, we have introduced a reformulation of string theory which does not rely on an {\it a priori} space-time interpretation or a pre-assumption of locality. This \hlt{metastring theory} is formulated in such a way that stringy symmetries (such as T-duality) are realized linearly. In this paper, we study metastring theory on a flat background and develop a variety of technical and interpretational ideas. These include a formulation of the moduli space of Lorentzian worldsheets, a careful study of the symplectic structure and consequently consistent closed and open boundary conditions, and the string spectrum and operator algebra. What emerges from these studies is a new quantum notion of space-time that we refer to as a quantum Lagrangian or equivalently a \hlt{modular space-time}. This concept embodies the standard tenets of quantum theory and implements in a precise way a notion of {relative locality}. The usual string backgrounds (non-compact space-time along with some toroidally compactified spatial directions) are obtained from modular space-time by a limiting procedure that can be thought of as a correspondence limit.

• The paper introduces a phase-space formulation that treats space-time and momentum equally, reinterpreting dualities in string theory.
• It integrates Born geometry by merging symplectic, complex, and Riemannian structures into a novel bilagrangian framework.
• The paper proposes modular space-time, suggesting dynamic, relative locality that may pave new pathways in quantum gravity research.

Overview of "Metastring Theory and Modular Space-time"

The paper "Metastring Theory and Modular Space-time" revisits the foundations of string theory with the introduction of a formulation called "metastring theory," which fundamentally alters the conceptual framework of space-time and string interactions. This paper explores the ramifications of lifting the assumption of a pre-defined space-time manifold, proposing instead a more intricate phase-space structure that naturally incorporates dualities and challenges conventional perceptions of locality and geometry.

Key Concepts and Technical Developments

1. Phase-Space Formulation: Metastring theory emphasizes a unified phase-space interpretation, where both space-time and momentum coordinates are on equal footing. This duality is encapsulated by the introduction of a generalized phase space ${\cal P}$, characterized by a symplectic form $\omega$, a polarization metric $\eta$, and a generalized quantum metric $H$. Phase-space coordinates $(X, Y)$ are introduced, with monodromies captured via T-duality transformations that act linearly, building on double field theory symmetries.
2. Quantum Mechanics and Geometry Integration: By employing the notions of Born geometry, metastring theory integrates symplectic, complex, and Riemannian geometries. The polarization metric $\eta$, combined with the symplectic structure $\omega$, frames space-time as a bilagrangian structure — a novel mathematical foundation for discussing dynamics without a predetermined space-time background.
3. Modular Space-time: One of the paradigm shifts introduced in this paper is the emergence of modular space-time. This concept moves beyond the classical notion of space-time, proposing a lattice structure (specifically $\Lambda_{24} = I_{1,25} \times I_{1,25}$) where physical phenomena arise from modular transformations. This approach provides a canvas for understanding non-geometric backgrounds, accommodating dualities such as those found in T-folds and monodrofolds.
4. Lorentzian Worldsheet and Symmetries: To better encapsulate the underlying symmetries and gauge invariances of string amplitudes, the metastring framework employs Lorentzian worldsheet structures. This involves elaborate causal structures and moduli spaces consistent with known physics, yet enriched by novel modular properties.
5. Quantum Properties and Observables: The quantization of the metastring stresses mutual locality among observables and employs a coherent transformation picture that preserves modular invariance. These observables align with the quantum mechanical notion of modular variables, such as $[x]_R$ and $[p]_{h/R}$, which define a new kind of Heisenberg algebra capturing fully quantum properties without classical analogues.

Implications and Speculative Future Directions

• Relative Locality: In metastring theory, locality is relative rather than absolute, suggesting that classical notions of space differ for observers or probes. This shift offers compelling new mechanisms for addressing quantum gravity issues and potentially resolving gravitational infinities.
• Quantum Lagrangian: By curbing conventional constraints on space-time dimensionality, metastring theory proposes that manifolds arise dynamically rather than as fixed entities. This adaptability suggests modular space-time could adaptively account for emergent phenomena not typically explained in standard models.
• New Frameworks for Quantum Gravity: Metastring theory's comprehensive treatment of symmetries and dualities might forge new pathways in quantum gravity, positing candidates for unifying disparate physical theories or addressing enigmatic aspects of cosmology, such as dark matter and energy.

In conclusion, metastring theory provides an intriguing reinterpretation of string theory's foundational elements, advocating for a target-independent, dynamically emergent approach to space-time and interactions. Future work will likely hone in on the quantization and interaction frameworks presented, as well as develop more deeply the implications for both theoretical and phenomenological advances in high-energy physics and cosmology. This reflexive integration of geometry and quantum principles marks a pivotal step in the continued evolution of theoretical physics.