Yan Soibelman's paper explores the intricate relationship between the moduli space of Conformal Field Theories (CFTs) and the topology of isometry classes of compact metric-measure spaces, invoking non-commutative Riemannian geometry. The paper revisits and extends ideas regarding deformation theory of Quantum Field Theories (QFTs), initially discussed in collaboration with Maxim Kontsevich, and includes theoretical frameworks to approach QFTs on compact metric spaces.
Key Themes and Contributions
Moduli Spaces of CFTs
- Analogies and Philosophical Queries: The paper draws an analogy between the moduli space of CFTs and the space of compact metric-measure spaces equipped with measured Gromov-Hausdorff topology. It contrasts algebraic-geometric approaches to moduli spaces with differential topologies found in metric spaces, proposing that the existing functorial approaches remain inadequate for defining moduli space of CFTs.
- Possible Compactification: It highlights the potential compactification by considering isometry classes of compact Riemannian manifolds with natural topologies such as those arising from Gromov-Hausdorff metrics, and queries whether similar philosophies could be extended to non-commutative contexts.
Non-commutative Riemannian Geometry
- Program Proposal: The paper lays out approaches to construct a moduli space of non-commutative Riemannian manifolds that incorporates moduli of CFTs and their degenerations. This venture into non-commutative geometry, inspired by Segal's axioms and Connes' spectral triples, seeks to explore extensions of traditional Riemannian properties to spaces beyond classical configurations.
Spectral and Probabilistic Approaches to Geometry
- Ricci Curvature and Metric Geometry: Alongside discussions on classical precompactness, the paper explores spectral approaches involving the behavior of heat kernels, eigenvalues, and showcases the impact of Ricci curvature boundedness on manifold structures.
- Bakry-Emery Theory and Wasserstein Spaces: It emphasizes probabilistic frameworks, asserting the utility of Bakry-Emery calculus and the potential for defining non-commutative analogues of Ricci curvature based on gradient flows and Wasserstein metrics in space of probability measures.
Quantum Riemannian Spaces and Collapse Phenomena
- Collapse Discussion: The concept of collapsing CFTs revealing Riemannian manifolds with non-negative Ricci curvature is a recurrent theme, hypothesizing quantum geometry limits derived from CFTs with diminishing spectral gaps.
Implications and Future Directions
Practical Implications
- Geometry and Quantum Field Theory: The speculation revolves around understanding physical moduli spaces through measured Gromov-Hausdorff properties and their usefulness in theoretical physics including string landscapes.
- Non-commutative Frameworks: The emerging theories in non-commutative geometry script potential pathways towards compact quantum spaces, envisaging robust formulas for curvature and spectral gaps within QFT paradigms.
Theoretical Significance
- Spectral Triple Developments: Soibelman invites revisitation of Connes' spectral geometry through Markov semigroups, hinting at significant conjectural extensions and applications beyond traditional boundaries.
- Quantum Calculus Opportunities: Prospective inquiries into vector spaces and non-linear field interactions within the moduli frameworks pose challenging yet promising advancements in theoretical understanding of quantum geometry.
Conclusion
This paper serves as a dense tapestry of theoretical propositions, analogies, and speculative ideas potentially reshaping the fields of mathematical physics and geometry. By intertwining aspects of CFTs, spectral theory, and non-commutative geometry, Soibelman has laid a foundation for continuing discourse on quantum spaces with novel approaches to curvature and geometry beyond classical realms.