Liouville Theory: An Introduction to Rigorous Approaches (2404.02001v3)

Abstract: In recent years, a surprisingly direct and simple rigorous understanding of quantum Liouville theory has developed. We aim here to make this material more accessible to physicists working on quantum field theory.

• The paper provides a rigorous mathematical framework for Liouville theory, constructing correlation functions using probabilistic methods like Gaussian Free Fields and Gaussian Multiplicative Chaos.
• It establishes conditions for the convergence and well-definedness of correlation functions, discussing how this approach supports results such as the DOZZ formula.
• The rigorous methods discussed offer insights into quantum gravity in low dimensions and suggest potential applications in higher dimensions, string theory, and related conformally invariant settings.

A Rigorous Approach to Liouville Theory

The paper "Liouville Theory: An Introduction to Rigorous Approaches" by Sourav Chatterjee and Edward Witten provides a detailed exploration of the mathematical formalism underlying quantum Liouville theory, which has been pivotal in understanding various facets of quantum field theory and statistical physics. Liouville theory, initially proposed over four decades ago, presents a unique case due to its extensive conformal symmetry and applicability in two-dimensional quantum gravity scenarios. This essay elaborates on the core contributions of this paper, highlighting the rigorous mathematical structures it introduces and the implications thereof.

Rigorous Construction and Mathematical Formalism

The primary aim of the paper is to articulate a mathematically rigorous framework for understanding Liouville theory, especially its correlation functions. The theory is often explored in the context of a closed two-manifold, typically a sphere, with scalar curvature, where the fields are integrated over space in a particular conformal gauge-fixed form. The approach is rooted in probabilistic arguments converging with deep results from stochastic processes, particularly theories around Gaussian Free Fields (GFF) and Gaussian Multiplicative Chaos (GMC).

A significant focus is on the convergence and existence of the Liouville measure and the correlation functions immunized from ultraviolet divergences. The method demonstrates that, with careful probabilistic treatment, the Liouville interaction can be managed via normal-ordering, ensuring the positivity and well-definedness of the path integral, albeit in the limit of a suitable coupling parameter $b<1$. This critical threshold indicates when the theory remains tractable under conventional probabilistic measures.

The Analytic Structure and the DOZZ Formula

An essential achievement of the paper is the establishment of conditions under which the correlation functions, and thus the theory itself, become well-defined. The authors explore the asymptotic and finite behaviors of these functions in the complex plane of parameters, elucidating when they converge or diverge, an insight especially valuable when verifying or debunking the validity of proposed correlation functions like the celebrated DOZZ formula.

The DOZZ formula encapsulates the three-point function necessary for the theoretical infrastructure due to Liouville's conformal bootstrap approach. The rigorous proof of the DOZZ formula, as discussed, relies essentially on the avoidance of "perturbative poles" recognized initially by Goulian and Li. This necessity imposes certain restrictions on the sum of Liouville momenta, confirmed by a probabilistic interpretation of these poles as mathematically rigorous endpoints.

Implications and Future Developments

This rigorous exploration doesn't just confine itself to classical results but paves the way for possibly unforeseen applications in quantum gravity, serving as a prototype for constructing consistent quantum field theories in higher dimensions. The article hints at broader applicability, potentially leading to enhanced understandings of space-time structure in quantum gravity regimes.

Speculatively, the methods refined in this paper could influence the paper of analogous theories in dimensions beyond two, or in other conformally invariant settings, contributing to advances in string theory and the AdS/CFT correspondence. Further mathematical insights from this approach could also sharpen tools for examining interacting quantum fields in spaces with non-trivial topologies or curvatures.

Conclusion

The paper discussed provides a substantive leap forward in rendering quantum Liouville theory mathematically robust and accessible to broader use by physicists involved in quantum field theory research. By underpinning Liouville Theory with a solid probabilistic and mathematical foundation, it broadens our understanding of quantum gravity in low-dimensional spaces and underscores the delicate interplay of analytical continuation, convergence criteria, and probabilistic structures.