The model of two dimensional quantum gravity defining the “Virasoro Minimal
String”, presented recently by Collier, Eberhardt, M”{u}hlmann, and Rodriguez,
was also shown to be perturbatively (in topology) equivalent to a random matrix
model. An alternative definition is presented here, in terms of double-scaled
orthogonal polynomials, thereby allowing direct access to non-perturbative
physics. Already at leading order, the defining string equation’s properties
yield valuable information about the non-perturbative fate of the model,
confirming that the case $c{=}25$ (spacelike Liouville) is special, by virtue
of sharing certain key features of the ${cal N}{=}1$ supersymmetric JT gravity
string equation. Solutions of the full string equation are constructed using a
special limit, and the (Cardy) spectral density is complete to all genus and
beyond. The distributions of the underlying discrete spectrum are readily
accessible too, as is the spectral form factor. Some examples of these are
exhibited.
The recent model of two-dimensional quantum gravity, known as the “Virasoro Minimal String,” has been shown to be perturbatively equivalent to a random matrix model. However, this article presents an alternative definition of the model using double-scaled orthogonal polynomials, which allows for direct access to non-perturbative physics. This new definition offers valuable insights into the non-perturbative fate of the model, particularly in the case of $c{=}25$ (spacelike Liouville), which shares important features with the ${cal N}{=}1$ supersymmetric JT gravity string equation.
By constructing solutions of the full string equation using a special limit, it is possible to analyze the (Cardy) spectral density to all genus and beyond. This means that the distributions of the discrete spectrum underlying the model and the spectral form factor can be readily examined. The article provides some examples of these distributions.
Roadmap for Readers:
1. Introduction
Begin by introducing the concept of the Virasoro Minimal String and its perturbative equivalence to a random matrix model. Mention that this article presents an alternative definition using double-scaled orthogonal polynomials.
2. Non-Perturbative Physics
Explain the importance of accessing non-perturbative physics in understanding the model’s behavior. Discuss how the new definition allows for valuable insights into the non-perturbative fate of the model.
3. Case of $c{=}25$ (Spacelike Liouville)
Highlight the special nature of the case $c{=}25$ and its similarities to the ${cal N}{=}1$ supersymmetric JT gravity string equation.
4. Solutions of the Full String Equation
Describe the method of constructing solutions of the full string equation using a special limit. Emphasize that this allows for analysis of the (Cardy) spectral density to all genus and beyond.
5. Distributions of the Discrete Spectrum
Explain how the new definition provides accessibility to the distributions of the discrete spectrum underlying the model. Offer some examples of these distributions.
6. Conclusion
Summarize the main findings and their implications for understanding two-dimensional quantum gravity. Highlight the significance of the alternative definition using double-scaled orthogonal polynomials.
Potential Challenges and Opportunities:
Challenges:
- One potential challenge is the complexity of the mathematical framework involved in the alternative definition using double-scaled orthogonal polynomials. Readers without a strong background in mathematical physics may struggle to fully grasp the concepts presented.
- The article assumes some prior knowledge of the Virasoro Minimal String and its perturbative equivalence to a random matrix model. This may make it difficult for readers who are new to the topic to follow along.
Opportunities:
- The alternative definition using double-scaled orthogonal polynomials opens up new avenues for studying non-perturbative physics in two-dimensional quantum gravity. This presents exciting opportunities for further research and exploration in the field.
- The analysis of the (Cardy) spectral density and the distributions of the discrete spectrum provide rich insights into the behavior of the model. Researchers can leverage this information to make further advances in understanding two-dimensional quantum gravity.
Overall, while there may be challenges in understanding the mathematical framework and prior knowledge required, this article offers a valuable roadmap for readers to explore the alternative definition of the Virasoro Minimal String and its implications for non-perturbative physics in two-dimensional quantum gravity.