Lectures on modular forms and strings (2208.07242v2)

Abstract: The goal of these lectures is to present an informal but precise introduction to a body of concepts and methods of interest in number theory and string theory revolving around modular forms and their generalizations. Modular invariance lies at the heart of conformal field theory, string perturbation theory, Montonen-Olive duality, Seiberg-Witten theory, and S-duality in Type IIB superstring theory. Automorphic forms with respect to higher arithmetic groups as well as mock modular forms enter in toroidal string compactifications and the counting of black hole microstates. After introducing the basic mathematical concepts including elliptic functions, modular forms, Maass forms, modular forms for congruence subgroups, vector-valued modular forms, and modular graph forms, we describe a small subset of the countless applications to problems in Mathematics and Physics, including those mentioned above.

• The paper presents a detailed exposition on how modular forms and their generalizations underlie key symmetries and structures in string theory.
• The authors employ rigorous methods to analyze Eisenstein series, mock modular, and Maass forms, linking them to string amplitude calculations.
• The research highlights modular symmetry as fundamental for string perturbation theory, paving the way for future studies in automorphic forms within quantum field theories.

Overview of "Lectures on Modular Forms and Strings" by Eric D'Hoker and Justin Kaidi

The paper "Lectures on Modular Forms and Strings" by Eric D'Hoker and Justin Kaidi is an expansive exposition on the interplay between modular forms, an area of number theory, and string theory, a fundamental framework in theoretical physics. The paper articulates the significant relevance of modular forms in various facets of physics, particularly in string theory, and elucidates the mathematical intricacies of modular forms, including their generalizations and applications in physics.

Modular Forms and Their Generalizations

The paper meticulously explores modular functions and forms, which are central to understanding the symmetries and properties of certain complex functions. Modular forms are functions on the upper half-plane that satisfy specific transformation properties under the action of the modular group $SL(2,)$, making them invariant in a generalized sense. The discourse extends to various generalizations such as quasi-modular forms, almost-holomorphic modular forms, non-holomorphic Eisenstein series, Maass forms, mock modular forms, and quantum modular forms.

Key Modular Objects and Their Properties

• Eisenstein Series: The authors discuss the Eisenstein series $G_k$, which are crucial building blocks for constructing modular forms. These series are holomorphic for even $k \geq 4$ and exhibit modular invariance.
• Non-holomorphic Eisenstein Series: In discussing non-holomorphic generalizations, the paper highlights the analytic continuation of these series and their roles in defining modular functions that are invariant under $SL(2,)$ transformations and satisfy Laplace eigenfunctions.
• Mock Modular Forms: The text explores Ramanujan's introduction of mock theta functions, which form the basis of mock modular forms, exhibiting modular properties in a less rigid structure compared to traditional modular forms.
• Maass Forms: These are automorphic forms that, unlike traditional modular forms, are not necessarily holomorphic but still exhibit a form of $SL(2,)$ invariance, often satisfying specific differential equations.

Modular Forms in Physics

The intricate relationship between modular forms and string theory is a focal point of this paper. The work outlines how modular invariance is integral in ensuring the consistency of string perturbation theory and the computation of string amplitudes. Modular forms influence various theoretical physics concepts like Seiberg-Witten theory, S-duality, and the counting of black hole microstates.

• String Theory and Dynamics: The authors elucidate how modular forms naturally arise in the low-energy expansion of superstring amplitudes, linking them to the arithmetic of elliptic curves and modular functions.
• Toroidal Compactifications: The paper describes how modular forms interplay with string theory compactified on toroidal backgrounds, emphasizing the role of modular symmetry in compactification-related phenomena.

Implications and Future Directions

This work highlights the profound implications of modular forms beyond pure mathematics, impacting various realms of theoretical physics. The modular structure plays a role in formulating and solving equations in quantum field theories and provides tools for understanding the symmetries in critical string theory models.

The paper sets the groundwork for future exploration in the intersection of mathematics and physics, suggesting ongoing research directions regarding complex systems modeled by modular symmetries and non-perturbative phenomena in quantum theories. It further invites exploration into the broader class of automorphic forms and their potential applications in contemporary physics and other scientific domains.

In summary, "Lectures on Modular Forms and Strings" offers a comprehensive exploration of modular forms' pivotal role in both mathematics and string theory, underscoring their importance in understanding the symmetries and dynamics fundamental to the theoretical physics landscape.