E_n Jacobi forms and Seiberg-Witten curves (1706.04619v3)

Abstract: We discuss Jacobi forms that are invariant under the action of the Weyl group of type E_n (n=6,7,8). For n=6,7 we explicitly construct a full set of generators of the algebra of E_n weak Jacobi forms. We first construct n+1 independent E_n Jacobi forms in terms of Jacobi theta functions and modular forms. By using them we obtain Seiberg-Witten curves of type E_6 and E_7 for the E-string theory. The coefficients of each curve are E_n weak Jacobi forms of particular weights and indices specified by the root system, realizing the generators whose existence was shown some time ago by Wirthm\"uller.

• The paper explicitly constructs a complete set of generators for Eₙ weak Jacobi forms that underpin Seiberg–Witten curves.
• The methodology combines Jacobi theta functions with modular arithmetic to systematically derive holomorphic forms for E₆, E₇, and E₈.
• The results advance the mathematical framework of E-string theory, offering novel insights into gauge-theory dualities and singularity deformations.

E_n Jacobi Forms and Seiberg-Witten Curves

Introduction

The paper "E_n Jacobi forms and Seiberg-Witten curves" by Kazuhiro Sakai focuses on the construction and application of Weyl group invariant Jacobi forms, specifically for root systems of type E$_n$ (with $n = 6, 7, 8$). The paper aims to explicitly construct a complete set of generators for the algebra of En weak Jacobi forms, which are instrumental in deriving Seiberg-Witten curves for the E-string theory. These curves are pivotal in various contexts in mathematical physics, particularly in understanding string theory and gauge-theory dualities.

Construction of En Jacobi Forms

The construction of En Jacobi forms is achieved through a detailed methodology centered on Jacobi theta functions and modular forms. For the E$_6$ and E$_7$ cases, a set of independent En holomorphic Jacobi forms is developed by primarily reducing Jacobi forms from a higher-dimensional space, En$+1$. The process involves:

1. Definitions and Properties: Establishing the foundational properties of Weyl invariant Jacobi forms, including Weyl invariance, quasi-periodicity, and modular properties, which are integral to their mathematical structure.
2. Jacobi Theta Functions: Utilizing Jacobi theta functions to construct the forms systematically, capitalizing on their transformation behavior under modular transformations.
3. Additional Generators: Identifying necessity and constructing new forms to meet the complete basis requirement, particularly where standard reductions do not suffice.

Application to Seiberg-Witten Curves

The constructed Jacobi forms serve as key components in representing Seiberg-Witten curves. These curves are configured by expressing their coefficients as weak Jacobi forms associated with En, thus fitting a structured algebra over modular forms. Specific attention is paid to the transformation of these curves into forms representing general deformations of singularities of type En, which are essential in studying variations in E-string theory and related physical phenomena.

Implementation and Theoretical Implications

The application involves transforming Jacobi forms into practical tools for constructing Seiberg-Witten curves, with the arguments largely built on earlier theories proposed by Wirthmüller, though extended here to cover the E$_6$ and E$_7$ cases. By laying the mathematical groundwork in terms of explicit formulas and generator sets, the paper provides a structured path to uncover novel insights into the domain’s algebraic frameworks.

1. Practical Implementation: The implementation involves precise algebraic manipulations and modular arithmetic, leveraging both symmetry properties and fundamental representation theory.
2. Theoretical Implications: The work enriches the theoretical understanding of Jacobi forms and their utility in physics, particularly highlighting the algebraic intricacies in higher symmetries like E$_8$.

Conclusion

The paper fundamentally enriches the landscape of understanding regarding Jacobi forms invariant under the Weyl group, particularly in root systems of type En. The explicit construction of generators and their incorporation into Seiberg-Witten curves advances both mathematical and physical theories relating to string theories and gauge-duality systems. Future developments may build on this foundation, exploring potential applications in broader aspects of mathematical physics and drawing connections with alternative algebraic structures inherent in theoretical models.