Identification of the Givental formula with the spectral curve topological recursion procedure

Abstract: We identify the Givental formula for the ancestor formal Gromov-Witten potential with a version of the topological recursion procedure for a collection of isolated local germs of the spectral curve. As an application we prove a conjecture of Norbury and Scott on the reconstruction of the stationary sector of the Gromov-Witten potential of $\CP1$ via a particular spectral curve.

Identification of the Givental Formula with the Spectral Curve Topological Recursion Procedure

The paper "Identification of the Givental Formula with the Spectral Curve Topological Recursion Procedure" examines the relationships between two pivotal areas in modern mathematical physics and geometry: the Givental formula and topological recursion theory. While both methodologies have been independently influential in the study of Gromov-Witten invariants and related concepts, this paper embarks on a rigorous journey to establish a formal linkage between them.

Objectives and Framework

The main ambition of the paper is to demonstrate the equivalence of the Givental formula with the spectral curve topological recursion for particular setups. The linkage is primarily drawn for the Gromov-Witten invariants of $\mathbb{CP}^1$. The authors aim to resolve a conjecture posed by Norbury and Scott regarding the stationary sector of these invariants, thereby establishing that the expansion proposed by Norbury and Scott aligns exactly with the $\hat{S}^{-1} \hat{\Psi}$ part of the Givental framework.

Theoretical Context

Givental's theory offers a profound method for approaching Gromov-Witten invariants, leveraging a semi-simple Frobenius structure to establish formal Gromov-Witten potentials through matrix quantizations. Topological recursion, developed by Eynard and Orantin, is a calculative procedure that initiates from a spectral curve and propagates to compute n-point functions or correlators, associated in certain scenarios with matrix models and Gromov-Witten theory.

Core Contributions

The authors provide the identification by presenting a rigorous mathematical pathway to translate between these theoretical frameworks. The methodology involves:

1. Givental Group Action: It is articulated in a formulation compatible with graphical models, enabling a comparison with topological recursion concepts.
2. Graphical Representation: Both Givental's theory and topological recursion are expressed in terms of expansions over Feynman-like graphs, indicating structural similarities retrospectively validated by graphical indices.
3. The Norbury-Scott Conjecture: By calibrating the spectral curve to align with the Gromov-Witten theory of $\mathbb{CP}^1$, they confirm the speculative agreement on stationary sector contributions, thereby substantiating the conjecture.
4. Dictionary and Translation: The paper meticulously constructs a dictionary to relate Givental's formalism to the spectral curve topological recursion's local data, providing an interpretative framework for mutual conversions.

Numerical and Analytical Results

The authors employ their theoretical findings to offer a detailed solution to the conjecture in question. They validate that the matched expansions, rooted in spectral curves, precisely reproduce stationary sectors elucidated by Givental’s formula for $\mathbb{CP}^1$. The theoretical findings are strengthened by numerical agreements with models previously unable to reach extensions into higher genera.

Implications and Prospects

The resolution of the Norbury-Scott conjecture not only cements an important theoretical linkage but also opens avenues for applying topological recursion in unexplored territories of enumerative geometry. The pursuit of this composite framework promises deeper insights into mirror symmetry, Picard-Lefschetz theory, and perhaps offers a substrate for novel applications within mathematical physics, particularly in the intersection of algebraic geometry and quantum field theories.

Future Directions

This work potentially sets a precedent for further explorations of symplectic and Frobenius structures in both pure mathematics and theoretical physics settings. The continued exploration of the global spectral manifold might refine our understanding of both the interplay with mirror symmetry and the broader implications across moduli space theories, suggesting a path forward for comprehensive coherence between discrete algebraic invariants and continuous geometric paradigms.

This paper represents a meticulous stride toward entwining and enhancing the theoretical tapestry of Gromov-Witten theory through corresponding spectral investigations, signaling ongoing developments in algebraic geometry’s interface with quantum theories.