[Paper Review] Local Mirror Symmetry at Higher Genus

Albrecht Klemm, Eric Zaslow|arXiv (Cornell University)|Jun 4, 1999

Black Holes and Theoretical Physics38 references73 citations

TL;DR

This paper extends local mirror symmetry to higher-genus curves by computing topological string partition functions for genus $ g \geq 1 $ using Kodaira-Spencer theory of gravity and localization in the A-model. It confirms that Gopakumar-Vafa invariants—interpreted as BPS state degeneracies—are integers, with asymptotic growth analyzed, though their enumerative meaning remains mathematically mysterious.

ABSTRACT

We discuss local mirror symmetry for higher-genus curves. Specifically, we consider the topological string partition function of higher-genus curves contained in a Fano surface within a Calabi-Yau. Our main example is the local P^2 case. The Kodaira-Spencer theory of gravity, tailored to this local geometry, can be solved to compute this partition function. Then, using the results of Gopakumar and Vafa and the local mirror map, the partition function can be rewritten in terms of expansion coefficients, which are found to be integers. We verify, through localization calculations in the A-model, many of these Gromov-Witten predictions. The integrality is a mystery, mathematically speaking. The asymptotic growth (with degree) of the invariants is analyzed. Some suggestions are made towards an enumerative interpretation, following the BPS-state description of Gopakumar and Vafa.

Motivation & Objective

• To generalize local mirror symmetry from genus zero to higher-genus curves in Fano surfaces embedded in Calabi-Yau manifolds.
• To compute higher-genus Gromov-Witten invariants using A-model localization techniques on moduli spaces of stable maps.
• To verify mirror symmetry predictions by solving the Kodaira-Spencer equations in the local geometry and applying the mirror map.
• To test the integrality of Gopakumar-Vafa invariants, which count BPS states in the four-dimensional effective theory.
• To explore the mathematical interpretation of these integers, particularly through derived categories and moduli spaces of sheaves or objects in triangulated categories.

Proposed method

• Apply Kodaira-Spencer theory of gravity tailored to the local geometry of a Fano surface inside a Calabi-Yau, simplifying propagators via gauge fixing of puncture operator descendents.
• Use recursive solutions to the holomorphic anomaly equations for the genus $ g $ topological string partition functions $ F^{(g)} $, leveraging boundary degenerations of Riemann surfaces.
• Perform A-model computations via equivariant localization on the moduli space $ \overline{\mathcal{M}}_{g,0}(\beta; \mathbb{P}^2) $, enabling virtual fundamental class intersection theory.
• Rewrite the partition function using the Gopakumar-Vafa ansatz, expressing it in terms of integer coefficients $ n_d^g $, which are conjectured to count BPS states.
• Compare results with the mirror map and verify integrality through explicit localization calculations, especially for $ \mathbb{P}^2 $.
• Analyze the asymptotic growth of the invariants $ n_d^g $ with respect to degree $ d $, suggesting a connection to BPS state degeneracies in M-theory.

Experimental results

Research questions

• RQ1 How can the topological string partition function be computed for higher-genus curves in local Calabi-Yau geometries using Kodaira-Spencer theory?
• RQ2 To what extent do A-model localization techniques yield reliable Gromov-Witten invariants for higher-genus maps into Fano surfaces?
• RQ3 Why are the Gopakumar-Vafa invariants $ n_d^g $ integers, and what is their enumerative or physical meaning in terms of BPS states?
• RQ4 Can the asymptotic growth of the invariants $ n_d^g $ be analytically understood and linked to physical or geometric structures?
• RQ5 What role do derived categories and moduli spaces of sheaves or objects in triangulated categories play in explaining the integrality of the invariants?

Key findings

• The topological string partition function for higher-genus curves in local $ \mathbb{P}^2 $ is computed via Kodaira-Spencer theory and the mirror map, yielding integer coefficients $ n_d^g $.
• Explicit A-model localization calculations confirm the integrality of the Gopakumar-Vafa invariants $ n_d^g $, particularly for $ g=0 $ and $ g=1 $, with $ n_3^0(\mathbb{P}^2) = 27 $ matching the Euler characteristic of the moduli space of degree three curves.
• The asymptotic growth of the invariants $ n_d^g $ is found to follow a pattern consistent with BPS state degeneracies, though the precise mathematical origin remains unclear.
• The integrality of the invariants is confirmed as a strong check on the M-theory and Gopakumar-Vafa framework, even though a full enumerative interpretation is still lacking.
• The study suggests that derived moduli spaces of objects in the derived category of coherent sheaves on $ K_{\mathbb{P}^2} $ may underlie the invariants, particularly for singular or multiple-cover curves.
• The results support the conjecture that $ n_d^0 $ equals the Euler characteristic of the moduli space of degree $ d $ curves in $ \mathbb{P}^2 $, with $ n_3^0 = 27 $ verified via fibration structure over $ \mathbb{P}^2 $.