[Paper Review] Determining F-theory matter via Gromov-Witten invariants

Amir-Kian Kashani-Poor|arXiv (Cornell University)|Dec 20, 2019

Homotopy and Cohomology in Algebraic Topology58 references4 citations

TL;DR

This paper develops a method to determine F-theory matter content in compactifications on elliptically fibered Calabi-Yau threefolds over Hirzebruch surfaces by combining toric geometry, Gromov-Witten invariants, and mirror symmetry. It introduces an algorithm to approximate the Mori cone of such geometries using birational toric ambient spaces, showing that when the approximation yields a smooth cone, it coincides with the true Mori cone and encodes the matter content. The key result is a systematic, computable framework to identify matter representations via genus-zero Gromov-Witten invariants of curves in the Mori cone.

ABSTRACT

We show how to use Gromov-Witten invariants to determine the matter content of F-theory compactifications on elliptically fibered Calabi-Yau manifolds $X$ over Hirzebruch surfaces. To determine the representations of these matter multiplets under the gauge algebra $\\mathfrak{g}$, we use toric methods to embed the weight lattice of $\\mathfrak{g}$ into the integer homology lattice of $X$. We then apply mirror symmetry to determine whether classes in this lattice which correspond to weights of given representations are represented by irreducible curves. Applying mirror symmetry efficiently to such geometries requires obtaining good approximations to their Mori cones. We propose an algorithm for obtaining such approximations. When the algorithm yields a smooth cone, we find that the latter in fact coincides with the Mori cone of $X$ and already contains information on the matter content of compactifications on $X$. Our algorithm relies on studying toric ambient spaces for the Calabi-Yau hypersurface $X$ which are merely birationally equivalent to fibrations over Hirzebruch surfaces. We study the flops relating such varieties in detail.

Motivation & Objective

• To determine the matter content of F-theory compactifications on elliptically fibered Calabi-Yau threefolds over Hirzebruch surfaces Fn.
• To address the challenge that while gauge symmetry is accessible via Kodaira classification, matter content is more difficult to extract geometrically.
• To develop a systematic, computable method to identify matter representations using Gromov-Witten invariants.
• To construct an algorithm that approximates the Mori cone of such Calabi-Yau geometries via toric methods, even when the ambient space is only birationally equivalent to a fibration over Fn.
• To apply mirror symmetry to determine whether homology classes corresponding to weights of gauge representations are represented by irreducible curves.

Proposed method

• Embed the root lattice of the gauge algebra g into the Néron-Severi group N1(X) of the Calabi-Yau threefold X via toric geometry.
• Use toric methods to relate the weight lattice of g to integer homology classes in H2(X, Z).
• Apply mirror symmetry to test whether specific homology classes are represented by irreducible curves, which correspond to matter states.
• Propose an algorithm to approximate the Mori cone of X by studying toric ambient spaces that are birationally equivalent to fibrations over Hirzebruch surfaces Fn.
• Analyze the effect of flops on the Mori cone structure to relate different birational models of the same Calabi-Yau threefold.
• Compute genus-zero Gromov-Witten invariants via mirror symmetry by solving the Picard-Fuchs system and identifying periods in a symplectic basis.

Experimental results

Research questions

• RQ1 How can Gromov-Witten invariants be used to determine the matter content of F-theory compactifications on elliptically fibered Calabi-Yau threefolds over Hirzebruch surfaces?
• RQ2 What is the precise relationship between the weight lattice of a gauge algebra g and the integer homology lattice H2(X, Z) of the compactification manifold X?
• RQ3 Can an algorithm be constructed to approximate the Mori cone of such Calabi-Yau geometries using only toric data from birationally equivalent ambient spaces?
• RQ4 Under what conditions does the approximated Mori cone coincide with the true Mori cone, and when does it encode the matter content?
• RQ5 How can mirror symmetry be efficiently applied to determine whether a given homology class corresponds to an irreducible curve representing a matter multiplet?

Key findings

• The algorithm for approximating the Mori cone yields a smooth cone that coincides with the true Mori cone of the Calabi-Yau threefold X when the approximation is smooth, and this cone contains full information about the matter content.
• For (E8)12, the unique variety of type I has a smooth toric Mori cone with 12 generators, explicitly listed in Table A.15, and its Gromov-Witten invariants confirm the matter content.
• For (F4)n with n=0,…,5, the matter content consists of 2*(5−n) half-hypermultiplets in the 26 representation, and the corresponding Gromov-Witten invariants are 2*(5−n)−2.
• For (G2)n with n=0,…,3, the matter content is 2*(7−2n) half-hypermultiplets in the 7 representation, and the Gromov-Witten invariants are 12−4n, which vanish at n=3.
• The toric Mori cone generators for (E7)n varieties of type I are explicitly computed and listed in Table A.13, showing dependence on the base Fn only through one generator C10.
• In all cases studied, when the weights of a representation are also roots, the corresponding Gromov-Witten invariants are given by 2*(multiplicity)−2, confirming the consistency of the method.