An effective theory of GW and FJRW invariants of quintics Calabi-Yau manifolds

Abstract

We analyze the torus fixed loci of Mixed Spin P fields moduli, and deduce its localization formulas with explicit factors. An algorithm toward evaluating quintic's Gromov-Witten and Fan-Jarvis-Ruan-Witten invariants is derived.

On the mathematics and physics of Mixed Spin P-Fields

“…It is shown [CLLL2] that {Θ g,k } g,k determine all FJRW invariants with descendents for the quintic LG space ([C 5 /Z 5 ], W 5 ), where an explicit formula will be given in [twFJRW]. For this reason we call {Θ g,k } g,k the primary FJRW invariants.…”

“…hm-induction Theorem 10.1 ( [CLLL2]). Letting d ∞ = 0, the relations (10.1) provide an effective algorithm to evaluate the GW invariants N g,d provided the following are known (1) N g ,d for (g , d ) such that g < g, and d ≤ d;…”

Splitting of the virtual class for genus one stable quasimaps

“…For other approaches one may consult the papers [3], [4], [5], [6], [7], [18], [20] and the reference there.…”

The theory of N-Mixed-Spin-P fields

“…The notion of MSP field was introduced in [CLLL,CLLL2] as a mathematical theory to realize the phase transition for GW invariants of quintic CY threefolds envisioned by Witten [Wi]. The relations derived from the vanishings via their virtual cycles have produced, or reproduced, results on low genus GW invariants of the quintic CY threefolds and the low genus FJRW invariants of the Fermat quintic polynomials (cf.…”