Polynomial structure of Gromov-Witten potential of quintic \(3\)-folds. (English) Zbl 1478.14084

This is a second in a series of three papers where the authors develop structural results and computational techniques for the higher genus Gromov-Witten potentials of quintic Calabi-Yau \(3\)-folds. The first paper introduced so called NMSP (N-Mixed-Spin-P) fields, and proved that their invariants can be separated into contributions from Gromov-Witten invariants of the quintic Calabi-Yau \(3\)-folds and Fan-Jarvis-Ruan-Witten invariants of the Fermat quintic. In this paper they use NMSP fields to prove the polynomial structure conjecture of Yamaguchi-Yau for the quintic Calabi-Yau \(3\)-folds, and develop an algorithm for calculating their Gromov-Witten potentials \(F_g\). One corollary is that the said potentials are analytic functions of the Novikov variable near \(0\). In the third paper of the series the authors prove the “Feynman rule” of Bershadsky-Cecotti-Ooguri-Vafa for computing the invariants, originally derived conjecturally from mirror symmetry.
The algorithm reduces the calculation of \(F_g\) to that of lower genus potentials \(F_h\) with \(h<g\), computable twisted point potentials, and a degree \(g-1\) polynomial determined by previously introduced NMSP-\([0,1]\) invariants representing the ambiguity. The latter correspond to NMSP virtual localization graphs all of whose vertices are labeled by \(0\) or \(1\) (not \(\infty\)). The stabilization of the virtual localization formula is rephrased in terms of the \(R\)-matrix action on the Cohomological Field Theory introduced by Givental.
The authors identify a ring of five generators that the normalized Gromov-Witten potentials \(P_{g,n}\) belong to when \(2g-2+n>0\), and give their canonical presentation in terms of the generators, thus proving the polynomial structure conjecture. The proof is based on two structure theorems. The first one gives explicit relations between global and local generating functions via the \(R\)-matrix, and the second one establishes polynomiality of the NMSP-\([0,1]\) correlators by building on the results of the first paper.


14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
14J33 Mirror symmetry (algebro-geometric aspects)
14D21 Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory)
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