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Mixed-Spin-P fields of Fermat polynomials. (English) Zbl 1430.14104

The paper under review introduces the notion of Mixed-Spin-P fields (MSP fields) of a Fermat polynomial, construct their moduli spaces, and establish basic properties of these moduli spaces. The result of this paper is used by Chang-Li-Li-Liu toward developing an effective theory evaluating all genera GW and FJRW invariants of the Fermat quintic threefold.
Fix a Fermat polynomial \(F_{N,r}(x)=x_1^r+\cdots+x_N^r\). An MSP field is defined to be a collection \(\xi = (\Sigma \subset C,\mathcal L, \mathcal N, \phi , \rho , \nu)\), consisting of a pointed twisted curve \(\Sigma \subset C\), invertible sheaves of \(\mathcal O_C\)-modules \(\mathcal L\) and \(\mathcal N\), fields \(\phi\in H^0(\mathcal L^{\oplus N})\) and \(\rho\in H^0 (\mathcal L^{*\otimes r}\otimes \omega^{\text{log}}_C)\), and a gauge field \(\nu = (\nu_1, \nu_2) \in H^0(\mathcal L \otimes \mathcal N) \oplus H^0(\mathcal N)\). The numerical invariants of \(\xi\) are the genus \(g\) of \(C\), the monodromy \(\gamma_i\) of \(\mathcal L\) at the \(i\)-th marking \(\Sigma_i\), and the bi-degrees \(d_0 = \deg(\mathcal L \otimes \mathcal N)\) and \(d_\infty = \deg \mathcal N\). For a choice of numerical invariants as above, \(\mathcal W_{g,\gamma_1,\dots, \gamma_n, d_0,d_\infty}\) denotes the moduli stack of the MSPs with these invariants. The main result of this paper proves that \(\mathcal W_{g,\gamma, d}\) is a separated DM stack, locally of finite type, equipped with a \(\mathbb C^*\)-action scaling the gauge field \(\nu_1\). Moreover, it has a \(\mathbb C^*\)-equivariant perfect obstruction theory and a cosection localized \(\mathbb C^*\)-equivariant virtual cycle \([\mathcal W_{g,\gamma, d}]^{\mathrm{vir}}\) lying in a proper substack of \(\mathcal W_{g,\gamma, d}\).

MSC:

14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
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