辛域论中接触同调的Nijenhuis算子和后裔递归
Nijenhuis operator in contact homology and descendant recursion in symplectic field theory
摘要：
本文研究了接触同调和有理辛域论中$S^1$参数化曲线模空间的一种新型相关器的代数结构。这种算子是非等变线性化接触同调微分（在Bourgeois-Oancea之后）的自然推广，并在接触同调中产生了不变的Nijenhuis（或遗传）算子（la Magri-Fuchssteiner），它从原始算子中恢复了后代理论。我们还概述了这种结构如何推广到（分次对称）双向量的完整SFT Poisson同调代数。后代汉密尔顿满足递归关系，类似于双汉密尔顿递归，相对于自然泊松结构在SFT和这样的双向量形成的对。当目标流形是乘积稳定的Hamilton结构$S^1\times M$，而$M$是辛流形时，其递归关系与$M$的Gromov-Witten理论中的亏格$0$拓扑递归关系相一致.
英文摘要：
In this paper we investigate the algebraic structure related to a new type of correlator associated to the moduli spaces of $S^1$-parametrized curves in contact homology and rational symplectic field theory. Such correlators are the natural generalization of the non-equivariant linearized contact homology differential (after Bourgeois-Oancea) and give rise to an invariant Nijenhuis (or hereditary) operator (à la Magri-Fuchssteiner) in contact homology which recovers the descendant theory from the primaries. We also sketch how such structure generalizes to the full SFT Poisson homology algebra to a (graded symmetric) bivector. The descendant hamiltonians satisfy to recursion relations, analogous to bihamiltonian recursion, with respect to the pair formed by the natural Poisson structure in SFT and such bivector. In case the target manifold is the product stable Hamiltonian structure $S^1\times M$, with $M$ a symplectic manifold, the recursion coincides with genus $0$ topological recursion relations in the Gromov-Witten theory of $M$.
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