Klein曲面的模空间及相关运算
Moduli spaces of Klein surfaces and related operads
摘要：
考虑经典的二维拓扑量子场论到允许不可定向曲面的Klein拓扑量子场论的推广。通过引入一个新的控制带对合的结合代数的运算，我们使用模运算的理论来处理这个问题。这个操作是Koszul和我们确定的对偶dg操作管理A-无穷代数与对合的莫比乌斯图，这是一个概括的带状图。然后将开拓扑共形域论推广到开Klein拓扑共形域论，并给出了开KTCFT运算的生成元和关系描述.我们推导出一个类似的带状图分解的模空间的黎曼曲面：莫比乌斯图分解的模空间的克莱因曲面（实代数曲线）。然后，Mobius图复形计算这些模空间的同调。我们还得到了一个不同的图复形计算可容许的稳定对称黎曼曲面的模空间的同调，这是部分紧化的模空间的Klein曲面。
英文摘要：
We consider the extension of classical 2-dimensional topological quantum field theories to Klein topological quantum field theories which allow unorientable surfaces. We approach this using the theory of modular operads by introducing a new operad governing associative algebras with involution. This operad is Koszul and we identify the dual dg operad governing A-infinity algebras with involution in terms of Mobius graphs which are a generalisation of ribbon graphs. We then generalise open topological conformal field theories to open Klein topological conformal field theories and give a generators and relations description of the open KTCFT operad. We deduce an analogue of the ribbon graph decomposition of the moduli spaces of Riemann surfaces: a Mobius graph decomposition of the moduli spaces of Klein surfaces (real algebraic curves). The Mobius graph complex then computes the homology of these moduli spaces. We also obtain a different graph complex computing the homology of the moduli spaces of admissible stable symmetric Riemann surfaces which are partial compactifications of the moduli spaces of Klein surfaces.
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