关于Frobenius流形的加变量及其推广
On adding a variable to a Frobenius manifold and generalizations
摘要：
令π：V \rightarrow M是一个（实的或全纯的）向量丛，其基具有几乎Frobenius结构（\circ_{M}，e_{M}，g_{M}），典型的纤维具有Frobenius代数的结构（\circ_{V}，e_{V}，g_{V}）。利用丛V上的联络D和态射α：V \rightarrow TM，构造了流形V上的几乎Frobenius结构（\circ，e_{V}，g），并研究了它何时是Frobenius.我们描述V上的所有（实）正定Frobenius结构，以这种方式获得，当M是一个半单Frobenius流形与非零旋转系数。在全纯情形下，我们在M上加上一个实结构k_{M}，在π的纤维上加上一个实结构k_{V}，并研究了流形V上的一个诱导实结构与几乎Frobenius结构（\circ，e_{V}，g）一起满足tt*-方程的情况.一路上，我们证明了各种性质的添加变量的弗罗贝纽斯流形，与勒让德变换和tt*-几何。
英文摘要：
Let π: V \rightarrow M be a (real or holomorphic) vector bundle whose base has an almost Frobenius structure (\circ_{M},e_{M}, g_{M}) and typical fiber has the structure of a Frobenius algebra (\circ_{V},e_{V},g_{V}). Using a connection D on the bundle V and a morphism α: V \rightarrow TM, we construct an almost Frobenius structure (\circ,e_{V},g) on the manifold V and we study when it is Frobenius. We describe all (real) positive-definite Frobenius structures on V, obtained in this way, when M is a semisimple Frobenius manifold with non-vanishing rotation coefficients. In the holomorphic setting we add a real structure k_{M} on M and a real structure k_{V} on the fibers of πand we study when an induced real structure on the manifold V, together with the almost Frobenius structure (\circ, e_{V}, g), satisfy the tt*-equations. Along the way, we prove various properties of adding variables to a Frobenius manifold, in connection with Legendre transformations and tt*-geometry.
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