Study on the anomaly of Spin^q manifolds

他先前的研究表明，自旋^q流形（M,g^M）具有正则CP^1-纤化，其总空间（Z,g^ Z =π^*g^M+g^<C_<P^1>>）称为扭转空间具有正则自旋结构。这个结构引出了Dirac算子【∂！】(1)无穷小手性异常的定义及研究内容：首先，它的无穷小变化δ_X【∂！/】，其中X是某个伴随束的横截面，它的异常对数det δ_X【∂！/】≡-∂/∂s | _ < s = 0 > ^ 1 /(2Γ(s)) f ^∞_0 t ^ s STr(δ值【∂/】【∂! /】e ^ < - t【∂!（研究者）从数学家的观点介绍了dt。在类比了物理的扭转理论和宇宙的创造理论之后，他考虑了将每根纤维坍缩成一点（回到宇宙前）的操作，即取绝热极限的操作，以产生其本质部分，表示为lim_<ε→0> log det δ_X【∂！/】_ε。调查极限是主要目的。(2)对Getzler变换的进一步研究：所谓的Getzler变换G_ε(【∂！^2是这项研究的有用工具。他证明了把它看作是两种变换的组合是非常有效的。通过对它们的密切调查，(1)中介绍的研究得到了很大的改进。(3)同步连接与Gilkey理论：他引入了一个同步连接的概念，表示为△^<g^Z>【对称】，这比Levi-Civita的概念要简单一些。据此，我们定义了同步测地线、坐标等，并研究了△^<g^Z>【对称】的曲率系数的一些不变多项式。不幸的是，它还没有完成，为了完成它，需要对典型的Spin^q流形进行进一步的研究。

His previous study says that a Spin^q manifold (M,g^M] possesses a canonical CP^1-fibration and its total space (Z,g^z=π^*g^M+g^<C_<P^1>>) called a twistor space has a canonical Spin structure. The structure induces the Dirac operator 【∂!/】 on Z.(1)The definition of the infinitesimal chiral anomaly and what should be studied : First, its infinitesimal variation δ_X【∂!/】 in the X-direction, where X is a cross-section of a certain adjoint bundle, and its anomaly log det δ_X【∂!/】≡-∂/∂s|_<s=0>^1/(2Γ(s)) f^∞_0 t^s STr(δ_X【∂!/】【∂!/】e^<-t【∂!/】^2>) dt were introduced (by the investigator) from the mathematician's viewpoint. After the analogy of the physical twistor theory and the creating theory of the universe, he considered the operation of collapsing each fiber into one point (returning to the pre-universe), i.e., the operation of taking the adiabatic limit, to produce its essential part denoted lim_<ε→0> log det δ_X【∂!/】_ε. To investigate the limit was the main purpose.(2)A closer investigation into the Getzler transformation : The so-called Getzler transformation G_ε(【∂!/】_ε)^2 is a useful tool for the study. He showed it very effective to consider it as a composition of two kinds of transformations. By investigating them closely, the study introduced in (1)was quite improved.(3)Synchronous connection and the Gilkey theory : He introduced a concept of synchronous connection denoted ▽^<g^Z>【symmetry】, which is a little bit simpler than the Levi-Civita one. Accordingly, we define synchronous geodesies, coordinates, etc., and come to the study some invariant polynomials of the curvatures coefficients of ▽^<g^Z>【symmetry】. Unfortunately, it is not yet finished and, to finish it, further invesatigation into typical Spin^q manifolds will be needed.