Study on the anomaly of Spin^q manifolds

他以前的研究表明，Spin^q流形（M，g^M）具有正则CP ^1-纤维化，并且其称为扭量空间的总空间（Z，g^z=π^*g^M+g^<C_<P^1>>）具有正则Spin结构。该结构诱导狄拉克算子[1/]在Z上（1）无穷小手征反常的定义及研究内容：首先，它的无穷小变差δ_X[X！/]其中X是某个伴随丛的横截面，其异常log det δ_X[δ！/] -| _<s=0>^1/（2Γ（s））f^∞_0 t^s STr（δ_X[π！/] [！/] e^<-t[！/]^ 2>）dt是从数学家的观点（由研究者）引入的。在物理扭量理论和宇宙创生理论的类比之后，他考虑了将每根纤维坍缩成一点（回到前宇宙）的操作，即，取绝热极限的操作，以产生其基本部分，表示为lim_<ε→0> log det δ_X[δ！/]_ε。研究极限是主要目的。(2)A Getzler变换的进一步研究：所谓的Getzler变换G_ε（[！/]_ε）^2是一个有用的研究工具。他表明，它非常有效地考虑它作为一个组成的两种转变。通过对它们的仔细研究，（1）中介绍的研究得到了很大的改进。（3）同步联络和Gilkey理论：他引入了同步联络的概念，记为对称性，比Levi-Civita的概念简单一些。因此，我们定义同步测地线、坐标等，进而研究了对称性的曲率系数的几种不变多项式。不幸的是，它还没有完成，为了完成它，需要进一步研究典型的Spin^q流形。

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.