Generalized Holomorphic Structures, Generalized Kahler Geometry, And An Application Of Whittaker Function

This thesis contains two independent parts. The (?)rst part mainly containsChapter 2, Chapter 3, Chapter 4, and Chapter 5, and the second part containsChapter 6.In Chapter 3, we investigate generalized holomorphic structures in generalizedcomplex geometry. We (?)nd that a generalized holomorphic vector bundle carriesa generalized complex structure on its total space if some additional conditionshold. We prove generalized holomorphicity is equivalent to the integrability of adistribution on the total space, and a family of linear Dirac structures associatedto this distribution is a generalized complex structure if a further condition holds.Under the same condition, we also prove that local generalized holomorphic framesexist around a regular point.In Chapter 4, what we call biholomorphic geometry on generalized K(|a¨)hler man-ifolds is investigated in the setting of generalized Riemannian geometry. The centralrole is played by what we call generalized Bismut connections. Via the connections,di(?)erential operators naturally arising in generalized K(|a¨)hler geometry can be ex-pressed explicitly, and interpreted as ordinary structures in complex geometry. J.Bismut’s local index theorem for non-K(|a¨)hler manifolds leads us to an approach toexpressing the index of the generalized Dolbeault operator on a generalized holo-morphic vector bundle, and we show this operator provides another solution of theMcKean-Singer problem. The index is just the Euler characteristic multiplied bythe rank of the bundle up to a sign.Chapter 5 contains a detailed study of generalized K(|a¨)hler geometry from theviewpoint of quantum 0+1-dimensional supersymmetricσ-model. Peierls bracketsrather than canonical quantization are used to quantize the superclassical system.Supercharges (or relevant di(?)erential operators) are expressed explicitly and covari-antly.In Chapter 6, we deal with a linear equation of in(?)nite dimension, (I + F)C =(?)I+F, which arises from conformal (?)eld theory. We show that, in a suitably chosen formalism from functional analysis, the solution C exists uniquely, and the content ofthe equation is precisely the Cayley transform. We prove that F can be realized byWhittaker function Wk,m(x). This realization leads us to new properties of Wk,m(x).We develop a method to compute the matrix elements of C, and generalize theproperties mentioned.