This document presents a systematic investigation of the geometric index theory of Dirac operators coupled superconnections. A local version of the index theorem for Dirac operators coupled to superconnection is proved, and extended to families. An [eta]-invariant is defined, and it is shown to satisfy an APS-like theorem. A geometric determinant line bundle with section, metric, and connection is associated to a family of Dirac operators coupled to superconnections, and its holonomy is calculated in terms of the [eta]-invariant. / text

Thesis (Ph. D.)--University of Texas at Austin, 2008. / Vita. Includes bibliographical references and index.

by Nga-Wai Liu. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1991. / Bibliography: leaves 161-166. / Chapter Chapter 0 --- Introduction / Chapter 0.1 --- Historical background I ´ؤ The Atiyah-Singer index theorem --- p.1 / Chapter 0.2 --- Historical background II ´ؤGauge theory --- p.3 / Chapter 0.3 --- Arrangement of the thesis --- p.5 / Chapter Chapter 1 --- Fredholm operators / Chapter 1.1 --- Basic propetries --- p.7 / Chapter 1.2 --- Compact operators --- p.8 / Chapter 1.3 --- Homotopy- invariance of the index --- p.9 / Chapter 1.4 --- Family of Fredholm operators ´ؤ Index bundle --- p.13 / Chapter 1.5 --- Wiener-Hopf operators --- p.19 / Chapter Chapter 2 --- K-theory / Chapter 2.1 --- K-theory of compact spaces --- p.24 / Chapter 2.2 --- K-theory with compact support --- p.28 / Chapter 2.3 --- Bott periodicity theorem --- p.32 / Chapter 2.4 --- Difference construction --- p.44 / Chapter 2.5 --- Thom isomorphism theorem on K-theory --- p.51 / Chapter Chapter 3 --- Operators on manifolds / Chapter 3.1 --- Differential operators on Euclidean spaces --- p.54 / Chapter 3.2 --- Differential operators on manifolds --- p.55 / Chapter 3.3 --- Pseudodifferential operators on Euclidean spaces --- p.58 / Chapter 3.4 --- Pseudodifferential operators on manifolds --- p.62 / Chapter 3.5 --- Elliptic operators --- p.70 / Chapter 3.6 --- Tensor products --- p.76 / Chapter Chapter 4 --- Atiyah-Singer index theorem / Chapter 4.1 --- The topological index --- p.84 / Chapter 4.2 --- The analytical index --- p.87 / Chapter 4.3 --- The Atiyah-Singer index theorem --- p.89 / Chapter 4.4 --- Characteristic classes --- p.95 / Chapter 4.5 --- Thorn isomorphisms --- p.98 / Chapter 4.6 --- Cohomological formulation of the topological index --- p.101 / Chapter Chapter 5 --- Geometric preliminaries / Chapter 5.1 --- "Connections on principal bundles, and associated bundles" --- p.104 / Chapter 5.2 --- Gauge transformations --- p.109 / Chapter 5.3 --- Riemannian geometry --- p.112 / Chapter 5.4 --- Bochner-Weitzenboch formula --- p.116 / Chapter 5.5 --- Characteristic classes via curvature forms --- p.121 / Chapter 5.6 --- Holonomy --- p.126 / Chapter Chapter 6 --- Gauge theory / Chapter 6.1 --- The Yang-Mills functionals --- p.128 / Chapter 6.2 --- Instantons on S4 --- p.131 / Chapter 6.3 --- Moduli of self-dual connections --- p.142 / Chapter 6.4 --- Manifold structure for Moduli of self-dual connections --- p.153 / References --- p.161

Thesis (Ph. D.)--University of Texas at Austin, 2008. / Vita. Includes bibliographical references.

We construct a geometric model for differential K-theory, and prove it is isomorphic to the model proposed in [25]. We construct differential K-orientations for families and elucidate the pushforward map given in [25] in detail. We prove a geometric index theorem for odd dimensional manifolds. Finally, using this index theorem and the holonomy theorem of Bismut and Freed from [10], we prove what may be considered a special case of a geometric refinement of the Aityah-Singer index theorem. / text

We introduce the formalism of supersymmetric quantum mechanics, including super-symmetry charges,Z2-graded Hilbert spaces, the chirality operator and the Wittenindex. We show that there is a one to one correspondence of fermions and bosons forenergies different than zero, which implies that the Witten index measures the dif-ference of fermions and bosons at the ground state. We argue that the Witten indexis the index of an elliptic operator. Quantization of the supersymmetric non-linearsigma model shows that the Witten index equals the Euler characteristic of the un-derlying Riemannian manifold over which the theory is defined. Finally, the pathintegral representation of the Witten index is applied to derive the Gauss-Bonnettheorem. Apart from this we introduce elementary mathematical background in thesubjects of topological invariance, Riemannian manifolds and index theory / Vi introducucerar formalismen f ̈or supersymmetrisk kvantmekanik, d ̈aribland super-symmetryladdningar,Z2-graderade Hilbertrum, kiralitetsoperatorn och Wittenin-dexet. Vi visar att det r ̊ader en till en-korrespondens mellan fermioner och bosonervid energiniv ̊aer skillda fr ̊an noll, vilket medf ̈or att Wittenindexet m ̈ater skillnadeni antal fermioner och bosoner vid nolltillst ̊andet. Vi argumenterar f ̈or att Wittenin-dexet ̈ar indexet p ̊a en elliptisk operator. Kvantisering av den supersymmetriskaicke-linj ̈ara sigmamodellen visar att Wittenindexet ̈ar Eulerkarakteristiken f ̈or denunderliggande Riemannska m ̊angfald ̈over vilken teorin ̈ar definierad. Slutligenapplicerar vi v ̈agintegralrepresentationen av Wittenindexet f ̈or att h ̈arleda Gauss-Bonnets sats. Ut ̈over detta introduceras ocks ̊a grundl ̈aggande matematisk bakrundi ämnena topologisk invarians, Riemmanska m ̊angfalder och indexteori.