Real secondary index theory

In this paper, we study the family index of a family of spin manifolds. In particular, we discuss to which extend the real index (of the Dirac operator of the real spinor bundle if the fiber dimension is divisible by 8) which can be defined in this case contains extra information over the complex index (the index of its complexification). We study this question under the additional assumption that the complex index vanishes on the k-skeleton of B. In this case, using local index theory we define new analytical invariants \hat c_k\in H^{k-1}(B;\reals/\integers). We then continue and describe these invariants in terms of known topological characteristic classes. Moreover, we show that it is an interesting new non-trivial invariant in many examples.Comment: LaTeX2e, 56 pages; v2: final version to appear in ATG, typos fixed, statement of 4.5.5 improve

Column Imprints: A Secondary Index Structure

Large scale data warehouses rely heavily on secondary indexes, such as bitmaps and b-trees, to limit access to slow IO devices. However, with the advent of large main memory systems, cache conscious secondary indexes are needed to improve also the transfer bandwidth between memory and cpu. In this paper, we introduce column imprint, a simple but efficient cache conscious secondary index. A column imprint is a collection of many small bit vectors, each indexing the data points of a single cacheline. An imprint is used during query evaluation to limit data access and thus minimize memory traffic. The compression for imprints is cpu friendly and exploits the empirical observation that data often exhibits local clustering or partial ordering as a side-effect of the construction process. Most importantly, column imprint compression remains effective and robust even in the case of unclustered data, while other state-of-the-art solutions fail. We conducted an extensive experimental evaluation to assess the applicability and the performance impact of the column imprints. The storage overhead, when experimenting with real world datasets, is just a few percent over the size of the columns being indexed. The evaluation time for over 40000 range queries of varying selectivity revealed the efficiency of the proposed index compar

The two definitions of the index difference

Given two metrics of positive scalar curvature metrics on a closed spin manifold, there is a secondary index invariant in real $K$-theory. There exist two definitions of this invariant, one of homotopical flavour, the other one defined by a index problem of Atiyah-Patodi-Singer type. We give a complete and detailed proof of the folklore result that both constructions yield the same answer. Moreover, we generalize this to the case of two families of positive scalar curvature metrics, parametrized by a compact space. In essence, we prove a generalization of the classical "spectral-flow-index theorem" to the case of families of real operators.Comment: Revised versio

M-Theory with Framed Corners and Tertiary Index Invariants

The study of the partition function in M-theory involves the use of index theory on a twelve-dimensional bounding manifold. In eleven dimensions, viewed as a boundary, this is given by secondary index invariants such as the Atiyah-Patodi-Singer eta-invariant, the Chern-Simons invariant, or the Adams e-invariant. If the eleven-dimensional manifold itself has a boundary, the resulting ten-dimensional manifold can be viewed as a codimension two corner. The partition function in this context has been studied by the author in relation to index theory for manifolds with corners, essentially on the product of two intervals. In this paper, we focus on the case of framed manifolds (which are automatically Spin) and provide a formulation of the refined partition function using a tertiary index invariant, namely the f-invariant introduced by Laures within elliptic cohomology. We describe the context globally, connecting the various spaces and theories around M-theory, and providing a physical realization and interpretation of some ingredients appearing in the constructions due to Bunke-Naumann and Bodecker. The formulation leads to a natural interpretation of anomalies using corners and uncovers some resulting constraints in the heterotic corner. The analysis for type IIA leads to a physical identification of various components of eta-forms appearing in the formula for the phase of the partition function

TOPYDE: A Tool for Physical Database Design

We describe a tool for physical database design based on a combination of theoretical and pragmatic approaches. The tool takes as input a relational schema, the workload defined on the schema, and some additional database characteristics and produces as output a physical schema. For the time being, the tool is tuned towards Ingres