I. Introduction: Transcending Nationalism: The European Community 1958-2000

We develop some of the ingredients needed for string theory on noncommutative spacetimes, proposing an axiomatic formulation of T-duality as well as establishing a very general formula for D-brane charges. This formula is closely related to a noncommutative Grothendieck-Riemann-Roch theorem that is proved here. Our approach relies on a very general form of Poincare duality, which is studied here in detail. Among the technical tools employed are calculations with iterated products in bivariant \(\mathsf K\)-theory and cyclic theory, which are simplified using a novel diagram calculus reminiscent of Feynman diagrams.

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D-Branes, RR-Fields and Duality on Noncommutative Manifolds

We review various K-theory classification conjectures in string theory Sen conjecture based proposals classify D-brane trajectories in backgrounds with no H flux, while Freed-Witten anomaly based proposals classify conserved RR charges and magnetic RR fluxes in topologically time-independent backgrounds In exactly solvable CFTs a classification of well-defined boundary states implies that there are branes representing every twisted K-theory class Some of these proposals fail to respect the self-duality of the RR fields in the democratic formulation of type II supergravity and none respect S-duality in type IIB string theory We discuss two applications The twisted K-theory classification has led to a conjecture for the topology of the T-dual of any configuration In the Klebanov-Strassler geometry twisted K-theory classifies universality classes of baryonic vacua

What Does(n't) K-theory Classify?

We review and elaborate on certain aspects of the connections between instanton counting in
maximally supersymmetric gauge theories and the computation of enumerative invariants of
smooth varieties. We study in detail three instances of gauge theories in six, four, and two
dimensions which naturally arise in the context of topological string theory on certain noncompact
threefolds. We describe how the instanton counting in these gauge theories is related
to the computation of the entropy of supersymmetric black holes and how these results are
related to wall-crossing properties of enumerative invariants such as Donaldson-Thomas and
Gromov-Witten invariants. Some features of moduli spaces of torsion-free sheaves and the
computation of their Euler characteristics are also elucidated.

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Instantons, Topological Strings and Enumerative Geometry

We describe a categorical framework for the classification of D-branes on non-commutative spaces using techniques from bivariant K-theory of C!-algebras. We present a new description of bivariant K-theory in terms of non-commutative correspondences which is nicely adapted to the study of T-duality in open string theory. We systematically use the diagram calculus for bivariant K-theory as detailed in our previous paper [12]. We explicitly work out our theory for a number of examples of noncommutative manifolds.

https://www.intlpress.com/site/pub/files/_fulltext/journals/atmp/2009/0013/0002/ATMP-2009-0013-0002-a004.pdf

Non-commutative correspondences, duality and D-branes in bivariant K-theory

We use the Baum-Douglas construction of K-homology to explicitly describe various aspects of D-branes in Type II superstring theory in the absence of background supergravity form fields. We rigorously derive various stability criteria for states of D-branes and show how standard bound state constructions are naturally realized directly in terms of topological K-cycles. We formulate the mechanism of flux stabilization in terms of the K-homology of non-trivial fibre bundles. Along the way we derive a number of new mathematical results in topological K-homology of independent interest.

Geometric K-Homology of Flat D-Branes

We study the classification of D-branes and Ramond-Ramond fields in Type I string theory by developing a geometric description of KO-homology. We define an analytic version of KO-homology using KK-theory of real C*-algebras, and construct explicitly the isomorphism between geometric and analytic KO-homology. The construction involves recasting the Cl(n)-index theorem and a certain geometric invariant into a homological framework which is used, along with a definition of the real Chern character in KO-homology, to derive cohomological index formulas. We show that this invariant also naturally assigns torsion charges to non-BPS states in Type I string theory, in the construction of classes of D-branes in terms of topological KO-cycles. The formalism naturally captures the coupling of Ramond-Ramond fields to background D-branes which cancel global anomalies in the string theory path integral. We show that this is related to a physical interpretation of bivariant KK-theory in terms of decay processes on spacetime-filling branes. We also provide a construction of the holonomies of Ramond-Ramond fields in Type II string theory in terms of topological K-chains.

KO-Homology and Type I String Theory

We study the classification of D-branes and Ramond–Ramond fields in Type I string theory by developing a geometric description of KO-homology. We define an analytic version of KO-homology using KK-theory of real C*-algebras, and construct explicitly the isomorphism between geometric and analytic KO-homology. The construction involves recasting the Cln-index theorem and a certain geometric invariant into a homological framework which is used, along with a definition of the real Chern character in KO-homology, to derive cohomological index formulas. We show that this invariant also naturally assigns torsion charges to non-BPS states in Type I string theory, in the construction of classes of D-branes in terms of topological KO-cycles. The formalism naturally captures the coupling of Ramond–Ramond fields to background D-branes which cancel global anomalies in the string theory path integral. We show that this is related to a physical interpretation of bivariant KK-theory in terms of decay processes on spacetime-filling branes. We also provide a construction of the holonomies of Ramond–Ramond fields in Type II string theory in terms of topological K-chains.

Ko-homology and type i string theory

http://export.arxiv.org/pdf/hep-th/0507043

Rui M. G. Reis

Papers

Geometric K-Homology of Flat D-Branes

KO-Homology and Type I String Theory

Ko-homology and type i string theory

Geometric K-Homology of Flat D-Branes