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Ramond-Ramond fields and twisted differential K-theory

Daniel Grady, Hisham Sati

Abstract

We provide a systematic approach to describing the Ramond-Ramond (RR) fields as elements in twisted differential K-theory. This builds on a series of constructions by the authors on geometric and computational aspects of twisted differential K-theory, which to a large extent were originally motivated by this problem. In addition to providing a new conceptual framework and a mathematically solid setting, this allows us to uncover interesting and novel effects. Explicitly, we use our recently constructed Atiyah-Hirzebruch spectral sequence (AHSS) for twisted differential K-theory to characterize the RR fields and their quantization, which involves interesting interplay between geometric and topological data. We illustrate this with the examples of spheres, tori, and Calabi-Yau threefolds.

Ramond-Ramond fields and twisted differential K-theory

Abstract

We provide a systematic approach to describing the Ramond-Ramond (RR) fields as elements in twisted differential K-theory. This builds on a series of constructions by the authors on geometric and computational aspects of twisted differential K-theory, which to a large extent were originally motivated by this problem. In addition to providing a new conceptual framework and a mathematically solid setting, this allows us to uncover interesting and novel effects. Explicitly, we use our recently constructed Atiyah-Hirzebruch spectral sequence (AHSS) for twisted differential K-theory to characterize the RR fields and their quantization, which involves interesting interplay between geometric and topological data. We illustrate this with the examples of spheres, tori, and Calabi-Yau threefolds.

Paper Structure

This paper contains 15 sections, 18 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. RR fields as twisted differential K-theory classes
  3. Differential K-theory
  4. The topological twisted K-theory
  5. The differential twisted K-theory
  6. Case 1: $X$ is a ${\rm Spin}^c$-manifold:
  7. Case 2: $X$ does not admit ${\rm Spin}^c$ structure:
  8. Lifting RR forms
  9. RR forms arising from differential K-theory
  10. RR forms arising from twisted differential K-theory
  11. Anomalies
  12. Explicit classification of RR fields in traditional backgrounds
  13. Twisted differential K-theory of spheres
  14. The (twisted) differential K-theory of tori
  15. The (twisted) differential K-theory of Calabi-Yau threefolds

Key Result

Proposition \oldthetheorem

Let $X$ be a CW-complex and let $h:X\to K(\mathbb Z,3)$ be a twist for K-theory. Let $K_h\to X$ and $H\mathbb Q[u,u^{-1}]_h\to X$ be the bundles of spectra representing $h$-twisted K-theory and rational cohomology (twisted by the post-composition of $h$ with the canonical map $K(\mathbb Z,3)\to K(\m inducing a twisted Chern character map ${\rm ch}_h:K_h(X)\to H_h(X;\mathbb Q)$, which reduces to t

Theorems & Definitions (37)

  • Definition \oldthetheorem: Smooth K-theory spectrum
  • Remark 1: Vector bundles with connections
  • Definition \oldthetheorem: Hopkins-Singer differential K-theory
  • Remark 2: Basic properties of $\widehat{\rm K}$
  • Proposition \oldthetheorem: Twisted Chern character
  • Proposition \oldthetheorem: Twisted differential Chern character
  • Remark 3: Refinement of the A-genus
  • Proposition \oldthetheorem: Torsion differentials in the $\widehat{\rm AHSS}$
  • Proposition \oldthetheorem: Degree two
  • Proposition \oldthetheorem: Degree four
  • ...and 27 more