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Type I anomaly cancellation revisited

Saghar S. Hosseini, Yuji Tachikawa, Hao Y. Zhang

Abstract

We revisit the issue of how the perturbative and global fermion anomaly of Type I string theory in ten dimensions is cancelled by the Green-Schwarz mechanism using the RR fields. This will be done by realising the RR fields as boundary modes of an eleven-dimensional bulk theory described in terms of a quadratic refinement of the differential KO-theory pairing. We will then generalise this analysis to Sugimoto's $usp(32)$ string and Sagnotti's $u(32)$ string. We also discuss in a more general setting the procedures which need to be followed when we try to cancel fermion anomalies in terms of $p$-form fields based on differential K-theory classes. This we illustrate by performing an analysis of the mod-2 anomaly cancellation in nine dimensions arising from the $S^1$ compactification of the Type I theory.

Type I anomaly cancellation revisited

Abstract

We revisit the issue of how the perturbative and global fermion anomaly of Type I string theory in ten dimensions is cancelled by the Green-Schwarz mechanism using the RR fields. This will be done by realising the RR fields as boundary modes of an eleven-dimensional bulk theory described in terms of a quadratic refinement of the differential KO-theory pairing. We will then generalise this analysis to Sugimoto's string and Sagnotti's string. We also discuss in a more general setting the procedures which need to be followed when we try to cancel fermion anomalies in terms of -form fields based on differential K-theory classes. This we illustrate by performing an analysis of the mod-2 anomaly cancellation in nine dimensions arising from the compactification of the Type I theory.
Paper Structure (56 sections, 252 equations, 4 figures)

This paper contains 56 sections, 252 equations, 4 figures.

Table of Contents

  1. Introduction and summary
  2. Some historical background
  3. Precise formulation of RR fields and the cancellation
  4. Comparison to the cancellation in the heterotic dual
  5. Organisation of the paper
  6. Invertible BF theories
  7. With differential ordinary cohomology
  8. With differential K or KO theory
  9. Defining features of differential K and KO theory classes
  10. Differential K or KO theory cocycles
  11. Integration
  12. The $\mathbb{R}/\mathbb{Z}$-valued pairing and BF theories
  13. The $\mathbb{Z}$-valued pairing and the quantisation of the fluxes
  14. Invertible Abelian Chern-Simons theories
  15. Quadratic refinements and invertible Abelian Chern-Simons theories
  16. ...and 41 more sections

Figures (4)

  • Figure 1: The hexagon diagram for the differential cohomology theory.
  • Figure 2: The hexagon diagram for generalised differential cohomology theory $\widecheck E$. We discuss the case $E=K$ and $E=KO$ more explicitly.
  • Figure 3: $p_s^*(\widecheck T)p_t^*(\widecheck T)\in \widecheck H^2(S^1\times S^1;\mathbb{Z})$ and the diagonal.
  • Figure 4: The choice of $\widecheck T_1$ and $\widecheck T_2$ on $W$.