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Fivebrane Structures

Hisham Sati, Urs Schreiber, Jim Stasheff

TL;DR

This work extends the established notion of String structures to a higher-dimensional setting, introducing Fivebrane structures as lifts through the 7-connected cover and identifying the obstruction by the fractional second Pontrjagin class $ rac{1}{6} p_2(TX)$. It connects anomaly cancellation in type II and heterotic string theories to these higher lifts, via dual Green-Schwarz mechanisms and degree-7 Chern-Simons forms, thereby linking worldvolume anomalies to topological constraints on tangent and gauge bundles. The paper also analyzes the cohomology of connected covers, torsion phenomena, and the congruence between different fractional Pontrjagin invariants, and it characterizes inequivalent Fivebrane structures as torsors under cohomology groups, laying groundwork for a differential-geometric realization in follow-up work. Overall, it provides a cohomological/topological framework for understanding fivebranes that complements and extends the String-structure perspective, with implications for anomaly cancellation and electric-magnetic duality in string/M-theory.

Abstract

We study the cohomological physics of fivebranes in type II and heterotic string theory. We give an interpretation of the one-loop term in type IIA, which involves the first and second Pontrjagin classes of spacetime, in terms of obstructions to having bundles with certain structure groups. Using a generalization of the Green-Schwarz anomaly cancelation in heterotic string theory which demands the target space to have a String structure, we observe that the "magnetic dual" version of the anomaly cancelation condition can be read as a higher analog of String structure, which we call Fivebrane structure. This involves lifts of orthogonal and unitary structures through higher connected covers which are not just 3- but even 7-connected. We discuss the topological obstructions to the existence of Fivebrane structures. The dual version of the anomaly cancelation points to a relation of String and Fivebrane structures under electric-magnetic duality.

Fivebrane Structures

TL;DR

This work extends the established notion of String structures to a higher-dimensional setting, introducing Fivebrane structures as lifts through the 7-connected cover and identifying the obstruction by the fractional second Pontrjagin class . It connects anomaly cancellation in type II and heterotic string theories to these higher lifts, via dual Green-Schwarz mechanisms and degree-7 Chern-Simons forms, thereby linking worldvolume anomalies to topological constraints on tangent and gauge bundles. The paper also analyzes the cohomology of connected covers, torsion phenomena, and the congruence between different fractional Pontrjagin invariants, and it characterizes inequivalent Fivebrane structures as torsors under cohomology groups, laying groundwork for a differential-geometric realization in follow-up work. Overall, it provides a cohomological/topological framework for understanding fivebranes that complements and extends the String-structure perspective, with implications for anomaly cancellation and electric-magnetic duality in string/M-theory.

Abstract

We study the cohomological physics of fivebranes in type II and heterotic string theory. We give an interpretation of the one-loop term in type IIA, which involves the first and second Pontrjagin classes of spacetime, in terms of obstructions to having bundles with certain structure groups. Using a generalization of the Green-Schwarz anomaly cancelation in heterotic string theory which demands the target space to have a String structure, we observe that the "magnetic dual" version of the anomaly cancelation condition can be read as a higher analog of String structure, which we call Fivebrane structure. This involves lifts of orthogonal and unitary structures through higher connected covers which are not just 3- but even 7-connected. We discuss the topological obstructions to the existence of Fivebrane structures. The dual version of the anomaly cancelation points to a relation of String and Fivebrane structures under electric-magnetic duality.

Paper Structure

This paper contains 28 sections, 4 theorems, 7 equations, 1 figure, 8 tables.

Table of Contents

  1. Introduction
  2. The context
  3. $\Sigma$-models
  4. Background fields
  5. Charges
  6. The Dual Green-Schwarz Mechanism and Higher Chern-Simons Forms
  7. Anomaly cancelation in string theory
  8. Anomalies in terms of $H_3$ and $H_7$
  9. The Chern-Simons forms
  10. Fivebrane Structures
  11. The homotopy groups of $SO(n)$
  12. String structures revisited
  13. Cohomology of the connected covers
  14. In characteristic 0
  15. Over the integers and the integers mod $p$.
  16. ...and 13 more sections

Key Result

Theorem 1

The obstruction to lifting a String structure on $X$ to a Fivebrane structure on $X$ is the fractional second Pontrjagin class $\frac{1}{6}p_2(T X)$.

Figures (1)

  • Figure 1: Homotopy groups of $\mathrm{O}(n)$ and its higher connected covers for $n > k+1$.

Theorems & Definitions (6)

  • Definition 1
  • Theorem 1
  • Definition 2
  • Proposition 1
  • Proposition 2
  • Proposition 3