Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Type II Actions from 11-Dimensional Chern-Simons Theories

Dmitriy M. Belov, Gregory W. Moore

TL;DR

This work presents a comprehensive action principle for Ramond-Ramond fields in type II supergravity by embedding the RR sector in an 11-dimensional Chern-Simons framework and quantizing via twisted differential K-theory. The RR field is organized as a self-dual object within a symplectic infinite-dimensional space, with the action depending on a choice of Lagrangian subspaces, which explains the non-uniqueness of the off-shell action. A key new result is a topological consistency condition: the fourth Wu class ν4 must admit a lift to H-twisted cohomology compatible with the RR/K-theory data, constraining consistent backgrounds. The formalism resolves previous ambiguities in B-field couplings, clarifies the coupling to D-branes, and provides a concrete path to compute partition functions, stress-energy, and quantum equations of motion in a Lorentz-covariant framework when considering off-shell and topologically nontrivial RR configurations.

Abstract

This paper continues the discussion of hep-th/0605038, applying the holographic formulation of self-dual theory to the Ramond-Ramond fields of type II supergravity. We formulate the RR partition function, in the presence of nontrivial H-fields, in terms of the wavefunction of an 11-dimensional Chern-Simons theory. Using the methods of hep-th/0605038 we show how to formulate an action principle for the RR fields of both type IIA and type IIB supergravity, in the presence of RR current. We find a new topological restriction on consistent backgrounds of type IIA supergravity, namely the fourth Wu class must have a lift to the H-twisted cohomology.

Type II Actions from 11-Dimensional Chern-Simons Theories

TL;DR

This work presents a comprehensive action principle for Ramond-Ramond fields in type II supergravity by embedding the RR sector in an 11-dimensional Chern-Simons framework and quantizing via twisted differential K-theory. The RR field is organized as a self-dual object within a symplectic infinite-dimensional space, with the action depending on a choice of Lagrangian subspaces, which explains the non-uniqueness of the off-shell action. A key new result is a topological consistency condition: the fourth Wu class ν4 must admit a lift to H-twisted cohomology compatible with the RR/K-theory data, constraining consistent backgrounds. The formalism resolves previous ambiguities in B-field couplings, clarifies the coupling to D-branes, and provides a concrete path to compute partition functions, stress-energy, and quantum equations of motion in a Lorentz-covariant framework when considering off-shell and topologically nontrivial RR configurations.

Abstract

This paper continues the discussion of hep-th/0605038, applying the holographic formulation of self-dual theory to the Ramond-Ramond fields of type II supergravity. We formulate the RR partition function, in the presence of nontrivial H-fields, in terms of the wavefunction of an 11-dimensional Chern-Simons theory. Using the methods of hep-th/0605038 we show how to formulate an action principle for the RR fields of both type IIA and type IIB supergravity, in the presence of RR current. We find a new topological restriction on consistent backgrounds of type IIA supergravity, namely the fourth Wu class must have a lift to the H-twisted cohomology.

Paper Structure

This paper contains 64 sections, 13 theorems, 195 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Self-duality constraints on RR fieldstrengths
  3. Symplectic structure.
  4. Self-duality of the Ramond-Ramond field.
  5. Complex structure.
  6. Formulating the partition function for topologically trivial fields
  7. "Democratic" formulation.
  8. Complexification of the gauge group.
  9. The partition function as a holomorphic section of a line bundle.
  10. Relation to Chern-Simons theory.
  11. Fieldspace and gauge transformations
  12. Differential K-group.
  13. Category of twisted differential K-cocycles.
  14. Twisted bundles and the twisted Chern character.
  15. $B$-field gauge transformations.
  16. ...and 49 more sections

Key Result

Lemma 7.1

$V_1$ defined by LdefV1 is an isotropic subspace in $V_{\mathbb R}$.

Figures (6)

  • Figure 1: The space of D-brane currents is fibered over the space $\check{Z}^2(X)$ of $B$-fields. On each connected component of $\check{Z}^2(B)$ there is a path connecting two points, say $\check{B}$ and $\check{B}+b$. There is a natural lift of this path into the total space: $(\check{B},\check{A})\mapsto (\check{B}+b,e^b\check{A})$. In addition we have a Chern-Simons line bundle over the space of D-brane currents. We now have to lift the path on $\check{Z}^2(X)$ to the total space of the line bundle such that it maps covariantly constant sections to covariantly constant sections.
  • Figure :
  • Figure :
  • Figure :
  • Figure :
  • ...and 1 more figures

Theorems & Definitions (18)

  • Lemma 7.1
  • Theorem 7.1
  • Theorem 7.2
  • Remark 7.1
  • Corollary 7.1
  • Remark 7.2
  • Theorem 7.3: Quantum equation of motion
  • proof
  • Lemma 8.1
  • proof
  • ...and 8 more