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Ramond-Ramond Fields, Cohomology and Non-Geometric Fluxes

Aaron Bergman, Daniel Robbins

TL;DR

This work develops a mathematical framework for Ramond-Ramond flux in type II string compactifications with nongeometric, T-fold fluxes. By combining Spin(d,d,ℤ) duality twists with base-space two-forms, the authors define a Z2-graded cohomology H_T^•(X,ℝ) that captures unquantized RR fields in nongeometric backgrounds and relate it to a generalized twisted cohomology H_H^• on a torus bundle T over X. The core result is an isomorphism H_T^•(X,ℝ) ≅ H_H^•(T,ℝ) under appropriate constructions of H from base flux data e^a, ω_a and the total-space three-form H, with T-duality acting as a gauge transformation on the underlying data. The paper also lays out a local-descriptor framework and discusses the route to quantization via twisted K-theory K_H^•(•), setting the stage for a subsequent, deeper treatment of RR flux quantization in nongeometric settings. Overall, it provides a rigorous bridge between nongeometric flux in string theory and twisted cohomology, with clear implications for constructing richer four-dimensional vacua and understanding duality-covariant RR flux classification.

Abstract

We consider compactifications of type II string theory in which a d-dimensional torus is fibered over a base X. In string theory, the transition functions of this fibration need not be simply diffeomorphisms of T^d but can involve elements of the T-duality group Spin(d,d,Z). We precisely define the notion of a T-fold with NSNS flux. Given such a T-fold, we define the Z_2-graded cohomology theory describing the unquantized RR field strengths and discuss how the data of a T-fold can be interpreted in terms of generalized NSNS fluxes and the twisted differential of Shelton-Taylor-Wecht.

Ramond-Ramond Fields, Cohomology and Non-Geometric Fluxes

TL;DR

This work develops a mathematical framework for Ramond-Ramond flux in type II string compactifications with nongeometric, T-fold fluxes. By combining Spin(d,d,ℤ) duality twists with base-space two-forms, the authors define a Z2-graded cohomology H_T^•(X,ℝ) that captures unquantized RR fields in nongeometric backgrounds and relate it to a generalized twisted cohomology H_H^• on a torus bundle T over X. The core result is an isomorphism H_T^•(X,ℝ) ≅ H_H^•(T,ℝ) under appropriate constructions of H from base flux data e^a, ω_a and the total-space three-form H, with T-duality acting as a gauge transformation on the underlying data. The paper also lays out a local-descriptor framework and discusses the route to quantization via twisted K-theory K_H^•(•), setting the stage for a subsequent, deeper treatment of RR flux quantization in nongeometric settings. Overall, it provides a rigorous bridge between nongeometric flux in string theory and twisted cohomology, with clear implications for constructing richer four-dimensional vacua and understanding duality-covariant RR flux classification.

Abstract

We consider compactifications of type II string theory in which a d-dimensional torus is fibered over a base X. In string theory, the transition functions of this fibration need not be simply diffeomorphisms of T^d but can involve elements of the T-duality group Spin(d,d,Z). We precisely define the notion of a T-fold with NSNS flux. Given such a T-fold, we define the Z_2-graded cohomology theory describing the unquantized RR field strengths and discuss how the data of a T-fold can be interpreted in terms of generalized NSNS fluxes and the twisted differential of Shelton-Taylor-Wecht.

Paper Structure

This paper contains 18 sections, 1 theorem, 116 equations.

Table of Contents

  1. Introduction
  2. Introduction for physicists
  3. Introduction for mathematicians
  4. Plan of the paper
  5. Clifford algebra actions
  6. Linear algebra of $V\oplus V^\ast$
  7. The Clifford algebra
  8. String theory context
  9. Three legs on the base
  10. One leg on the base
  11. Two legs on the base
  12. The one dimensional case
  13. Higher dimensions
  14. Combining two legs and one leg
  15. Local description
  16. ...and 3 more sections

Key Result

Theorem 1

Given the data defined as above, we have the following isomorphism: Furthermore, in the case where we can choose $H = 0$, $H^\bullet_\mathcal{T}(X,\mathbb{Z})$ can be defined over the integers and we have

Theorems & Definitions (2)

  • Definition 1
  • Theorem 1