Abstract
We investigate the patching of double and exceptional field theories. In double field theory the patching conditions imposed on the spacetime after solving the strong section condition imply that the 3-form field strength $H$ is exact. A similar conclusion can be reached for the form field strengths of exceptional field theories after some plausive assumptions are made on the relation between the transition functions of the additional coordinates and the patching data of the form field strengths. We illustrate the issues that arise, and explore several alternative options which include the introduction of C-folds and of the topological geometrisation condition.
Highlights
Similar suggestions have emerged in the context of string theory and M-theory following the early works of [1,2,3,4]
We shall demonstrate that the transformations of [22, 31] that solve the strong section condition after interpreting them as patching conditions following [32] imply that the NS-NS 3-form field strength H which arises in string theory is an exact 3-form
We have shown that the patching conditions of double field theory (DFT) as arise from generalize coordinate transformations after solving the strong section condition imply that the NS-NS 3-form field is an exact 3-form
Summary
We proceed to the patching of k-forms and the introduction of new coordinates, it is instructive to explain the patching of closed 3-forms ω3 For this let M be a n-dimensional manifold with a good cover {Uα}α∈I and a partition of unity {ρα}α∈I subordinate to. Note that the patching data a1αβ, a0αβγ, nαβγδ are skewsymmetric in the interchange of any two of the open set labels, i.e. a1βγ = −a1γβ and for the rest. The patching data a1βγ of ω3 at double overlaps are not uniquely defined either. They are defined up to a gauge transformation a1αβ → a1αβ − χ1α + χ2β + dψα0β ,. Anything else is inconsistent with the identification of ω3 as a closed 3-form on M
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