Abstract
In this work the exceptional field theory formulation of supergravity with SL(5) gauge group is considered. This group appears as a U-duality group of $D=7$ maximal supergravity. In the formalism presented the hidden global duality group is promoted into a gauge group of a theory in dimensions 7+number of extended directions. This work is a continuation of the series of works for $E_{8,7,6}, SO(5,5)$ and $SL(3)\times SL(2)$ duality groups.
Highlights
String and M-theoretical extended objects can be understood as wave-like or monopole-like solutions of Exceptional Field Theory
EFT is formulated as a non-compactified theory, one has always keep in mind an additional condition, that forces us to either drop all or a subset of the extended coordinates returning effectively to the conventional supergravity or its toroidal compactifications, or to perform a generalized Scherk-Schwarz reduction, leading to themaximal gauged supergravity [17,18,19, 21]. This condition, named section condition, appears in the algebra of generalized diffeomorphisms given by the so-called generalized Lie derivative [38]
Transformation of covariant objects in Exceptional Field Theory is defined by the usual rule δΛV M = (LΣV )M = (LΛV )M + YKMLN ∂N ΛK V L ≡ [Λ, V ]M D, (3.1)
Summary
Let us briefly review the EFT construction and setup our conventions. Transformation of covariant objects in Exceptional Field Theory is defined by the usual rule δΛV M = (LΣV )M = (LΛV )M + YKMLN ∂N ΛK V L ≡ [Λ, V ]M D ,. Rewriting the projector P explicitly as in the appendix B one obtains the following transformation of a field in the fundamental 5 representation (LΛU )m. It is important to mention the section condition YKMLN ∂M ⊗ ∂N = 0, that for the case of SL(5) U-duality group can be written as ǫimnkl∂mn ⊗ ∂kl = 0 In this form the section condition implies existence of trivial generalized transformation given by Λm0 n = ǫmnklp∂klξp, i.e. δΛ0V M = 0 up to the section condition. Where the identity (C.5) has been used Note that these have precisely the same form is in the D = 7 maximal gauged supergravity up to the following mnemonic replacements of derivatives along extended coordinates by components Ymn and Zmn,k of embedding tensor.
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