U-duality and Courant Algebroid in Exceptional Field Theory

Abstract

In this paper, we study the field transformation under U-duality in exceptional field theories. Take $SL(5)$ and $SO(5,5)$ exceptional field theories as examples, we explicitly show that the U-duality transformation is governed by the differential geometry of a corresponding Courant algebroid structure. The field redefinition specified by $SL(5)$ and $SO(5,5)$ transformations can be realized by Courant algebroid anchor mapping. Based on the existence of Courant algebroid in $E_d$ exceptional field theory, we expect that the Courant algebroid anchor mapping also exist in exceptional field theories with higher dimensional exceptional groups, such as $E_6$ and $E_7$. Intriguingly, the U-dual M2-brane and M5-brane can be realized with the same structure of Courant algebroid in exceptional field theory. Since in each exceptional field theory, all the involved fields can be mapped with the same anchor, the full Lagrangian is governed by the Courant algebroid anchor mapping. In particular, this is realized by the U-duality mapping in extended generalised geometry, from the extended bundle $E=TM \oplus \Lambda^2 T^*M \oplus \Lambda^5 T^*M\oplus \Lambda^6 TM$ to the U-dual bundle $E^*=T^*M \oplus \Lambda^2 TM \oplus \Lambda^5 TM \oplus \Lambda^6 T^*M$ under the global $E_d$ symmetry. From M-theory point of view, a U-dual effective theory of M-theory is expected from Courant algebroid anchor mapping in such a global manner via U-duality.