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U-duality and Courant Algebroid in Exceptional Field Theory

Rui Sun
Nov 2022
摘要
In this paper, we study the field transformation under U-duality inexceptional field theories. Take $SL(5)$ and $SO(5,5)$ exceptional fieldtheories as examples, we explicitly show that the U-duality transformation isgoverned by the differential geometry of a corresponding Courant algebroidstructure. The field redefinition specified by $SL(5)$ and $SO(5,5)$transformations can be realized by Courant algebroid anchor mapping. Based onthe existence of Courant algebroid in $E_d$ exceptional field theory, we expectthat the Courant algebroid anchor mapping also exist in exceptional fieldtheories 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 samestructure of Courant algebroid in exceptional field theory. Since in eachexceptional field theory, all the involved fields can be mapped with the sameanchor, the full Lagrangian is governed by the Courant algebroid anchormapping. In particular, this is realized by the U-duality mapping in extendedgeneralised 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 isexpected from Courant algebroid anchor mapping in such a global manner viaU-duality.
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