On discrete U-duality in M-theory

Abstract

We give a complete set of generators for the discrete exceptional U-duality groups of toroidal compactified type II theory and M-theory in d 3. For this, we use the DSZ quantization in d = 4 as originally proposed by Hull and Townsend, and determine the discrete group inducing integer shifts on the charge lattice. It is generated by fundamental unipotents, which are constructed by exponentiating the Chevalley generators of the corresponding Lie algebra. We then extend a method suggested by the above authors and used by Sen for the heterotic string to obtain the discrete U-duality group in d = 3, thereby obtaining a quantized symmetry in d = 3 from a d = 4 quantization condition. This is studied first in a toy model, corresponding to d = 5 simple supergravity, and then applied to M-theory. It turns out that, in the toy model, the resulting U-duality group in d = 3 is strictly smaller than the one generated by the fundamental unipotents corresponding to all Chevalley generators. However, for M-theory, both groups agree. We illustrate the compactification to d = 3 by an embedding of d = 4 particle multiplets into the d = 3 theory.

Black holes inN=8supergravity from SO(4,4) hidden symmetries

“…Explicit algorithms for particular cosets have been developed extensively over the years, starting from the pioneering work on Einstein gravity [8,9], understood in terms of an SL(2, R) coset [48], and on Einstein-Maxwell theory [49], understood in terms of an SU(2, 1) coset [50][51][52][53]. Other theories considered are Kaluza-Klein theory, understood in terms of an SL(3, R) coset [54]; the particular Einstein-Maxwell-dilaton-axion theory used for generating solutions of N = 4 supergravity written in terms of a Sp(4, R) coset [55][56][57][58][59][60][61][62]; 5d minimal supergravity, which admits G 2(2) symmetries in [47,[63][64][65][66][67][68][69][70]; and ST U supergravity in 4 and 5 dimensions, which admits SO(4, 4) symmetries [71][72][73][74][75]. For the full N = 8 supergravity, reduction to 3 Euclidean dimensions gives the maximal N = 16 supergravity theory [31] with 128 scalars parameterizing the coset E 8(8) /SO * (16) [32,76,77].…”

FlippedSL(2,R)duality in five-dimensional supergravity

“…A similar analysis for the case when the two Killing vectors are both spacelike was already done in [27].…”

“…The dimensional reduction of the bosonic sector of five-dimensional minimal supergravity to four dimensions leads to a theory with a massless axion and a dilaton coupled to gravity and two Uð1Þ gauge fields with Chern-Simons coupling [15,[26][27][28]. As was shown in Ref.…”

F-theory family unification

“…In fact, the string junctions representing the E 8 roots are most conveniently described in terms of Freudenthal's realization of E 8 [117][118][119]; the exceptional Lie algebra E 8 is known to be generated by traceless E I J (I, J = 1, . .…”