On Exotic Six-Dimensional Supergravity Theories
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HYSICS OF ELEMENTARY PARTICLES AND ATOMIC NUCLEI. THEORY
On Exotic Six-Dimensional Supergravity Theories G. Galatia, * and F. Riccionib, ** a
bINFN
SISSA, Trieste, 34136 Italy Sezione di Roma, Dipartimento di Fisica, Università di Roma “La Sapienza”, Roma, 00185 Italy *e-mail: [email protected] **e-mail: [email protected] Received November 15, 2019; revised January 15, 2020; accepted February 28, 2020
Abstract—The exotic 1 = (3, 1) and 1 = (4, 0) supergravity theories in six dimensions are theories describing potentials with mixed-symmetry indices satisfying a second order self-duality relation, and giving rise to maximal supergravity in five dimensions upon dimensional reduction. After reviewing how the linearised equations are constructed, we show how introducing additional gauge symmetries one can derive the same equations from first-order self-duality relations. We discuss how this result can be used to obtain a covariant lagrangian. DOI: 10.1134/S1547477120050155
1. INTRODUCTION Among supersymmetric theories, the maximal ones, i.e. the ones with 32 supercharges, are special for two reasons. First of all, they always contain gravity, which implies that supersymmetry becomes a gauge symmetry. Besides, if one ignores gaugings (that is the possibility to promote the global symmetries that rotate the abelian gauge vectors to local ones consistently with supersymmetry), these theories are unique, with the notable exception of the ten-dimensional case, in which one has the possibility of having two supercharges of opposite chirality, corresponding to the type-IIA theory, or of the same chirality, corresponding to type IIB. The highest dimension in which such theories exist is eleven. The dimensional reduction of the eleven-dimensional theory gives type IIA, and the lower-dimensional theories can be obtained by further reductions of this theory as well as from dimensional reduction of type IIB. Supersymmetric multiplets are representations of supersymmetry algebras, and in particular the maximal supergravity multiplets are the only massless representations containing fields of at most spin 21 of the Poincaré superalgebras with 32 supercharges. On the other hand, in six dimensions, together with the 1 = (2, 2) superalgebra, whose only massless representation is the maximal six-dimensional supergravity multiplet, there exist two additional Poincaré superalgebras, namely the 1 = (3, 1) and 1 = (4, 0) superalgebras, whose unique massless representations con-
taining fields of at most spin 2 are multiplets describing exotic mixed-symmetry gauge potentials instead of gravity [1]. These are the multiplets that we will discuss in the following. We recall that in six dimensions the superalgebras are in general denoted with 1 = ( p, q), where the supercharges are p spinors and one chirality and q spinors of the opposite chirality. This means that the 1 = (3,1) and 1 = (4, 0) superalgebras are chiral. The R-symmetry is USp(2 p) × USp(2q), and the supercharges belong to the fundamentals of the symplectic groups
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