Hodge duality transformations in tensor-hierarchy formulations

Abstract

Abstract We show in arbitrary ( D − 1 ) + 1 space-time dimensions that duality-transformations are possible in general tensor-hierarchy with field-strengths of arbitrary ranks. We first consider the non-Abelian tensor field-strength G μ 1 ⋯ μ n I of the potential B μ 1 ⋯ μ n − 1 I , and its compensator field-strength H μ 1 ⋯ μ n − 1 I of the potential C μ 1 ⋯ μ n − 2 I , where I is the adjoint index. We next perform duality transformations into their respective Hodge-dual field-strength M μ 1 ⋯ μ D − n I of the potential K μ 1 ⋯ μ D − n − 1 I and the field-strength N μ 1 ⋯ μ D − n + 1 I of the potential L μ 1 ⋯ μ D − n I . As a typical application, we show this to take place in D = 3 + 1 dimensions with an N = 1 supersymmetric set consisting of a Yang–Mills vector-multiplet ( A μ I , λ I ) , an extra vector-multiplet ( B μ I , χ I ) , and a compensating chiral multiplet ( C I , ρ I , E I ) . The λ I and χ I are Majorana spinors, and B μ I is an extra vector with the Proca–Stückelberg scalar CI , while EI is a pseudo-scalar. We perform duality-transformations on B μ I and   C I to their Hodge-dual fields   K μ I and   L μ ν I , respectively, ending up with the new set of multiplets   ( A μ I , λ I ) ,   ( K μ I , χ I ) and   ( L μ ν I , ρ I , E I ) . Another example is the set of multiplets   ( A μ I , λ I , B μ ν ρ I ) and   ( C μ ν I , χ I , φ I ) in   D = 3 + 1 . Even though   B μ ν ρ I has the maximal 4th rank field-strength, its supersymmetric dual system exists with   ( A μ I , λ I ) and   ( L I , χ I , φ I ) after a duality-transformation. Our duality-transformations provide previously-unknown-links among supersymmetric tensor-hierarchy formulations.