PASCOS 2012 Non-Abelian Tensor Multiplet in Four Dimensions Subhash R AJPOOT 1) Department of Physics & Astronomy California State University 1250 Bellflower Boulevard Long Beach, CA 90840 This work is done in collaboration with Dr. H. Nishino. I will present the work as follows. • STATEMENT OF THE PROBLEM • THE SOLUTION • SUPERSYMMETRY • SUPERFIELD LANGUAGE • DISCUSS RELEVANCE OF THE WORK a. STANDARD MODEL b. ON THE QUEST FOR THE UNIFICATION OF FUNDAMENTAL FORCES PACS: 11.15.-q, 11.30.Pb, 12.60.Jv Key Words: Non-Abelian Tensor, N = 1 Supersymmetry, Tensor Multiplet, Vector Field in Non-Trivial Representation, Consistency of Field Equations and Couplings. 1) E-Mail: rajpoot@csulb.edu 1
1. The Problem The basic problem with a non-Abelian tensor, when it has its own kinetic term, is easily seen as follows. Let I be the adjoint index of a non-Abelian group G , and let a non-Abelian vector field A µI couple minimally to the antisymmetric tensor B µνI . Consider the most conventional field strength I ≡ +3 D ⌊ I ≡ +3( ∂ ⌊ I + gf IJK A ⌊ G (0) J B νρ ⌋ K ) , ⌈ µ B νρ ⌋ ⌈ µ B νρ ⌋ (1 . 1) ⌉ ⌉ ⌈ µ ⌉ µνρ where D µ is the usual gauge-covariant derivative with the minimal coupling with the f IJK coupling constant g and the structure constant of the group G . Consider a � d 4 x L 0 with the lagrangian 2) tentative action I 0 ≡ I ) 2 − 1 I ) 2 , L 0 ≡ − 1 12 ( G (0) 4 ( F µν (1 . 2) µνρ with F µνI ≡ 2 ∂ ⌊ ⌉ I + gf IJK A µJ A νK . Obviously, the B -field equation is 3) ⌈ µ A ν ⌋ 2 D ρ G (0) µνρI . δ L 0 δB µνI = + 1 = 0 . (1 . 3) The problem is that the divergence of this B -field equation does not vanish: � δ L 0 � J G (0) µνρK � = 0 , ? = + 1 4 gf IJK F νρ 0 = D ν (1 . 4) δB µνI unless F µνI or G (0) I vanishes trivially. This inconsistency problem is already at the clas- µνρ sical level before quantization. This is also one of the reasons, why topological formulations with vanishing field strength F µνI . = 0 such as [1] are easier to formulate for non-Abelian tensors. An additional problem is related to the so-called local tensorial gauge transformation of the B -field: I = + D ⌊ I − D ⌊ I , δ β B µν ⌈ µ β ν ⌋ ⌈ ν β µ ⌋ (1 . 5) ⌉ ⌉ because the field strength G µνI is not invariant under δ β : I = +3 gf IJK F ⌊ K � = 0 . δ β G (0) J β ρ ⌋ (1 . 6) ⌈ µν ⌉ µνρ This further implies the non -invariance of the action: δ β I 0 � = 0. These two problems are mutually related, because the non-vanishing of (1.4) is also interpreted as the action non- invariance δ β I 0 � = 0. 2) We use the signature ( − , + , + , +) for four dimensions (4D) in this paper. 3) The symbol . = stands for a field equation, to be distinguished from an algebraic identity. We also ? use the symbol = for an equality under question. 2
2. The Solution to Problem The solution to the problem above is to introduce a non-trivial Chern-Simons (CS) term into the G -field strength: I ≡ +3 D ⌊ I ≡ +3( ∂ ⌊ I + gf IJK A ⌊ J B νρ ⌋ K ) − 3 f IJK C ⌊ J F νρ ⌋ K G µνρ ⌈ µ B νρ ⌋ ⌈ µ B νρ ⌋ ⌉ ⌉ ⌈ µ ⌉ ⌈ µ ⌉ I − 3 f IJK C ⌊ K , ≡ + G (0) J F νρ ⌋ (2 . 1) ⌈ µ ⌉ µνρ where C µI is a ‘compensator’ vector field, also carrying the adjoint index. The field strength for C is defined by I ≡ + D ⌊ I − D ⌊ I + gB µν I . H µν ⌈ µ C ν ⌋ ⌈ ν C µ ⌋ (2 . 2) ⌉ ⌉ Now these field strengths G and H are invariant under the δ β -transformation I = + D ⌊ I − D ⌊ I δ β B µν ⌈ µ β ν ⌋ ⌈ ν β µ ⌋ (2 . 3a) ⌉ ⌉ I = − gβ µ I , δ β C µ (2 . 3b) which is the ‘proper’ gauge transformation for B µνI , and δ γ -transformations I = − f IJK F µν J γ K , δ γ B µν (2 . 4a) I = D µ γ I . δ γ C µ (2 . 4b) is the ‘proper’ gauge transformation for C µI . The role played by the C ∧ F -term in (2.1) is to cancel the unwanted term in (1.6). The C -field itself should have its own ‘gauge’ transformation as the covariant gradient (2.4b). ⌉ I ) in (2.2) is cancelled by the contribution of δ γ ( gB µνI ), so The contribution of δ γ (2 D ⌊ ⌈ µ C ν ⌋ that δ γ H µνI = 0. In other words, we have the total invariances I = 0 , I = 0 , δ β G µνρ δ β H µν (2 . 5a) I = 0 , I = 0 . δ γ G µνρ δ γ H µν (2 . 5b) Accordingly, we also have the consistency problem (1.4) solved. Consider the kinetic terms for the B, C and A -fields: 1 I ) 2 − 1 I ) 2 − 1 I ) 2 . L 1 ≡ − 12 ( G µνρ 4 ( H µν 4 ( F µν (2 . 6) 3
The total action is also invariant δ β I 1 = δ γ I 1 = 0. The new field equations for B and C -fields are 2 gH µν I . δ L 1 2 D ρ G µνρ I − 1 δB µνI = + 1 = 0 , (2 . 7a) J G µρσ K . δ L 1 δC µI = − D ν H µν I + 1 2 f IJK F ρσ = 0 , (2 . 7b) The divergence of the B -field equation vanishes now: � . � δ L 1 � δ L 1 � ? = + 1 0 = D ν 2 g = 0 , (2 . 8) δB µνI δC µI where the last equality holds because of the C -field equation. In other words, the unwanted FG -term in (1.4) is now cancelled by the contribution of the C -field equation. This has solved the previous problem (1.4). Relevantly, the divergence of (2.10) also vanishes, as it should: � . � δ L 1 � δ L 1 � ? = + f IJK F µν J 0 = D µ = 0 , (2 . 9) δC µI δB µνK without any inconsistency. We emphasize repeatedly that these invariances have never been accomplished without the peculiar CS terms both in (2.1) and (2.2). Recently, the long-standing problem with non-Abelian tensors [2] has been solved by de Wit, Samtleben, and Nicolai [3][4]. The original motivation in [3] was to generalize the tensor and vector field interactions in manifestly E 6(+6) -covariant formulation of five-dimensional (5D) maximal supergravity by gauging non-Abelian sub-groups. In [4], this work was further related to M-theory [5] by confirming the representation assignments under the duality group of the gauge charges. The underlying hierarchies of these tensor and vector gauge fields are presented with the consistency of general gaugings. The hierarchy in [3][4] has been further applied to the conformal supergravity in 6D [6]. In ref. [6], the ‘minimal tensor hierarchy’ as a special case of the more general hierarchy in A µr [3][4] has been discussed. This hierarchy consists of and two-form gauge potentials B µνI , with two labels r and I . Also introduced is the 3-form gauge potentials C µνρ r with 4
the index r is dual to r of A µr . The field strengths of vector and two-form gauge potentials are defined by [6] r ≡ 2 ∂ ⌊ r + h I I , r B µν F µν ⌈ µ A ν ⌋ (1 . 1a) ⌉ I ≡ 3 D ⌊ I + 6 d rs s − 2 f pq q + g Ir C µνρr . I A ⌊ r ∂ ν A ρ ⌋ s d rs I A ⌊ r A ν p A ρ ⌋ H µνρ ⌈ µ B νρ ⌋ (1 . 1b) ⌉ ⌈ µ ⌉ ⌈ µ ⌉ The prescription for tensor-vector system, which we will be based upon, is described with eq. (3.22) in [6]. To be more specific, we consider in the present paper the product of two identical gauge groups G × G [7], whose adjoint indices are respectively r, s, ··· and r ′ , s ′ , ··· . Accordingly, we use the coefficients f rs ′ t ′ = − f s ′ r t ′ = + 1 2 f rs ′ t ′ , t = f rs t , f rs (1 . 2a) 2 f rs ′ t , h r ′ s = δ r ′ d t rs ′ = d t s ′ r = − 1 , (1 . 2b) s where f rst is the structure constant of a non-Abelian gauge group. We use the same field content arising by this prescription. Since the outstanding paper [6] gives the extensive details of how to get our system from [3][4][7], there is nothing new to explain, except for our notational preparation. In our notation, the field strengths of the B and C -fields are respectively G and H defined by I ≡ +3 D ⌊ I − 3 f IJK C ⌊ K , J F νρ ⌋ G µνρ ⌈ µ B νρ ⌋ (1 . 3a) ⌉ ⌈ µ ⌉ I ≡ +2 D ⌊ I + gB µν I . H µν ⌈ µ C ν ⌋ (1 . 3b) ⌉ The gauge transformations for B, C and A -fields are I , C µ I , A µ I ) = ( − f IJK α J B µν K , − f IJK α J C µ K , + D µ α I ) , δ α ( B µν (1 . 4a) I , C µ I , A µ I ) = ( +2 D ⌊ I , − gβ µ I , 0) , δ β ( B µν ⌈ µ β ν ⌋ (1 . 4b) ⌉ I , C µ I , A µ I ) = ( − f IJK F µν J γ K , D µ γ I , 0) . δ γ ( B µν (1 . 4c) C µI As (1.3b) or (1.4b) shows, is a vectorial Stueckelberg field, absorbed into the lon- B µνI . Due to the general hierarchy [3][4], all field strengths are gitudinal component of invariant: I , H µν I , F µν I ) = − f IJK α J ( G µνρ K , H µν K , F µν K ) , δ α ( G µνρ (1 . 5a) I , H µν I , F µν I ) = 0 , I , H µν I , F µν I ) = 0 . δ β ( G µνρ δ γ ( G µνρ (1 . 5b) 5
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