Bispinor Auxiliary Fields in Duality-Invariant Electrodynamics Revisited: The U(N) Case
aa r X i v : . [ h e p - t h ] A ug Bispinor Auxiliary Fields in Duality-InvariantElectrodynamics Revisited: The U ( N ) Case
E.A. Ivanov, B.M. Zupnik
Bogoliubov Laboratory of Theoretical Physics, JINR,141980 Dubna, Moscow Region, Russia [email protected] , [email protected] ABSTRACT
We update and detail the formulation of the duality-invariant systems of N interacting abelian gauge fields with N auxiliary bispinor fields added. Inthis setting, the self-duality amounts to U ( N ) invariance of the nonlinearinteraction of the auxiliary fields. The U ( N ) self-dual Lagrangians arise aftersolving the nonlinear equations of motion for the auxiliary fields. We alsoelaborate on a new extended version of the bispinor field formulation involvingsome additional scalar auxiliary fields and study U ( N ) invariant interactionswith derivatives of the auxiliary bispinor fields. Such interactions generatehigher-derivative U ( N ) self-dual theories.PACS: 11.15.-q, 03.50.-z, 03.50.DeKeywords: Electrodynamics, duality, auxiliary fields Introduction
The U ( N ) duality property is inherent to nonlinear interactions of N abelian gauge fieldstrengths F kαβ , ¯ F k ˙ α ˙ β , ( k = 1 , . . . , N ) [1]-[5]. The notorious examples of these duality in-variant systems are multi-field generalizations of the Born-Infeld (BI) theory. So far, theconstruction of such generalized BI systems was based on introducing some auxiliary ma-trix scalar fields χ with bilinear algebraic relations between these fields and the scalarcombinations of the gauge field strengths [4]. The corresponding nonlinear Lagrangiansas functions of F kαβ , ¯ F k ˙ α ˙ β arose after substituting the perturbative solution for χ .The recent revival of interest in the duality-invariant systems was mainly triggered bythe hypothesis that the duality considerations could play the decisive role in checking theconjectured ultraviolet finiteness of the maximally extended N = 8 , d = 4 supergravity(see, e.g., [6, 7, 8]). In theories of this kind there simultaneously appear a few gauge fields,so it is just U ( N ) duality that is of relevance to this circle of problems.The auxiliary bispinor field formulation of the U ( N ) duality was introduced in [9]as a natural generalization of the analogous approach to the U (1) duality-symmetric (orself-dual) systems [10, 11]. The U ( N ) self-duality is equivalent to the manifest U ( N )invariance of the interaction Lagrangian for the auxiliary fields.In this paper we further detail the bispinor field formulation and consider several newexamples of the U ( N ) duality-invariant models. It is a continuation of our recent paper[12], where the bispinor auxiliary field formulation of the U (1) duality was renewed andrelated to the latest developments in this area.We start, in Section 2, by recalling the standard setting for the U ( N ) self-dual theoriesin terms of N Maxwell gauge field strengths F kαβ and then turn to the ( F, V ) representationof these theories with N bispinor auxiliary fields V kαβ added. By construction, the U ( N )self-dual ( F, V ) Lagrangian satisfies the Gaillard-Zumino (GZ) representation with anarbitrary invariant interaction E ( V ) [9]. We study the general parametrization of thescalar U ( N ) invariants constructed out of the auxiliary fields. The equations of motionfor the auxiliary fields are in one-to-one correspondence with “the deformed twisted self-duality constraints” (for the U ( N ) duality group) proposed in [8]. We also introduceadditional scalar matrix auxiliary fields µ and consider an alternative formalism involvingtwo types of the auxiliary fields. This µ representation simplifies solving the auxiliary-fieldequations and constructing the self-dual Lagrangians.Section 3 is devoted to examples of the U ( N ) self-dual theories, including U ( N ) gen-eralizations of the BI theory. The first type of the U ( N ) BI models is the real form ofthe U ( N ) × U ( N ) generalization considered in Ref. [4]. We translate this model into our µ representation with the special U ( N ) invariant auxiliary interaction. We also proposean alternative U ( N ) generalization of the BI theory and construct the perturbative La-grangian for this model. Some other examples of the U ( N ) self-dual models are as wellconsidered. One example corresponds to the simplest quartic interaction of the auxil-iary fields, and another one is constructed by analogy with the new exact U (1) self-dualLagrangian given in [12].The U ( N ) self-dual models with higher derivatives are studied in Section 4. In theAppendix we rewrite our ( F k , V k ) Lagrangian and the self-duality equation in the tensor1ormalism.Throughout the paper we basically use the notations and abbreviations of Ref. [12]. U ( N ) duality Our starting point is the nonlinear Lagrangian with N abelian gauge field strengths F iαβ , ¯ F i ˙ α ˙ β , ( i = 1 , . . . , N ) L ( F k , ¯ F l ) = −
12 [( F k F k ) + ( ¯ F k ¯ F k )] + L int ( F k , ¯ F l ) . (2.1)It is manifestly invariant under the real O ( N ) transformation δ ξ F kαβ = ξ kl F lαβ , δ ξ ¯ F k ˙ α ˙ β = ξ kl ¯ F k ˙ α ˙ β , ξ kl = − ξ lk . (2.2)It is convenient to define N ( N + 1) complex scalar variables and to consider the La-grangian as a real function of these variables ϕ kl = ϕ lk = ( F k F l ) , ¯ ϕ kl = ( ¯ F k ¯ F l ) , (2.3) L ( F k , ¯ F k ) = L ( ϕ kl , ¯ ϕ kl ) . (2.4)The nonlinear equations of motion E kα ˙ α = ∂ ˙ βα ¯ P k ˙ α ˙ β ( F ) − ∂ β ˙ α P kαβ ( F ) = 0 (2.5)involve the dual nonlinear field strenghts P kαβ ( F ) = i ∂L∂F kαβ = 2 iF lαβ ∂L∂ϕ kl , and c . c . . (2.6)The gauge field strengths F kαβ , ¯ F k ˙ α ˙ β obey the standard Bianchi identities B kα ˙ α = ∂ ˙ βα ¯ F k ˙ α ˙ β − ∂ β ˙ α F kαβ = 0 . (2.7)The on-shell duality transformations are realized as δ η F kαβ = η kl P lαβ , δ η P kαβ = − η kl F lαβ , (2.8) In the tensor notation one deals with the self-dual field G + kmn , P kαβ ( F ) = 18 ( σ m ¯ σ n − σ n ¯ σ m ) αβ G + kmn . η kl = η lk are N ( N + 1) real parameters. These transformations extend the O ( N )group (2.2) to the group U ( N ) and so reside in the coset U ( N ) /O ( N ). The equations ofmotion (2.5) together with the Bianchi identities (2.7) are covariant under (2.8), δ η E kα ˙ β = − η kl B kα ˙ β , δ η B kα ˙ β = η kl E kα ˙ β , (2.9)provided that the generalized consistency conditions hold:( P k P l ) + ( F k F l ) − c . c . = 0 , ( F k P l ) − ( F l P k ) − c . c . = 0 . (2.10)These conditions are U ( N ) covariant on their own right. The symmetric conditions canbe rewritten as the matrix differential equations for the Lagrangian ϕ − L ϕ ϕL ϕ = ¯ ϕ − L ¯ ϕ ¯ ϕL ¯ ϕ . (2.11) ( F, V ) representation The generalized auxiliary field representation of the O ( N ) invariant Lagrangian [9] con-tains N complex auxiliary O ( N ) vector fields V kαβ , ¯ V k ˙ α ˙ β L ( F k , V k ) = L ( F k , V k ) + E [( V k V l ) , ( ¯ V k ¯ V l )] , (2.12) L ( F k , V k ) = 12 [( F k F k ) + ( ¯ F k ¯ F k )] − F k V k ) + ( ¯ F k ¯ V k )]+ ( V k V k ) + ( ¯ V k ¯ V k ) . (2.13)Here, E is an O ( N ) invariant real interaction depending on N ( N + 1) scalar complexvariables ν kl = ( V k V l ) , ¯ ν kl = ( ¯ V k ¯ V l ) . (2.14)The interaction E ( ν kl , ¯ ν kl ) is assumed to be regular at the origin, so that it admits expan-sion in power series. The dynamical equation of motion following from this Lagrangianhas the form ∂ ˙ βα ¯ P k ˙ α ˙ β ( F, V ) − ∂ β ˙ α P kαβ ( F, V ) = 0 , (2.15)where P kαβ ( F, V ) = i ( F k − V k ) αβ . (2.16)The U ( N ) /O ( N ) duality transformations are implemented as δ η F kαβ = η kl P lαβ = iη kl ( F l − V l ) αβ , δ η P kαβ = − η kl F lαβ . (2.17)The corresponding η kl transformations of the auxiliary fields follow from (2.17) and thedefinition (2.16). The full U ( N ) transformations of the auxiliary fields can be written as δV kαβ = ( ξ kl − iη kl ) V lαβ , δ ¯ V k ˙ α ˙ β = ( ξ kl + iη kl ) ¯ V l ˙ α ˙ β . (2.18)3he SU ( N ) subgroup is singled out by the condition η kk = Tr η = 0 .The U ( N ) transformations of the scalar variables ν kl and ¯ ν kl can be presented in thematrix form as δν = [ ξ, ν ] − i { η, ν } , δ ¯ ν = [ ξ, ¯ ν ] + i { η, ¯ ν } . (2.19)Then, we define the Hermitian matrix variables a kl = (¯ νν ) kl , ¯ a kl = ( ν ¯ ν ) kl = (¯ νν ) lk , (2.20) δa = i [ ξ, a ] + i [ η, a ] , δ ¯ a = i [ ξ, ¯ a ] − i [ η, ¯ a ] . (2.21)We also define the matrix monomials a n with the following properties( a n ¯ ν ) kl = (¯ ν ¯ a n ) kl = ( a n ¯ ν ) lk , ( νa n ) kl = (¯ a n ν ) kl = ( νa n ) lk . (2.22)From these monomials one can construct N independent real U ( N ) invariants A n , ( n =1 , , . . . , N ) : A n = 1 n Tr a n , dA n = Tr ( daa n − ) , ∂A n ∂a kl = ( a n − ) kl . (2.23)An alternative choice of the U ( N ) invariants is connected with the spectrum λ ( A , . . . , A N ) , . . . , λ N ( A , . . . , A N ) of the Hermitian matrix a [13, 14]. This spectrum can be found bysolving the characteristic equation A ( a ) = ( a − λ )( a − λ ) · · · ( a − λ N ) = 0 . (2.24)Like in the U (1) case [10, 11, 12], the dynamical equations of motion (2.15), togetherwith the Bianchi identities (2.7) and the algebraic equations of motion for the auxiliaryfields V kαβ , ¯ V k ˙ α ˙ β following from (2.12), are covariant under the U ( N ) duality transforma-tions, provided that the interaction function E ( ν kl , ¯ ν kl ) in (2.12) is U ( N ) invariant [9], E ( ν kl , ¯ ν kl ) ⇒ E ( A , . . . , A N ) , (2.25)with E ( A , . . . , A N ) being an analytic function.The more convenient representation for the interaction Lagrangian E ( A n ) is throughthe matrix function E ( a ) E = Tr E ( a ) . (2.26)The derivative matrix function E a is defined as follows d E = dA n E n = Tr ( daE a ) = ( da lk E kla ) , E n = ∂ E ∂A n , (2.27)whence ( E a ) kl = E δ kl + E a kl + E a kj a jl + . . . . (2.28)4sing this representation and the relations (2.22), we define the holomorphic derivatives E kl := ∂ E ∂ν kl = ( E a ) kr ¯ ν rl = E ¯ ν kl + E a kr ¯ ν rl + E a kj a jr ¯ ν rl + . . . , (2.29)¯ E kl := ∂ E ∂ ¯ ν kl = ν kr ( E a ) rl = E ν kl + E ν kr a rl + E ν kr a rj a jl + . . . . (2.30)The basic algebraic equations of the U ( N ) duality-invariant models are obtained byvarying, with respect to V kαβ and ¯ V k ˙ α ˙ β , the Lagrangian (2.12), in which the general function E is substituted by the U ( N ) invariant one E defined in (2.25):( F k − V k ) αβ = E kl V lαβ = ( E a ) kr ¯ ν rl V lαβ , ( ¯ F k − ¯ V k ) ˙ α ˙ β = ¯ E kl ¯ V l ˙ α ˙ β = ν kr ( E a ) rl ¯ V l ˙ α ˙ β . (2.31)Equations of motion (2.31) are equivalent to the nonlinear twisted self-duality constraintswhich were postulated in [8, 15]. The important corollaries of (2.31) are the scalar matrixalgebraic equation ϕ kl = [ δ kr + E kr ] ν rs [ δ sl + E sl ] (2.32)and its conjugate.By analogy with the U (1) case [9, 12], the general solution of the algebraic equations(2.31) can be written in the following concise form: V kαβ = F lαβ G kl ( ϕ, ¯ ϕ ) , (2.33) G kl = [ δ kl + E kl ] − = 12 δ kl − ∂L∂ϕ kl , (2.34) P kαβ = 2 iF lαβ ∂L∂ϕ kl = iF lαβ [ δ kl − G kl ] . (2.35)Using (2.33), (2.34), we can uniquely restore the Lagrangian L ( ϕ kl , ¯ ϕ kl ) by its holomorphicderivatives: dL = dϕ lk ∂L∂ϕ kl + c.c. . (2.36)Note that Eqs. (2.31), (2.32) are simplified under the particular choices of E , e.g., for E = E ( A ): F kαβ = V rαβ [ δ kr + E ¯ ν kr ] , (2.37) ϕ kl = ν kl + E [¯ ν kr ν rl + ν ks ¯ ν sl ] + E ¯ ν kr ν rs ¯ ν sl . (2.38)The famous GZ representation of the U ( N ) self-dual Lagrangians has the followingform in the ( F, V ) representation L ( F k , V k ) = i P k ( F, V ) ¯ F k − P k ( F, V ) F k ] + [( V k V k ) − ( F k V k )]+ [( ¯ V k ¯ V k ) − ( ¯ F k ¯ V k )] + E , (2.39)5here ( V k V k ) − ( F k V k ) is the complex bilinear U ( N ) invariant. Using Eqs. (2.31), wecan also prove that the U ( N ) self-duality conditions (2.10) in the ( F, V ) representationare none other than the conditions of U ( N ) invariance of the auxiliary interaction E [9]( P k P l ) + ( F k F l ) − c . c . = (cid:18) V l ∂ E ∂V k (cid:19) + (cid:18) V k ∂ E ∂V l (cid:19) − c . c . = 0 , (2.40)( F k P l ) − ( F l P k ) − c . c . = i (cid:18) V k ∂ E ∂V l (cid:19) − i (cid:18) V l ∂ E ∂V k (cid:19) + c . c . = 0 . (2.41) µ representation In the U (1) case, there is a more convenient parametrization of the duality-invariant auxil-iary interaction, the “ µ representation” [11, 12]. Making use of it essentially simplifies theroad from the auxiliary-field equations to the final nonlinear duality-invariant Lagrangian.In the U ( N ) case, the µ representation is set up in terms of the matrix variables µ kl = ∂ E ( a ) ∂ν kl = ( E a ) kr ¯ ν rl , ¯ µ kl = ν kr ( E a ) rl , (2.42) b kl = µ ks ¯ µ sl = ( E a ¯ ννE a ) kl = ( aE a ) kl , b lk = ¯ b kl = ¯ µ kr µ rl , (2.43)where the relations (2.22) were used. These newly defined matrix variables possess thefollowing transformation laws: δµ = [ ξ, µ ] + i { η, µ } , δ ¯ µ = [ ξ, ¯ µ ] − i { η, ¯ µ } , δb = [ ξ, b ] + i [ η, b ] (2.44)and reveal the properties( b n µ ) kl = ( µ ¯ b n ) kl = ( b n µ ) lk , (¯ µb n ) kl = (¯ b n ¯ µ ) kl = (¯ µb n ) lk . (2.45)From them one can construct N independent U ( N ) invariants B n = 1 n Tr b n , (2.46)which are going to be the arguments of the µ representation analog of the invariantfunction E ( A n ) .The connection with the basic objects of the original ν representation is establishedthrough the Legendre transformation I ( B n ) = Tr I ( b ) := E − ν kl µ kl − ¯ ν kl ¯ µ kl = Tr [ E − aE a ] , (2.47) ν kl = − ∂ I ∂µ kl = − (¯ µI b ) kl = − ( ¯ I ¯ b ¯ µ ) kl , (2.48) d I = Tr ( dbI b ) = Tr ( d ¯ b ¯ I ¯ b ) , (2.49)where I ( B n ) is a real analytic invariant function which is the interaction in the µ represen-tation. We introduced the covariant matrix functions I ( b ) and I b which are representable6s formal series over the powers of the matrix b , with the coefficients being functions ofthe invariants B n , e.g., I ( b ) = X k k ! I ( k ) ( B n ) b k , δI ( b ) = [( ξ + iη ) , I ( b )] . (2.50)They are related to the matrix functions E ( a ) and E a by the covariant matrix equations I ( b ) = E ( a ) − aE a , E ( a ) = I ( b ) − bI b , E a = − I − b , a = bI b , b = aE a . (2.51)Eqs. (2.32) can be rewritten in the µ representation as ϕ kl = [ δ kr + E kr ] ν rs [ δ sl + E sl ] = − ( δ kr + µ kr ) ∂ I ∂µ rs ( δ sl + µ sl )= − [( + µ )¯ µI b ( + µ )] kl , (2.52)or ϕ = − [( + µ )¯ µI b ( + µ )] , ¯ ϕ = − ( + ¯ µ ) I b µ ( + ¯ µ ) , (2.53)where the relations (2.45) are used and denotes the unit matrix.In the particular representation, I b can be chosen as the matrix power series expansionwith the numerical coefficients i k ( I b ) kl = − δ kl + i b kl + 12 i ( b ) kl + . . . . (2.54)Then we can write the following recursive complex matrix equation for µ : µ = 12 ¯ ϕ − µ ¯ µ − ¯ µµ − ¯ µµ ¯ µ + 12 i µ ¯ µµ + 12 i ¯ µµ ¯ µµ + 12 i µ ¯ µµ ¯ µ + 12 i ¯ µµ ¯ µµ ¯ µ + 14 i µ ¯ µµ ¯ µµ + O ( µ ) . (2.55)Solving it, e.g., for µ as µ = µ kl ( ϕ, ¯ ϕ ) , we can reconstruct the holomorphic derivatives ofthe Lagrangian from Eqs. (2.34) and (2.42), (cid:18) + µ (cid:19) kl = 12 δ kl − ∂L∂ϕ kl , (2.56)and finally restore the nonlinear perturbative Lagrangian in the F -representation L = 12 Tr (cid:20) − ( ϕ + ¯ ϕ ) + ϕ ¯ ϕ − ϕ ¯ ϕ − ϕ ¯ ϕ (cid:21) + 18 Tr (cid:20) ϕ ¯ ϕ + ϕ ¯ ϕ + (2 + 14 i ) ϕ ¯ ϕϕ ¯ ϕ + 2 ϕ ¯ ϕ (cid:21) + O ( ϕ ) , (2.57)where the single-trace matrix terms of higher orders are omitted.7ike in the U (1) case [12], one can define a combined ( F, V, µ ) off-shell representationfor the U ( N ) self-dual Lagrangians, treating µ ik , ¯ µ ik as independent auxiliary fields L ( V k , F k , µ kl ) = 12 [( F k F k ) + ( ¯ F k ¯ F k )] − V k · F k ) + ( ¯ V k · ¯ F k )]+ ( V k V l )( δ kl + µ kl ) + ¯ V k ¯ V l ( δ kl + ¯ µ kl ) + I ( B n ) , (2.58)where B n are the invariants (2.46). Eliminating the V k variables from this Lagrangian, V kαβ = (cid:2) ( + µ ) − (cid:3) kl F lαβ , and c . c . , (2.59)we arrive at the ( F, µ ) representation of the Lagrangian:˜ L ( F k , µ kl ) = 12 ( F k F l )[( µ − )( + µ ) − ] kl + c.c. + I ( B n ) . (2.60)Varying this Lagrangian with respect to µ kl we obtain the matrix auxiliary equation whichis equivalent to Eq. (2.53).In the specific examples we can exploit the similarity between the U (1) interactionfunction I ( b ) [12] and the matrix function I ( b ) of the U ( N ) case, although solving thematrix equations is the much more difficult task. For the simple particular U ( N ) inter-action presented by a one-argument function E ( A ) , A = a kk = ν kl ¯ ν lk , we find, e.g., µ kl = E ¯ ν kl , ¯ µ kl = E ν kl , ν kl = −I ¯ µ kl , (2.61) b kl = E a kl , B = b kk = E A , E = −I − . (2.62)The corresponding interaction function in the µ -representation involves only the trace B , I ( B ) = E ( A ) − A E , ν kl = −I ¯ µ kl , a kl = I b kl , E = −I − . (2.63)The equation (2.53) has the following form in this case: ϕ = −I ( + µ )¯ µ ( + µ ) . (2.64)We consider the representation I ( B ) = − B + 12 j B + 16 j B + . . . , I = − j B + 12 j B + . . . (2.65)( j , j , . . . are some constants) and the corresponding recursion relations for µ ( ϕ, ¯ ϕ ). The4-th order term in the corresponding self-dual Lagrangian L (4) ( ϕ, ¯ ϕ ) = 18 Tr (cid:2) ϕ ¯ ϕ + ϕ ¯ ϕ + 2 ϕ ¯ ϕϕ ¯ ϕ + 2 ϕ ¯ ϕ (cid:3) + 132 j [Tr( ϕ ¯ ϕ )] . (2.66)contains the double-trace term. The subsequent recursions give terms with several traces.8 Examples of the U ( N ) self-dual models Here we present some examples of U ( N ) self-dual models with actions involving no higherderivatives. Basically, these are generalizations of the U (1) examples considered in [12].Similarly to the U (1) case, the corresponding interactions written in terms of the auxiliaryvariables can be chosen in a closed form, while the equivalent on-shell expressions, withthe auxiliary variables being eliminated in terms of the Maxwell field strengths, can begiven only as infinite series in powers of the field strengths. An important difference fromthe U (1) case is that there exist several inequivalent U ( N ) duality-invariant models whichare reduced to the same U (1) model in the one field-strength limit. For instance, thereare few U ( N ) duality-invariant extensions of the standard BI theory. As distinct fromthe latter, the Lagrangians of such generalized BI theories seem not to admit a closedrepresentation in terms of the Maxwell field strengths (even for the simplest non-trivial U (2) case). U ( N ) generalizations of the Born-Infeld model The U ( N ) × U ( N ) generalization of the BI theory proposed in [4] deals with the Hermitianscalar matrix fields ˆ α lk = 14 ( F k ¯ F l ) , ˆ β kl = 14 (˜ F k ¯ F l ) (3.1)constructed out of N complex field-strengths F kmn and their conjugates ¯ F kmn (with ˜ F kmn = ε mnst F k st ). The basic complex scalar auxiliary field χ kl = χ lk of this model satisfies thematrix equation χ kl + 12 χ kr ¯ χ lr = ˆ α lk + i ˆ β kl = ˆ ϕ kl , (3.2)with the solution representable as the matrix power series.The U ( N ) generalization of the BI theory we are interested in corresponds to imposingthe reality condition F k = ¯ F k and using the symmetric matricesˆ α kl → t kl = 14 η mr η ns F kmn F lrs , ˆ β kl → z kl = 18 ε mnrs F kmn F lrs , ˆ ϕ kl → ϕ kl = ( F k F l ) . (3.3)In this notation, we consider the following nonlinear Lagrangian of N abelian gauge fieldstrengths F kαβ L ABMZ ( ϕ, ¯ ϕ ) = Tr (cid:20) −
12 ( ϕ + ¯ ϕ ) + 12 ϕ ¯ ϕ −
14 ( ϕ ¯ ϕ + ϕ ¯ ϕ ) (cid:21) + 18 Tr (cid:2) ϕ ¯ ϕ + 2 ϕ ¯ ϕ + ϕ ¯ ϕϕ ¯ ϕ + ϕ ¯ ϕ (cid:3) + O ( ϕ ) . (3.4)The U ( N ) duality condition (2.11) can be directly proved for this Lagrangian.9ur interpretation of this model makes use of the following exact invariant interactionin the matrix µ representation (2.60): I = Tr I ( b ) , I ( b ) = 2 b ( b − ) , I b = − b − ) . (3.5)We can calculate L ( ϕ, ¯ ϕ ) as the power series, based on the expansion I ( b ) = − b − b − b − b − . . . . (3.6)For the proper choice of the numerical coefficients in (2.57), i = − , . . . , we can reproducethe Lagrangian (3.4).By analogy with the U (1) case, we can come back to the original ( F, V ) formulation,defining the matrix variable a by the relation a = 4 b ( − b ) . (3.7)The auxiliary interaction has the single-trace form E ( a ) = Tr E ( a ). The matrix relationsbetween various quantities in the ( F, V ) and µ representations are similar to those validin the U (1) duality case for the BI theory [11, 12] E ( a ) = 2 b ( a )[ + b ( a )][ − b ( a )] = 2[2 t ( a ) + 3 t ( a ) + ] , (3.8) t + t − a = 0 , t = 1 b − , (3.9)2 E a = [ − aE a ] , E a = 12 [ b ( a ) − ] . (3.10)Solving these equations, one can find closed expressions for both t ( a ) and the single-traceinteraction E ( a ). They look rather bulky and so it is not too illuminating to present themhere. Up to the 3-d order in a : E = Tr (cid:20) a − a + 332 a + O ( a ) (cid:21) . (3.11)An alternative U ( N ) generalization of the BI Lagrangian proceeds from the µ repre-sentation with the invariant auxiliary interaction of the simple form I ( B ) = 2 B B − , B = Tr µ ¯ µ . (3.12)This interaction corresponds to the choice j = − , . . . in (2.65). We can obtain the self-dual nonlinear Lagrangian in the F representation, using the recursion equation (2.64).The ( F, V ) representation of the same model deals with the invariant interaction whichis a function of the single U ( N ) invariant variable A = ( V k V l )( ¯ V l ¯ V k ): E ( A ) = 12 A − A + 332 A + O ( A ) . (3.13)10 .2 Other examples of U ( N ) self-dual theories The simplest quartic U ( N ) invariant interaction E SI = 12 ( V k V l )( ¯ V k ¯ V l ) = 12 Tr a = 12 A (3.14)produces the self-dual model, which is U ( N ) generalization of the “simplest interaction U (1) self-dual model” of refs. [11, 12] (it was rediscovered in [8, 15]). In this case, thebasic polynomial auxiliary equation F kαβ = [ δ kl + 12 ( ¯ V k ¯ V l )] V lαβ (3.15)has the perturbative solution for the function G kl = δ kl − ∂L SI /∂ϕ kl (2.34). The corre-sponding power-series Lagrangian reads L SI = Tr[ −
12 ( ϕ + ¯ ϕ ) + 12 ϕ ¯ ϕ −
14 ( ϕ ¯ ϕ + ϕ ¯ ϕ )+ 18 ( ϕ ¯ ϕ + 2 ϕ ¯ ϕ + 2 ϕ ¯ ϕϕ ¯ ϕ + ϕ ¯ ϕ )] + . . . . (3.16)We can also consider the µ representation for this model I ( B ) = − B . (3.17)Some other U (1) examples considered in [11, 12] also admit extensions to the U ( N )duality case. For instance, we can choose the interaction I ( B ) = 2 ln(1 − B ) = − B + 12 B + 13 B + . . . ) , I = 2 B − F -representation, using the perturbativesolution of Eq. (2.64) specialized to this case. The generalized U ( N ) self-dual Lagrangians with higher derivatives [16] in the formulationthrough bispinor auxiliary fields are constructed in the close analogy with the U (1) case[12]. They involve the same bilinear term (2.13) and the U ( N ) invariant interaction E Kder containing derivatives of the auxiliary fields L der ( F k , V k ) = L ( F k , V k ) + E K ( V, ∂V, ∂ V, . . . ∂ K V ) , (4.1)where K denotes the maximal total degree of derivatives. Terms with derivatives in E K contain the coupling constant c of dimension − U ( N )-covariant local equations of motion for the auxiliary fields in this casecontain the Lagrangian derivative of the invariant auxiliary interaction( V k − F k ) αβ + 12 ∆ E K ∆ V kαβ = 0 , (4.2)where we consider the Lagrange derivative∆ E K ∆ V k = ∂ E K ∂V k − ∂ m ∂ E K ∂∂ m V k + ∂ m ∂ n ∂ E K ∂∂ m ∂ n V k + . . . . The equivalent twisted self-duality relations for higher-derivative theories were postulatedin [8, 16].The consistency condition( P k P l ) + ( F k F l ) = 2( F k V l ) + 2( F l V k ) − V k V l ) = (cid:18) V l ∆ E K ∆ V k (cid:19) + (cid:18) V k ∆ E K ∆ V l (cid:19) = (cid:20)(cid:18) V k ∂ E K ∂V l (cid:19) + (cid:18) ∂ m V k ∂ E K ∂ ( ∂ m V l ) (cid:19) + (cid:18) ∂ m ∂ n V k ∂ E K ∂ ( ∂ m ∂ n V l ) (cid:19) + . . . (cid:21) + [ k ↔ l ]+total derivatives (4.3)is evidently valid for the derivative interaction E K . Then the η kl invariance of the inter-action E K is equivalent to the following integral form of the self-duality condition η kl Z d x [( P k P l ) + ( F k F l ) − ( ¯ P k ¯ P l ) − ( ¯ F k ¯ F l )] = Z d x [ δ η E K + derivatives] = 0 . (4.4)The O ( N ) ξ -invariance of L der ( F k , V k ) is manifest, as in the case without derivatives.Solving Eq. (4.2), we obtain the perturbative solution V kαβ ( F k ), which involves boththe field strengths and their derivatives.The simplest bilinear invariant with two derivatives E ( V, ∂V ) = ca ∂ ˙ ββ V kαβ ∂ ˙ ξα ¯ V k ˙ β ˙ ξ (4.5)makes the fields V and ¯ V propagating.An example of the nonlinear interaction with two derivatives (still with the standardbilinear terms) corresponds to the choice E = b c∂ m ( V k V l ) ∂ m ( ¯ V k ¯ V l ) . (4.6)The basic auxiliary equation in this case reads F kαβ = V lαβ [ δ kl − cb ✷ ( ¯ V k ¯ V l )] , (4.7)and it can be recursively solved for V lαβ in terms of F kαβ and its derivatives. The corre-sponding Lagrangian in the F -representation is given by the formal series L = −
12 ( ϕ kk + ¯ ϕ kk ) + cb ∂ m ϕ kl ∂ m ¯ ϕ kl − c b ( ϕ kl ✷ ¯ ϕ lr ✷ ¯ ϕ rk + ¯ ϕ kl ✷ ϕ lr ✷ ϕ rk ) + O ( c ) . (4.8)12he number of derivatives increases with each recursion.An example of the U ( N ) invariant auxiliary interaction involving four derivatives is E = gc ( ∂ m V k ∂ n V l )( ∂ m ¯ V k ∂ n ¯ V l ) , (4.9)where g is a dimensionless coupling constant. The corresponding auxiliary field equationreads F kαβ = V kαβ − gc ∂ m (cid:2) ∂ n V lαβ ( ∂ m ¯ V k ∂ n ¯ V l ) (cid:3) . (4.10)It is not difficult to recursively solve this equation too and to construct the correspondingself-dual Lagrangian with higher derivatives.All these examples are U ( N ) generalizations of the U (1) self-dual Lagrangians withhigher derivatives presented in [12]. Like in the case without higher derivatives, the set ofinequivalent U ( N ) duality-invariant models of this kind is much richer compared to their U (1) prototypes due to the proliferation of the Maxwell field strengths and the associatedauxiliary tensorial fields. In this paper, we further elaborated on the formalism with the bispinor (tensor) auxiliaryfields for the U ( N ) self-dual abelian gauge theories initiated in [9]. The general Lagrangianof the U ( N ) self-dual model is parametrized by the invariant interaction of the auxiliaryfields. The U ( N ) covariant local twisted self-duality condition arises in this formulation asthe equation of motion for the bispinor auxiliary fields. As compared to [9], we presentedan alternative formulation of the U ( N ) self-dual theories which makes use of the matrixscalar auxiliary fields in parallel with the bispinor ones, discussed a few new examples andshowed how to generate U ( N ) self-dual theories with higher derivatives in the consideredsetting.In a recent paper [17] we gave basic elements of N = 1 supersymmetric generalizationof the U ( N ) self-dual bosonic actions, using the auxiliary chiral superfields. The N = 1supersymmetrization of the bispinor auxiliary field formalism for the U (1) case, throughenhancing this field to an auxiliary chiral N = 1 superfield, was earlier accomplished byKuzenko [18] within a more general framework of the superfield N = 1 supergravity .The U ( N ) self-duality formulations detailed here seem to admit rather straightforwardsupersymmetric extensions (with both rigid and local supersymmetries) along the lines ofthese works.We also note that the bispinor auxiliary field formulation can be set up as well forself-dual abelian gauge theories in the d = 4 Euclidean space [19] and the space withthe signature (2 , V iαβ and ¯ V i ˙ α ˙ β are substituted by two sets of realindependent fields transforming as SO L (3) and SO R (3) vectors in the Euclidean case, oras SO L (1 ,
2) and SO R (1 ,
2) vectors for the signature (2 , In [18], N = 2 supersymmetrization was also considered.
13s non-compact, and it is the general linear group GL ( N ) (it is reduced to dilatations L (1) ∼ SO (1 ,
1) in the N = 1 case).Finally, it is worthwhile to mention that the notion of self-duality can be defined fortheories with p -form gauge fields, not only for p = 1, and in diverse dimensions, notonly for d = 4 (see, e.g., [3] and references therein). It would be tempting to introducethe appropriate tensorial auxiliary fields for this web of generalized self-dualities andto see how they could help in constructing the relevant actions and understanding theinterrelations between various types of such dualities . Acknowledgements
We acknowledge a partial support from the RFBR grants Nr.12-02-00517, Nr.11-02-90445,the grant DFG LE 838/12-1 and a grant of the Heisenberg-Landau program. E.I. thanksthe Organizers of the Workshop “Higher Spins, Strings and Duality” (Galileo Galilei Insti-tute for Theoretical Physics, Florence, March 18 - May 10, 2013) for the kind hospitalityat the final stage of this work, and the participants for useful discussions.
A. Spinor and tensor notations in self-dual theories
Our bispinor U (1) formalism translated to the tensor notation was considered in [12].Here we present the basic formulas of the tensor reformulation of this approach for thecase of U ( N ) self-dual theories.The vectors in the spinor and tensor notations are related as A kα ˙ β = ( σ m ) α ˙ β A km , (A.1)where k = 1 , , . . . N . The same correspondence for N abelian field strengths is given bythe relations F k βα ( A ) = 14 ( σ m ¯ σ n ) βα F kmn = 18 ( σ m ¯ σ n − σ n ¯ σ m ) βα F + kmn , (A.2)¯ F k ˙ β ˙ α = 14 (¯ σ n ) ˙ ββ ( σ m ) β ˙ α F kmn = −
18 (¯ σ m σ n − ¯ σ n σ m ) ˙ β ˙ α F − kmn , (A.3)where F kmn = ∂ m A kn − ∂ n A km , ˜ F kmn = 12 ε mnrs F krs , F + kmn = 12 F kmn + i F kmn , (A.4) g F + kmn = − i F + kmn , F − kmn = 12 F kmn − i F kmn , g F − kmn = i F − kmn . (A.5)Thus, F kαβ is the equivalent bispinor notation for the self-dual tensor field F + kmn , and ¯ F k ˙ α ˙ β amounts to the anti-self-dual tensor F − kmn . It was shown, e.g., in [20] that the d = 4 BI theory can be obtained by dimensional reduction froma self-dual theory in d = 6. ϕ kl = F kαβ F lαβ = t kl + iz kl = 12 ( F + k F + l ) , ¯ ϕ kl = ¯ F k ˙ α ˙ β ¯ F l ˙ α ˙ β = t kl − iz kl = 12 ( F − k F − l ) . (A.6)The bispinor and tensor representations of the dual field strengths appearing in thenonlinear equations of motion are related as P kβα ( F ) = 18 ( σ m ¯ σ n − σ n ¯ σ m ) βα G + kmn , ¯ P k ˙ β ˙ α = −
18 (¯ σ m σ n − ¯ σ n σ m ) ˙ β ˙ α G − kmn , (A.7) G ± kmn = 12 G kmn ± i G kmn , ˜ G kmn = 2 ∆ L ∆ F kmn , (A.8)where we employed the Lagrange derivative.The similar relations are valid for the auxiliary fields V kβα = 18 ( σ m ¯ σ n − σ n ¯ σ m ) βα V + kmn , ¯ V k ˙ β ˙ α = −
18 (¯ σ m σ n − ¯ σ n σ m ) ˙ β ˙ α V − kmn . (A.9)The scalar variable ν can be expressed as ν kl = V kαβ V lαβ = 12 ( V + k V + l ) , ¯ ν kl = ¯ V k ˙ α ˙ β ¯ V l ˙ α ˙ β = 12 ( V − k V − l ) . (A.10)The one-to-one correspondence between the specific variables used in [8, 15, 16] andour variables in the tensor notation is as follows : T k = F k − i G k , ˜ T k = ˜ F k − i ˜ G k ,T ∗ k = F k + i G k , g T ∗ k = ˜ F k + i ˜ G k , (A.11) δ ω T k = iω kl T l , δ ω ˜ T k = iω kl ˜ T l , δ ω T ∗ k = − iω kl T ∗ l , δ ω g T ∗ k = − iω kl f T ∗ l , (A.12)where ω kl = ξ kl + iη kl are the U ( N ) parameters. The relations between the self-dual (andanti-self-dual) parts of these two sets of complex variables can be listed as T + k = 12 ( T + i ˜ T ) k = ( F − V ) k + i (˜ F − ˜ V ) k = 2( F + k − V + k ) , (A.13) T − k = 12 ( T − i ˜ T ) k = 2 V − k , (A.14) T ∗ + k ≡ ¯ T + k = ( T − ) ∗ k = 12 ( T ∗ + i f T ∗ ) k = 2 V + k , (A.15) T ∗− k ≡ ¯ T − k = ( T + ) ∗ k = 12 ( T ∗ − i f T ∗ ) k = 2( F − k − V − k ) . (A.16)Being cast in the tensor notations, our Lagrangian (2.12) becomes: L ( F , V ) = −
14 [( F + k F + k ) + ( F − k F − k )] + 12 [( V + k − F + k ) + ( V − k − F − k ) ]+ E ( A n ) , (A.17) Sometimes, for brevity, we omit the antisymmetric tensor indices. E is defined in terms of the matrix a kl = 14 ( V − k V − r )( V + r V + l ) , A n = 1 n Tr a n . (A.18)The tensor form of our algebraic equation of motion (2.31) reads( F + − V + ) kmn = ∂ E ∂ V + kmn . (A.19)After passing to the T -tensor notation according to Eqs. (A.14), (A.15), we can rewritethe same equation as T + kmn = 4 ∂ E ∂ ¯ T + kmn , (A.20)that precisely coincides with the general twisted self-duality condition postulated in [8, 15].Our interaction function E proves to be identical to the “deformation function” I (1) ofthis approach. Note that the variables T k , T ∗ k are entirely equivalent to our variableson mass shell, when the auxiliary fields are traded for the Maxwell field strengths by theirequations of motion, while off shell it is more convenient to deal with the variables F k , V k . References [1] M.K. Gaillard and B. Zumino,
Duality rotations for interacting fields , Nucl. Phys. B (1981) 221.[2] M.K. Gaillard and B. Zumino,
Self-duality in nonlinear electromagnetism , In: Su-persymmetry and quantum field theory, eds. J. Wess and V.P. Akulov, p. 121-129,Springer-Vellag, 1998, hep-th/9705226 ;M.K. Gaillard and B. Zumino,
Nonlinear electromagnetic self-duality and Legendretransform , In: Duality and Supersymmetric Theories, eds. D.I. Olive and P.C. West,p. 33, Cambridge University Press, 1999, hep-th/9712103 .[3] S.M. Kuzenko and S. Theisen,
Supersymmetric duality rotation , JHEP (2000)034, hep-th/0001068 ;S.M. Kuzenko and S. Theisen,
Nonlinear self-duality and supersymmetry , Fortsch.Phys. (2001) 273, hep-th/0007231 .[4] P. Aschieri, D. Brace, B. Morariu, B. Zumino, Nonlinear self-duality in even dimen-tions , Nucl. Phys. B (2000) 571, hep-th/9909021 ;P. Aschieri, D. Brace, B. Morariu, B. Zumino,
Proof of a symmetrized trace con-jecture for the abelian Born-Infeld Lagrangian , Nucl. Phys. B (2000) 521, hep-th/0003228 .[5] P. Aschieri, S. Ferrara and B. Zumino,
Duality rotations in nonlinear electrodynam-ics and extended supergravity , Riv. Nuovo Cim. (2008) 625, arXiv:0807.4039[hep-th] . 166] G. Bossard, C. Hillmann, H. Nicolai, E symmetry in perturbatively quantised N =8 supergravity , JHEP (2010) 052, arXiv:1007.5472 [hep-th] .[7] R. Kallosh, E symmetry and finiteness of N = 8 supergravity , JHEP (2012)083, arXiv:1103.4115 [hep-th] ; R. Kallosh, N = 8 counterterms and E currentconservation , JHEP (2011) 073, arXiv:1104.5480 [hep-th] .[8] G. Bossard, H. Nicolai, Counterterms vs. dualities , JHEP (2011) 074, arXiv:1105.1273 [hep-th] .[9] E.A. Ivanov, B.M. Zupnik,
New representation for Lagrangians of self-dual nonlinearelectrodynamics , In: Supersymmetries and quantum symmetries, eds. E. Ivanov etal , p. 235, Dubna, 2002, hep-th/0202203 .[10] E.A. Ivanov, B.M. Zupnik, N = 3 supersymmetric Born-Infeld theory , Nucl. Phys.B (2001) 3, hep-th/0110074 .[11] E.A. Ivanov, B.M. Zupnik, New approach to nonlinear electrodynamics: dualities assymmetries of interaction , Yader. Fiz. (2004) 2212 [Phys. Atom. Nucl. (2004)2188], hep-th/0303192 .[12] E.A. Ivanov, B.M. Zupnik, Bispinor auxiliary fields in duality-invariant electrody-namics revisited , Phys. Rev. D (2013) 065023, arXiv:1212.6637 [hep-th] .[13] F.R. Gantmakher, Theory of matrices , Moscow, Nauka, 1967 (in Russian).[14] B.M.Zupnik, V.I.Ogievetsky,
Investigation of nonlinear realizations of chiral groupsby generating-functions method , Theor. Math. Phys. (1969) 14.[15] J.J.M. Carrasco, R. Kallosh, R. Roiban, Covariant procedure for perturbative non-linear deformation of duality-invariant theories , Phys. Rev. D (2012) 025007, arXiv:1108.4390 [hep-th] .[16] W. Chemissany, R. Kallosh, T. Ortin, Born-Infeld with higher derivatives , Phys. Rev.D (2012) 046002, arXiv:1112.0332 [hep-th] .[17] E. Ivanov, O. Lechtenfeld, B. Zupnik, Auxiliary superfields in N = 1 supersymmetricself-dual electrodynamics , JHEP (2013) 133, arXiv:1303.5962 [hep-th] .[18] S.M. Kuzenko, Duality rotations in supersymmetric nonlinear electrodynamics revis-ited , JHEP (2013) 153, arXiv:1301.5194 [hep-th] .[19] G.W. Gibbons, K. Hashimoto,
Non-Linear Electrodynamics in Curved Backgrounds ,JHEP (2000) 013, hep-th/0007019 .[20] D. Berman,
SL(2,Z) duality of Born-Infeld theory from non-linear self-dual electro-dynamics in 6 dimensions , Phys. Lett. B (1997) 153, hep-th/9706208hep-th/9706208