SPECIAL INTERACTIONS BETWEEN A DFLG IN TERMS OF A MIXED SYMMETRY TENSOR FIELD (k,1) AND A TOPOLOGICAL BF MODEL

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1 SPECIL INTERCTIONS ETWEEN DFLG IN TERMS OF MIXED SYMMETRY TENSOR FIELD k,1 ND TOPOLOGICL F MODEL C. IZDDE, M.T. MIUT, S.O. SLIU, L. STNCIU-OPREN Department of Physics, University of Craiova, 13 l. I. Cuza Street, Craiova , Romania bizdadea@central.ucv.ro; mtudristioiu@central.ucv.ro; osaliu@central.ucv.ro; lstanciu@central.ucv.ro Received February 20, 2013 ll consistent couplings in D = k + 3 spacetime dimensions between a topological F model with a maximal field spectrum and a dual formulation of linearized gravity DFLG in terms of a massless tensor field with the mixed symmetry k,1 have been generated using the method of deforming the solution to the master equation by specific cohomological techniques. ll consistent cross-couplings have been computed and classified. s a result of the deformation procedure all the ingredients of the emerging coupled theory are strongly modified compared to their free limit. Key words: consistent interactions in gauge field theory, local RST cohomology, topological F models, mixed symmetry-type tensor fields. PCS: Ef. 1. INTRODUCTION Topological F field theories are important in view of the fact that pure three-dimensional gravity is just a F theory and, moreover, in higher dimensions general relativity and supergravity in shtekar formalism may also be formulated as topological F theories with some extra constraints. On the other hand, tensor fields in exotic representations of the Lorentz group, characterized by a mixed Young symmetry type [2 5], held the attention lately on some important issues, like the dual formulation of field theories of spin two or higher [6 9], the impossibility of consistent cross-interactions in the dual formulation of linearised gravity [10], or the derivation of some exotic gravitational interactions [11, 12]. The purpose of this paper is to report and classify all consistent cross-couplings in D = k + 3 between a dual formulation of linearised gravity DFLG in terms of a massless tensor gauge field with the mixed symmetry k,1 and an belian F model with a maximal field spectrum. The method employed at the construction of Paper presented at The 8 th Workshop on Quantum Field Theory and Hamiltonian Systems, September 19-22, 2012, Craiova, Romania. RJP Rom. 58Nos. Journ. Phys., 5-6, Vol , Nos. 5-6, 2013 P , c ucharest, 2013

2 2 Special interactions between DFLG and topological F models 447 interactions is based on the deformation of the solution to the classical master equation [13, 14] with the help of cohomological techniques based on the computation of local RST cohomology [15 17]. ll couplings have been obtained under some general hypotheses from field theory: analyticity in the coupling constant, space-time locality, Lorentz covariance, and Poincaré invariance of the deformations, combined with the preservation of the number of derivatives on each field. This paper generalizes our previous results from [18, 19]. Under the same hypotheses it has been shown that the massless tensor field with the mixed symmetry k,1 allows no selfinteractions in D = k + 3 [20]. ll consistent self-interactions in an arbitrary dimension D that can be added to a topological F model with a maximal field spectrum have been presented in [21]. The general procedure was exemplified on two models: k = 4 also considered in [19] and k = 5, approached for the first time in this work. The method from [13,14] has been widely used in the literature at the construction of various interacting models, such as F models [22], tensor fields of degree two [23], or D = 11 SUGR [24]. The main characteristics of the interacting model can be synthesized as follows: 1 the Lagrangian action and its gauge symmetries ends at order two in the coupling constant λ; 2 all cross-couplings lie at order one in λ; 3 there appear new self-interactions among the F fields at order two in λ, strictly linked to the presence of the DFLG sector; 4 the consistency of the entire gauge structure of the coupled theory is dictated by a set of algebraic and differential equations that involve all the functions of the undifferentiated scalar field from the F sector that parametrize the self-interactions and cross-couplings at order one in λ, called consistency equations; 5 the gauge structure of the coupled models strongly depends on the precise nontrivial solutions to the consistency equations. The new major results regard the full classification of all cross-couplings into three categories, performed in terms of the number of form gauge fields from the F sector of strictly positive form degree. 2. MIN RESULT. CLSSIFICTION OF CROSS-COUPLING VERTICES The starting point is a free theory in D = k + 3, k 2, with the Lagrangian action written as the sum between the Lagrangian action of a topological F model with a maximal field spectrum that contains two sorts of fields, [m], [m+1] m=0,ik, and the Lagrangian action of a free, massless tensor field with the mixed symmetry k,1 t µ1...µ k α meaning it is antisymmetric in its first k indices and fulfils the identity t [µ1...µ k α] 0 S L [Φ α 0 ] = d k+3 x [ Ik 1 m+1 m=0 [m+1] µ 1...µ m+1 [m] [µ1 µ2...µ m+1 ]

3 448 C. izdadea et al. 3 ] 1 2 k+1! F µ1...µ k+1 αf µ 1...µ k+1 α k + 1F µ1...µ k F µ 1...µ k 1 S L,F [ [m] µ1...µ m, [m+1] µ 1...µ m+1 ] + S L,t [t µ1...µk α]. The field spectrum, denoted by Φ α 0, contains two types of p-forms from the F sector and the tensor field with the mixed symmetry k,1 Φ α [m] 0 = µ1...µ m, [m+1] µ 1...µ m+1,t µ1...µ k α, 2 m=0,ik where Ik the maximum value of m, equal to Ik = [k/2] + 1, with [k/2] the integer part of k/2. For the fields from the F sector we use an overscript which represents the form degree [m] is a m-form and [m+1] a m + 1-form. The notation [µ 1 µ 2...µ k+1 ] signifies complete antisymmetry with respect to the indices between brackets, with the convention that the minimum number of terms is always used. In this paper we work with the Minkowski metric of mostly plus signature σ µν = σ µν = diag + + and with the Levi Civita symbol ε µ 1...µ k+3 defined according to the convention ε 01...k+2 = ε 01...k+2 = 1. The tensor field F µ1...µ k+1 α from 1 displays the mixed symmetry k + 1,1 and reads as F µ1 µ k+1 α = [µ1 t µ2...µ k+1 ] α, F µ1 µ k = σ µ k+1α F µ1 µ k+1 α. 3 The Lagrangian action 1 is invariant under a generating set of gauge symmetries of the form [0] [m] [m 1] δ = 0, δ µ1...µ m = [µ1 ɛ m,0µ 2...µ m], m = 1,Ik, 4 [m+1] µ 1...µ m+1 [m+2] ρµ 1...µ m+1 δ = m + 2 ρ m+1,0, m = 0,Ik, 5 δ t µ1...µ k α = [µ1 χ µ2...µ k ] α + [µ1 θ µ2...µ k ]α + k+1 k α θ µ1...µ k. 6 The gauge parameters from the F sector are denoted by ɛ s and s while for the mixed symmetry tensor field we used χ and θ, and we collectively denoted all the gauge parameters by. ll the gauge parameters are bosonic and completely antisymmetric where applicable, excepting χ µ1...µ k 1 α, which displays the mixed symmetry k 1, 1. Related to the F sector, the over-script represents the form degree, while the other two lower indices between parentheses signify the form field to which a certain gauge parameter is associated with and respectively the reducibility level. The above gauge transformations are belian and off-shell, k + 1-order reducible. The Lagrangian formulations of a free F model with a maximal field spectrum in an arbitrary space-time dimension D and respectively of a massless tensor field with the mixed symmetry k,1 can be found in detail in [21, 25].

4 4 Special interactions between DFLG and topological F models 449 It has been shown in the literature [13, 14] that it is possible to reformulate the problem of constructing consistent interactions as a deformation problem of the solution to the classical master equation corresponding to a given free theory. This procedure can be solved with the help of the local RST cohomology [15 17]. Using the above technique we completely computed the fully deformed solution of the master equation, which complies with all the working hypotheses. It stops at order two in the coupling constant and contains all possible interactions vertices: direct self-interactions in the F sector at order one, cross-couplings between F and DFLG theories at order one, and indirect self-interactions in the F sector at order two, generated only by the presence of DFLG. There appear no self-interactions in the k,1 sector [20] since on the one hand they require k = 2m, D = 4m, and, on the other hand, the DFLG setting imposes the restriction D = k +3 2m+3, so they cannot overlap. The detailed structure of direct self-interactions in the F sector at order one can be found in [21]. They are parametrized by some functions denoted by Z and W that may depend at most on the undifferentiated scalar field [0] ϕ from the F sector. The cross-couplings between F and DFLG together with the indirect self-interactions in the F sector are parametrized in terms of some other functions, denoted by U and V, also depending at most of ϕ. We have shown that the consistency of the deformed solution to the master equation to all orders in the coupling constant require that all these parametrizing functions satisfy a set of algebraic and differential equations, called consistency equations. Once we have computed the fully deformed solution to the master equation, we are able to extract from it the entire gauge formulation of the model that couples a topological F model to a k,1-type dual formulation of linearized gravity in D = k + 3. The Lagrangian action of this coupled model can be written as S L [Φ α 0 ] = S L [Φ α 0 ] + d k+3 x [ λ a F 0 + a int 0 + λ 2 ] b 0, 7 where S L [Φ α 0 ] is the free action 1 and a F 0 provides the F self-interactions at order one in the coupling constant λ [see [21], formulas 6 9]. The piece a int 0 contains all the cross-couplings at order one in λ and reads as a int 0 = M µ1...µ k F µ 1...µ k, 8

5 450 C. izdadea et al. 5 where F µ 1...µ k is given in 3. The compact notation M µ1...µ k signifies M µ1...µ k = k ε µ1...µ k µ k+1 µ k+2 µ k+3 V µ k+1µ k+2 µ k+3 Ik + + V m 2 m=3 N=1 V k V n1,n2,...,nn n 1,n 2,...,n N N [ n 1 ] [ n 2 ] U n1, n 2,..., n N [µ1 N=1 n 1, n 2,..., n N N [n 1 ] [ρ1 [n 2 ] [n N ]...ρm 2 ] [m+1] ρ 1...ρ m 2 µ k+1 µ k+2 µ k+3 [ n N ]...µk ]. 9 The two sums after n 1, n 2,...,n N N and respectively n 1, n 2,..., n N N from 9 are restricted to satisfy the conditions n 1 + n n N = m 2, 1 n 1 n 2... n N Ik, n i = n j n i = 2p, where m = 3,Ik, N = 1,V m 2, and respectively n 1 + n n N = k, 1 n 1 n 2... n N Ik, n i = n j n i = 2p, with N = 1,V k. The sum limits V k and V m 2 are respectively defined by V k = [ ] k 2 + k mod2, Vm 2 = [ ] m mmod2. 12 The component b 0 from 7 is nothing but the Lagrangian density of the coupled model at order two in the coupling constant. Its existence is dictated by the RST deformation procedure and follows from the consistency of the first-order deformation of the solution to the master equation at order two in λ. We have proved by specific cohomological computations that this consistency issue of order two is completely equivalent to a set of algebraic and differential equations that must be satisfied by all the parametrizing functions mentioned in the above of the type Z, W, U, and V, to be called consistency equations. ssuming these equations possess solutions, b 0 can be expressed only in terms of 9 like b 0 = k!3 4 M µ 1...µ k M µ 1...µ k. 13 It is clear from 9 that 13 indeed describes only F self-interactions, as announced earlier. Let us argue why these self-interactions are strictly implemented by the presence of the cross-couplings to the k,1 sector. Indeed, the vanishing of 8 is equivalent to the vanishing of 9, which further implies the annihilation of 13. On the other hand, the condition M µ1...µ k = 0 is completely equivalent to the vanishing of all Us and V s or, in other words, of all functions that parametrize the cross-couplings

6 6 Special interactions between DFLG and topological F models 451 The fact that the fully deformed solution to the master equation ends at order two in λ and hence 7 behaves in the same manner follows from the result that all the deformations of order three or higher of the solution to the master equation are trivial provided the consistency equations are fulfilled and therefore all components of order three or higher in λ from 7 can be taken to vanish under the same assumption. We remark, from 8 and 9, that there are three types of cross-couplings between a topological F model and a particular dual linearised gravity in D = k+3 dimension, parametrized by the smooth functions V, V n1,n 2,...,n N, and U n1, n 2,..., n N of the scalar field ϕ. s we can see, only the last type of couplings is PT-invariant, the other two breaking this invariance. Recall that these functions are not arbitrary, but together with the functions of the type Z and W that parametrize the F selfinteractions are subject to the consistency equations. nalysing 7 and then 8 we observe that all cross-couplings lie at order one in λ, are linear in the tensor F µ 1...µ k from the k,1 sector, and can be appropriately classified according to the number of - and -type F forms as follows: 1 none with more then one ; 2 a single kind with one and no s, present in any D 5 k 2 and containing only the three-form, 3 at least one vertex with a single and at least one, present earliest in D = 6 k = 3, and 4 without s, but al least one. The last type of couplings enables a single -type form only in D = 5 k = 2, otherwise for higher values of k, at least two s are required. Finally, we mention that the gauge structure of the interacting theory is highly deformed with respect to that of the starting belian model: the gauge algebra becomes open and the reducibility relations only hold on-shell. Their concrete expressions cannot be output in the general setting discussed here since they strongly depend on the solutions to the consistency equations. 3. EXMPLES In this section we particularize the general results from the previous section to k = 4 and k = 5. The topological F model involved has in both cases the same maximal field spectrum, namely one scalar field, two types of one-forms, two kinds of two-forms, two sorts of three-forms, and one four-form. The F self-interactions in D = 7 and respectively D = 8 are given in detail in [21]. In what follows we analyse only the cross-couplings and associated F self-interactions at order two in the coupling constant. The main task is to find all the distinct solutions to conditions 10 and 11 in order to obtain the concrete form of 9 for each of these situations RESULTS FOR k = 4, LIS D = 7 In this situation I4 = 3 and m involved in conditions 10 is single-valued, m = 3. On behalf of 12, it follows V 1 = 1, and hence N = 1, such that 10 possess

7 452 C. izdadea et al. 7 a unique solution: n 1 = 1. ccordingly, there is a single parametrizing function, V 1. Related to conditions 11, we have that N takes values within the range 2,V 4. s the first relation in 12 gives V 4 = 2, it follows that N = 2. There are two distinct solutions to 11: n 1 = 1, n 2 = 3 and respectively n 1 = n 2 = 2. The parametrizing functions will consequently be denoted by U 1,3 and U 2,2, but for notational simplicity we switch to U 1,3 U 1, U 2,2 U 2. This particular model has been exposed also in [19]. Under these considerations, 9 and the coupled Lagrangian action for k = 4 become respectively M µ1 µ 2 µ 3 µ 4 =ε µ1 µ 2 µ 3 µ 4 µ 5 µ 6 µ 7 [ S L [Φ α 0 ] = S L [Φ α 0 ] + λ d 7 x Mϕ + + W 4 [µ1 µ2 µ 3 ] + ε µ1...µ 7 Z 2 µ 1 µ 2 µ 3 V µ 5 µ 6 µ 7 + V 1 ρµ 5 µ 6 µ 7 ρ + U 1 [µ1 µ2 µ 3 µ 4 ] + U 2 [µ1 µ 2 µ3 µ 4 ], 14 + µ 1 µ 2 µ 3 µ 4 µ 5 µ 6 µ 7 µ1 µ 1 + W 2 µ1 µ 2 µ 1 µ 2 W 5 [µ1 µ 2 µ3 µ 4 ] + W 6 [µ1 µ2 µ 3 µ 4 ] Y 2 + M µ1 µ 2 µ 3 µ 4 F µ 1µ 2 µ 3 µ 4 ] + 18λ 2 µ 1 µ 2 µ 3 µ 4 µ 5 µ 6 µ ! Y 3 + W 3 µ1 µ 2 µ 2 µ 1 µ 2 µ 3 µ 4 µ 1 µ 2 µ 3 µ 1 µ 2 µ 3 µ 4 µ 5 µ 6 µ 7 d 7 xm µ1 µ 2 µ 3 µ 4 M µ 1µ 2 µ 3 µ fter long and tedious computations we inferred the consistency conditions, mentioned in general in the previous section, as: dw 2 + 3W 4W 2 = 0, dw 3 + 4W 6 3W 4 W 3 = 0, 16 dz 2 3W 4 + 4W 6 Z 2 = 0, dy W 4 + 2W 6 Y 2 W 3 Y 3 = 0, 17 dw W dm 4 2W 6 W Z 2 Y 3 = 0, = 0, 18 W 2 W 4 + W 3 W 5 24Z 2 Y 2 = 0, 2W 2 W 6 + 3W 3 W Z 2 Y 2 = 0, 19 W 2 Y 3 +9W 5 Y 2 = 0, W 2 = 0, W 2 W 3 = 0, W 2 Z 2 = 0, W 3 Z 2 = 0, 20 du 2 + 2[ 2W 5U 1 + 3W 4 U Y 3 V Y 2 V 1 ] = 0, 21 dv 3W 4V + W 3 V 1 4Z 2 U 1 = 0, 2W 2 U 1 + 3W 3 U Y 2 V = 0, 22 W 2 V 1 + 6W 5 V Z 2 U 2 = 0, W 2 V = 0, 23

8 8 Special interactions between DFLG and topological F models 453 where all the functions of the type W, Z, Y, or M are specific to first-order F selfinteractions. They are far more complex than the consistency equations satisfied in the case of a pure F model, which follow from the above ones if we set zero all the Us and V s. The deformed generating set of gauge transformations follows from the components of anti-ghost number one present in the deformed solution to the master equation and takes the concrete form δ ϕ = λ δ self ϕ, 24 δ µ1 µ 2 = [µ1 ɛ 2,0µ2 ] + λ δ self δ µ1 = µ1 [0] ɛ 1,0 + λ δ self δ µ1 µ 2 µ 3 = [µ1 ɛ 3,0µ2 µ 3 ] + λ δ self µ1, 25 µ1 µ λv ε µ 1 µ 2 µ 3...µ 7 [µ 3 θ µ 4...µ 7 ], 26 µ1 µ 2 µ 3 4λV 1 ε µ1 µ 2 µ 3 µ 4...µ 7 ρ [ρ θ µ 4...µ 7 ] λε µ1 µ 2 µ 3 µ 4...µ 7 V 1 F µ 4...µ λM µ 4...µ 7 [0] ɛ 1,0, 27 ρµ 1 µ 1 δ = 2 ρ 1,0 + λ δ Ω self µ 1 α 1 du 1 4λ [µ2 µ3 µ 4 µ 5 ] + du λε ν 1 ν 2...ν 7 dv µ 1 ν 1 ν 2 + dv 1 [ + λf µ 1µ 2 µ 3 µ λM µ 1µ 2 µ 3 µ du 2 [µ2 µ 3 ɛ 2,0µ4 ] du 1 dv [5] 1 [µ2 ε µ3 µ 4 ]ν 1...ν 5 ν 1...ν 5 4,0 ρµ 1 µ 2 ] [µ2 µ 3 µ4 µ 5 ] [µ2 µ 1 µ 2 ν 1 ν 2 µ2 4 du 1 [µ 1 θ µ 2µ 3 µ 4 µ 5 ] [ν 3 θ ν 4...ν 7 ] µ2 µ 3 µ 4 dv 1 ε µ 2...µ 8 ɛ 3,0µ3 µ 4 ] + dv ε µ 2...µ 8 µ 1 µ 2 δ = 3 ρ 2,0 + λ δ Ω self 1 µ 2 α 1 µ 4λ µ 5...µ 8 µ 5...µ 8 3,0 [0] ɛ 1,0, 28 4U 1 µ3 µ 4 µ 5 [µ 1 θ µ 2µ 3 µ 4 µ 5 ] 3 5 V 1 µ 2 ν 1 ν 2 1ε ν1 ν 2...ν 7 µ [ν 3 θ ν 4...ν 7 ] 3λF µ 1µ 2 µ 3 µ λM µ 1µ 2 µ 3 µ 4 [5] ν 1...ν 5 4U 1 ɛ 3,0µ3 µ 4 + V 1 ε µ3 µ 4 ν 1...ν 5, 29 4,0

9 454 C. izdadea et al. 9 ρµ 1 µ 2 µ 3 µ 1 µ 2 µ 3 δ = 4 ρ 3,0 + λ δ Ω self µ 1 µ 2 µ 3 α 1 24λU 2 µ4 µ 5 [µ 1 θ µ 2µ 3 µ 4 µ 5 ] + 12λU 2 F µ 1µ 2 µ 3 µ λM µ 1µ 2 µ 3 µ 4 ɛ 2,0µ4, 30 ρµ 1 µ 2 µ 3 µ 4 µ 1 µ 2 µ 3 µ 4 [5] δ = 5 ρ 4,0 + λ δ Ω self µ 1 µ 2 µ 3 µ 4 α λU 1 µ5 [µ 1 θ µ 2µ 3 µ 4 µ 5 ] + 4λU 1 F µ 1µ 2 µ 3 µ λM µ 1µ 2 µ 3 µ 4 [0] ɛ 1,0, 31 δ t µ1 µ 2 µ 3 µ 4 α = [µ1 χ µ2 µ 3 µ 4 ] α + [µ1 θ µ2 µ 3 µ 4 ]α 4 α θ µ1 µ 2 µ 3 µ 4 [ ] + 12λσ α[µ1 U 1 2 [0] µ2 µ 3 µ 4 ] ɛ 1,0 µ2 ɛ 3,0µ3 µ 4 ] + U 2 µ2 µ 3 ɛ 2,0µ4 ] + 3λσ α[µ1 ε µ2 µ 3 µ 4 ]ν 1 ν 2 ν 3 ν 4 4V ν 1 ν 2 ν 3 ν 4 3,0 V 1 ν 1 ν 2 ν 3 ν 4 ɛ 1,0 5V 1 [5] ρν 1 ν 2 ν 3 ν 4 ρ 4,0 ll components carrying the index self stem from F self-interactions at order one in λ. Their structure is detailed in [21]. n important remark is that that also the gauge transformations of the k,1 tensor field are deformed with respect to the initial ones. The entire gauge structure of the interacting model being controlled by the functions Mϕ, W i ϕ i=1,6, Z 2 ϕ, and Y j ϕ j=2,3, which are restricted to satisfy the consistency equations 16 23, our procedure is valid provided these equations possess solutions. We list below one class of solutions, suggestively resembling to Liouville-type field theories: W 2 = W 3 = Y 2 = Y 3 = 0, W 4 = α, W 5 = k 2 exp[ 23α 2βϕ], W 6 = β, M = k 1, Z 2 = k 3 exp[3α + 4βϕ], U 1 = θ, U 2 = k 2 exp 6αϕ 32 ϑ k3 + θ β exp4βϕ, V = k 3 k 2 U 2 exp9αϕ. In the above and V 1 are arbitrary functions and α, β, θ, k 1, k 2, k 3, and ϑ are some arbitrary, non-vanishing real constants. Replacing back the above solution into action 15 and its gauge transformations 24 32, we are then able to determine the entire structure of this particular kind of F model coupled to DFLG in D = RESULTS FOR k = 5, LIS D = 8 This situation amounts to I5 = 3 and m is again single valued, m = 3, like for k = 4. Conditions 10 exhibit exactly the same solutions, m 1 = 1. This observation holds in general, for any two models with consecutive values k and k + 1 if k is even, in which case Ik and Ik + 1 are equal, so both models present the same structure of the cross-coupling vertices that break the PT invariance they may differ by a phase factor. nalysing now conditions 11, we find that N = 2,V5, while 12 yields V 5 = 3. For each allowed value of N, the set 11 inherits a unique.

10 10 Special interactions between DFLG and topological F models 455 solution: n 1 = 2, n 2 = 3 for N = 2; n 1 = 1, n 2 = n 3 = 2 for N = 3. The associated parametrizing functions are U 2,3 and U 1,2,2, to be turned into U 2,3 U 3 and respectively U 1,2,2 U 4 for notational simplicity purposes. Consequently, 9 and the coupled Lagrangian action display the forms M µ1 µ 2 µ 3 µ 4 µ 5 = ε µ1 µ 2 µ 3 µ 4 µ 5 µ 6 µ 7 µ 8 S L [Φ α 0 ] =S L [Φ α 0 ] + λ + V µ 6 µ 7 µ 8 + V 1 ρµ 6 µ 7 µ 8 ρ + U 3 [µ1 µ 2 µ3 µ 4 µ 5 ] + U 4 [µ1 µ2 µ 3 µ4 µ 5 ], 33 [ d 8 x Mϕ + W 3 µ1 µ 2 µ 2 + W 4 [µ1 µ2 µ 3 ] µ1 µ 1 + W 2 µ 1 µ 2 µ µ 2 µ 3 µ 4 1 +W 6 [µ1 µ2 µ 3 µ 4 ] µ + ε µ1...µ R 1 2 Z 1 µ1 µ 2 µ 1 µ 2 W 5 [µ1 µ 2 µ3 µ 4 ] µ 1...µ 4 µ 5...µ 8 µ 1 µ 2 µ 3 µ 4 µ 5 6 µ 7 µ 8 µ + 1 4! R µ 1 µ 2 µ 3 µ 4 µ 5 µ 6 µ 7 µ M µ1...µ 5 F µ 1...µ 5 ] + 90λ 2 d 8 xm µ1...µ 5 M µ 1...µ The key point of our procedure the consistency conditions, have been computed and are listed below: dw 2 + 3W 4W 2 = 0, dw 3 + 4W 6 3W 4 W 3 = 0, 35 dz 1 8W 6Z 1 = 0, dr W 4R 2 3W 5 R 1 = 0, 36 dw 5 + 2[3W 4 2W 6 W Z 1 R 1 ] = 0, dm = 0, 37 W 2 W 4 + W 3 W 5 = 0, 2W 2 W 6 + 3W 3 W 5 = 0, 3W 2 R 1 + W 3 R 2 = 0, 38 W 2 = 0, W 2 W 3 = 0, W 3 Z 1 = 0, 39 du 3 + 3W 4 + 4W 6 U W 3 U R 1 V = 0, 40 dv 3W 4V + W 3 V 1 = 0, W 2 U 4 + 2W 5 U 3 + 4R 2 V = 0, 41 W 2 V W 5 V 4Z 1 U 3 = 0, W 2 V = The completely deformed solution to the master equation offer the associated

11 456 C. izdadea et al. 11 gauge symmetries of action 34: δ ϕ = λ δ self ϕ, 43 [0] δ µ1 = µ1 ɛ 1,0 + λ δ self µ1, 44 δ µ1 µ 2 = [µ1 ɛ 2,0µ2 ] + λ δ self µ1 µ λv ε µ 1 µ 2...µ 8 [µ 3 θ µ 4...µ 8 ], 45 δ µ1 µ 2 µ 3 = [µ1 ɛ 3,0µ2 µ 3 ] + λ δ self µ1 µ 2 µ 3 5λV 1 ε µ1 µ 2 µ 3 µ 4...µ 8 ρ [ρ θ µ 4...µ 8 ] λε µ1 µ 2 µ 3 µ 4...µ 8 V 1 F µ 4...µ λM µ 4...µ 8 [0] ɛ 1,0, 46 δ ρµ 1 µ 1 = 2 ρ du 3 [µ2 µ 3 1,0 + λ δ self µ 1 5λ µ4 µ 5 µ 6 ] + du λε ν 1...ν 8 dv µ 1 ν 1 ν 2 + dv 1 + 5λF µ 1µ 2 µ 3 µ 4 µ λM µ 1µ 2 µ 3 µ 4 µ 5 1 dv 1 4 ε µ 2...µ 5 ν 1 ν 2 ν 3 ν 4 du 3 ν 1 ν 2 ν 3 ν 4 [µ2 µ 3 µ 4 + du 4 + dv ε µ 2...µ 5 ν 1 ν 2 ν 3 ν 4 [µ2 ν 1 ν 2 ν 3 ν 4 µ3 µ 4 3,0 + dv 1 [µ2 µ3 µ 4 µ5 µ 6 ] µ 1 µ 2 ν 1 ν 2 µ2 [0] ɛ 1,0 [ [µ 1 θ µ 2...µ 6 ] [ν 3 θ ν 4...ν 8 ] du 4 [µ2 µ 3 µ4 µ 5 ] ɛ 2,0µ5 ] + du 3 [5] µ2 ε µ3 µ 4 µ 5 ν 1...ν 5 [µ2 µ 3 ɛ 3,0µ4 µ 5 ] ν 1...ν 5 4,0 ], 47 δ 5λ µ 1 µ 2 = 3 ρ 5U 4 [µ3 µ 4 ρµ 1 µ 2 2,0 + λ δ self µ 1 µ 2 µ5 µ 6 ] [µ 1 θ µ 2...µ 6 ] 1 2 V µ 1 µ 2 ν 1 ν 2 1ε ν1...ν 8 [ν 3 θ ν 4...ν 8 ] 5λF µ 1µ 2 µ 3 µ 4 µ λM µ 1µ 2 µ 3 µ 4 µ 5 [5] 4U 4 [µ3 µ 4 ɛ 2,0µ5 ] V 1 ε µ3 µ 4 µ 5 ν 1...ν 5 ν 1...ν 5 4,0, 48

12 12 Special interactions between DFLG and topological F models 457 ρµ 1 µ 2 µ 3 µ 1 µ 2 µ 3 µ 1 µ 2 µ 3 δ = 4 ρ 3,0 + λ δ self 50λ U 3 µ4 µ 5 µ 6 + U 4 [µ4 µ5 µ 6 ] [µ 1 θ µ 2µ 3 µ 4 µ 5 µ 6 ] + 30λF µ 1µ 2 µ 3 µ 4 µ λM µ 1µ 2 µ 3 µ 4 µ 5 U 4 µ4 µ 5 [0] ɛ 1,0 U 4 [µ4 ɛ 2,0µ5 ] + U 3 ɛ 3,0µ4 µ 5, 49 δ µ 1 µ 2 µ 3 µ 4 [5] = 5 ρ ρµ 1 µ 2 µ 3 µ 4 4,0 + λ δ self µ 1 µ 2 µ 3 µ 4 50λU 3 µ5 µ 6 [µ 1 θ µ 2µ 3 µ 4 µ 5 µ 6 ] 20λU 3 F µ 1µ 2 µ 3 µ 4 µ λM µ 1µ 2 µ 3 µ 4 µ 5 ɛ 2,0µ5, 50 δ t µ1 µ 2 µ 3 µ 4 µ 5 α = [µ1 χ µ2 µ 3 µ 4 µ 5 ] α + [µ1 θ µ2 µ 3 µ 4 µ 5 ]α + 5 α θ µ1 µ 2 µ 3 µ 4 µ 5 [ + 60λσ α[µ1 U 3 µ2 µ 3 µ 4 ɛ 2,0µ5 ] µ2 µ 3 ɛ 3,0µ4 µ 5 ] ] [0] U 4 µ2 µ 3 µ4 µ 5 ] ɛ 1,0 µ2 µ3 µ 4 ɛ 2,0µ5 ] + 15λσ α[µ1 ε µ2 µ 3 µ 4 µ 5 ]ν 1...ν 4 4V ν 1...ν 4 3,0 + V 1 ν 1...ν 4 [0] ɛ 1,0 + 5V 1 [5] ρν 1...ν 4 ρ 4,0 Finally, we give several solutions to the consistency equations 35 42, which involve all the parameterizing functions. first class of solutions, suggestively resembling to Liouville-type field theories, is: W 2 = W 3 = 0, W 4 = α, W 5 = 48ρ 3α+2β Z 1, W 6 = β, M = k 1, Z 1 = k 2 exp8βϕ, R 1 = ρ, R 2 = 432ρ2 Z 3α+2β 2 1, U 3 = 36ρ 3α+2β V, V = k 3 exp3αϕ, where, U 4, and V 1 are arbitrary functions, with α, β, ρ, k 1, k 2, and k 3 are arbitrary, non-vanishing constants satisfying 3α + 2β 0. second class of solutions is represented by: = W 2 = W 5 = Z 1 = R 2 = 0, W 4 = 4 3 W 6, U 3 = αw 3, U 4 = 8α 2 W βR 1, V = βw 3, V 1 = 4βW 6, with M, W 3, W 6, and R 1 arbitrary functions and α, β non-vanishing constants. The third class reads as: = W 2 = W 3 = Z 1 = U 3 = V = 0, W 4 = 2 3 W 6, R 1 = αw 6, R 2 = 9α 2 W 5, where M, W 5, W 6, U 4, and V 1 are arbitrary functions and α, β denote some non-vanishing constants. The fourth and last class of solutions corresponds to the choice: = W 2 = W 3 = W 4 = W 5 = W 6 = Z 1 = R 1 = R 2 = 0, while M and V i i=1,4 remain arbitrary. Each of the above solutions replaced in 34 and

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