BF MODELS 1. INTRODUCTION

BF MODELS C. BIZDADEA1, C. C. CIOBÎRCÃ2, E. M. CIOROIANU3, S. O. SALIU4, S. C. SÃRARU5 Faculty of Physics, University of Craiova, A. I. Cuza Str., Cra...
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BF MODELS C. BIZDADEA1, C. C. CIOBÎRCÃ2, E. M. CIOROIANU3, S. O. SALIU4, S. C. SÃRARU5 Faculty of Physics, University of Craiova, A. I. Cuza Str., Craiova 200585, Romania Received February 21, 2005

The problem of investigating consistent interactions that can be added to a set of scalar fields, two collections of one-forms and a system of two-forms, described in the free limit by a sum of abelian BF models, is reviewed .

1. INTRODUCTION

A big step in the progress of the BRST formalism was its cohomological understanding [1], which allowed, among others, a useful investigation of many interesting aspects related to the perturbative renormalization problem [2, 3], anomaly-tracking mechanism [3, 4], simultaneous study of local and rigid invariances of a given theory [5], as well as to the reformulation of the construction of consistent interactions in gauge theories [6] in terms of the deformation theory [7], or, actually, in terms of the deformation of the solution to the master equation. Joint to these topics, the problem of obtaining consistent deformations has naturally found its extension at the Hamiltonian level by means of local BRST cohomology [8]. There is a large variety of models of interest in theoretical physics, which have been investigated in the light of the BRST symmetry [9, 10]. Some of them focus on the class of BF-like theories [11]. On the one hand, interacting BF theories are related to Chern-Simons-Witten gravity or topological two-branes with nonzero three-form. On the other hand, such theories are important in view of their relationship with Poisson Sigma Models, which are known to explain interesting aspects of two-dimensional gravity, including the study of classical solutions [12]. The purpose of this paper is to review some of our results [13] regarding the BRST approach to the problem of constructing consistent interactions for certain classes of BF models. All the interactions were obtained under the general assumptions of smoothness of the deformations in the coupling constant, 1

e-mail address: [email protected] e-mail address: [email protected] 3 e-mail address: [email protected] 4 e-mail address: [email protected] 5 e-mail address: [email protected] 2

Rom. Journ. Phys., Vol. 50, Nos. 3– 4 , P. 241–248, Bucharest, 2005

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C. Bizdadea, C. C. Ciobîrcã, E. M. Cioroianu, S. O. Saliu, S. C. Sãraru

2

locality, (background) Lorentz invariance, Poincaré invariance, PT invariance and preservation of the number of derivatives on each field. 2. FREE MODEL

Our starting point is a free, topological field theory of BF-type, that involves two types of one-forms, a collection of scalar fields, and a system of two-forms, described by the Lagrangian action







S0 ⎣⎢⎡ Aµa , Hµa , ϕa , Baµν ⎤⎦⎥ = d D x ⎜ Hµa ∂ µ ϕa + 1 Baµν ∂[µ Aνa] ⎟ . 2 ⎝ ⎠

(1)

The gauge structure of the theory (1) essentially depends on the spacetime dimension. In D = 2 dimensions the above action is invariant under the gauge transformations

δε Aµa = ∂ µε a ,

a , δε Hµa = ∂ νεµν

δε ϕa = 0,

δε Baµν = 0,

(2)

while in D ≥ 3 dimensions the gauge transformations take the form

δε Aµa = ∂ µε a ,

a , δε Hµa = ∂ νεµν

δε ϕa = 0,

δε Baµν = ∂ ρεaµνρ .

(3)

a and ε µνρ are bosonic, the last two sets being The gauge parameters ε a , εµν a

completely antisymmetric. In all the situations the gauge algebra is Abelian. The gauge transformations (2) (D = 2) are independent or, in other words, are irreducible. In the general case of an arbitrary D > 2 dimension the gauge transformations (3) are off-shell, (D – 2)-stage reducible. This means that the gauge generators from (3) possess null vectors. Moreover, the null vectors are no longer independent and so on. The structure of the reducibility relations is important from the point of view of the BRST symmetry as it requires the introduction of a tower of ghosts for ghosts, as well as of their antifields. 3. CONSTRUCTION OF CONSISTENT INTERACTIONS 3.1. SETTING THE PROBLEM

We begin with a “free” gauge theory, described by a Lagrangian action ⎤ S0 Φ α 0 ⎥⎥⎦ , which is assumed to be invariant under some gauge transformations ⎡ ⎢ ⎢⎣

δε Φ α 0 = Z αα 0 ( Φ ) ε α1 , 1

such that the gauge algebra reads as

δS0 α 0 Z ( Φ ) = 0, δΦ α 0 α1

(4)

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BF models

Z αβ ( Φ )

δZβα 0 ( Φ )

− Zββ0 ( Φ )

243

δZ αα 0 ( Φ )

= δΦβ0 δS = Cαλ1β ( Φ ) Z λα 0 ( Φ ) + Mαα 0ββ0 ( Φ ) β0 . 1 1 1 1 1 δΦ 0 1

1

δΦβ0

1

1

(5)

In (4–5) δS0 /δΦ α 0 denotes the Euler-Lagrange derivatives of S0 ⎢⎣⎢ Φ α 0 ⎥⎦⎥ . If some ⎡



of the functions Mαα 0ββ0 are not vanishing, we say that the gauge algebra (5) is 1 1

open or, in other words, that the gauge algebra only closes on-shell. We consider the problem of constructing consistent interactions among the fields Φ α 0 such that the couplings preserve the field spectrum and the original number of gauge symmetries. In view of this, we deform the original action S0 (1)

(2)

S0 ⎯→ S0 = S0 + g S0 + g2 S0 + …

(6)

and the original gauge symmetries,

Z αα 0 1

⎯→

Z αα 0 1

=

Z αα 0 1

+

(1)α 0 g Z α1

+

g2

(2)α 0 Z α1

+…

(7)

in such a way that the new gauge transformations δε Φ α 0 = Z αα 0 εα1 are indeed 1

gauge symmetries of the full action (6) (1) (1) ⎛ ⎞ δ ⎜ S0 + g S0 + g2 S0 + … ⎟ α0 (1)α 0 ⎞ ⎝ ⎠ ⎛⎜ Z α 0 + g (1) 2 + + Z g Z … ⎟ = 0. α α 1 1 α ⎜ 1 ⎟ δΦ α 0 ⎝ ⎠

(8)

In the above g stands for the deformation parameter, also known as the coupling constant. In the case where the original gauge transformations are reducible, one should also demand that (7) remain reducible. By projecting the equation (8) on the various powers in the coupling constant we obtain an equivalent tower of equations. The equation corresponding to the order g0 is satisfied by assumption, being nothing but the second relation in (4). The higher-order equations are rather intricate because they are non-linear and involve simultaneously not only the deformed action, but also all the deformed gauge generators. As it will be seen below, a more convenient way to construct the consistent interactions relies on the cohomological approach, based on the BRST symmetry. The cohomological approach systematizes the recursive construction to co-cycles of the BRST differential. Finally, by reformulating the problem of consistent interactions at a cohomological level, one can bring in the powerful tools of homological algebra.

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3.2. COHOMOLOGICAL REFORMULATION

At the level of the BRST formalism, the entire gauge structure of a theory is completely captured by the BRST differential, s. The main feature of s is its nilpotency, s2 = 0. Denoting by ( , ) the antibracket, and by S the canonical generator of the Lagrangian BRST symmetry sF = ( F , S )

(9)

the nilpotency of s is equivalent to the classical master equation

( S, S ) = 0.

(10)

In agreement with the structure (4–5) of the gauge algebra, the solution to the master equation (10) starts like ⎛ ⎞ S = S0 + Φ∗α 0 Zαα 0 ηα1 + 1 ⎜⎜⎜ η∗λ1 Cαλ1β − 1 Φ∗α 0 Φβ∗ 0 Mαα 0ββ0 ⎟⎟⎟ ηα1 ηβ1 + … , 1 1 1 1 1 2⎝ 2 ⎠

(11)

where Φ∗α 0 represents the antifields associated with the original fields, ηα1 are

the ghosts corresponding to the gauge parameters ε α1 , and η∗λ1 denotes the antifields of the ghosts. Due to the fact that the solution to the master equation contains all the information on the gauge structure of a given theory, we can reformulate the problem of introducing consistent interactions as a deformation problem of the solution to the master equation corresponding to the “free” theory. If an interacting gauge theory can be consistently constructed, then the solution S to the master equation associated with the “free” theory can be deformed into a solution S S → S = S + gS1 + g2 S2 + … (12) = S + g d D x a + g 2 d D x b + …,





of the master equation for the deformed theory

( S , S ) = 0,

(13)

such that both the ghost and antifield spectra of the initial theory are preserved. The equation (13) splits, according to the various orders in g, into

( S, S ) = 0,

(14)

2 ( S1 , S ) = 0,

(15)

2 ( S2 , S ) + ( S1 , S1 ) = 0,

(16)

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BF models

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( S3 , S ) + ( S1 , S2 ) = 0,

(17)

The equation (14) is fulfilled by hypothesis. The next one requires that the first-order deformation of the solution to the master equation, S1, is a co-cycle of the “free” BRST differential. However, only cohomologically non-trivial solutions to (15) should be taken into account, as the BRST-exact ones (BRST co-boundaries) correspond to trivial interactions. This means that S1 pertains to the ghost number zero cohomological space of s, H 0 ( s ) , which is generically nonempty due to its isomorphism to the space of physical observables of the “free” theory. It has been shown (on behalf of the triviality of the antibracket map in the cohomology of the BRST differential) that there are no obstructions in finding solutions to the remaining equations ((16–17), etc.). However, the resulting interactions may be nonlocal, and there might even appear obstructions if one insists on their locality. The analysis of these obstructions can be done with the help of cohomological techniques. 4. RESULTS

There are three main types of consistent interactions that can be added to a given gauge theory: (i) the first type deforms only the Lagrangian action, but not its gauge transformations; (ii) the second kind modifies both the action and its transformations, but not the gauge algebra; (iii) the third, and certainly most interesting category, changes everything, namely, the action, its gauge symmetries and the accompanying algebra. From the full deformed solution to the master equation we can identify the interacting theory, as well as its gauge structure. In what follows we list only the final results concerning the Lagrangian description of the interacting gauge theories. The self-interactions among the BF-fields lead to the deformed Lagrangian action a , S0 ⎡⎣⎢ Aµa , H µa , ϕa , Baµν ⎤⎦⎥ = d D x Hµa D µ ϕa + 1 Baµν Fµν (18) 2 where a = ∂ A a + g δWbc Ab Ac . D µ ϕa = ∂ µ ϕa + gWab Abµ , Fµν (19) [µ ν ] δϕa µ ν

)

(



The functions Wab are antisymmetric, depend only on the undifferentiated scalar fields ϕa and, furthermore, satisfy the equation

Wa[ b

δWcd ] δϕa

= 0.

(20)

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In D = 2 dimensions the action (18) is invariant under the deformed gauge transformations

( )b ε b ,

δε Aµa = Dµ ⎛ δW bc ⎜ δϕa ⎝

b − g⎜ δε Hµa = ( D ν ) εµν ⎜ a b

δε ϕa = − gWabε b ,

a

Hµc −

(21)

⎞ δ2Wbc Bdµν Acν ⎟⎟ ε b , ⎟ δϕa δϕd ⎠

(22)

δWab µν b ⎞⎟ B ε ⎟, δϕc c ⎠

(23)



δε Baµν = g ⎜⎜ Wabε bµν − ⎝

where

Wbc c Aµ . ( Dµ )b = δab∂µ − g δδϕ a a

(24)

Clearly, the gauge transformations (21–23) are irreducible. The deformed gauge transformations of the action (18) for D ≥ 3 read as

( )b ε b ,

δε Aµa = Dµ b −g δε Hµa = ( D ν ) εµν a

b

a

(25)

δWbc c b H ε + δϕa µ

δ2Wbc g δ2Wcd cν dρ +g Bdµν Acνε b + A A εbµνρ , 2 δϕa δϕb δϕa δϕd δε ϕa = − gWabε b ,

(27)

Wab µν b ⎟ Bc ε ⎟ , ( )a εbµνρ + g ⎜⎜⎝ Wabε bµν − δδϕ c ⎠

δε Baµν = Dρ



b

(26)



(28)

with Wac c Aρ . ( Dρ )a = δba∂ρ + g δδϕ b b

(29)

It is easy to see that (25–28) reduce to (21–23) if we take εaµνρ = 0. In all cases (D ≥ 2) the deformed gauge algebra is open, by contrast with the original one, which is Abelian. The gauge transformations (25–28) remain (D – 2)-stage reducible, but the reducibility relations only hold on-shell (where on-shell means here on the surface of field equations for the action (18)). We do not give the explicit form of the reducibility relations, but they can be found in [13]. So far, it is clear that the entire deformation is controlled by Wab since if we set Wab = 0 we recover the initial free topological field theory. As shown in [13], the antisymmetric functions Wab ( ϕ ) that satisfy (20) can be interpreted as

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the Poisson tensor on a target Poisson manifold locally parametrized by the dynamical scalar fields {ϕa } . 5. CONCLUSION

In this paper we have discussed a cohomological approach to the problem of constructing consistent interactions in some classes of BF models. Under the general assumptions of smoothness of the deformations in the coupling constant, locality, (background) Lorentz invariance, Poincaré invariance, PT invariance and preservation of the number of derivatives on each field, we have exhausted the self-interactions for some pure BF model. We have also given references to papers where the above mentioned interactions are completely analyzed along cohomological lines. Acknowledgment. This paper has been supported from the type A grant 304/2004 with the Romanian National Council for Academic Scientific Research (CNCSIS) and the Romanian Ministry of Education and Research (MEC).

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