ULB–PMIF–93/01

{centering}

Consistent couplings between fields with a gauge
freedom and deformations of the master equation

Glenn Barnich and Marc Henneaux

Faculté des Sciences, Université Libre de Bruxelles,

Campus Plaine C.P. 231, B-1050 Bruxelles, Belgium

The antibracket in BRST theory is known to define a map

associating with two equivalence classes of BRST invariant observables of respective ghost number p and q an equivalence class of BRST invariant observables of ghost number p+q+1. It is shown that this map is trivial in the space of all functionals, i.e., that its image contains only the zeroth class. However it is generically non trivial in the space of local functionals.

Implications of this result for the problem of consistent interactions among fields with a gauge freedom are then drawn. It is shown that the obstructions to constructing such interactions lie precisely in the image of the antibracket map and are accordingly inexistent if one does not insist on locality. However consistent local interactions are severely constrained. The example of the Chern-Simons theory is considered. It is proved that the only consistent, local, Lorentz covariant interactions for the abelian models are exhausted by the non-abelian Chern-Simons extensions.

()Aspirant au Fonds National de la Recherche
Scientifique (Belgium)

()Also at Centro de Estudios
Científicos de Santiago, Chile.

## 1 Introduction

The antifield formalism [1, 2] appears to be one of the most powerful and elegant methods for quantizing arbitrary gauge theories. Originally presented as a set of efficient working rules, its physical foundations have been gradually clarified by showing how gauge invariance is completely captured by BRST cohomology [3, 4, 5]. Some of its geometrical aspects (Schouten bracket, role of Stokes theorem in the proof of the gauge independence of the path integral) have been developed in [6] and more recently in [7, 8, 9, 10]. The somewhat magic importance of the antifield formalism in string field theory [11, 12, 13, 14, 15, 16] and its remarkable underlying algebraic structure [17, 18, 19, 20, 21] have attracted further considerable attention (see also [22]). It is fair to believe that more interesting results are still to come.

The purpose of this letter is to reanalyze the long-standing problem of
constructing
consistent interactions among fields with a gauge freedom in the light of
the antibracket formalism.
We point out that this problem can be economically reformulated as a
deformation problem
in the sense of deformation theory [23], namely that of
deforming
consistently the master equation. We then show, by using the properties of
the antibracket,
that there is no obstruction to constructing interactions that consistently
preserve
the gauge symmetries of the free theory if one allows the interactions to be
non local^{1}^{1}1These results are in line with the light-front analysis
of [24],
as well as with the work of [25] where the role of the master
equation is also
strongly stressed..
Obstructions arise only if one insists on locality. We provide a
reformulation of the deformation of
the master equation that takes locality into account, and illustrate the new
features
to which this leads by considering the three dimensional
Chern-Simons theory. We show that the only local, Lorentz covariant,
consistent
interactions for free
(abelian) Chern-Simons models are given by the non-abelian Chern-Simons
theories. We also
establish the rigidity of the non-abelian Chern-Simons models with a simple
gauge group.
A fuller account of our results will be reported elsewhere [26].

## 2 The master equation and the antibracket map

We first recall some basic properties of the antifield formalism.
The starting point is the action in Lagrangian form , with
gauge
symmetries^{2}^{2}2We shall follow the presentation of the antifield
formalism
given in [7] (chapters 15, 17 and 18). We refer the reader to
that reference for more
information.

(1) |

Given , one can, by introducing ghosts and antifields, construct the solution of the master equation,

(2) |

(3) |

where denotes collectively the original fields, the ghosts and the ghosts of ghosts if necessary, while stands for the antifields. The solution of the master equation captures all the information about the gauge structure of the theory. The existence of reflects the consistency of the gauge transformations. The Noether identities, the (on-shell) closure of the gauge transformations and the higher order gauge identities are contained in the master equation . The original gauge invariant action itself and the gauge transformations can be recovered from by setting the antifields equal to zero in or in ,

(4) |

(5) |

The BRST differential in the algebra of the fields and the antifields is generated by through the antibracket,

(6) |

The BRST cohomology is denoted by . It is easy to verify that the antibracket induces a well defined map in cohomology,

(7) |

(8) |

where denotes the cohomological class of the BRST-closed element . We call (7) “the antibracket map”. If one takes in (8), one gets a map from to sending on .

It is sometimes useful to introduce auxiliary fields in a given theory, namely, fields that can be eliminated by means of their own equations of motion. This may, for instance, simplify the gauge structure and the geometric interpretation of the theory. One then has various equivalent formulations and a natural question to ask is : what is the relationship between the BRST cohomologies and the antibracket map of these equivalent formulations ? Not surprisingly, one has

###### Theorem 1

the BRST cohomologies and associated with two formulations of a theory differing in the auxiliary field content are isomorphic. Furthermore, the isomorphism i : commutes with the antibracket map.

Proof : the proof is direct and based on the explicit relationship between the solutions of the master equation of both formulations worked out in [27]. We leave it as an exercise to the reader.

Using theorem 1, one can now establish the crucial result that the
antibracket map is
trivial^{3}^{3}3The proof assumes spacetime to be of the product form
where is some -dimensional spatial
manifold. It is also assumed
that the Lagrangian fulfills the standard regularity conditions that
guarantee
the existence of the reduced phase space, in terms of which the Cauchy
problem admits a unique
solution (see e.g. [7]). This implies in particular the
existence of proper
gauge fixings..

###### Theorem 2

the antibracket map is trivial, i.e., the antibracket of two BRST-closed functionals is BRST-exact.

Proof : the proof consists in two steps : (i) One adds auxiliary fields and fixes the gauge in such a way that (a) the gauge fixed equations of motion are of first order in the time derivatives and can be solved for ; and (b) the BRST variation of the fields depends only on the fields and not on their time derivatives or on the antifields. This can be done for instance by going to the Hamiltonian formalism, and, as we have seen, modifies neither the BRST cohomology nor the antibracket map. (ii) By expressing the fields in terms of initial data on a Cauchy hypersurface, one proves the existence, in each BRST cohomological class, of a representative that does not involve the antifields. More precisely, let be a solution of and let be the functional of the free initial data that coincides with on-shell. One easily verifies that . Furthermore, and are in the same cohomological class due to general properties of the antifield formalism [7]. For representatives that do not involve the antifields, vanishes identically and not just in cohomology. This proves the theorem (A more detailed analysis will be given in [26]).

## 3 Higher order maps

The triviality of the antibracket map enables one to define higher order operations in cohomology. For example, if , one can define a squared map as follows : the antibracket is a coboundary. Accordingly, there exist a functional of degree such that . The functional is defined up to a cocycle. Now is easily verified to be BRST-closed and the cohomological class of does not depend on the ambiguity in . Furthermore, if . Hence, the application that maps on is well-defined. In our case, however, the squared map and all the other higher order maps that can be defined in a similar fashion are trivial since one can choose representatives in such that and hence both strictly vanish.

## 4 Constructing consistent couplings as a deformation problem

We now turn to the problem of introducing consistent interactions for a “free” action with “free” gauge symmetries

(9) |

(10) |

We want to modify

(11) |

in such a way that one can consistently deform the original gauge symmetries,

(12) |

By “consistently”, we mean that the deformed gauge transformations are indeed gauge symmetries of the full action (11),

(13) |

This implies automatically that the modified gauge transformations close on-shell for the interacting action (see [7], chapter 3). In the case where the original gauge transformations are reducible, one should also demand that (12) remain reducible. Interactions fulfilling these requirements are called “consistent”. [It may be necessary to add further consistency requirements, but this will not be considered here].

A trivial type of consistent interactions is obtained by making field redefinitions . One gets

(14) |

Interactions that can be eliminated by field redefinitions are usually thought of as being no interactions. We shall say that a theory is rigid if the only consistent deformations are proportional to up to field redefinitions. In that case, the interactions can be summed as

(15) |

and simply amount to a change of the coupling constant in front of the unperturbed action.

The problem of constructing consistent interactions is a complicated one because one must simultaneously modify and in such a way that (13) is valid order by order in . It has been studied for lower spins by many authors (see for instance [28, 29, 30] and references therein) and some aspects of the algebraic structure underlying the construction were clarified in [31]. One can reformulate more economically the problem in terms of the solution of the master equation. Indeed, if the interactions can be consistently constructed, then the solution of the master equation for the free theory can be deformed into the solution of the master equation for the interacting theory

(16) |

(17) |

The master equation guarantees that the consistency requirements on and are fulfilled.

There is a definite advantage in reformulating the problem of consistent
interactions as the
problem of deforming the master equation^{4}^{4}4Deforming the master
equation also appears
in renormalization theory where (17) is replaced by
the equation
for the generating function of proper vertices
[32]..
It is that one can bring in the cohomological techniques of deformation
theory.
The master equation for splits according to the deformation parameter
as

(18) | |||||

(19) | |||||

The first equation is satisfied by assumption, while the second implies that is a cocycle for the free differential . Suppose that is a coboundary, . This corresponds to a trivial deformation because is then modified as in (14)

(21) |

(the other modifications induced by affect the higher order structure functions which carry some intrinsic ambiguity [5]). Hence, non trivial deformations are determined by the zeroth cohomological space of the undeformed theory. This space is generically non-empty : it is isomorphic to the space of observables [3, 4, 7].

The next equation (19) implies that should be such that is trivial in . But we have seen that the map induced by the antibracket is trivial and so, this requirement is automatically satisfied. Similarily, the higher order maps are also trivial, which guarantees that the next terms exist. Thus given an initial element of , there is no obstruction in continuing the construction to get the complete . The next terms are determined up to an element of , i.e., up to a gauge invariant function. At each order in there is the freedom of adding to the interaction an arbitrary element of .

We can thus conclude that in the absence of particular requirements on the form of the interactions such as spacetime locality or manifest Lorentz covariance, there is no obstruction to constructing interactions that preserve the initial gauge symmetries as in (13). In orther words, there is no “no-go theorem”.

## 5 Spacetime locality of the deformation - The example of free abelian Chern-Simons models

The above construction does not yield, in general, a local action and is somewhat formal. In practice, it is usually demanded that the deformation be local in spacetime, i.e., that be local functionals. This leads to interesting developments.

In order to implement locality in the above analysis, we recall that if is a local functional which vanishes for all allowed field configurations, , then, the -form is a “total derivative”, ,where is the spacetime exterior derivative and is such that (see e.g. [7] chapter 12). That is, one can “desintegrate” equalities involving local functionals but the integrands are determined up to -exact terms.

Let where is a n-form depending on the variables and a finite number of their derivatives, and let be the antibracket for such -forms, i.e.,

(22) |

if and . [Because is a local functional, there exists such that (22) holds, but is defined only up to -exact terms. This ambiguity plays no role in the subsequent developments]. The equations (18-4) for read

(23) | |||||

in terms of the integrands . The equation (23) expresses that should be BRST closed modulo and again, it is easy to see that a BRST-exact term modulo corresponds to trivial deformations. Non trivial local deformations of the master equation are thus determined by . [Note that an element of yields upon integration an element of only if appropriate surface terms vanish. We shall not investigate this question here and work with all the elements of ].

Now, while is always cohomologically trivial, it is not true, in general, that it is the BRST variation of a local functional. Hence, may not be BRST-exact modulo , and the map

(25) |

defined by the antibracket appears to possess a lot of structure. Furthermore, even when the image of is trivial in , so that the squared map can be defined, there is no guarantee that this squared map is trivial. For this reason, the construction of local, consistent interactions is a problem that is quite constrained.

To illustrate this point, we shall analyze the case of the abelian Chern-Simons models in three dimensions.

The action is given by

(26) |

where is a non degenerate, symmetric and constant matrix. The equations of motion imply . An irreducible set of gauge transformations can be taken to be

(27) |

The minimal solution to the classical master equation is

(28) |

and the local version of the BRST symmetry is then

(29) |

with and . As we have pointed out, the perturbation should obey (23),

(30) |

i.e., should define an element of . The equation (30) can be analyzed along lines familiar from the algebraic study of anomalies. Indeed, one gets from (30) a set of “descent equations” [33, 34]

(31) | |||

(32) | |||

(33) | |||

(34) |

To solve (30), one needs to find the most general element at the bottom of the ladder that can be lifted all the way up to yield an element of . This is the procedure followed in [35]. Now, the last element of a descent belongs to , and must be a polynomial in the ghosts . [Because the equations of motion imply , is trivial in cohomology]. Thus

(35) |

where is completely antisymmetric. This implies

(36) |

where belongs to and is a constant -form. By Lorentz covariance, this term must be zero. This leads then to

(37) |

and finally to

(38) |

It should be noted that is
-trivial in the space of all functionals.
Indeed, assuming that the fields decrease at infinity^{5}^{5}5Different boundary conditions or a non trivial spacetime topology
would require a more sophisticated treatement., one can decompose
and
as

(39) |

Because by the equations of motion, one finds that vanishes on-shell,

(40) |

This implies that is BRST-exact ([7]) and indeed

(41) |

with

(42) |

However, is a non-local functional of the fields and thus, cannot be eliminated by local redefinitions. One then computes . One finds

(43) |

The integrand of this expression is a -cocycle modulo because the Jacobi identity for the local antibracket holds modulo . In order to construct a non trivial local interaction, this cocycle must be trivial in . Because the image of and contains no terms without derivatives, a necessary and sufficient condition for this cocycle to be -trivial modulo is that it vanishes. This is the case if and only if the constants verify the Jacobi identity, even though is BRST-exact in the space of all functionals for arbitrary choices of thanks to (41). This implies that is a solution to our deformation problem which corresponds of course to the well-known non-abelian Chern-Simons theories :

(44) |

Accordingly, the only consistent, Lorentz covariant couplings of abelian Chern-Simons models are the non-abelian extensions.

## 6 Rigidity of non-abelian Chern-Simons theory

We close this letter by proving the rigidity of the Chern-Simons theory with a simple gauge group. The descent equations for are identical to (31)-(34), but this time, the -form of ghost number should be closed for the non-abelian BRST differential. The only non trivial element of is the primitive form , where is a priori an invariant polynomial in the non-abelian field strength , which, however can be set equal to zero, since the equations of motion are . The primitive form can be lifted as in the familiar Yang-Mills case [35] to yield times the Chern-Simons action. Since there is no cohomology in ghost degree , or (apart from the irrelevant constants), there is no other element that can be lifted to yield another solution from a shorter descent. This proves the rigidity of the Chern-Simons action.

## 7 Conclusion

Reformulating the problem of consistent interactions in terms of deformations of the master equation allows the use of powerful BRST cohomological techniques. The triviality of the antibracket map in cohomology in the space of all functionals allows to built consistent interactions from any gauge invariant functionals of the undeformed theory. However, these interactions may be non local and obstructions on consistent local couplings do exist in practise. The study of these obstructions require additional tools familiar from the study of anomalies. The analysis has been illustrated for the Chern-Simons models.

## 8 Acknowledgements

We acknowledge fruitful discussions with Michel Dubois-Violette, Marc Knecht, Jim Stasheff and Claudio Teitelboim. This work has been supported in part by the “Fonds National de la Recherche Scientifique (Belgium)” and by a research contract with the Commission of the European Communities.

## References

- [1] I.A. Batalin and G.A. Vilkovisky, Phys. Lett. B102 (1981) 27.
- [2] I.A. Batalin and G.A. Vilkovisky, Phys. Rev D28 (1983) 2567.
- [3] J. Fisch, M. Henneaux, J.D. Stasheff and C. Teitelboim, Commun. Math. Phys. 120 (1989) 379.
- [4] J.M.L. Fisch and M. Henneaux, Commun. Math. Phys. 128 (1990) 627.
- [5] M. Henneaux, Nucl. Phys. B (Proc. Suppl.) 18A (1990) 47.
- [6] E. Witten, Mod. Phys. Lett. A5 (1990) 487.
- [7] M. Henneaux and C. Teitelboim, “Quantization of Gauge Systems”, Princeton University Press (Princeton : New Jersey 1992).
- [8] A. Schwarz “The Geometry of Batalin-Vilkovisky Quantization”, Commun. Math. Phys. (to appear), preprint hep-th 9205088
- [9] A. Schwarz “Semi-classical Approximation in Batalin-Vilkovisky Formalism”, preprint hep-th 9210115.
- [10] O.M. Khudaverdian and A.P. Nersessian, “On the Geometry of the Batalin-Vilkovisky Formalism”, Genève preprint UGVA-DPT 1993/03-807.
- [11] M. Bochicchio, Phys. Lett. B193 (1987) 31.
- [12] C.B. Thorn, Phys. Rep. 174 (1989) 1.
- [13] B. Zwiebach, Nucl. Phys. B390 (1993) 33.
- [14] E. Witten, Phys. Rev. D46 (1992) 5467.
- [15] E. Verlinde, Nucl. Phys. B381 (1992) 141.
- [16] H. Hata and B. Zwiebach, “Developing the covariant Batalin-Vilkovisky approach to string theory”, preprint hep-th 9301097.
- [17] J.D. Stasheff, Trans. Amer. Math. Soc. 108 (1963) 293.
- [18] E. Getzler and J.D.S. Jones, “n-Algebras and Batalin-Vilkovisky Algebras”, preprint (1992).
- [19] B.H. Lian and G.J. Zuckerman, “New Perspectives on the BRST-algebraic structure of string theory”, preprint hep-th 9211072.
- [20] E. Getzler, “Batalin-Vilkovisky algebras and two dimensional topological field theories”, preprint hep-th 9212043.
- [21] M. Penkava and A. Schwarz, “On some algebraic structures arising in string theory”, preprint hep-th 9212072 .
- [22] J. Alfaro and P.H. Damgaard, “Origin of antifields in the Batalin-Vilkovisky Lagrangian formalism”, preprint CERN-TH-6788-93, hep-th 9301103.
- [23] M. Gerstenhaber, Ann. Math. 79 (1964) 59.
- [24] A.K.H. Bengtsson, I. Bengtsson and L. Brink, Nucl. Phys. B277 (1983) 31, 41 ; see also L. Brink in E.S. Fradkin’s festschrift, I.A. Batalin, C.J. Isham and G.A. Vilkovisky eds., Adam Hilger (1987).
- [25] F. Fougère, M. Knecht and J.Stern, “Algebraic construction of higher spin interaction vertices”, preprint IPNO-TH-91-44; L. Capiello, M. Knecht, S. Ouvry and J. Stern, Ann. Phys. (N.Y.) 193 (1989) 10.
- [26] G. Barnich and M. Henneaux, in preparation.
- [27] M. Henneaux, Phys. Lett. B238 (1990) 299.
- [28] R. Arnowitt and S. Deser, Nucl. Phys. 49 (1963) 133.
- [29] F.A. Berends, G.H. Burgers and H. Van Dam, Nucl. Phys. B260 (1985) 295; A.K.H. Bengtsson, Phys. Rev D32 (1985) 2031.
- [30] P.C. Argyres and C.R. Nappi, Phys. Lett. B224 (1989) 89 ; Nucl. Phys. B.
- [31] T. Lada and J.D. Stasheff, “Introduction to sh-Lie algebras for physicists”, UNC preprint UNC-MATH-92/2.
- [32] J. Zinn-Justin “Renormalization of Gauge Theories” in ”Trends in Elementary Particle Theory” (Springer 1975).
- [33] R. Stora, “Continuum gauge theories” in “New developments in quantum field theory and statistical mechanics” (Cargèse 1976) eds. M. Lévy and P. Mitter (Plenum, New York, 1977); “Algebraic structure and topological origin of anomalies” in “Progress in Gauge Field Theory” (Cargèse 1983) eds. G. ‘t Hooft et al. (Plenum, New York, 1984).
- [34] B. Zumino, “Chiral anomalies and differential geometry” in “Relativity, groups and topology II. Les Houches Lectures 1983.” eds. B.S. De Witt and R. Stora (North-Holland, Amsterdam, 1984)
- [35] M. Dubois-Violette, M. Talon and C.M. Viallet, Commun. Math. Phys. 102 (1985) 105.