Göteborg ITP 97-03

April 1997

hep-th/9704209

(Appendix B inserted May 1997 and a new introduction August 1997)

Antisymplectic gauge theories

Igor Batalin
^{1}^{1}1On leave of absence from
P.N.Lebedev Physical Institute, 117924 Moscow, Russia

E-mail:
and Robert Marnelius
^{2}^{2}2E-mail:

Institute of Theoretical Physics

Chalmers University of Technology

S-412 96 Göteborg, Sweden

A general field-antifield BV formalism for antisymplectic first class constraints is proposed. It is as general as the corresponding symplectic BFV-BRST formulation and it is demonstrated to be consistent with a previously proposed formalism for antisymplectic second class constraints through a generalized conversion to corresponding first class constraints. Thereby the basic concept of gauge symmetry is extended to apply to quite a new class of gauge theories potentially possible to exist.

## 1 Introduction

The field-antifield BV formalism [1] is a Lagrangian path integral method to quantize general gauge theories (important early contributions are [2]-[4]). It has been shown to work for an ever increasing number of models. In the BV formalism one introduces antifields with opposite Grassmann parities to all field and ghost variables. It involves in a crucial way also an antibracket operation and a nilpotent differential -operator. The understanding of the formalism was further deepened in [5]-[9] and in [10]-[14]. In the approach of the latter papers a coordinate invariant general covariant formulation was developed. The field-antifield variables are here considered as arbitrary coordinates on an antisymplectic manifold . (The standard BV formalism may then be viewed as formulated in terms of antisymplectic Darboux coordinates.) In this formalism the geometric coordinate invariant properties is formally demonstrated. Furthermore, the formalism specifies the conditions a consistent invariant measure density has to satisfy. It involves also new ingredients like a hypergauge formulation [10, 14] and a multilevel formalism [11]. Gauge invariance is demonstrated in general terms. Among the further generalizations are deformed -operators in [12] and an Sp(2) version in [14, 15]. In [10] it was also shown how antisymplectic second class constraints may be introduced and treated consistently within this formalism. In this paper we continue this set of formal generalizations with still another one. Here we show that the path integral may be formulated on a large antisymplectic manifold also in the presence of antisymplectic first class constraints. All required conditions are shown to be formally satisfied. This is therefore a major further generalization of the general covariant BV formalism. The beautiful general mathematical structure of the BV formalism is thereby further extended. However, it remains to demonstrate the existence of examples which satisfy the generalizations suggested by the obtained formal results. Although we expect them to exist this is certainly a nontrivial issue. Anyway the formal results suggest alternative formulations which could turn out to be useful. Particularly the results of the present paper could allow for formulations with specific global symmetries which are preferable for some reasons. At a more speculative level our results show the existence of new types of gauge theories in an antisymplectic quantum theory in the spirit of [5]-[8].

In section 2 we recapitulate some basic properties of the general covariant BV formalism. In section 3 and appendices A and B we present our formulation and its formal properties. In section 4 and appendix C we consider then a generalized conversion of antisymplectic second class constraints into corresponding first class ones by means of an extension of the field-antifield manifold . This provides for an explicit formal verification of the formalism. Throughout the paper we make use of deWitt’s compact notation which reduces the treatment to a finite dimensional one. In principle all functionals may be either local or nonlocal.

## 2 Basics of general covariant BV formalism

The basic object in field-antifield quantization is the nilpotent fermionic second order differential operator

(1) |

where are derivatives with respect to local coordinates , , on an antisymplectic manifold . Their Grassmann parities are . ( are generalized fields and antifields.) is a measure density and an odd metric tensor with the properties: and . Another basic object in the field-antifield formalism is the antibracket given by

(2) |

for arbitrary functions on . It satisfies

(3) |

The measure density satisfies also

(4) |

All these properties follow from the nilpotency of the -operator 1̊, i.e. .

On we assume the existence of a quantum master action satisfying

(5) |

In terms of there is another nilpotent second order differential operator defined by [6]

(6) |

which satisfies and

(7) |

Given two solutions and of the master equation 5̊, one has

(8) |

The path integral in this generalized BV formalism is given by

(9) |

where is the above quantum master action and a hyper gauge-fixing master action which also satisfies the quantum master equation 5̊. are second level Lagrange multipliers [12, 13] with no corresponding antifields. This means that is viewed as containing first level Lagrange multipliers and their antifields () with the Grassmann parities . The actions and have then the form

(10) |

where are hyperconstraints that fixes the antifields in . Also satisfies the master equation 5̊. Under these conditions one may show that the path integral 8̊ is independent of the precise form of the hypergauge conditions (-independence) [14].

If we introduce some constraints , , on such that is invertible, then we may define a “Dirac” antibracket by the expression [10]

(11) |

where is the invers to . Obviously

(12) |

Since for any the metric is degenerate on . However, even in terms of such a metric there is a consistent path integral and it is given by [10]

(13) |

where now and satisfy the quantum master equations 5̊ with replaced by

(14) |

Thus, for antisymplectic second class constraints there is a consistent formulation already. We shall now propose a consistent formulation for corresponding first class constraints.

## 3 Field-antifield formalism with first class constraints

Let us call antisymplectic first class constraints provided satisfy

(15) |

In the presence of such constraints we propose the following path integral

(16) | |||||

where is a gauge independent measure density, and where are gauge-fixing conditions to , i.e. is required to be invertible. The Grassmann parities of the field variables in 1̊5 are

(17) |

The first class constraints are in addition to 1̊4 required to satisfy

(18) |

which also may be viewed as conditions on and . A general representation of the path integral 1̊5 is given in appendix A.

The path integral is invariant under the supertransformation

(19) |

where is an odd constant. It leads to

(20) |

and

(21) |

from 1̊6. Furthermore, it gives rise to the following Jacobian

(22) |

All these contributions from the transformation 1̊7 in the integrand of are compensated by the transformations

(23) | |||||

together with the resulting contributions from the corresponding Jacobians

(24) |

The path integral is also independent of the gauge-fixing functions . To see this consider the shift

(25) |

in . It is in fact exactly compensated by the transformation 1̊7 with the choice

(26) |

since when compared to the previous transformation with constant this choice gives rise to the following additional contribution to the Jacobian

(27) | |||||

## 4 Conversion and the Abelian case

We shall now apply and verify the general formulation above. We consider then a generalized conversion of antisymplectic second class constraints into corresponding first class ones. Within the ordinary Hamiltonian formalism the conversion mechanism has been formulated in general terms in [16, 17] (see also [18]). It has been applied to many models. One interesting application is the new approach to geometric quantization in [19] which is mainly based on [17]. In the following application to the field-antifield formalism a new ingredient appears since we not only have antibrackets, which corresponds to Poisson brackets, but also the nilpotent differential -operator.

Consider the second class constraints , , on which by definition are such that is invertible. We now convert these constraints into abelian first class constraints by extending the original antisymplectic manifold . Introduce therefore the additional field-antifield coordinates with the Grassmann parities . On the resulting extended manifold, , we define then an extended antibracket with the extended metric

(28) |

where is an odd invertible constant matrix. On we may then define first class constraints satisfying

(29) |

These functions may be constructed perturbatively with the ansatz

(30) |

We may furthermore construct gauge invariant functions to any function by the conditions

(31) |

Also these conditions may be solved perturbatively with an ansatz of the form

(32) |

In appendix B it is shown that

(33) |

where the right-hand side is the Dirac antibracket 1̊0. One may also show that (see below)

(34) |

provided

(35) |

is the corresponding -operator to 1̊ on the extended manifold with the metric 2̊6 and with a measure density satisfying 3̊6. To show 3̊1 it is sufficient to solve 2̊8 and 3̊0 up to the first order in as is shown in appendix B, while 3̊2 requires a solution up to second order.

The gauge independent path integral 1̊5 is here given by

where and satisfy the quantum master equations

(37) |

According to 1̊6 we have additional conditions like (see appendix A)

(38) |

They are easy to solve for measure densities satisfying 3̊6 since these conditions then reduce to

(39) |

This implies that and in this case are gauge invariant extensions of and defined by

(40) |

which means that and . It follows now that the path integral 3̊3 in e.g. the gauge reduces to

(41) |

which when compared to the second class expression in 1̊2 requires the boundary condition

(42) |

where is the first order coefficient in the expansion 2̊8. That and satisfy the appropriate master equations follows from 3̊1 and 3̊2.

Another equivalent but more explicit and transparent way to derive the equivalence between 1̊2 and 3̊3 is to first construct gauge invariant coordinates defined by

(43) |

which also may be solved by a perturbative ansatz like 3̊0. Then we have for any gauge invariant function and

(44) |

Thus, the gauge invariant functions lives on the submanifold of spanned by . The same is true for the -operator as will be shown below.

It is convenient to change coordinates on from to . In terms of these coordinates we have

(45) |

where are derivatives with respect to . Furthermore, we have

(46) |

where are derivatives with respect to while keeping fixed, and where is related to through the formula

(47) |

The “bar”-metric in 4̊4 is given in 4̊3 and

(48) |

Since the Jacobi identities yield

(49) |

we have also

(50) |

due to 4̊2. The -operator expression 4̊4 may be decomposed as follows

(51) |

Obviously for any gauge invariant function . Furthermore, since the condition implies that

(52) |

for any function , we have

(53) |

Therefore, if we restrict such that only depends on then also only depends on due to 4̊9. The nilpotency of requires then that is nilpotent which in turn implies [13]

(54) |

This result may equivalently be expressed as follows: In order for the measure density to satisfy 3̊6 it should be such that

(55) |

for any measure density where satisfies

(56) |

We assert that there is a solution of the form

(57) |

where also the function is to be determined (see appendix C). Obviously in agreement with the assertion 3̊2, provided

(58) |

where is the Dirac measure density in 1̊2 and 1̊3. Thus, the -operator is just a gauge invariant extension of on . It should also be mentioned that the transformation 5̊21 with the identification 5̊3 is consistent with the boundary condition 4̊0 due to the relation

(59) |

which follows from 5̊22 and 5̊23 to lowest order in (see formula c̊9 in appendix C).

Acknowledgements

I.A.B. thanks Klaus Bering and Poul Damgaard
for stimulating discussions at an early
stage of this work. I.A.B. would also like to thank
Lars Brink for very
warm hospitality at the Institute of Theoretical
Physics, Chalmers and Göteborg
University. The work is partially supported by grant INTAS-RFBR 95-0829.
The work of I.A.B. is also
supported by grants
INTAS 93-2058, INTAS 93-0633, RFBR 96-01-00482,
RFBR 96-02-17314, and NorFa 97.40.002-O.

Appendix A

Invariant formulation of the path integral

Let us extend the antisymplectic manifold
in section 2 by three sets of field-antifield pairs:
. Their
Grassmann parities are

(A.1) |

On this extended manifold, , we define the bosonic charge

(A.2) |

where satisfies the algebra 1̊4 and where the dots indicate terms independent of and such that satisfies

(A.3) |

where the antibracket from now on is the extended one on . An invariant path integral may then be written as

(A.4) |

where and satisfy the extended master equations

(A.5) |

where in turn is the nilpotent -operator 1̊ extended to with . The objects , and are in addition required to satisfy

(A.6) |

The gauge-fixing charge is odd and is of the form

(A.7) |

The solutions of å5 are

(A.8) |

The equations for and reduce to 5̊ if and only if . Otherwise and generalize to satisfy the modified equations with supertrace “anomalies” in their right-hand sides (see appendix B). The formulation given in section 2 corresponds therefore to the first case. Notice that eq.1̊6 is obtained from å7 when å6 is inserted, and that with the choice å8 the path integral å4 reduces to 1̊5 after the identifications , , and provided and only depends on .

The path integral å4 is invariant under the following transformation

(A.9) |

where and where is an odd constant. The contribution to the Jacobian

(A.10) |

is compensated by corresponding terms from å7. The path integral å4 is also independent of since is compensated by the additional contribution to the Jacobian from the transformation å9 with the choice . Furthermore, å4 is independent of which contains the hypergauge-fixing. To see this consider the transformation

(A.11) |

where is an odd infinitesimal function satisfying the condition

(A.12) |

The Jacobian of å11 is

(A.13) |

Summing up the total contribution from å11 in å4 one finds after use of the master equations å5 that what remains may be viewed as the following transformations

(A.14) |

However, since is independent of as was shown above only remains. Notice that in consistency with å7.

One may also notice that å4 is invariant under general anticanonical transformations of the form

(A.15) |

for any odd infinitesimal function provided and transform according to

(A.16) |

This may be used to demonstrate the existence of the above formulation. Consider the abelian case

(A.17) |

Let the master actions here satisfy and . (This implies that å7 reduces to 3̊5.) We define then , ,