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BRST - BFV analysis of anomalies in bosonic string theory interacting with background gravitational field

The general BRST-BFV analysis of anomaly in the string theory coupled to background fields is carried out. An exact equation for c-valued symbol of anomaly operator is found and structure of its solutions is studied.

1. All conventional string models are anomalous at the quantum level and only under certain conditions on parameters of a theory anomalies can be cancelled. In theories of a string interacting with background fields the role of parameters is playing by these fields and conditions of anomaly cancellation are interpreted as effective equations of motion for them [1][2][3][4]. Standard covariant method for deriving equations of motion for background fields in string theories is based on the principle of quantum Weyl invariance which demands that renormalized operator of the energy-momentum tensor trace vanishes. General structure of such an operator in the theory of a string coupled to massless background fields was studied in details in refs. [5,6] (see also the reviews [7,8]).
More consistent (from the theoretical point of view) method of investigation of anomalous theories is formulated within the framework of BRST -BFV quantization [9][10][11][12] (see also the book [17]). The BRST -BFV quantization method, or the method of generalized canonical quantization provides a powerful and universal approach to formulation of gauge theories. Condition of anomaly cancellation in this approach consists in preserving of gauge algebra of the theory at the quantum level. In the case of string theories it reduces to the nilpotency condition for the fermionic operator Ω generating quantum gauge algebra.
The explicit form of quantum operators Ω and Ω 2 in free string theory was constructed and critical values of parameters (such as space-time dimensions) were obtained from the nilpotency condition in refs. [13,14](see also [15,16]). Generalization of this approach to string models with background fields faces problems connected with very complicated structure of these theories. A string interacting with background fields is discribed by a σ-model type action that is essentially non-linear and so requires a suitable perturbation scheme. An attempt to construct such a scheme was undertaken in ref. [18] where the nilpotency condition was stadied for bosonic string coupled to massless background fields within the expansion both in powers ofh and in normal coordinates. The lowest order was considered in details and correspondence with covariant approach was also discussed.
Some general features of the anomaly can be found using only general properties of the operator Ω without explicit computation of Ω 2 and so without construction of any perturbation scheme. In refs. [19,20] general structure of quantum anomaly in theories of free bosonic and fermionic strings was derived from the Jacobi (super)identity for the operator Ω and dimensional considerations. It his paper we consider on the same base the theory of bosonic string coupled to background gravitational field. Being a theory with non-polynomial interaction, this model provides a number of new aspects and details, but as we show the structure of anomaly can be determined up to a function of string coordinates and two constants.
2. The closed bosonic string interacting with background gravitational field is described by the action Here G µν (x) is the metric of d−dimensional space-time with coordinates x µ ; µ, ν = 0, 1, ..., d−1. g ab is the metric of the two-dimensional world sheet of the string, σ a = (τ, σ) are coordinates on the world sheet; a, b = 0, 1.
The theory possesses two first class constraints [18] satisfying Virasoro algebra (in terms of Poisson brackets) p µ (σ) are momenta conjugated to x µ (σ) and x ′µ = ∂ σ x µ . Let us introduce the canonical BRST charge Here η, P andη,P are ghost fields corresponding to constraints L andL respectively. The charge Ω is a generator of BRST transformation δB = {B, Ω}ǫ where ǫ is a fermionic parameter and B is an arbitrary functional on extended phase space. In the case of the theory under consideration it leads to the following transformation properties for the fundamental variables: Here Γ µ αβ are Christoffel symbols corresponding to the metric G µν and In terms of Y µ ,Ȳ µ the constraints (2) have the form The fields Y µ andȲ µ are the generalization of known Fubini fields very convenient for further analysis and will be.
3. Canonical quantization of the theory (1) was considered in ref. [18]. Operators of the constraints L andL represent complicated functions of creation and annihilation operators for oscillating modes of the string and coordinate and momentum operators for zero modes. We will assume that the operators L andL are expressed in some normal form (for example, Wick ordering for oscillating modes and Weyl ordering for zero ones [12,18]). The operator Ω is given by the eq.(4) where L andL are operators of the constraints and η, P,η,P are ghost operators satisfying known commutation relations.
Within the BRST-BFV method the theory is anomalous if the following relations are fulfilled Here [ , ] is supercommutator, H T = H 0 + [Ψ, Ω] is the total hamiltonian and Ψ is a gauge fermion [9][10][11][12]. If A and A H are equal to zero the theory is called anomaly free. Operators A and A H are called anomalies. Obviously, the operator Ω must obey the operatorical (super)Jacobi identity regardless of whether the theory is anomalous or not: The eqs. (7,9) allow to write the equation for anomaly A: [ Ω, A ] = 0.
As for the eq.(8), in string theories A H does not represent an independent anomaly because the canonical hamiltonian H 0 equals zero and so Therefore, the vanishing of A H (8) is a consequence of the vanishing of A and the identity (9). We will show that the eq.(10) is very powerful restriction on the form of anomaly A and allows to write down the A explicitly up to an arbitrary function having a definite tensor structure and depending only on x µ . Using eq.(4,10) one can get the anomaly A as follows We see that the form of anomaly is defined by the commutators It is convenient to carry out further analysis in terms of symbols of operators. Let B and C are some operators depending on operators of canonical variables and written in some normal form. Any operator can be associated with a c−valued function of classical arguments Γ M ≡ (x µ (σ), p µ (σ)) called the normal symbol of the operator (see for details [12,21]). Let B(Γ) and C(Γ) be symbols of the operators B and C. Symbol corresponding to the product BC is called * -product of the symbols B(Γ) and C(Γ) and has the form Here Γ ′M ≡ (x µ (σ ′ ), p µ (σ ′ )); the sum over M, N implies the summation over µ, ν and integration over σ, σ ′ . The contraction in (13) corresponds to the choosed type of ordering. The symbol of commutators Let us consider * -commutator corresponding to the commutator [L(σ), L(σ ′ )]. Taking into account eq.(13) one obtains The function A (1) (σ, σ ′ ) appends when we expand the exponential in eq.(13) in series in power of contractions. Since every contractions containsh, * -commutator should be in general case an infinite series in powers ofh. The first term in (13) is proportional to classical Poisson bracket, the following terms correspond to loop corrections. In free string theory the constraints are quadratic function of Γ M . It means that the series inh terminetes of the second order and we get the known result for the free string anomaly.
In the string theory interacting with arbitrary background fields the series for A (1) (σ, σ ′ ) contains all orders inh and its straightforward calculation is difficult and complicated enough problem. Symbols corresponding to commutators [L(σ),L(σ ′ )] and [L(σ),L(σ ′ )] have structure analogous to (14). For further analysis it is useful to take into account the dimensions of the basic objects of the theory Then Let the functions A (2) (σ, σ ′ ) and A (3) (σ, σ ′ ) are given by the equations and the functions A (1) (σ, σ ′ ) is given by the eq. (14). Let A is a symbol corresponding to the operator of anomaly A. Taking into account the eq.(12) we obtain Here dim A 1 = dim A 2 = dim A 3 = 4 and η(σ),η(σ ′ ) are classical anticommuting functions. The eqs. (14.17)

It allows to write
where arbitrary functions As to the function A (3) (σ, σ ′ ) it has more general structure where arbitrary functions h k (k = 0, 1, 2, 3) have dimensions dim h k = 3 − k.
Using the eqs. (19,20) we can write the anomaly A as follows A = dσdσ ′ ih 2 13 12π (ηη ′′′ +ηη ′′′ ) + a 0 + a 1 + a 2 + a 3 , Thus we obtained the anomaly up to unknown functions h 0 , f 1 , h 1 , g 1 , h 2 , f 3 , h 3 , g 3 . However we have not yet used the eq.(10). Being rewritten in terms of symbols of operators the eq.(10) leads to equation of the following typê where the operatorδ has the structurê Here δ 0 = δ is the operator of classical BRST-transformations (5). An explicit form of δ n at n > 0 is unessential for our purposes. The functional A can be represented as follows where A n are quantum correction to the anomaly. It is easy to show that the eq.(22-24) lead to The eq.(25) is one for the first (one-loop) quantum correction, the eq.(26) allows to find higher quantum corrections to the anomaly. 4. Our aim is to investigate the structure of the eqs.(25,26) solutions. We start with eq.(25) defining the one-loop contribution to the anomaly.
As a result of above discussion the eqs.(26) defining higher quantum correction to the anomaly are reduced to δA n = 0.
But the equation of this type has been studied in previous section. Therefore we are able to write down the most general form of the anomaly where Here β (n) µν (x) are arbitrary functions and β (n) ,β (n) are the arbitrary constants. Thus we see that the general form of anomaly (41) can be established up to above functions and constants using only the generic capabilities of generalized canonical quantization. Certainly, to find these functions and constants we should perform some kind of loop calculation. The example of such calculations was given in ref. [18]. It is evident that the found general form of anomaly (41) in any case allows to simplify the process of real loop calculations.
The dimensions of θ and ρ allow to write where Φ ≡ (M,M , N,N, P,P ) and χ ≡ (K,K, L, Q,Q, V,V , Z,Z) are some functions with definite dimensions. A straightforward calculation shows that ∂ σ ρ have the ghost structure given by the right hand side of the eq.(43) only if χ = 0 and hence ρ = 0. Now let as consider the structure of δθ. According to the eq.(44) one gets dim M = dimM = 2, dim N = dimN = 1 and dim P = dimP = 0. Therefore M,M can be expanded in the basis of Y µȲ ν , Y µ Y ν ,Ȳ µȲ ν , Y ′µ ,Ȳ ′ν ; the N,N can be expanded in the basis of Y µ ,Ȳ µ and P,P are arbitrary functions of x. Taking into account the properties of Φ, calculating δθ and demanding that δθ has the ghost structure given by the right hand side of eq.(43) one gets where ξ µ (x) is an arbitrary vector field satisfying the relation ∇ µ ξ ν (x) = ∇ ν ξ µ (x). Here ∇ µ ξ ν (x) is a covariant derivative in terms of background metric G µν (x). The eqs.(43, 45) together with ρ = 0 lead tõ β µν (x) = β µν (x) + ∇ µ ξ ν (x), β = β,β =β.
The eqs.(46) define the admissible ambiguity in general structure of the anomaly (41). 7. We have considered the problem of anomaly in string theory interacting with background gravitational field in framework of generalized canonical quantization procedure. This problem has been formulated in terms of symbols of operators as a problem of solution of the equation δA(η,η, x, p) = 0 where δ is the canonical BRST -transformation (5) and A is a symbol of the anomaly operator. A general solution of above equation has been obtained and an admissible arbitrariness in the solution has been described. We see, in principle, that the anomaly problem is reduced to a cohomology problem for the operator δ (5) acting in space of c-valued functions depending on coordinates of the extended phase space.
Taking into account the general structure of the anomaly (41) one can immediately write the conditions of anomaly cancellation in the form β µν (x) = 0, β = 0,β = 0. As we have already mentioned the functions β µν (x) and the constants β,β are not defined within the approach under consideration. Therefore, we should develop a calculational technique allowing to compute the function β µν and the constants β,β only on the base of the generalized canonical quantization scheme. The first step towards such a technique was undertaken in ref. [18].
The approach general and allows to investigate various string models in background fields. In particular, inclusion of massless antisymmetric tensor field in to the theory will not lead to any problems and will not demand essential modification of the formalism.