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On the Problems with Background Independence in String Theory

The problems with background independence are discussed in the example of open string theory. Based on the recent proposal by Witten I calculate the String Field Theory action in conformal perturbation theory to second order and demonstrate that the proper treatment of contact terms leads to nontrivial equations of motion. I conjecture the form of the field theory action to all orders.

In contrast with difficult problems in realistic 4d Field Theory models, where the theory is defined and an explicit analytic solution is not yet known, String Theory isn't yet even defined. In many cases what we have is just a number of S-matrices for the processes when the background is fixed by our choice of conformal field theory in the first quantized formulation for amplitudes. Satisfactory formulation of String Theory would have been a formulation where we don't need to refer to any particular classical background and these "classical backgrounds" are given by solutions of some equations. The latter statement is very vague, because unfortunately it is not even clear (at least to the author) what should be the right terminology to address the question. It is believed, by analogy with the second quantized description of ordinary quantum field theory, that the understanding of vacuum structure of string theory as well as the nonperturbative character, can be achieved by developing the field theory language to describe the target space theory. It might well be that the procedure that allows us to construct second quantized field theory from Feynman sum over trajectories directly applied to string theory is not the best way to approach the problem and some other new ideas should be introduced. One of the most important ingredients of any construction has to be a background independence.
In this paper I will address the question of background independent formulation of string theory in the example of open string theory recently suggested by Witten [1]  I'll show that the integrals of total derivatives do not decouple inside the correlation functions that defines the String Field Theory action in the formalism of [1] due to contact term contributions. I'll explain that these contributions have universal character and can't be removed by change of renormalization scheme (this statement has the same origin as the one for gauge anomalies in field theory). This fact leads to slight modifications of the assumptions made in original paper [1] and also in [3]. It was shown in [3] that under the key assumption of decoupling of total derivatives, plus the requirement that BRST operator on the boundary is coupling constant independent, theory has a linear character.
I'll demonstrate here that including the contribution of total derivatives one also has to properly define "BRST" operator on the boundary, which now necessarily should depend on couplings to satisfy the consistency condition. I'll discuss the ambiguities related to this issue and will make a particular choice. In this setup I'll calculate the field theory action in the lines of [3] using the conformal perturbation theory around some fixed point and demonstrate the existence of following relation : up to the second order in coupling constant. Here β is the world-sheet β-function and Z is a partition function. I think that (1) is true to all orders, but calculations beyond second order, as usual, are very complicated. It was discovered in [4] from very general arguments that action should have the above form with first term in (1) given by some vector field. Thus, this vector field is identified with β-function in our approach. This identification is consistent with the statement of [4], that zeros of vector field are the classical equations of motion (CFT on world-sheet). 2 We will give an alternative way of explaining this statement in the section 2.
For the reasons that we are dealing with interacting field theory, we are forced to loose the background independence during the calculation of the action in the perturbation theory, so this formalism doesn't achieve the final goal; the approach is also coordinate system dependent. In fact, the latter makes it difficult to reconstruct the final answer globally, once it is computed in the perturbation theory. But, if (1) is correct to all orders, formal background independence is preserved. At the same time any approach to write down the expressions for β and Z should appeal to perturbation expansion.
I'll not discuss the important issue of gauge symmetries for (1), but as it follows from (1), all symmetries of partition function are automatically the symmetries of action S.
I think that the general transformation properties could be written using the results of Section 2 in the lines of [1].
The form of (1) is quite general one could conjecture that it should be true also for the closed string theory even the analog of [1] (the corresponding space of 2d field theories together with background independent formulation) is not yet known. The expression (1) is simple, but the objects that enter, β and Z, usually are impossible to calculate in closed form. As a result this formalism doesn't avoid the usual technical difficulties that are present in any other formulation of string field theory, although it is formally background independent and contains all string modes.

Boundary Problem and Open String Field Theory
In the beginning of this section we first will describe the construction of [1] with the emphasis of places where some assumptions are made. The idea of the construction in [1] is based on BV formalism. Let M be a supermanifold which is equipped with closed, nondegenerate odd simplectic structure ω and U (1) symmetry, called ghost number U . This means that in Darboux coordinates ψ, θ on M with ψ fermionic and θ bosonic ω = dψdθ and ω has ghost number 1. In analogy with ordinary (bosonic) simplectic manifolds one can define the Poisson Brackets, antibracket, with One can show that the following two simple facts take place: i. If V is a vector field that generates the symmetries of ω, which means that (di V + i V d)ω = 0, then there exists a function S that ii. Vector field defined by (1.2) generates a symmetry of ω for any function S. In the quantization of gauge theories the action S is given on subspace of M with U = 0, S 0 , and we have to find S. That is in fact what the Faddeev-Popov procedure does in the case of Gauge Theories. In the case of String Field Theory, the idea of [1] was to identify the antibracket and vector field in terms of the world-sheet theory and thus identify S as the action of the corresponding target space theory. It was claimed in [1] that this can be done in the case of Open String in a background independent way. The following identifications were proposed: where < ... > formally is defined through the world-sheet theory given by path integral corresponding to the 2d action Here, the first term is the closed string background and the second term describes an arbitrary boundary interaction, parametrised by coupling constants t i (in general there are infinite number of coupling constants), with the condition that the boundary operator V has the form with O being a general operator of ghost number 1. Q is a BRST operator defined by BRST current: Q = C dσJ BRST with contour C approaching the boundary ∂Σ 3 and with v i being the killing vector that generates the rotation of disc. This world-sheet action is also equipped with an ultraviolet cutoff a.
From the above identifications we have the definition of string field theory action: The fact that ω defined by (1.6) is closed follows from the identity: Here we use the definition of b −1 and take two limits: first we shrink the contour C to a point and get zero; second we take the limit when the contour approaches the boundary ∂Σ and get the right hand side in (1.11). Thus, in the notation and the last step follows from (1.11).
BRST invariance of (1.6) is equivalent to exactness of the right hand side in (1.10).
This follows from the simple observation that because the transformation law of ω is If we use the identity (1.11) for the first term in (1.13) and the definition (1.9) we get: (1.14) We are considering a deformation of the critical string, so we can drop all terms of the type < {Q, ...} > 0 , where subscript 0 means the expectation value in the unperturbed theory of some number of operators, using the argument of the contour deformation in the definition for Q, and thus in the last term of (1.14) we can integrate by parts in the path 4 One should note that the action defined by (1.10) differs from Zamolodchikov's c-function integral to obtain + < {Q, (dO) 2 } >; the same is true for the last term in (1.13), which leads to + 1 2 < {Q, (dO) 2 } >. The first two terms in (1.14) are equal and contribute as 2 < (b −1 dO)dO{Q, O} >. Combining all the terms we get after some cancellations: here, in (1.17), the first identity means that L 0 is a generator of the rotation of circle, and the second identity assumes that total derivatives decouple inside the correlation functions. is to deform this subalgebra by including the contributions of boundary term. In the next section we will evaluate the right hand side in (1.10) and formulate this consistency condition in more clear terms for the case when the boundary interaction doesn't mixes ghosts and matter.
At the end of this section as an illustration I would like to discuss a known example of perturbation of a conformal field theory (closed string) by dimension one operator, where the decoupling of total derivatives doesn't takes place and the obstruction is a β-function [10]. Similar calculations will be performed in the next section for open strings.
Consider some CFT perturbed by a dimension one operator V i t i . 6 We will denote the correlation functions in the perturbed theory by << ... >> and those in the unperturbed theory by < .. >; so, the partition function for can be written as << 1 >>, or < exp(i V ) >. One can calculate the trace of stress-tensor in the perturbed theory in the following way. We start from the expectation value of the holomorphic part of stress tensor << T (z,z) >> and use the operator expansion algebra (here we used the fact that V has dimension one in the CFT that we are perturbing; we are speaking about closed string) to deduce that expectation value of the stress tensor is given by total derivatives, integrated over the points where the operator is inserted: If we claim that the contribution of the total derivative in the right hand side is zero, we will conclude that the expectation values of stress tensor is zero; but the latter is wrongwe know that the following relation holds: Thus the obstruction for decoupling is the β-function, so we have to calculate the contribution of contact terms. We have to use the operator expansion algebra for V i to proceed further: contour surrounding each point w j ; denoting w i = w j + ρe iθ with small ρ , we get: (1.22) If we expand the denominator in (1.22) in the powers of ρ and integrate over θ, we see that only second term contributes, with final answer: Thus, we obtain the desired formula (1.20) in the second order for β-function.

Modifications and Perturbative Calculation
In this section we consider the situation when ghosts and matter are decoupled. This means that the boundary interaction V is a functional purely of the matter coordinates X and the operators O are just O = cV . The following calculations were largely already presented in [3], but it was assumed that (1.16) and (1.17) are valid assumptions. As a result the calculations have captured only the linear part of β in (1) and the conclusion was that the theory has a linear equation of motion. Below I will modify the construction by using the parallel transport of Q for to satisfy the consistency condition (1.15). This procedure is generally ambiguous and I will suggest the possible criteria. I would like to note that there has to be a relation of this procedure with the issues discussed in [11].
On the world-sheet we are dealing with interacting field theory and thus only the way to perform the calculations is to use the perturbation theory. This means that in any calculation we will loose background independence (formally), but the goal is to get final expressions in the invariant terms that don't appeal to a particular background. Thus, we will consider the perturbation theory near a fixed point t 0 , which corresponds to some conformal field theory with stress tensor T and corresponding BRST operator Q. When the contour C approaches the boundary of the disc the operator is ambiguous, or it is better to say, it needs to be defined. We have to understand what the correct Q is and make sure that the consistency condition is satisfied.
We could write the left hand side of equation (1.10) in the following form: (2. 2) The ghost correlation functions in (2.2) is easy to evaluate and it is given by a standard formula, because we consider the case when ghosts and matter are decoupled. For the general 3-point function we have: Let us treat (2.2) term by term. The expression for the first term is simple: (2.4) If we use the notation L n = 2π 0 dθe inθ T m (θ), the second term in (2.2) can be written as a combination of three expressions: (2.5) One can simplify (2.5) first noting that: (2.7) We need two more identities: and finally for the second term in (2.8) (2.9) Here we used the notation where L int is the boundary interaction term, and dL −1 = dt i ∂ ∂t i L −1 , assuming that L −1 might depend on the couplings. Now we see that the difference between (2.6) and (2.7), which enters in (2.5) is given by: (2.10) The last term in (2.10) could be dropped because it is an expectation value of operator where we denote by one form X the following expression: (2.12) Now I would like to require that under proper definition of renormalization scheme and the generators of SL(2, R) subalgebra, the object X is identically zero. One should note that the consistency condition (1.15) requires that X is just an exact form, thus there is an ambiguity in the definition of Q and equivalently the symmetries of (1.6). Moreover it is not guaranteed that in the deformed theory the stress tensor, that enters in the definition of Q, is deformed accordingly to this requirement. Below I will give arguments that it is indeed the case and that the vanishing of X is a natural choice. They couldn't be considered as a rigorous proof to all orders in t; they are just arguments and most likely they can lead to such a proof. Before I turn to this very important question let us evaluate the first term in (2.12), to be sure that the total derivative doesn't make it zero. We have in the lowest order in t: Here we had used the operator expansion algebra for V i , with ∆ i being the dimension of V i in the CFT corresponding to the coupling constant t 0 , and integrated over θ 2 from θ 1 + a to θ 1 + 2π − a. The term in c with the "resonance term", is convergent in the limit when we remove the cutoff, and others diverge. These divergent terms can be removed by redefinition of couplings or the same, by a subtraction procedure (see below), while the constant term is universal and can't be removed. Also, there should be a higher order correction in couplings in (2.13).
We see that, like in the example for closed string at the end of the previous section, total derivative in X doesn't decouple and is proportional to β-function coefficient C k ij ; thus (1.17) fails, so does (1.16).
Until now we had avoided the question about the transformation properties of boundary operator. To proceed further it is necessary to know the action of L −1 , L 0 and L 1 on V 's. What we need to cancel the contribution of (2.13) in X is: This leads to a suggestion (that has to be verified) that the proper deformation of the action of SL(2, R) algebra is given by: and γ is the matrix of anomalous dimensions, which for operators V are simply given by [8] γ j i (t) = δ j i − ∂β j ∂t i = δ j i ∆ i + 4c j ik t k + ...., (2.18) and β j is the β-function for operator V j . This deformation is a very natural one. 7 In fact what we need is (2.17) in the first nontrivial order in couplings to compare with (2.13), because the latter we are able to calculate only up to this order. Here are the arguments in support of (2.17): consider the auotomorphizms of disc: or infinitesimally with |z 0 | < 1 and |α| = 1. The requirement, that corresponding P SL(2, R) subalgebra of Virasoro algebra is not broken in perturbed theory, fixes the transformation law (2.17 in the perturbation theory (or its modification in higher orders) and make sure that X vanishes to all orders. In fact, there are ambiguities related to both calculations, caused by renormalization scheme dependence beyond second order.
Our final expression (2.21) can be written in the form announced in the introduction: The linear term for β i is given by (1 − ∆ i )t i , thus the resonance condition in the second order is the one we had written before (2.15): 1 − ∆ k = 1 − ∆ i + 1 − ∆ j . They correspond to finite terms in (2.13) and can't be removed by coordinate transformation. This in fact proves that the non-vanishing of total derivative term in S is universal.
The expression (2.22) shows that for exactly marginal deformations of base point (which is a particular CFT) action S is the same as partition function and this confirms the statement of [4]. Obviously, assuming that β = 0 (∆ i = 1 and c k ij = 0 in the perturbation theory) from the begining and going through our calculations again we get dS = 0. Because any attempt to calculate string field theory action in the present approach should use the perturbation theory we can't make any statement about global properties of action.
Also, it is difficult to check above statement about equation of motion directly from final expression (2.22) without going to world-sheet and using the identities described in this Section; it would be very interesting to find such procedure. Last important comment related to these questions: we couldn't compare two actions (note that dependance on t's enter through β) if they are calculated in perturbation theory arround different points in space of t's. For this we will need the natural paralel transport in the space of theories that we unfortunately don't have; we only know the deformed relations for SL(2, R) generators in perturbation theory. Thus, the result is not truly background independent even it looks so formally.